[STATEMENT]
lemma prj_chine:
    shows "\<tau>\<mu>.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0"
    and "\<tau>\<mu>.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 &&& t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have 1: "arrow_of_spans_of_maps V H \<a> \<i> src trg
                 (s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) (r\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>1) (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) (t\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>1) chine \<and>
               (((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine)) \<cdot> the_\<nu> =
               ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) chine \<and> (((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine)) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
using chine_is_induced_map
[PROOF STATE]
proof (prove)
using this:
is_induced_map chine

goal (1 subgoal):
 1. arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) chine \<and> (((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine)) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) chine \<and> (((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine)) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?u\<^sub>\<tau> = "u \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?w\<^sub>\<tau> = "\<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?w\<^sub>\<tau>' = "\<tau>\<mu>.p\<^sub>1 \<star> chine"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have u\<^sub>\<tau>: "ide ?u\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
using \<chi>.u_simps(3)
[PROOF STATE]
proof (prove)
using this:
trg s\<^sub>0 = src u

goal (1 subgoal):
 1. ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have w\<^sub>\<tau>: "ide ?w\<^sub>\<tau> \<and> is_left_adjoint ?w\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (simp add: \<omega>.is_map \<rho>.T0.antipar(1) left_adjoints_compose)
[PROOF STATE]
proof (state)
this:
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have w\<^sub>\<tau>': "ide ?w\<^sub>\<tau>' \<and> is_left_adjoint ?w\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
by (simp add: is_map left_adjoints_compose)
[PROOF STATE]
proof (state)
this:
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?\<theta>\<^sub>\<tau> = "\<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                   \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?\<theta>\<^sub>\<tau>' = "(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                   ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
let ?\<beta>\<^sub>\<tau> = "\<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have \<theta>\<^sub>\<tau>: "\<guillemotleft>?\<theta>\<^sub>\<tau> : t\<^sub>0 \<star> ?w\<^sub>\<tau> \<Rightarrow> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1) \<omega>.the_\<theta>_in_hom \<chi>.u_simps(3)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0
\<guillemotleft>\<omega>.the_\<theta> : src r\<^sub>0 \<rightarrow> trg r\<^sub>0\<guillemotright>
\<guillemotleft>\<omega>.the_\<theta> : t\<^sub>0 \<star> \<omega>.chine \<Rightarrow> r\<^sub>0\<guillemotright>
trg s\<^sub>0 = src u

goal (1 subgoal):
 1. \<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro comp_in_homI, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have \<theta>\<^sub>\<tau>': "\<guillemotleft>?\<theta>\<^sub>\<tau>' : t\<^sub>0 \<star> ?w\<^sub>\<tau>' \<Rightarrow> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof (intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (6 subgoals):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> ?b4\<guillemotright>
 2. \<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : ?b4 \<Rightarrow> ?b3\<guillemotright>
 3. \<guillemotleft>(\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine : ?b3 \<Rightarrow> ?b2\<guillemotright>
 4. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ?b2 \<Rightarrow> ?b1\<guillemotright>
 5. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : ?b1 \<Rightarrow> ?b\<guillemotright>
 6. \<guillemotleft>u \<star> the_\<theta> : ?b \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] : t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine \<Rightarrow> (t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
using t\<^sub>0u\<^sub>1.p\<^sub>1_simps assoc'_in_hom
[PROOF STATE]
proof (prove)
using this:
trg t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] : src ?h \<rightarrow> trg ?f\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>\<^sup>-\<^sup>1[?f, ?g, ?h] : local.dom ?f \<star> local.dom ?g \<star> local.dom ?h \<Rightarrow> (cod ?f \<star> cod ?g) \<star> cod ?h\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>

goal (5 subgoals):
 1. \<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> ?b3\<guillemotright>
 2. \<guillemotleft>(\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine : ?b3 \<Rightarrow> ?b2\<guillemotright>
 3. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ?b2 \<Rightarrow> ?b1\<guillemotright>
 4. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : ?b1 \<Rightarrow> ?b\<guillemotright>
 5. \<guillemotleft>u \<star> the_\<theta> : ?b \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine \<Rightarrow> (u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
using \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>

goal (4 subgoals):
 1. \<guillemotleft>(\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ?b2\<guillemotright>
 2. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ?b2 \<Rightarrow> ?b1\<guillemotright>
 3. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : ?b1 \<Rightarrow> ?b\<guillemotright>
 4. \<guillemotleft>u \<star> the_\<theta> : ?b \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>(\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine : (u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine \<Rightarrow> ((u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>(\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>

goal (3 subgoals):
 1. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ?b1\<guillemotright>
 2. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : ?b1 \<Rightarrow> ?b\<guillemotright>
 3. \<guillemotleft>u \<star> the_\<theta> : ?b \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine : ((u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine \<Rightarrow> (u \<star> u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
using assoc_in_hom
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>[?f, ?g, ?h] : src ?h \<rightarrow> trg ?f\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>[?f, ?g, ?h] : (local.dom ?f \<star> local.dom ?g) \<star> local.dom ?h \<Rightarrow> cod ?f \<star> cod ?g \<star> cod ?h\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine : ((u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>

goal (2 subgoals):
 1. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>u \<star> the_\<theta> : ?b \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] : (u \<star> u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : (u \<star> u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>u \<star> the_\<theta> : u \<star> (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>u \<star> the_\<theta> : u \<star> (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine \<Rightarrow> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>u \<star> the_\<theta> : u \<star> (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>u \<star> the_\<theta> : u \<star> (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have \<beta>\<^sub>\<tau>: "\<guillemotleft>?\<beta>\<^sub>\<tau> : t\<^sub>1 \<star> ?w\<^sub>\<tau> \<Rightarrow> t\<^sub>1 \<star> ?w\<^sub>\<tau>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
proof (intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (4 subgoals):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b2\<guillemotright>
 2. \<guillemotleft>local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b2 \<Rightarrow> ?b1\<guillemotright>
 3. \<guillemotleft>the_\<nu> : ?b1 \<Rightarrow> ?b\<guillemotright>
 4. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] : t\<^sub>1 \<star> ?w\<^sub>\<tau> \<Rightarrow> (t\<^sub>1 \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (3 subgoals):
 1. \<guillemotleft>local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b1\<guillemotright>
 2. \<guillemotleft>the_\<nu> : ?b1 \<Rightarrow> ?b\<guillemotright>
 3. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1 : (t\<^sub>1 \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<omega>.the_\<nu>_props \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>
local.iso \<omega>.the_\<nu>
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (2 subgoals):
 1. \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>the_\<nu> : r\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
using the_\<nu>_in_hom(2)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] : (t\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>1 \<star> ?w\<^sub>\<tau>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
using t\<^sub>0u\<^sub>1.p\<^sub>1_simps assoc_in_hom
[PROOF STATE]
proof (prove)
using this:
trg t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>[?f, ?g, ?h] : src ?h \<rightarrow> trg ?f\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<guillemotleft>\<a>[?f, ?g, ?h] : (local.dom ?f \<star> local.dom ?g) \<star> local.dom ?h \<Rightarrow> cod ?f \<star> cod ?g \<star> cod ?h\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
define LHS where "LHS = (t \<star> ?\<theta>\<^sub>\<tau>) \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>)"
[PROOF STATE]
proof (state)
this:
LHS = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have LHS: "\<guillemotleft>LHS : t\<^sub>1 \<star> ?w\<^sub>\<tau> \<Rightarrow> t \<star> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof (unfold LHS_def, intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. \<guillemotleft>\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b1\<guillemotright>
 2. \<guillemotleft>\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b1 \<Rightarrow> ?b\<guillemotright>
 3. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<tau> \<star> ?w\<^sub>\<tau> : t\<^sub>1 \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> (t \<star> t\<^sub>0) \<star> ?w\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by (intro hcomp_in_vhom, auto)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (2 subgoals):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] : (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[t, t\<^sub>0, ?w\<^sub>\<tau>] : (t \<star> t\<^sub>0) \<star> ?w\<^sub>\<tau> \<Rightarrow> t \<star> t\<^sub>0 \<star> ?w\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] : (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] : (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] : (t \<star> t\<^sub>0) \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>t \<star> ?\<theta>\<^sub>\<tau> : t \<star> t\<^sub>0 \<star> ?w\<^sub>\<tau> \<Rightarrow> t \<star> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
have "src t = trg (t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. src t = trg (t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (metis \<chi>.u_simps(3) \<mu>.ide_base \<sigma>.ide_leg0 \<sigma>.leg1_simps(3) \<tau>\<mu>.composable
                \<theta>\<^sub>\<tau> arrI assoc_simps(3) r\<^sub>0s\<^sub>1.ide_u r\<^sub>0s\<^sub>1.p\<^sub>0_simps trg_vcomp vconn_implies_hpar(2))
[PROOF STATE]
proof (state)
this:
src t = trg (t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
src t = trg (t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using \<theta>\<^sub>\<tau>
[PROOF STATE]
proof (prove)
using this:
src t = trg (t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t \<star> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
define RHS where "RHS = ((t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>')) \<cdot> ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (state)
this:
RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have RHS: "\<guillemotleft>RHS : t\<^sub>1 \<star> ?w\<^sub>\<tau> \<Rightarrow> t \<star> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
unfolding RHS_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : ?b \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>?\<beta>\<^sub>\<tau> : t\<^sub>1 \<star> ?w\<^sub>\<tau> \<Rightarrow> t\<^sub>1 \<star> ?w\<^sub>\<tau>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
using \<beta>\<^sub>\<tau>
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>(t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>') : t\<^sub>1 \<star> ?w\<^sub>\<tau>' \<Rightarrow> t \<star> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>') : t\<^sub>1 \<star> ?w\<^sub>\<tau>' \<Rightarrow> t \<star> t\<^sub>0 \<star> ?w\<^sub>\<tau>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
using \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>t \<star> ?\<theta>\<^sub>\<tau>' : t \<star> t\<^sub>0 \<star> ?w\<^sub>\<tau>' \<Rightarrow> t \<star> ?u\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using w\<^sub>\<tau>' \<theta>\<^sub>\<tau>' \<tau>.leg0_simps(2) \<tau>.leg0_simps(3) hseqI' ideD(1) t\<^sub>0u\<^sub>1.p\<^sub>1_simps
                  trg_hcomp \<tau>.base_in_hom(2) hcomp_in_vhom
[PROOF STATE]
proof (prove)
using this:
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
arr t\<^sub>0
trg t\<^sub>0 = src t
\<lbrakk>arr ?\<mu>; arr ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> hseq ?\<nu> ?\<mu>
ide ?a \<Longrightarrow> arr ?a
trg t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0
hseq ?\<mu> ?\<nu> \<Longrightarrow> trg (?\<mu> \<star> ?\<nu>) = trg ?\<mu>
\<guillemotleft>t : t \<Rightarrow> t\<guillemotright>
\<lbrakk>\<guillemotleft>?\<mu> : ?f \<Rightarrow> ?g\<guillemotright>; \<guillemotleft>?\<nu> : ?h \<Rightarrow> ?k\<guillemotright>; src ?h = trg ?f\<rbrakk> \<Longrightarrow> \<guillemotleft>?\<nu> \<star> ?\<mu> : ?h \<star> ?f \<Rightarrow> ?k \<star> ?g\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<guillemotleft>t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t \<star> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have eq: "LHS = RHS"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) = \<Delta>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) =
                \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                  (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                       (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot>
                  \<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot>
                  (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
unfolding LHS_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             (t \<star> \<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot>
                             \<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                             \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                       (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] =
                  (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                       (t \<star> \<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
using whisker_left \<tau>.ide_base \<theta>\<^sub>\<tau> arrI seqE
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
ide t
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>?f : ?a \<Rightarrow> ?b\<guillemotright> \<Longrightarrow> arr ?f
\<lbrakk>seq ?g ?f; \<lbrakk>arr ?f; arr ?g; local.dom ?g = cod ?f\<rbrakk> \<Longrightarrow> ?T\<rbrakk> \<Longrightarrow> ?T

goal (1 subgoal):
 1. t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by (metis (full_types))
[PROOF STATE]
proof (state)
this:
t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             (t \<star> \<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot>
                             \<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot>
                             ((\<tau> \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(\<tau> \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] =
                  \<a>[t \<star> t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using assoc_naturality
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?\<mu>; arr ?\<nu>; arr ?\<tau>; src ?\<mu> = trg ?\<nu>; src ?\<nu> = trg ?\<tau>\<rbrakk> \<Longrightarrow> \<a>[cod ?\<mu>, cod ?\<nu>, cod ?\<tau>] \<cdot> ((?\<mu> \<star> ?\<nu>) \<star> ?\<tau>) = (?\<mu> \<star> ?\<nu> \<star> ?\<tau>) \<cdot> \<a>[local.dom ?\<mu>, local.dom ?\<nu>, local.dom ?\<tau>]

goal (1 subgoal):
 1. (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (metis \<omega>.w_simps(2-6) \<rho>.leg1_simps(3) \<rho>\<sigma>.leg1_simps(2) \<tau>.tab_simps(1)
                  \<tau>.tab_simps(2,4-5) hseqE r\<^sub>0s\<^sub>1.leg1_simps(5) r\<^sub>0s\<^sub>1.leg1_simps(6))
[PROOF STATE]
proof (state)
this:
(\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             ((t \<star> \<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot>
                             (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                             (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<cdot>
                    \<a>[t \<star> t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] =
                  \<a>[t, t\<^sub>0 \<star> \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by (simp add: \<rho>.T0.antipar(1))
[PROOF STATE]
proof (state)
this:
seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using pentagon \<rho>.T0.antipar(1) iso_hcomp
                      invert_side_of_triangle(1)
                        [of "\<a>[t, t\<^sub>0, \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
                            "t \<star> \<a>[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
                            "\<a>[t, t\<^sub>0 \<star> \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> \<rho>\<sigma>.p\<^sub>1)"]
[PROOF STATE]
proof (prove)
using this:
seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
\<lbrakk>ide ?f; ide ?g; ide ?h; ide ?k; src ?f = trg ?g; src ?g = trg ?h; src ?h = trg ?k\<rbrakk> \<Longrightarrow> (?f \<star> \<a>[?g, ?h, ?k]) \<cdot> \<a>[?f, ?g \<star> ?h, ?k] \<cdot> (\<a>[?f, ?g, ?h] \<star> ?k) = \<a>[?f, ?g, ?h \<star> ?k] \<cdot> \<a>[?f \<star> ?g, ?h, ?k]
trg r\<^sub>0\<^sup>* = src r\<^sub>0
\<lbrakk>local.iso ?\<mu>; local.iso ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> local.iso (?\<nu> \<star> ?\<mu>)
\<lbrakk>seq \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]; (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]; local.iso (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])\<rbrakk> \<Longrightarrow> seq (local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) (\<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<and> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = local.inv (t \<star> \<a>[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t \<star> t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             \<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                             (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1)) \<cdot>
                             (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(t \<star> \<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] =
                  \<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using assoc_naturality [of t \<omega>.the_\<theta> \<rho>\<sigma>.p\<^sub>1] \<omega>.\<theta>_simps(3) \<rho>\<sigma>.leg1_simps(2) hseq_char
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr t; arr \<omega>.the_\<theta>; arr r\<^sub>0s\<^sub>1.p\<^sub>1; src t = trg \<omega>.the_\<theta>; src \<omega>.the_\<theta> = trg r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>[cod t, cod \<omega>.the_\<theta>, cod r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[local.dom t, local.dom \<omega>.the_\<theta>, local.dom r\<^sub>0s\<^sub>1.p\<^sub>1]
trg \<omega>.the_\<theta> = src t
hseq r\<^sub>1 r\<^sub>0s\<^sub>1.p\<^sub>1
hseq ?\<nu> ?\<mu> = (arr ?\<mu> \<and> arr ?\<nu> \<and> src ?\<nu> = trg ?\<mu>)

goal (1 subgoal):
 1. (t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> ((t \<star> \<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[t, t\<^sub>0 \<star> \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             \<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using whisker_right \<rho>.T0.antipar(1) \<omega>.\<Delta>_simps(1) \<omega>.\<Delta>_naturality comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
trg r\<^sub>0\<^sup>* = src r\<^sub>0
seq (\<omega> \<star> r\<^sub>0) \<rho>
(\<omega> \<star> r\<^sub>0) \<cdot> \<rho> = (t \<star> \<omega>.the_\<theta>) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine] \<cdot> (\<tau> \<star> \<omega>.chine) \<cdot> \<omega>.the_\<nu>
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (((t \<star> \<omega>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>[t, t\<^sub>0, \<omega>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((\<tau> \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (t \<star> (\<chi> \<star> s\<^sub>0) \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             \<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
using whisker_left whisker_right \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by (metis (full_types) \<chi>.\<Delta>_simps(1) \<tau>.ide_base \<theta>\<^sub>\<tau> arrI r\<^sub>0s\<^sub>1.ide_u seqE)
[PROOF STATE]
proof (state)
this:
t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (t \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             \<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> \<rho>\<sigma>.p\<^sub>0) =
                  t \<star> \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> \<rho>\<sigma>.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
using \<chi>.in_hom whisker_left
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)

goal (1 subgoal):
 1. (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = t \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
using assoc_naturality [of \<chi> s\<^sub>0 \<rho>\<sigma>.p\<^sub>0] \<chi>.in_hom
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr \<chi>; arr s\<^sub>0; arr r\<^sub>0s\<^sub>1.p\<^sub>0; src \<chi> = trg s\<^sub>0; src s\<^sub>0 = trg r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk> \<Longrightarrow> \<a>[cod \<chi>, cod s\<^sub>0, cod r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[local.dom \<chi>, local.dom s\<^sub>0, local.dom r\<^sub>0s\<^sub>1.p\<^sub>0]
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>

goal (1 subgoal):
 1. t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = (t \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
have "seq (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
using \<chi>.in_hom
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>

goal (1 subgoal):
 1. seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
apply (intro seqI hseqI)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> arr \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
 2. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> \<guillemotleft>s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?a3 \<rightarrow> ?b3\<guillemotright>
 3. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> \<guillemotleft>\<chi> : ?b3 \<rightarrow> ?c3\<guillemotright>
 4. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = cod \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
apply auto
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> \<guillemotleft>\<chi> : src u \<rightarrow> ?c3\<guillemotright>
 2. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = s \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> \<guillemotleft>\<chi> : src u \<rightarrow> ?c3\<guillemotright>
 2. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = s \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
show "\<guillemotleft>\<chi> : src u \<rightarrow> trg \<chi>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<chi> : src u \<rightarrow> trg \<chi>\<guillemotright>
[PROOF STEP]
by (metis \<chi>.\<Delta>_simps(1) \<chi>.u_simps(3) hseqE in_hhom_def seqE)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<chi> : src u \<rightarrow> trg \<chi>\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright> \<Longrightarrow> local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = s \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
show "dom (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) = s \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = s \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by (metis \<Delta>_simps(1) \<chi>.in_hom hcomp_simps(1,3) hseq_char in_homE seqE
                      u_simps(4))
[PROOF STATE]
proof (state)
this:
local.dom (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = s \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
using whisker_left
[PROOF STATE]
proof (prove)
using this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)

goal (1 subgoal):
 1. t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> (\<chi> \<star> s\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (t \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             (\<a>[t, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> \<rho>\<sigma>.p\<^sub>1)) \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using whisker_right comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<cdot> \<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (t \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot>
                             (\<omega> \<star> r\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using assoc_naturality [of \<omega> r\<^sub>0 \<rho>\<sigma>.p\<^sub>1] \<omega>.in_hom \<rho>.T0.antipar(1) comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr \<omega>; arr r\<^sub>0; arr r\<^sub>0s\<^sub>1.p\<^sub>1; src \<omega> = trg r\<^sub>0; src r\<^sub>0 = trg r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk> \<Longrightarrow> \<a>[cod \<omega>, cod r\<^sub>0, cod r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>[local.dom \<omega>, local.dom r\<^sub>0, local.dom r\<^sub>0s\<^sub>1.p\<^sub>1]
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
trg r\<^sub>0\<^sup>* = src r\<^sub>0
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<a>[t, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> ((\<omega> \<star> r\<^sub>0) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (t \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>1) = \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi>
[PROOF STEP]
using comp_cod_arr comp_arr_dom \<omega>.in_hom interchange comp_ide_arr
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
\<lbrakk>seq ?\<nu> ?\<mu>; seq ?\<tau> ?\<sigma>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?\<tau> \<cdot> ?\<sigma> = (?\<nu> \<star> ?\<tau>) \<cdot> (?\<mu> \<star> ?\<sigma>)
\<lbrakk>ide ?b; seq ?b ?f\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f

goal (1 subgoal):
 1. (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi>
[PROOF STEP]
by (metis \<tau>.base_in_hom(2) \<tau>.ide_base r\<^sub>0s\<^sub>1.\<phi>_simps(1) r\<^sub>0s\<^sub>1.\<phi>_simps(4) seqI')
[PROOF STATE]
proof (state)
this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi>

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi>

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = (\<omega> \<star> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi> = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)
[PROOF STEP]
using r\<^sub>0s\<^sub>1.\<phi>_in_hom comp_cod_arr comp_arr_dom \<omega>.in_hom interchange
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : src r\<^sub>0s\<^sub>1.p\<^sub>0 \<rightarrow> trg s\<^sub>1\<guillemotright>
\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
\<lbrakk>seq ?\<nu> ?\<mu>; seq ?\<tau> ?\<sigma>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?\<tau> \<cdot> ?\<sigma> = (?\<nu> \<star> ?\<tau>) \<cdot> (?\<mu> \<star> ?\<sigma>)

goal (1 subgoal):
 1. \<omega> \<star> r\<^sub>0s\<^sub>1.\<phi> = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)
[PROOF STEP]
by (metis in_homE)
[PROOF STATE]
proof (state)
this:
\<omega> \<star> r\<^sub>0s\<^sub>1.\<phi> = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)
[PROOF STEP]
have "(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)"
[PROOF STATE]
proof (prove)
using this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)

goal (1 subgoal):
 1. (t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((t \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> (\<omega> \<star> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             (\<omega> \<star> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using whisker_left \<rho>.T0.antipar(1) \<rho>\<sigma>.composable \<chi>.in_hom comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
trg r\<^sub>0\<^sup>* = src r\<^sub>0
src r = trg s
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (t \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "(t \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0) =
                   \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
have "\<guillemotleft>(\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) : s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0 \<Rightarrow> u \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>(\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
using \<omega>.in_hom \<chi>.in_hom
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>(\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by force
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. (t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>(\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. (t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by (metis (no_types) \<omega>.in_hom comp_arr_dom comp_cod_arr in_homE
                      interchange)
[PROOF STATE]
proof (state)
this:
(t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((t \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (\<omega> \<star> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) =
                  (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
have "seq (\<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) (\<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (\<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
using \<chi>.in_hom
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>

goal (1 subgoal):
 1. seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (\<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
by force
[PROOF STATE]
proof (state)
this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (\<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (\<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
using comp_arr_dom comp_cod_arr \<omega>.in_hom \<chi>.in_hom interchange
[PROOF STATE]
proof (prove)
using this:
seq (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) (\<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>
\<lbrakk>seq ?\<nu> ?\<mu>; seq ?\<tau> ?\<sigma>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?\<tau> \<cdot> ?\<sigma> = (?\<nu> \<star> ?\<tau>) \<cdot> (?\<mu> \<star> ?\<sigma>)

goal (1 subgoal):
 1. \<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
by (metis in_homE)
[PROOF STATE]
proof (state)
this:
\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> (\<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<sigma> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<rho> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) =
                  ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
using assoc_naturality \<omega>.in_hom \<chi>.in_hom
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?\<mu>; arr ?\<nu>; arr ?\<tau>; src ?\<mu> = trg ?\<nu>; src ?\<nu> = trg ?\<tau>\<rbrakk> \<Longrightarrow> \<a>[cod ?\<mu>, cod ?\<nu>, cod ?\<tau>] \<cdot> ((?\<mu> \<star> ?\<nu>) \<star> ?\<tau>) = (?\<mu> \<star> ?\<nu> \<star> ?\<tau>) \<cdot> \<a>[local.dom ?\<mu>, local.dom ?\<nu>, local.dom ?\<tau>]
\<guillemotleft>\<omega> : r \<Rightarrow> t\<guillemotright>
\<guillemotleft>\<chi> : s \<Rightarrow> u\<guillemotright>

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by (metis \<rho>\<sigma>.leg0_simps(3) assoc'_naturality hcomp_in_vhomE in_hom in_homE
                  u_simps(2) u_simps(4) u_simps(5))
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<omega> \<star> \<chi> \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
have "... = \<Delta>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
using whisker_left \<rho>\<sigma>.tab_def comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (r \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) =
                \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                  (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                         (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                         (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                    \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine) \<cdot>
                    \<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                    ((\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
unfolding RHS_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                                     (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                                     (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                               \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine) \<cdot>
                               \<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "the_\<nu> \<cdot> (inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                    ((\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) =
                  the_\<nu> \<cdot> (inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                    ((t\<^sub>1 \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_inv_arr \<rho>.T0.antipar(1) comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
inverse_arrows ?f ?g \<Longrightarrow> ?g \<cdot> ?f = local.dom ?f
trg r\<^sub>0\<^sup>* = src r\<^sub>0
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h

goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "... = the_\<nu> \<cdot> (inv \<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using comp_cod_arr \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> ((t\<^sub>1 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "... = the_\<nu> \<cdot> (r\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using whisker_right [of \<rho>\<sigma>.p\<^sub>1] r\<^sub>0s\<^sub>1.ide_leg1 \<omega>.the_\<nu>_props(2) \<omega>.the_\<nu>_simps(4)
                    \<rho>.leg1_simps(2) comp_inv_arr'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide r\<^sub>0s\<^sub>1.p\<^sub>1; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = (?\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (?\<mu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide r\<^sub>0s\<^sub>1.p\<^sub>1
local.iso \<omega>.the_\<nu>
local.dom \<omega>.the_\<nu> = r\<^sub>1
arr r\<^sub>1
local.iso ?f \<Longrightarrow> local.inv ?f \<cdot> ?f = local.dom ?f

goal (1 subgoal):
 1. the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "... = the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>
[PROOF STEP]
using comp_arr_dom
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; local.dom ?f = ?a\<rbrakk> \<Longrightarrow> ?f \<cdot> ?a = ?f

goal (1 subgoal):
 1. the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
the_\<nu> \<cdot> (r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = the_\<nu>
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                             (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                             \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine) \<cdot>
                             \<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "arr ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                       ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
using \<theta>\<^sub>\<tau>'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "arr (\<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                                 ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "arr ((\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                                 ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "arr (((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
have "arr ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
[PROOF STEP]
using calculation
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
[PROOF STEP]
have "t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                    ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] =
                  (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot>
                    (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                    (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine])"
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]

goal (1 subgoal):
 1. t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
using whisker_left \<rho>.T0.antipar(1) \<tau>.ide_base
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> the_\<theta>) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] ((\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) (((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
seq (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
trg r\<^sub>0\<^sup>* = src r\<^sub>0
ide t

goal (1 subgoal):
 1. t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (t \<star> u \<star> the_\<theta>) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             (t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                             (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot>
                             ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu>
[PROOF STEP]
using assoc'_naturality [of t u the_\<theta>] \<tau>\<mu>.composable \<theta>_simps(3) comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr t; arr u; arr the_\<theta>; src t = trg u; src u = trg the_\<theta>\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod t, cod u, cod the_\<theta>] \<cdot> (t \<star> u \<star> the_\<theta>) = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom t, local.dom u, local.dom the_\<theta>]
src t = trg u
trg the_\<theta> = src (t \<star> u)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (t \<star> u \<star> the_\<theta>)) \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             (t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                             (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot>
                             \<a>[t \<star> t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(\<tau> \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, \<tau>\<mu>.p\<^sub>1, chine] =
                  \<a>[t \<star> t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)
[PROOF STEP]
using assoc_naturality
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?\<mu>; arr ?\<nu>; arr ?\<tau>; src ?\<mu> = trg ?\<nu>; src ?\<nu> = trg ?\<tau>\<rbrakk> \<Longrightarrow> \<a>[cod ?\<mu>, cod ?\<nu>, cod ?\<tau>] \<cdot> ((?\<mu> \<star> ?\<nu>) \<star> ?\<tau>) = (?\<mu> \<star> ?\<nu> \<star> ?\<tau>) \<cdot> \<a>[local.dom ?\<mu>, local.dom ?\<nu>, local.dom ?\<tau>]

goal (1 subgoal):
 1. (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)
[PROOF STEP]
by (metis \<tau>.leg1_simps(3) \<tau>.tab_simps(1,2,4) \<tau>.tab_simps(5) \<tau>\<mu>.leg0_simps(2)
                  \<tau>\<mu>.leg1_simps(2) hseqE src_hcomp t\<^sub>0u\<^sub>1.leg1_simps(3,5-6) w_simps(2)
                  w_simps(4-6))
[PROOF STATE]
proof (state)
this:
(\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             (t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot>
                             ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot>
                    \<a>[t \<star> t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] =
                  \<a>[t, t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine)
[PROOF STEP]
using pentagon t\<^sub>0u\<^sub>1.p\<^sub>1_simps uw\<theta> \<tau>.T0.antipar(1) iso_hcomp
                    comp_assoc_assoc'
                    invert_side_of_triangle(1)
                      [of "\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]"
                          "t \<star> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]"
                          "\<a>[t, t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine)"]
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; ide ?g; ide ?h; ide ?k; src ?f = trg ?g; src ?g = trg ?h; src ?h = trg ?k\<rbrakk> \<Longrightarrow> (?f \<star> \<a>[?g, ?h, ?k]) \<cdot> \<a>[?f, ?g \<star> ?h, ?k] \<cdot> (\<a>[?f, ?g, ?h] \<star> ?k) = \<a>[?f, ?g, ?h \<star> ?k] \<cdot> \<a>[?f \<star> ?g, ?h, ?k]
trg t\<^sub>0u\<^sub>1.p\<^sub>1 = src t\<^sub>0
ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
trg t\<^sub>0\<^sup>* = src t\<^sub>0
\<lbrakk>local.iso ?\<mu>; local.iso ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> local.iso (?\<nu> \<star> ?\<mu>)
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
\<lbrakk>seq \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]; (t \<star> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) = \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]; local.iso (t \<star> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])\<rbrakk> \<Longrightarrow> seq (local.inv (t \<star> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine])) (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<and> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) = local.inv (t \<star> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]

goal (1 subgoal):
 1. (t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> \<a>[t \<star> t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             ((t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot>
                             ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1, chine] =
                  \<a>[t, u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)
[PROOF STEP]
using assoc_naturality [of t t\<^sub>0u\<^sub>1.\<phi> chine] t\<^sub>0u\<^sub>1.cospan
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr t; arr t\<^sub>0u\<^sub>1.\<phi>; arr chine; src t = trg t\<^sub>0u\<^sub>1.\<phi>; src t\<^sub>0u\<^sub>1.\<phi> = trg chine\<rbrakk> \<Longrightarrow> \<a>[cod t, cod t\<^sub>0u\<^sub>1.\<phi>, cod chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) = (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[local.dom t, local.dom t\<^sub>0u\<^sub>1.\<phi>, local.dom chine]
trg u\<^sub>1 = trg t\<^sub>0

goal (1 subgoal):
 1. (t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>[t, t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                             ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0, chine] =
                  \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> ((t \<star> (\<mu> \<star> \<tau>\<mu>.p\<^sub>0)) \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)
[PROOF STEP]
using assoc_naturality [of t "\<mu> \<star> \<tau>\<mu>.p\<^sub>0" chine]
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr t; hseq \<mu> t\<^sub>0u\<^sub>1.p\<^sub>0; arr chine; src t = trg (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0); src (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) = trg chine\<rbrakk> \<Longrightarrow> \<a>[cod t, cod (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0), cod chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[local.dom t, local.dom (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0), local.dom chine]

goal (1 subgoal):
 1. (t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)
[PROOF STEP]
by (simp add: \<tau>\<mu>.composable)
[PROOF STATE]
proof (state)
this:
(t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>[t, u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                             ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot>
                             ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine)) \<cdot>
                    ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) =
                  ((t \<star> ((u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0)) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)
[PROOF STEP]
using whisker_right whisker_left [of t "\<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]" "\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]"]
                    \<tau>\<mu>.composable comp_assoc_assoc'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<lbrakk>ide t; seq \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] = (t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0])
src t = trg u
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h

goal (1 subgoal):
 1. (((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine)

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "... = (t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine
[PROOF STEP]
using comp_cod_arr \<tau>\<mu>.composable
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
src t = trg u

goal (1 subgoal):
 1. ((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t \<star> (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine
[PROOF STEP]
have "(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine)) \<cdot>
                            ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) =
                          (t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine"
[PROOF STATE]
proof (prove)
using this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine

goal (1 subgoal):
 1. (((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) = (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                             ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine)) \<cdot>
                             ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "((\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine)) \<cdot>
                    ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) =
                  ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) = (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine
[PROOF STEP]
using comp_inv_arr' comp_cod_arr \<tau>\<mu>.composable comp_assoc_assoc'
                    whisker_right [of chine "\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0]" "\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0]"]
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> local.inv ?f \<cdot> ?f = local.dom ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
src t = trg u
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>[?f, ?g, ?h] \<cdot> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = ?f \<star> ?g \<star> ?h
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] \<cdot> \<a>[?f, ?g, ?h] = (?f \<star> ?g) \<star> ?h
\<lbrakk>ide chine; seq \<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> \<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<cdot> \<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine = (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)

goal (1 subgoal):
 1. ((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) = (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) = (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) = (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) = (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                             ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot>
                             ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (((\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine)) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot>
                             (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot>
                             (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                             ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine)) \<cdot>
                             (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot>
                             ((t \<star> \<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot>
                             (\<a>[t, t\<^sub>0, \<tau>\<mu>.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) =
                  \<tau>\<mu>.tab \<star> chine"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) = \<tau>\<mu>.tab \<star> chine
[PROOF STEP]
using uw\<theta> whisker_right [of chine]
[PROOF STATE]
proof (prove)
using this:
ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<lbrakk>ide chine; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> chine = (?\<nu> \<star> chine) \<cdot> (?\<mu> \<star> chine)

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) = \<tau>\<mu>.tab \<star> chine
[PROOF STEP]
by (metis \<tau>\<mu>.tab_def \<tau>\<mu>.tab_in_vhom' arrI seqE)
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) = \<tau>\<mu>.tab \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) = \<tau>\<mu>.tab \<star> chine

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) = \<tau>\<mu>.tab \<star> chine
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> ((t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<star> chine) \<cdot> (\<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<star> chine) \<cdot> ((\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine)) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]) \<cdot>
                    (t \<star> \<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot>
                    ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0] \<star> chine) =
                  \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot>
                   (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot>
                   ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot>
                   (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
using \<a>'_def \<alpha>_def \<tau>\<mu>.composable
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
src t = trg u

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace>

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace>

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "... = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>\<tau>\<mu>.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
using \<tau>\<mu>.composable
[PROOF STATE]
proof (prove)
using this:
src t = trg u

goal (1 subgoal):
 1. \<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace>
[PROOF STEP]
apply (intro E.eval_eqI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. src t = trg u \<Longrightarrow> E.VPar (\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)) \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]
 2. src t = trg u \<Longrightarrow> \<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<^bold>\<rfloor>
[PROOF STEP]
by simp_all
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace>

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, (\<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]) \<^bold>\<cdot> (\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> \<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, (\<^bold>\<langle>u\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>] \<^bold>\<cdot> ((\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[\<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>]) \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>) \<^bold>\<cdot> (\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle>, \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>\<^bold>] \<^bold>\<star> \<^bold>\<langle>chine\<^bold>\<rangle>)\<rbrace> = \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace>

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
have "... = \<a>[t \<star> u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using \<a>'_def \<alpha>_def \<tau>\<mu>.composable
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[?\<mu>, ?\<nu>, ?\<tau>] \<equiv> \<alpha>' (?\<mu>, ?\<nu>, ?\<tau>)
\<alpha> ?\<mu>\<nu>\<tau> \<equiv> \<a>[fst ?\<mu>\<nu>\<tau>, fst (snd ?\<mu>\<nu>\<tau>), snd (snd ?\<mu>\<nu>\<tau>)]
src t = trg u

goal (1 subgoal):
 1. \<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrace>\<^bold>\<a>\<^bold>[\<^bold>\<langle>t\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>u\<^bold>\<rangle>, \<^bold>\<langle>u\<^sub>0\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>t\<^sub>0u\<^sub>1.p\<^sub>0\<^bold>\<rangle>, \<^bold>\<langle>chine\<^bold>\<rangle>\<^bold>]\<rbrace> = \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> (\<a>\<^sup>-\<^sup>1[t, u, (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine] \<cdot> (t \<star> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>[t, (u \<star> u\<^sub>0) \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> ((t \<star> \<a>\<^sup>-\<^sup>1[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<star> chine) \<cdot> (\<a>[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "... = \<Delta>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
using \<Delta>_naturality
[PROOF STATE]
proof (prove)
using this:
((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu>

goal (1 subgoal):
 1. ((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t \<star> u) \<star> the_\<theta>) \<cdot> \<a>[t \<star> u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<tau>\<mu>.tab \<star> chine) \<cdot> the_\<nu> = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab

goal (1 subgoal):
 1. ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) =
                      \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
hence "(LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1) =
               (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>1)"
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. (LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
using u\<^sub>\<tau> r\<^sub>0s\<^sub>1.ide_u LHS RHS iso_is_section [of "\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0]"] section_is_mono
                monoE \<tau>\<mu>.composable comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
ide r\<^sub>0s\<^sub>1.p\<^sub>0
\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
local.iso \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<Longrightarrow> section \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]
section ?g \<Longrightarrow> local.mono ?g
\<lbrakk>local.mono ?g; seq ?g ?f; seq ?g ?f'; ?g \<cdot> ?f = ?g \<cdot> ?f'\<rbrakk> \<Longrightarrow> ?f' = ?f
src t = trg u
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (metis (no_types, lifting) \<Delta>_simps(1) \<mu>.ide_base
              \<open>\<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) =
               ((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab\<close>
              \<tau>.ide_base hseq_char ideD(1) ide_u iso_assoc')
[PROOF STATE]
proof (state)
this:
(LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
hence 1: "LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] = RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
using this:
(LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
using epiE LHS RHS iso_is_retraction retraction_is_epi \<tau>\<mu>.composable
                \<omega>.the_\<nu>_props iso_hcomp
[PROOF STATE]
proof (prove)
using this:
(LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = (RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
\<lbrakk>epi ?f; seq ?g ?f; seq ?g' ?f; ?g \<cdot> ?f = ?g' \<cdot> ?f\<rbrakk> \<Longrightarrow> ?g = ?g'
\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
local.iso ?f \<Longrightarrow> retraction ?f
retraction ?g \<Longrightarrow> epi ?g
src t = trg u
\<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>
local.iso \<omega>.the_\<nu>
\<lbrakk>local.iso ?\<mu>; local.iso ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> local.iso (?\<nu> \<star> ?\<mu>)

goal (1 subgoal):
 1. LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by (metis \<Delta>_simps(1) \<omega>.the_\<nu>_simps(2)
              \<open>((\<omega> \<star> \<chi>) \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<rho>\<sigma>.tab =
               \<a>\<^sup>-\<^sup>1[t, u, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)\<close>
              \<rho>.leg1_simps(3) ide_is_iso local.comp_assoc r\<^sub>0s\<^sub>1.ide_leg1 r\<^sub>0s\<^sub>1.p\<^sub>1_simps seqE)
[PROOF STATE]
proof (state)
this:
LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
show "LHS = RHS"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
have "epi \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
using iso_is_retraction retraction_is_epi \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> retraction ?f
retraction ?g \<Longrightarrow> epi ?g
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
have "seq LHS \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
using LHS \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
have "seq RHS \<a>[t\<^sub>1, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
using RHS \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>RHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
using epiE 1
[PROOF STATE]
proof (prove)
using this:
epi \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq LHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
seq RHS \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
\<lbrakk>epi ?f; seq ?g ?f; seq ?g' ?f; ?g \<cdot> ?f = ?g' \<cdot> ?f\<rbrakk> \<Longrightarrow> ?g = ?g'
LHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = RHS \<cdot> \<a>[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]

goal (1 subgoal):
 1. LHS = RHS
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
LHS = RHS

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
LHS = RHS

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
LHS = RHS

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have 1: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright> \<and> ?\<beta>\<^sub>\<tau> = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
using LHS_def RHS_def u\<^sub>\<tau> w\<^sub>\<tau> w\<^sub>\<tau>' \<beta>\<^sub>\<tau> \<theta>\<^sub>\<tau> \<theta>\<^sub>\<tau>' eq \<tau>.T2 [of ?w\<^sub>\<tau> ?w\<^sub>\<tau>' ?\<theta>\<^sub>\<tau> ?u\<^sub>\<tau> ?\<theta>\<^sub>\<tau>' ?\<beta>\<^sub>\<tau>]
[PROOF STATE]
proof (prove)
using this:
LHS = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
LHS = RHS
\<lbrakk>ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1); ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine); \<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
obtain \<gamma>\<^sub>\<tau> where \<gamma>\<^sub>\<tau>: "\<guillemotleft>\<gamma>\<^sub>\<tau> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright> \<and> ?\<beta>\<^sub>\<tau> = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> ?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<tau>. \<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 1
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<tau>. \<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have iso_\<beta>\<^sub>\<tau>: "iso ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
using uw\<theta> \<beta>\<^sub>\<tau> the_\<nu>_props \<omega>.the_\<nu>_props hseqI' iso_assoc' \<omega>.hseq_leg\<^sub>0
[PROOF STATE]
proof (prove)
using this:
ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
\<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
local.iso the_\<nu>
\<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>
local.iso \<omega>.the_\<nu>
\<lbrakk>arr ?\<mu>; arr ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> hseq ?\<nu> ?\<mu>
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[?f, ?g, ?h]
hseq t\<^sub>0 \<omega>.chine

goal (1 subgoal):
 1. local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (intro isos_compose)
[PROOF STATE]
proof (prove)
goal (7 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
 3. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
 4. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso the_\<nu>
 5. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq the_\<nu> ((local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
 6. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 7. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (metis \<omega>.is_ide \<rho>\<sigma>.leg1_simps(2) \<tau>.ide_leg1 \<tau>.leg1_simps(2)
                           \<tau>.leg1_simps(3) hseqE r\<^sub>0s\<^sub>1.ide_leg1 hcomp_simps(1) vconn_implies_hpar(3))
[PROOF STATE]
proof (prove)
goal (6 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
 3. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso the_\<nu>
 4. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq the_\<nu> ((local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
 5. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 6. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (metis \<rho>\<sigma>.leg1_simps(2) hseqE ide_is_iso r\<^sub>0s\<^sub>1.ide_leg1 src_inv iso_inv_iso
                          iso_hcomp vconn_implies_hpar(1))
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso the_\<nu>
 3. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq the_\<nu> ((local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
 4. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 5. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso the_\<nu>
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq the_\<nu> ((local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
 3. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 4. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq the_\<nu> ((local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 3. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> local.iso \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]
 2. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
apply (metis \<tau>.ide_leg1 \<tau>.leg1_simps(3) hseqE ide_char iso_assoc t\<^sub>0u\<^sub>1.ide_leg1
                      t\<^sub>0u\<^sub>1.p\<^sub>1_simps w\<^sub>\<tau>')
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>; \<guillemotleft>the_\<nu> : r\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>; local.iso the_\<nu>; \<guillemotleft>\<omega>.the_\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>; local.iso \<omega>.the_\<nu>; \<And>\<mu> \<nu>. \<lbrakk>arr \<mu>; arr \<nu>; src \<nu> = trg \<mu>\<rbrakk> \<Longrightarrow> hseq \<nu> \<mu>; \<And>f g h. \<lbrakk>ide f; ide g; ide h; src f = trg g; src g = trg h\<rbrakk> \<Longrightarrow> local.iso \<a>\<^sup>-\<^sup>1[f, g, h]; hseq t\<^sub>0 \<omega>.chine\<rbrakk> \<Longrightarrow> seq \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] (the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
hence eq': "((t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>')) =
                  ((t \<star> ?\<theta>\<^sub>\<tau>) \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>)) \<cdot> inv ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
using this:
local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<Longrightarrow> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
have "seq ((t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>')) ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
using LHS RHS_def eq
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
LHS = RHS

goal (1 subgoal):
 1. seq ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
seq ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<Longrightarrow> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
hence "(t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>') =
               (((t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>')) \<cdot> ?\<beta>\<^sub>\<tau>) \<cdot> inv ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
using this:
seq ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by (meson invert_side_of_triangle(2) iso_\<beta>\<^sub>\<tau>)
[PROOF STATE]
proof (state)
this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<Longrightarrow> (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
using LHS_def RHS_def eq
[PROOF STATE]
proof (prove)
using this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = (((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
LHS = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
LHS = RHS

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by argo
[PROOF STATE]
proof (state)
this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have 2: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright> \<and> inv ?\<beta>\<^sub>\<tau> = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
using u\<^sub>\<tau> w\<^sub>\<tau> w\<^sub>\<tau>' \<beta>\<^sub>\<tau> \<theta>\<^sub>\<tau> \<theta>\<^sub>\<tau>' eq' \<tau>.T2 [of ?w\<^sub>\<tau>' ?w\<^sub>\<tau> ?\<theta>\<^sub>\<tau>'?u\<^sub>\<tau> ?\<theta>\<^sub>\<tau> "inv ?\<beta>\<^sub>\<tau>"] iso_\<beta>\<^sub>\<tau> comp_assoc
[PROOF STATE]
proof (prove)
using this:
ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
\<lbrakk>ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine); ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1); \<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) : t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>; (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
obtain \<gamma>\<^sub>\<tau>' where
        \<gamma>\<^sub>\<tau>': "\<guillemotleft>\<gamma>\<^sub>\<tau>' : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright> \<and> inv ?\<beta>\<^sub>\<tau> = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> ?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<tau>'. \<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 2
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<tau>'. \<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have "inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "\<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. \<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> ?w\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
using \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' whisker_left \<beta>\<^sub>\<tau> iso_\<beta>\<^sub>\<tau> comp_inv_arr'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
local.iso (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
local.iso ?f \<Longrightarrow> local.inv ?f \<cdot> ?f = local.dom ?f

goal (1 subgoal):
 1. t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
by (metis (no_types, lifting) \<tau>.ide_leg1 calculation in_homE)
[PROOF STATE]
proof (state)
this:
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
have "?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
using \<gamma>\<^sub>\<tau>
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
have "... = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
using \<gamma>\<^sub>\<tau>' comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
have "... = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
using \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' whisker_left
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)

goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
by (metis (full_types) \<tau>.ide_leg0 seqI')
[PROOF STATE]
proof (state)
this:
(\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have
          "\<And>\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> ?w\<^sub>\<tau> \<and> ?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = ?w\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright> \<and> t\<^sub>1 \<star> ?w\<^sub>\<tau> = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have "(t \<star> ?\<theta>\<^sub>\<tau>) \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>) =
                  ((t \<star> ?\<theta>\<^sub>\<tau>) \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> ?w\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
[PROOF STEP]
by (metis LHS LHS_def comp_arr_dom in_homE)
[PROOF STATE]
proof (state)
this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>\<tau> \<theta>\<^sub>\<tau> \<omega>.w_simps(4) \<tau>.leg1_in_hom(2) \<tau>.leg1_simps(3) hcomp_in_vhom ideD(1)
                    trg_hcomp ide_in_hom(2) \<tau>.T2
[PROOF STATE]
proof (prove)
using this:
(t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> (t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
trg \<omega>.chine = src t\<^sub>0
\<guillemotleft>t\<^sub>1 : t\<^sub>1 \<Rightarrow> t\<^sub>1\<guillemotright>
src t\<^sub>1 = src t\<^sub>0
\<lbrakk>\<guillemotleft>?\<mu> : ?f \<Rightarrow> ?g\<guillemotright>; \<guillemotleft>?\<nu> : ?h \<Rightarrow> ?k\<guillemotright>; src ?h = trg ?f\<rbrakk> \<Longrightarrow> \<guillemotleft>?\<nu> \<star> ?\<mu> : ?h \<star> ?f \<Rightarrow> ?k \<star> ?g\<guillemotright>
ide ?a \<Longrightarrow> arr ?a
hseq ?\<mu> ?\<nu> \<Longrightarrow> trg (?\<mu> \<star> ?\<nu>) = trg ?\<mu>
ide ?f \<Longrightarrow> \<guillemotleft>?f : ?f \<Rightarrow> ?f\<guillemotright>
\<lbrakk>ide ?w; ide ?w'; \<guillemotleft>?\<theta> : t\<^sub>0 \<star> ?w \<Rightarrow> ?u\<guillemotright>; \<guillemotleft>?\<theta>' : t\<^sub>0 \<star> ?w' \<Rightarrow> ?u\<guillemotright>; \<guillemotleft>?\<beta> : t\<^sub>1 \<star> ?w \<Rightarrow> t\<^sub>1 \<star> ?w'\<guillemotright>; (t \<star> ?\<theta>) \<cdot> \<a>[t, t\<^sub>0, ?w] \<cdot> (\<tau> \<star> ?w) = ((t \<star> ?\<theta>') \<cdot> \<a>[t, t\<^sub>0, ?w'] \<cdot> (\<tau> \<star> ?w')) \<cdot> ?\<beta>\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w \<Rightarrow> ?w'\<guillemotright> \<and> ?\<beta> = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta> = ?\<theta>' \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
thus "\<And>\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau> \<Rightarrow> ?w\<^sub>\<tau>\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> ?w\<^sub>\<tau> \<and> ?\<theta>\<^sub>\<tau> = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = ?w\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
by (metis \<theta>\<^sub>\<tau> comp_arr_dom ide_in_hom(2) in_homE w\<^sub>\<tau>)
[PROOF STATE]
proof (state)
this:
\<guillemotleft>?\<gamma>4 : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>?\<gamma>4 : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
\<guillemotleft>?\<gamma>4 : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
have "\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = ?w\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
\<guillemotleft>?\<gamma>4 : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (1 subgoal):
 1. \<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (2 subgoals):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
 2. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
thus "ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)"
[PROOF STATE]
proof (prove)
using this:
\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
using w\<^sub>\<tau>
[PROOF STATE]
proof (prove)
using this:
\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau> = \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide (\<gamma>\<^sub>\<tau>' \<cdot> \<gamma>\<^sub>\<tau>)

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "\<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
using \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. \<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> ?w\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
by (metis \<beta>\<^sub>\<tau> \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' \<tau>.ide_leg1 calculation comp_arr_inv' in_homE iso_\<beta>\<^sub>\<tau> whisker_left)
[PROOF STATE]
proof (state)
this:
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
using \<gamma>\<^sub>\<tau>'
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "... = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
using \<gamma>\<^sub>\<tau> comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "... = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
using \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' whisker_left \<tau>.ide_leg0
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
ide t\<^sub>0

goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>') = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
have "\<And>\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> ?w\<^sub>\<tau>' \<and> ?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>)
                              \<Longrightarrow> \<gamma> = ?w\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
have "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright> \<and> t\<^sub>1 \<star> ?w\<^sub>\<tau>' = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have "(t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>') =
                  ((t \<star> ?\<theta>\<^sub>\<tau>') \<cdot> \<a>[t, t\<^sub>0, ?w\<^sub>\<tau>'] \<cdot> (\<tau> \<star> ?w\<^sub>\<tau>')) \<cdot> (t\<^sub>1 \<star> ?w\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
have 1: "t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by (meson \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' \<tau>.ide_leg1 seqI' whisker_left)
[PROOF STATE]
proof (state)
this:
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
have "((LHS \<cdot> inv ?\<beta>\<^sub>\<tau>) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = LHS \<cdot> inv ?\<beta>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
using LHS_def RHS_def \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>' eq eq'
[PROOF STATE]
proof (prove)
using this:
LHS = (t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
LHS = RHS
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. ((LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
[PROOF STEP]
by argo
[PROOF STATE]
proof (state)
this:
((LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = LHS \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
unfolding LHS_def
[PROOF STATE]
proof (prove)
using this:
((((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
using 1
[PROOF STATE]
proof (prove)
using this:
((((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>)) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>') = ((t \<star> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> \<a>[t, t\<^sub>0, \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)) \<cdot> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1])
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>) \<cdot> (t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>')

goal (1 subgoal):
 1. (t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
[PROOF STEP]
by (simp add: calculation(2) eq' comp_assoc)
[PROOF STATE]
proof (state)
this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>\<tau>' \<theta>\<^sub>\<tau>' \<omega>.w_simps(4) \<tau>.leg1_in_hom(2) \<tau>.leg1_simps(3) hcomp_in_vhom ideD(1)
                    trg_hcomp ide_in_hom(2) \<tau>.T2 \<tau>.T0.antipar(1) t\<^sub>0u\<^sub>1.base_simps(2)
                    t\<^sub>0u\<^sub>1.leg1_simps(4)
[PROOF STATE]
proof (prove)
using this:
(t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> (t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
trg \<omega>.chine = src t\<^sub>0
\<guillemotleft>t\<^sub>1 : t\<^sub>1 \<Rightarrow> t\<^sub>1\<guillemotright>
src t\<^sub>1 = src t\<^sub>0
\<lbrakk>\<guillemotleft>?\<mu> : ?f \<Rightarrow> ?g\<guillemotright>; \<guillemotleft>?\<nu> : ?h \<Rightarrow> ?k\<guillemotright>; src ?h = trg ?f\<rbrakk> \<Longrightarrow> \<guillemotleft>?\<nu> \<star> ?\<mu> : ?h \<star> ?f \<Rightarrow> ?k \<star> ?g\<guillemotright>
ide ?a \<Longrightarrow> arr ?a
hseq ?\<mu> ?\<nu> \<Longrightarrow> trg (?\<mu> \<star> ?\<nu>) = trg ?\<mu>
ide ?f \<Longrightarrow> \<guillemotleft>?f : ?f \<Rightarrow> ?f\<guillemotright>
\<lbrakk>ide ?w; ide ?w'; \<guillemotleft>?\<theta> : t\<^sub>0 \<star> ?w \<Rightarrow> ?u\<guillemotright>; \<guillemotleft>?\<theta>' : t\<^sub>0 \<star> ?w' \<Rightarrow> ?u\<guillemotright>; \<guillemotleft>?\<beta> : t\<^sub>1 \<star> ?w \<Rightarrow> t\<^sub>1 \<star> ?w'\<guillemotright>; (t \<star> ?\<theta>) \<cdot> \<a>[t, t\<^sub>0, ?w] \<cdot> (\<tau> \<star> ?w) = ((t \<star> ?\<theta>') \<cdot> \<a>[t, t\<^sub>0, ?w'] \<cdot> (\<tau> \<star> ?w')) \<cdot> ?\<beta>\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w \<Rightarrow> ?w'\<guillemotright> \<and> ?\<beta> = t\<^sub>1 \<star> \<gamma> \<and> ?\<theta> = ?\<theta>' \<cdot> (t\<^sub>0 \<star> \<gamma>)
trg t\<^sub>0\<^sup>* = src t\<^sub>0
hseq t\<^sub>0\<^sup>* u\<^sub>1
trg t\<^sub>0u\<^sub>1.p\<^sub>1 = trg (t\<^sub>0\<^sup>* \<star> u\<^sub>1)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
thus "\<And>\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<tau>' \<Rightarrow> ?w\<^sub>\<tau>'\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> ?w\<^sub>\<tau>' \<and> ?\<theta>\<^sub>\<tau>' = ?\<theta>\<^sub>\<tau>' \<cdot> (t\<^sub>0 \<star> \<gamma>)
                         \<Longrightarrow> \<gamma> = ?w\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine = t\<^sub>1 \<star> \<gamma> \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<And>\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> \<gamma> = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>) \<Longrightarrow> \<gamma> = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
by (metis \<theta>\<^sub>\<tau>' comp_arr_dom ide_in_hom(2) in_homE w\<^sub>\<tau>')
[PROOF STATE]
proof (state)
this:
\<guillemotleft>?\<gamma>4 : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>?\<gamma>4 : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
\<guillemotleft>?\<gamma>4 : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
have "\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = ?w\<^sub>\<tau>'"
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
\<guillemotleft>?\<gamma>4 : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> t\<^sub>1 \<star> ?\<gamma>4 = t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> ?\<gamma>4) \<Longrightarrow> ?\<gamma>4 = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. \<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
thus "ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')"
[PROOF STATE]
proof (prove)
using this:
\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
using w\<^sub>\<tau>'
[PROOF STATE]
proof (prove)
using this:
\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>' = t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)

goal (1 subgoal):
 1. ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide (\<gamma>\<^sub>\<tau> \<cdot> \<gamma>\<^sub>\<tau>')

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'

goal (2 subgoals):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
 2. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
thus "\<tau>\<mu>.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1"
[PROOF STATE]
proof (prove)
using this:
inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
using w\<^sub>\<tau> w\<^sub>\<tau>' \<gamma>\<^sub>\<tau> isomorphic_symmetric isomorphic_def
[PROOF STATE]
proof (prove)
using this:
inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'
ide (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<and> is_left_adjoint (\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)
ide (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
?f \<cong> ?g \<Longrightarrow> ?g \<cong> ?f
(?a \<cong> ?a') = (\<exists>f. \<guillemotleft>f : ?a \<Rightarrow> ?a'\<guillemotright> \<and> local.iso f)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<cong> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have iso_\<gamma>\<^sub>\<tau>: "iso \<gamma>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.iso \<gamma>\<^sub>\<tau>
[PROOF STEP]
using \<open>inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'\<close>
[PROOF STATE]
proof (prove)
using this:
inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'

goal (1 subgoal):
 1. local.iso \<gamma>\<^sub>\<tau>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
local.iso \<gamma>\<^sub>\<tau>

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have \<gamma>\<^sub>\<tau>'_eq: "\<gamma>\<^sub>\<tau>' = inv \<gamma>\<^sub>\<tau>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<gamma>\<^sub>\<tau>' = local.inv \<gamma>\<^sub>\<tau>
[PROOF STEP]
using \<open>inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'\<close> inverse_unique
[PROOF STATE]
proof (prove)
using this:
inverse_arrows \<gamma>\<^sub>\<tau> \<gamma>\<^sub>\<tau>'
inverse_arrows ?f ?g \<Longrightarrow> local.inv ?f = ?g

goal (1 subgoal):
 1. \<gamma>\<^sub>\<tau>' = local.inv \<gamma>\<^sub>\<tau>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<gamma>\<^sub>\<tau>' = local.inv \<gamma>\<^sub>\<tau>

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?w\<^sub>\<mu> = "\<tau>\<mu>.p\<^sub>0 \<star> chine"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?w\<^sub>\<mu>' = "\<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?u\<^sub>\<mu> = "s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?\<theta>\<^sub>\<mu> = "the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?\<theta>\<^sub>\<mu>' = "(\<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
let ?\<beta>\<^sub>\<mu> = "\<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot>
                  \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot>
                  (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have w\<^sub>\<mu>: "ide ?w\<^sub>\<mu> \<and> is_left_adjoint ?w\<^sub>\<mu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
using is_map left_adjoints_compose
[PROOF STATE]
proof (prove)
using this:
is_left_adjoint chine
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; src ?f = trg ?f'\<rbrakk> \<Longrightarrow> is_left_adjoint (?f \<star> ?f')

goal (1 subgoal):
 1. ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have w\<^sub>\<mu>': "ide ?w\<^sub>\<mu>' \<and> is_left_adjoint ?w\<^sub>\<mu>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
using \<chi>.is_map left_adjoints_compose
[PROOF STATE]
proof (prove)
using this:
is_left_adjoint \<chi>.chine
\<lbrakk>is_left_adjoint ?f; is_left_adjoint ?f'; src ?f = trg ?f'\<rbrakk> \<Longrightarrow> is_left_adjoint (?f \<star> ?f')

goal (1 subgoal):
 1. ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by (simp add: is_map left_adjoints_compose)
[PROOF STATE]
proof (state)
this:
ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have 1: "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<mu> \<Rightarrow> ?w\<^sub>\<mu>'\<guillemotright> \<and> ?\<beta>\<^sub>\<mu> = u\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<mu> = ?\<theta>\<^sub>\<mu>' \<cdot> (u\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have \<theta>\<^sub>\<mu>: "\<guillemotleft>?\<theta>\<^sub>\<mu> : u\<^sub>0 \<star> ?w\<^sub>\<mu> \<Rightarrow> ?u\<^sub>\<mu>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have \<theta>\<^sub>\<mu>': "\<guillemotleft>?\<theta>\<^sub>\<mu>' : u\<^sub>0 \<star> ?w\<^sub>\<mu>' \<Rightarrow> ?u\<^sub>\<mu>\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>(\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : u\<^sub>0 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<guillemotleft>(\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : u\<^sub>0 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have \<beta>\<^sub>\<mu>: "\<guillemotleft>?\<beta>\<^sub>\<mu> : u\<^sub>1 \<star> ?w\<^sub>\<mu> \<Rightarrow> u\<^sub>1 \<star> ?w\<^sub>\<mu>'\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
proof (intro comp_in_homI)
[PROOF STATE]
proof (state)
goal (9 subgoals):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> ?b7\<guillemotright>
 2. \<guillemotleft>local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : ?b7 \<Rightarrow> ?b6\<guillemotright>
 3. \<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b6 \<Rightarrow> ?b5\<guillemotright>
 4. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : ?b5 \<Rightarrow> ?b4\<guillemotright>
 5. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b4 \<Rightarrow> ?b3\<guillemotright>
 6. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b3 \<Rightarrow> ?b2\<guillemotright>
 7. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 8. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 9. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine] : u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0 \<star> chine \<Rightarrow> (u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine\<guillemotright>

goal (8 subgoals):
 1. \<guillemotleft>local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> ?b6\<guillemotright>
 2. \<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : ?b6 \<Rightarrow> ?b5\<guillemotright>
 3. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : ?b5 \<Rightarrow> ?b4\<guillemotright>
 4. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b4 \<Rightarrow> ?b3\<guillemotright>
 5. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b3 \<Rightarrow> ?b2\<guillemotright>
 6. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 7. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 8. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine \<Rightarrow> (t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
using t\<^sub>0u\<^sub>1.\<phi>_uniqueness(2) hcomp_in_vhom
[PROOF STATE]
proof (prove)
using this:
local.iso t\<^sub>0u\<^sub>1.\<phi>
\<lbrakk>\<guillemotleft>?\<mu> : ?f \<Rightarrow> ?g\<guillemotright>; \<guillemotleft>?\<nu> : ?h \<Rightarrow> ?k\<guillemotright>; src ?h = trg ?f\<rbrakk> \<Longrightarrow> \<guillemotleft>?\<nu> \<star> ?\<mu> : ?h \<star> ?f \<Rightarrow> ?k \<star> ?g\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>
[PROOF STEP]
by (simp add: t\<^sub>0u\<^sub>1.\<phi>_in_hom(2) w_in_hom(2))
[PROOF STATE]
proof (state)
this:
\<guillemotleft>local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine : (u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine\<guillemotright>

goal (7 subgoals):
 1. \<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> ?b5\<guillemotright>
 2. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : ?b5 \<Rightarrow> ?b4\<guillemotright>
 3. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b4 \<Rightarrow> ?b3\<guillemotright>
 4. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b3 \<Rightarrow> ?b2\<guillemotright>
 5. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 6. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 7. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] : (t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
using \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : (t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine \<Rightarrow> t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>

goal (6 subgoals):
 1. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> ?b4\<guillemotright>
 2. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : ?b4 \<Rightarrow> ?b3\<guillemotright>
 3. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b3 \<Rightarrow> ?b2\<guillemotright>
 4. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 5. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 6. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<gamma>\<^sub>\<tau> iso_\<gamma>\<^sub>\<tau>
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
local.iso \<gamma>\<^sub>\<tau>

goal (1 subgoal):
 1. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)
local.iso \<gamma>\<^sub>\<tau>
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (5 subgoals):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b3\<guillemotright>
 2. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : ?b3 \<Rightarrow> ?b2\<guillemotright>
 3. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 4. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 5. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> (t\<^sub>0 \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (4 subgoals):
 1. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b2\<guillemotright>
 2. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : ?b2 \<Rightarrow> ?b1\<guillemotright>
 3. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 4. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1 : (t\<^sub>0 \<star> \<omega>.chine) \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> r\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>1\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
using \<rho>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
trg r\<^sub>0\<^sup>* = src r\<^sub>0

goal (1 subgoal):
 1. \<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 : (t\<^sub>0 \<star> \<omega>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright>

goal (3 subgoals):
 1. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> ?b1\<guillemotright>
 2. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : ?b1 \<Rightarrow> ?b\<guillemotright>
 3. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (2 subgoals):
 1. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> ?b\<guillemotright>
 2. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : ?b \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0 : s\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>0 \<Rightarrow> (u\<^sub>1 \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> (u\<^sub>1 \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 : s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> (u\<^sub>1 \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : (u\<^sub>1 \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
show "\<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] : (u\<^sub>1 \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0\<guillemotright>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : (u\<^sub>1 \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : (u\<^sub>1 \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
have eq\<^sub>\<mu>: "(u \<star> ?\<theta>\<^sub>\<mu>) \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>) =
                   ((u \<star> ?\<theta>\<^sub>\<mu>') \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>'] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>')) \<cdot> ?\<beta>\<^sub>\<mu>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
let ?LHS = "(u \<star> ?\<theta>\<^sub>\<mu>) \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>)"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
let ?RHS = "((u \<star> ?\<theta>\<^sub>\<mu>') \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>'] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>')) \<cdot> ?\<beta>\<^sub>\<mu>"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "?RHS = (u \<star> (\<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                         \<a>[u, u\<^sub>0, \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                         \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                         (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                         \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (u \<star> \<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             \<a>[u, u\<^sub>0, \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "u \<star> (\<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] =
                  (u \<star> \<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
using whisker_left \<mu>.ide_base \<theta>\<^sub>\<mu>'
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
ide u
\<guillemotleft>(\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : u\<^sub>0 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>

goal (1 subgoal):
 1. u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = ((u \<star> \<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (\<mu> \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0])
                      (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "src u = trg \<a>[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
have "(u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0] =
                             \<a>[u, u\<^sub>0 \<star> \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                               \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
using pentagon hseqI' comp_assoc
                    invert_opposite_sides_of_square
                      [of "u \<star> \<a>[u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
                          "\<a>[u, u\<^sub>0 \<star> \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0)"
                          "\<a>[u, u\<^sub>0, \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0]" "\<a>[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"]
                    inv_hcomp \<chi>.is_ide \<chi>.w_simps(3) \<chi>.w_simps(4) \<mu>.base_simps(2) \<mu>.ide_base
                    \<mu>.ide_leg0 \<mu>.leg0_simps(2) \<mu>.leg0_simps(3) \<sigma>.leg1_simps(3)
                    assoc'_eq_inv_assoc ide_hcomp r\<^sub>0s\<^sub>1.ide_u r\<^sub>0s\<^sub>1.p\<^sub>0_simps hcomp_simps(1)
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0))
src u = trg \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) = u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])
local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
\<lbrakk>ide ?f; ide ?g; ide ?h; ide ?k; src ?f = trg ?g; src ?g = trg ?h; src ?h = trg ?k\<rbrakk> \<Longrightarrow> (?f \<star> \<a>[?g, ?h, ?k]) \<cdot> \<a>[?f, ?g \<star> ?h, ?k] \<cdot> (\<a>[?f, ?g, ?h] \<star> ?k) = \<a>[?f, ?g, ?h \<star> ?k] \<cdot> \<a>[?f \<star> ?g, ?h, ?k]
\<lbrakk>arr ?\<mu>; arr ?\<nu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> hseq ?\<nu> ?\<mu>
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>seq (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)); (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]; local.iso (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]); local.iso \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]\<rbrakk> \<Longrightarrow> seq (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) (local.inv \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<and> seq (local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0])) \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<and> (\<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> local.inv \<a>[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = local.inv (u \<star> \<a>[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]
\<lbrakk>local.iso ?\<nu>; local.iso ?\<mu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> local.inv (?\<nu> \<star> ?\<mu>) = local.inv ?\<nu> \<star> local.inv ?\<mu>
ide \<chi>.chine
src \<chi>.chine = src s\<^sub>0
trg \<chi>.chine = src u\<^sub>0
arr u
ide u
ide u\<^sub>0
arr u\<^sub>0
trg u\<^sub>0 = src u
src s\<^sub>1 = src s\<^sub>0
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = local.inv \<a>[?f, ?g, ?h]
\<lbrakk>ide ?\<nu>; ide ?\<mu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> ide (?\<nu> \<star> ?\<mu>)
ide r\<^sub>0s\<^sub>1.p\<^sub>0
trg r\<^sub>0s\<^sub>1.p\<^sub>0 = src s\<^sub>1
hseq ?\<nu> ?\<mu> \<Longrightarrow> src (?\<nu> \<star> ?\<mu>) = src ?\<mu>

goal (1 subgoal):
 1. (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (\<mu> \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot>
                             \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "(u \<star> \<chi>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] =
                  \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
using assoc_naturality [of u \<chi>.the_\<theta> \<rho>\<sigma>.p\<^sub>0] \<chi>.\<theta>_simps(3)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr u; arr \<chi>.the_\<theta>; arr r\<^sub>0s\<^sub>1.p\<^sub>0; src u = trg \<chi>.the_\<theta>; src \<chi>.the_\<theta> = trg r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk> \<Longrightarrow> \<a>[cod u, cod \<chi>.the_\<theta>, cod r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[local.dom u, local.dom \<chi>.the_\<theta>, local.dom r\<^sub>0s\<^sub>1.p\<^sub>0]
trg \<chi>.the_\<theta> = src u

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u, u\<^sub>0 \<star> \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot>
                             (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> \<rho>\<sigma>.p\<^sub>0) =
                  ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
using assoc'_naturality [of \<mu> \<chi>.chine \<rho>\<sigma>.p\<^sub>0]
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr \<mu>; arr \<chi>.chine; arr r\<^sub>0s\<^sub>1.p\<^sub>0; src \<mu> = trg \<chi>.chine; src \<chi>.chine = trg r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod \<mu>, cod \<chi>.chine, cod r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<mu>, local.dom \<chi>.chine, local.dom r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot>
                             (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = \<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                             (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "(\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, \<rho>\<sigma>.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) =
                  \<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
using comp_inv_arr' comp_cod_arr
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> local.inv ?f \<cdot> ?f = local.dom ?f
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f

goal (1 subgoal):
 1. (\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<a>\<^sup>-\<^sup>1[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (\<a>[u, s\<^sub>0, \<rho>\<sigma>.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot>
                             (\<omega>.the_\<theta> \<star> \<rho>\<sigma>.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, \<rho>\<sigma>.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "arr ((u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. seq (u \<star> \<chi>.the_\<theta>) (\<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)
[PROOF STEP]
using \<chi>.\<theta>_simps(3)
[PROOF STATE]
proof (prove)
using this:
trg \<chi>.the_\<theta> = src u

goal (1 subgoal):
 1. seq (u \<star> \<chi>.the_\<theta>) (\<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
seq (u \<star> \<chi>.the_\<theta>) (\<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
hence "((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                     ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) =
                   (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> \<chi>.the_\<theta>) (\<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
using whisker_right
[PROOF STATE]
proof (prove)
using this:
seq (u \<star> \<chi>.the_\<theta>) (\<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>)
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
using \<chi>.\<Delta>_naturality
[PROOF STATE]
proof (prove)
using this:
(\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> = (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu>

goal (1 subgoal):
 1. (u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(u \<star> \<chi>.the_\<theta>) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine] \<cdot> (\<mu> \<star> \<chi>.chine) \<cdot> \<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
have "((u \<star> \<chi>.the_\<theta>) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot>
                            ((\<mu> \<star> \<chi>.chine) \<star> \<rho>\<sigma>.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> \<rho>\<sigma>.p\<^sub>0) =
                          (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> \<rho>\<sigma>.p\<^sub>0"
[PROOF STATE]
proof (prove)
using this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. ((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = (\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (((u \<star> \<chi>.the_\<theta>) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<a>[u, u\<^sub>0, \<chi>.chine] \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> ((\<mu> \<star> \<chi>.chine) \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (?\<theta>\<^sub>\<tau> \<cdot> (t\<^sub>0 \<star> inv \<gamma>\<^sub>\<tau>)) \<cdot>
                             \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = ?\<theta>\<^sub>\<tau>' \<cdot> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine] \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using \<gamma>\<^sub>\<tau>' \<gamma>\<^sub>\<tau>'_eq
[PROOF STATE]
proof (prove)
using this:
\<guillemotleft>\<gamma>\<^sub>\<tau>' : t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<guillemotright> \<and> local.inv (\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau>' \<and> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] = (\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>')
\<gamma>\<^sub>\<tau>' = local.inv \<gamma>\<^sub>\<tau>

goal (1 subgoal):
 1. ((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]) \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>)) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot>
                             ((\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                             (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             ((\<mu> \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                    (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine] =
                  \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((\<a>\<^sup>-\<^sup>1[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot> \<a>[t\<^sub>0, \<tau>\<mu>.p\<^sub>1, chine]) \<cdot>
                      (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine] =
                    ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>1) \<star> chine) \<cdot>
                      (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_inv_arr' \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> local.inv ?f \<cdot> ?f = local.dom ?f
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_cod_arr t\<^sub>0u\<^sub>1.\<phi>_uniqueness
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
\<guillemotleft>?\<mu> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = t\<^sub>0u\<^sub>1.\<phi>
local.iso t\<^sub>0u\<^sub>1.\<phi>

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> ((t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1) \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (t\<^sub>0u\<^sub>1.\<phi> \<cdot> inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using whisker_right t\<^sub>0u\<^sub>1.\<phi>_uniqueness
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?\<nu> \<cdot> ?\<mu> \<star> ?f = (?\<nu> \<star> ?f) \<cdot> (?\<mu> \<star> ?f)
\<guillemotleft>?\<mu> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = t\<^sub>0u\<^sub>1.\<phi>
local.iso t\<^sub>0u\<^sub>1.\<phi>

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = ((u\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_arr_inv' \<tau>.T0.antipar(1) t\<^sub>0u\<^sub>1.\<phi>_uniqueness
[PROOF STATE]
proof (prove)
using this:
local.iso ?f \<Longrightarrow> ?f \<cdot> local.inv ?f = cod ?f
trg t\<^sub>0\<^sup>* = src t\<^sub>0
\<guillemotleft>?\<mu> : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<Rightarrow> u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0\<guillemotright> \<Longrightarrow> ?\<mu> = t\<^sub>0u\<^sub>1.\<phi>
local.iso t\<^sub>0u\<^sub>1.\<phi>

goal (1 subgoal):
 1. (t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(t\<^sub>0u\<^sub>1.\<phi> \<cdot> local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = \<a>\<^sup>-\<^sup>1[u\<^sub>1, \<tau>\<mu>.p\<^sub>0, chine]"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_cod_arr \<tau>.T0.antipar(1)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr ?f; cod ?f = ?b\<rbrakk> \<Longrightarrow> ?b \<cdot> ?f = ?f
trg t\<^sub>0\<^sup>* = src t\<^sub>0

goal (1 subgoal):
 1. ((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
using comp_assoc
[PROOF STATE]
proof (prove)
using this:
((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> ((t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> (\<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine)) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine) \<cdot>
                             \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
using assoc'_naturality [of \<mu> \<tau>\<mu>.p\<^sub>0 chine] comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>arr \<mu>; arr t\<^sub>0u\<^sub>1.p\<^sub>0; arr chine; src \<mu> = trg t\<^sub>0u\<^sub>1.p\<^sub>0; src t\<^sub>0u\<^sub>1.p\<^sub>0 = trg chine\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[cod \<mu>, cod t\<^sub>0u\<^sub>1.p\<^sub>0, cod chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<mu>, local.dom t\<^sub>0u\<^sub>1.p\<^sub>0, local.dom chine]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
using uw\<theta> pentagon comp_assoc
                  invert_opposite_sides_of_square
                    [of "u \<star> \<a>[u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine]"
                        "\<a>[u, u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, \<tau>\<mu>.p\<^sub>0] \<star> chine)" "\<a>[u, u\<^sub>0, ?w\<^sub>\<mu>]"
                        "\<a>[u \<star> u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine]"]
                  \<mu>.base_simps(2) \<mu>.ide_base \<mu>.ide_leg0 \<mu>.leg0_simps(2) assoc'_eq_inv_assoc
                  ide_hcomp hcomp_simps(1) t\<^sub>0u\<^sub>1.ide_u
[PROOF STATE]
proof (prove)
using this:
ide chine \<and> \<guillemotleft>the_\<theta> : (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<lbrakk>ide ?f; ide ?g; ide ?h; ide ?k; src ?f = trg ?g; src ?g = trg ?h; src ?h = trg ?k\<rbrakk> \<Longrightarrow> (?f \<star> \<a>[?g, ?h, ?k]) \<cdot> \<a>[?f, ?g \<star> ?h, ?k] \<cdot> (\<a>[?f, ?g, ?h] \<star> ?k) = \<a>[?f, ?g, ?h \<star> ?k] \<cdot> \<a>[?f \<star> ?g, ?h, ?k]
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f
\<lbrakk>seq (u \<star> \<a>[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)); (u \<star> \<a>[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) = \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> \<a>[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]; local.iso (u \<star> \<a>[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]); local.iso \<a>[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]\<rbrakk> \<Longrightarrow> seq (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) (local.inv \<a>[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<and> seq (local.inv (u \<star> \<a>[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<and> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine)) \<cdot> local.inv \<a>[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = local.inv (u \<star> \<a>[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine]
arr u
ide u
ide u\<^sub>0
arr u\<^sub>0
\<lbrakk>ide ?f; ide ?g; ide ?h; src ?f = trg ?g; src ?g = trg ?h\<rbrakk> \<Longrightarrow> \<a>\<^sup>-\<^sup>1[?f, ?g, ?h] = local.inv \<a>[?f, ?g, ?h]
\<lbrakk>ide ?\<nu>; ide ?\<mu>; src ?\<nu> = trg ?\<mu>\<rbrakk> \<Longrightarrow> ide (?\<nu> \<star> ?\<mu>)
hseq ?\<nu> ?\<mu> \<Longrightarrow> src (?\<nu> \<star> ?\<mu>) = src ?\<mu>
ide t\<^sub>0u\<^sub>1.p\<^sub>0

goal (1 subgoal):
 1. (u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
by force
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta>) \<cdot> (\<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u \<star> u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
have "... = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<tau>\<mu>.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, ?w\<^sub>\<mu>] \<cdot> (\<mu> \<star> ?w\<^sub>\<mu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
using whisker_left comp_assoc
[PROOF STATE]
proof (prove)
using this:
\<lbrakk>ide ?f; seq ?\<nu> ?\<mu>\<rbrakk> \<Longrightarrow> ?f \<star> ?\<nu> \<cdot> ?\<mu> = (?f \<star> ?\<nu>) \<cdot> (?f \<star> ?\<mu>)
(?h \<cdot> ?g) \<cdot> ?f = ?h \<cdot> ?g \<cdot> ?f

goal (1 subgoal):
 1. ((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
((u \<star> the_\<theta>) \<cdot> (u \<star> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine])) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)

goal (1 subgoal):
 1. (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
(u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
show "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : ?w\<^sub>\<mu> \<Rightarrow> ?w\<^sub>\<mu>'\<guillemotright> \<and> ?\<beta>\<^sub>\<mu> = u\<^sub>1 \<star> \<gamma> \<and> ?\<theta>\<^sub>\<mu> = ?\<theta>\<^sub>\<mu>' \<cdot> (u\<^sub>0 \<star> \<gamma>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
using w\<^sub>\<mu> w\<^sub>\<mu>' \<theta>\<^sub>\<mu> \<theta>\<^sub>\<mu>' \<beta>\<^sub>\<mu> eq\<^sub>\<mu> \<mu>.T2 [of ?w\<^sub>\<mu> ?w\<^sub>\<mu>' ?\<theta>\<^sub>\<mu> ?u\<^sub>\<mu> ?\<theta>\<^sub>\<mu>' ?\<beta>\<^sub>\<mu>]
[PROOF STATE]
proof (prove)
using this:
ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
\<guillemotleft>the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>(\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : u\<^sub>0 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
\<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>
(u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]
\<lbrakk>ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine); ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0); \<guillemotleft>the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>(\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] : u\<^sub>0 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] : u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> u\<^sub>1 \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; (u \<star> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]) \<cdot> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine] \<cdot> (\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) = ((u \<star> (\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> \<a>[u, u\<^sub>0, \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<mu> \<star> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)) \<cdot> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine]\<rbrakk> \<Longrightarrow> \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. \<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)
[PROOF STEP]
by fast
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
obtain \<gamma>\<^sub>\<mu> where \<gamma>\<^sub>\<mu>: "\<guillemotleft>\<gamma>\<^sub>\<mu> : ?w\<^sub>\<mu> \<Rightarrow> ?w\<^sub>\<mu>'\<guillemotright> \<and> ?\<beta>\<^sub>\<mu> = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> ?\<theta>\<^sub>\<mu> = ?\<theta>\<^sub>\<mu>' \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<mu>. \<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using 1
[PROOF STATE]
proof (prove)
using this:
\<exists>!\<gamma>. \<guillemotleft>\<gamma> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>)

goal (1 subgoal):
 1. (\<And>\<gamma>\<^sub>\<mu>. \<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>) \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
show "?w\<^sub>\<mu> \<cong> ?w\<^sub>\<mu>'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
using w\<^sub>\<mu> w\<^sub>\<mu>' \<gamma>\<^sub>\<mu> BS3 [of ?w\<^sub>\<mu> ?w\<^sub>\<mu>' \<gamma>\<^sub>\<mu> \<gamma>\<^sub>\<mu>] isomorphic_def
[PROOF STATE]
proof (prove)
using this:
ide (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine) \<and> is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine)
ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)
\<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright> \<and> \<a>[u\<^sub>1, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (\<chi>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (t\<^sub>0 \<star> local.inv \<gamma>\<^sub>\<tau>) \<cdot> \<a>[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> (local.inv t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = u\<^sub>1 \<star> \<gamma>\<^sub>\<mu> \<and> the_\<theta> \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0, chine] = ((\<chi>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> \<a>\<^sup>-\<^sup>1[u\<^sub>0, \<chi>.chine, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (u\<^sub>0 \<star> \<gamma>\<^sub>\<mu>)
\<lbrakk>is_left_adjoint (t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine); is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0); \<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>; \<guillemotleft>\<gamma>\<^sub>\<mu> : t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<Rightarrow> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>\<rbrakk> \<Longrightarrow> local.iso \<gamma>\<^sub>\<mu> \<and> local.iso \<gamma>\<^sub>\<mu> \<and> \<gamma>\<^sub>\<mu> = \<gamma>\<^sub>\<mu>
(?a \<cong> ?a') = (\<exists>f. \<guillemotleft>f : ?a \<Rightarrow> ?a'\<guillemotright> \<and> local.iso f)

goal (1 subgoal):
 1. t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t\<^sub>0u\<^sub>1.p\<^sub>0 \<star> chine \<cong> \<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0

goal:
No subgoals!
[PROOF STEP]
qed