[STATEMENT]
lemma kruskal_edge_arc:
  assumes "equivalence F"
      and "forest w"
      and "arc e"
      and "regular F"
      and "F \<le> forest_components (F \<sqinter> w)"
      and "regular w"
      and "w * e\<^sup>T = bot"
      and "e * F * e = bot"
      and "e\<^sup>T \<le> w\<^sup>\<star>"
    shows "arc (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * -F)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. arc (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)
[PROOF STEP]
proof (unfold arc_expanded, intro conjI)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?E = "top * e\<^sup>T * w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?F = "F * e\<^sup>T * top"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?G = "top * e * -F"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?FF = "F * e\<^sup>T * e * F"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?GG = "-F * e\<^sup>T * e * -F"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
let ?w = "forest_components (F \<sqinter> w)"
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (simp add: assms(5) inf.coboundedI1)
[PROOF STATE]
proof (state)
this:
F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)
[PROOF STEP]
apply (rule inf_star_import)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. (F \<sqinter> w)\<^sup>T \<le> w\<^sup>T
 2. univalent (w\<^sup>T)
 3. reflexive ((F \<sqinter> w)\<^sup>\<star>)
 4. regular (w\<^sup>T)
[PROOF STEP]
apply (simp add: conv_isotone)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. univalent (w\<^sup>T)
 2. reflexive ((F \<sqinter> w)\<^sup>\<star>)
 3. regular (w\<^sup>T)
[PROOF STEP]
apply (simp add: assms(2))
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. reflexive ((F \<sqinter> w)\<^sup>\<star>)
 2. regular (w\<^sup>T)
[PROOF STEP]
apply (simp add: star.circ_reflexive)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. regular (w\<^sup>T)
[PROOF STEP]
by (metis assms(6) conv_complement)
[PROOF STATE]
proof (state)
this:
forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
forest_components (F \<sqinter> w) \<sqinter> w\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)
[PROOF STEP]
using comp_inf.mult_left_isotone mult_right_isotone star_isotone
[PROOF STATE]
proof (prove)
using this:
?x \<le> ?y \<Longrightarrow> ?x \<sqinter> ?z \<le> ?y \<sqinter> ?z
?x \<le> ?y \<Longrightarrow> ?z * ?x \<le> ?z * ?y
?x \<le> ?y \<Longrightarrow> ?x\<^sup>\<star> \<le> ?y\<^sup>\<star>

goal (1 subgoal):
 1. (F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F \<sqinter> w)\<^sup>T\<^sup>\<star> * ((F \<sqinter> w)\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (F \<sqinter> w)\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) = (F \<sqinter> w)\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (simp add: assms(2) acyclic_star_inf_conv)
[PROOF STATE]
proof (state)
this:
(F \<sqinter> w)\<^sup>T\<^sup>\<star> * (w\<^sup>\<star> \<sqinter> w\<^sup>T\<^sup>\<star>) = (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>
[PROOF STEP]
have "w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
using this:
F \<sqinter> w\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (1 subgoal):
 1. w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e
[PROOF STEP]
by (simp add: mult_left_isotone mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e = w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e
[PROOF STEP]
by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint sup_commute)
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e = w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * e\<^sup>T * e = w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e = w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e
[PROOF STEP]
by (simp add: assms(7))
[PROOF STATE]
proof (state)
this:
w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e = w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * e\<^sup>T * e \<squnion> w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e = w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>+"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>+
[PROOF STEP]
by (metis assms(3) arc_univalent mult_assoc mult_1_right mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>+

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>+ * e\<^sup>T * e \<le> w * (F \<sqinter> w)\<^sup>T\<^sup>+

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (F \<sqinter> w)\<^sup>T\<^sup>+ \<le> w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (simp add: comp_associative conv_isotone mult_left_isotone mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>+ \<le> w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w)\<^sup>T\<^sup>+ \<le> w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>
[PROOF STEP]
using assms(2) coreflexive_comp_top_inf inf.sup_right_divisibility
[PROOF STATE]
proof (prove)
using this:
forest w
coreflexive ?x \<Longrightarrow> ?x * top \<sqinter> ?y = ?x * ?y
(?y \<le> ?x) = (\<exists>z. z \<sqinter> ?x = ?y)

goal (1 subgoal):
 1. w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * w\<^sup>T * (F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> (F \<sqinter> w)\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> F\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (simp add: conv_dist_inf star_isotone)
[PROOF STATE]
proof (state)
this:
(F \<sqinter> w)\<^sup>T\<^sup>\<star> \<le> F\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F\<^sup>T\<^sup>\<star>
[PROOF STEP]
have 1: "w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F"
[PROOF STATE]
proof (prove)
using this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F\<^sup>T\<^sup>\<star>

goal (1 subgoal):
 1. w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F
[PROOF STEP]
by (metis assms(1) order.antisym mult_1_left mult_left_isotone star.circ_plus_same star.circ_reflexive star.left_plus_below_circ star_left_induct_mult_iff)
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "F * e\<^sup>T * e \<le> forest_components (F \<sqinter> w) * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e \<le> forest_components (F \<sqinter> w) * e\<^sup>T * e
[PROOF STEP]
by (simp add: assms(5) mult_left_isotone)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e \<le> forest_components (F \<sqinter> w) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e \<le> forest_components (F \<sqinter> w) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> forest_components w * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. forest_components (F \<sqinter> w) * e\<^sup>T * e \<le> forest_components w * e\<^sup>T * e
[PROOF STEP]
by (simp add: comp_isotone conv_dist_inf star_isotone)
[PROOF STATE]
proof (state)
this:
forest_components (F \<sqinter> w) * e\<^sup>T * e \<le> forest_components w * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
forest_components (F \<sqinter> w) * e\<^sup>T * e \<le> forest_components w * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>T\<^sup>\<star> * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. forest_components w * e\<^sup>T * e = w\<^sup>T\<^sup>\<star> * e\<^sup>T * e
[PROOF STEP]
by (metis (no_types) assms(7) comp_associative conv_bot conv_dist_comp conv_involutive conv_star_commute star_absorb)
[PROOF STATE]
proof (state)
this:
forest_components w * e\<^sup>T * e = w\<^sup>T\<^sup>\<star> * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
forest_components w * e\<^sup>T * e = w\<^sup>T\<^sup>\<star> * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T\<^sup>\<star> * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (metis assms(3) arc_univalent mult_assoc mult_1_right mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>\<star> * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>
[PROOF STEP]
have 2: "F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>

goal (1 subgoal):
 1. F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "w * F * e\<^sup>T * e \<le> w * F * e\<^sup>T * e * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * F * e\<^sup>T * e \<le> w * F * e\<^sup>T * e * e\<^sup>T * e
[PROOF STEP]
using comp_associative ex231c mult_right_isotone
[PROOF STATE]
proof (prove)
using this:
?x * ?y * ?z = ?x * (?y * ?z)
?x \<le> ?x * ?x\<^sup>T * ?x
?x \<le> ?y \<Longrightarrow> ?z * ?x \<le> ?z * ?y

goal (1 subgoal):
 1. w * F * e\<^sup>T * e \<le> w * F * e\<^sup>T * e * e\<^sup>T * e
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w * F * e\<^sup>T * e \<le> w * F * e\<^sup>T * e * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * F * e\<^sup>T * e \<le> w * F * e\<^sup>T * e * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * F * e\<^sup>T * e * e\<^sup>T * e = w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e
[PROOF STEP]
using 2
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>

goal (1 subgoal):
 1. w * F * e\<^sup>T * e * e\<^sup>T * e = w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e
[PROOF STEP]
by (simp add: comp_associative inf.absorb1)
[PROOF STATE]
proof (state)
this:
w * F * e\<^sup>T * e * e\<^sup>T * e = w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * F * e\<^sup>T * e * e\<^sup>T * e = w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e
[PROOF STEP]
by (metis assms(3) arc_univalent mult_assoc mult_1_right mult_right_isotone mult_left_isotone inf.sup_left_isotone)
[PROOF STATE]
proof (state)
this:
w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * (F * e\<^sup>T * e \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F
[PROOF STEP]
using 1
[PROOF STATE]
proof (prove)
using this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F

goal (1 subgoal):
 1. w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w * F * e\<^sup>T * e \<le> F
[PROOF STEP]
have 3: "w * F * e\<^sup>T * e \<le> F"
[PROOF STATE]
proof (prove)
using this:
w * F * e\<^sup>T * e \<le> F

goal (1 subgoal):
 1. w * F * e\<^sup>T * e \<le> F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w * F * e\<^sup>T * e \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "e\<^sup>T * e * F * w\<^sup>T \<le> F"
[PROOF STATE]
proof (prove)
using this:
w * F * e\<^sup>T * e \<le> F

goal (1 subgoal):
 1. e\<^sup>T * e * F * w\<^sup>T \<le> F
[PROOF STEP]
by (metis assms(1) conv_dist_comp conv_dist_inf conv_involutive inf.absorb_iff1 mult_assoc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * top \<sqinter> F"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * e * F * w\<^sup>T \<le> F

goal (1 subgoal):
 1. e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * top \<sqinter> F
[PROOF STEP]
by (simp add: comp_associative mult_right_isotone)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * top \<sqinter> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * top \<sqinter> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> e\<^sup>T * e * F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T * top \<sqinter> F \<le> e\<^sup>T * e * F
[PROOF STEP]
by (metis conv_involutive dedekind_1 inf_top_left mult_assoc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top \<sqinter> F \<le> e\<^sup>T * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F
[PROOF STEP]
have 4: "e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F

goal (1 subgoal):
 1. e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(top * e)\<^sup>T * (?F \<sqinter> w\<^sup>T\<^sup>\<star>) = e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) = e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (metis assms(1) comp_inf.star.circ_decompose_9 comp_inf.star_star_absorb conv_dist_comp conv_involutive conv_top covector_inf_comp_3 vector_top_closed mult_assoc)
[PROOF STATE]
proof (state)
this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) = e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) = e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star> = e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (simp add: assms(3) arc_top_edge)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star> = e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top * e * F * w\<^sup>T\<^sup>\<star> = e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> e\<^sup>T * e * F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star> \<le> e\<^sup>T * e * F
[PROOF STEP]
using 4 star_right_induct_mult
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * e * F * w\<^sup>T \<le> e\<^sup>T * e * F
?y * ?x \<le> ?y \<Longrightarrow> ?y * ?x\<^sup>\<star> \<le> ?y

goal (1 subgoal):
 1. e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star> \<le> e\<^sup>T * e * F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star> \<le> e\<^sup>T * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F * w\<^sup>T\<^sup>\<star> \<le> e\<^sup>T * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T * e * F \<le> F
[PROOF STEP]
by (metis assms(3) arc_injective conv_involutive mult_1_left mult_left_isotone)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * e * F \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F
[PROOF STEP]
have 5: "(top * e)\<^sup>T * (?F \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F"
[PROOF STATE]
proof (prove)
using this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F

goal (1 subgoal):
 1. (top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(?F \<sqinter> w) * w\<^sup>T\<^sup>+ = ?F \<sqinter> w * w\<^sup>T\<^sup>+"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ = F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+
[PROOF STEP]
by (simp add: vector_export_comp)
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ = F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ = F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> ?F \<sqinter> w\<^sup>T\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+ \<le> F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>
[PROOF STEP]
by (metis assms(2) comp_associative inf.sup_right_isotone mult_left_isotone star.circ_transitive_equal star_left_unfold_equal sup.absorb_iff2 sup_monoid.add_assoc)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+ \<le> F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * w\<^sup>T\<^sup>+ \<le> F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have 6: "... \<le> top * e * F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star> \<le> top * e * F
[PROOF STEP]
using 5
[PROOF STATE]
proof (prove)
using this:
(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F

goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star> \<le> top * e * F
[PROOF STEP]
by (metis assms(3) shunt_mapping conv_dist_comp conv_involutive conv_top)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star> \<le> top * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F
[PROOF STEP]
have 7: "(?F \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F"
[PROOF STATE]
proof (prove)
using this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F

goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "e\<^sup>T * top * e \<le> 1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. coreflexive (e\<^sup>T * top * e)
[PROOF STEP]
by (simp add: assms(3) point_injective)
[PROOF STATE]
proof (state)
this:
coreflexive (e\<^sup>T * top * e)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
coreflexive (e\<^sup>T * top * e)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. reflexive F
[PROOF STEP]
by (simp add: assms(1))
[PROOF STATE]
proof (state)
this:
reflexive F

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
e\<^sup>T * top * e \<le> F
[PROOF STEP]
have 8: "e * -F * e\<^sup>T \<le> bot"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top * e \<le> F

goal (1 subgoal):
 1. e * - F * e\<^sup>T \<le> bot
[PROOF STEP]
by (metis p_antitone p_antitone_iff p_bot regular_closed_bot schroeder_3_p schroeder_4_p mult_assoc)
[PROOF STATE]
proof (state)
this:
e * - F * e\<^sup>T \<le> bot

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "?FF \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> ?GG) * w\<^sup>T \<le> ?F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
using comp_inf.mult_left_isotone mult_isotone mult_assoc
[PROOF STATE]
proof (prove)
using this:
?x \<le> ?y \<Longrightarrow> ?x \<sqinter> ?z \<le> ?y \<sqinter> ?z
\<lbrakk>?w \<le> ?y; ?x \<le> ?z\<rbrakk> \<Longrightarrow> ?w * ?x \<le> ?y * ?z
?a * ?b * ?c = ?a * (?b * ?c)

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> ?F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> ?G) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T
[PROOF STEP]
by (metis assms(3) arc_top_edge comp_inf.star.circ_decompose_9 comp_inf_covector inf.sup_right_isotone inf_le2 mult_left_isotone mult_right_isotone vector_top_closed mult_assoc)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (?F \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> ?G) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T
[PROOF STEP]
by (simp add: vector_export_comp)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (?F \<sqinter> w) * w\<^sup>T\<^sup>+ * (?G\<^sup>T \<sqinter> w\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)
[PROOF STEP]
by (simp add: covector_comp_inf covector_comp_inf_1 covector_mult_closed)
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * (w\<^sup>T\<^sup>+ \<sqinter> top * e * - F) * w\<^sup>T = (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> top * e * F * (?G\<^sup>T \<sqinter> w\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)
[PROOF STEP]
using 7 mult_left_isotone
[PROOF STATE]
proof (prove)
using this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ \<le> top * e * F
?x \<le> ?y \<Longrightarrow> ?x * ?z \<le> ?y * ?z

goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> w) * w\<^sup>T\<^sup>+ * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> top * e * F * ?G\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * (top * e * - F)\<^sup>T
[PROOF STEP]
by (simp add: mult_right_isotone)
[PROOF STATE]
proof (state)
this:
top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * (top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * e * F * ((top * e * - F)\<^sup>T \<sqinter> w\<^sup>T) \<le> top * e * F * (top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * e * -F * e\<^sup>T * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * e * F * (top * e * - F)\<^sup>T = top * e * - F * e\<^sup>T * top
[PROOF STEP]
by (metis assms(1) conv_complement conv_dist_comp conv_top equivalence_comp_left_complement mult_assoc)
[PROOF STATE]
proof (state)
this:
top * e * F * (top * e * - F)\<^sup>T = top * e * - F * e\<^sup>T * top

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> top * e * - F * e\<^sup>T * top
[PROOF STEP]
have 9: "?FF \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> ?GG) * w\<^sup>T = bot"
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> top * e * - F * e\<^sup>T * top

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
[PROOF STEP]
using 8
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> top * e * - F * e\<^sup>T * top
e * - F * e\<^sup>T \<le> bot

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
[PROOF STEP]
by (metis comp_associative covector_bot_closed le_bot vector_bot_closed)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence 10: "?FF \<sqinter> w * (w\<^sup>+ \<sqinter> ?GG) * w\<^sup>T = bot"
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
[PROOF STEP]
using assms(1) comp_associative conv_bot conv_complement conv_dist_comp conv_dist_inf conv_star_commute star.circ_plus_same
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
equivalence F
?x * ?y * ?z = ?x * (?y * ?z)
symmetric bot
(- ?x)\<^sup>T = - ?x\<^sup>T
(?x * ?y)\<^sup>T = ?y\<^sup>T * ?x\<^sup>T
(?x \<sqinter> ?y)\<^sup>T = ?x\<^sup>T \<sqinter> ?y\<^sup>T
?x\<^sup>\<star>\<^sup>T = ?x\<^sup>T\<^sup>\<star>
?x\<^sup>\<star> * ?x = ?x\<^sup>+

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G) * top * (w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G)\<^sup>T = (?F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G)) * top * ((w \<sqinter> ?E \<sqinter> ?G)\<^sup>T \<sqinter> ?F\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T)
[PROOF STEP]
by (simp add: conv_dist_inf inf_commute inf_left_commute)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (?F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G)) * top * (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T \<sqinter> ?F\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T
[PROOF STEP]
using covector_comp_inf vector_conv_covector vector_mult_closed vector_top_closed
[PROOF STATE]
proof (prove)
using this:
covector ?z \<Longrightarrow> ?x * (?y \<sqinter> ?z) = ?x * ?y \<sqinter> ?z
vector ?v = covector (?v\<^sup>T)
vector ?y \<Longrightarrow> vector (?x * ?y)
surjective top

goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) = (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * top * (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T \<sqinter> ?F\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T
[PROOF STEP]
by (simp add: vector_export_comp)
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?F \<sqinter> top * e * F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * top * (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T
[PROOF STEP]
by (simp add: assms(1) conv_dist_comp inf_assoc inf_commute mult_assoc)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T = F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?F * e * F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * top * (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top \<sqinter> top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * (top * (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T)
[PROOF STEP]
using assms(3) arc_top_edge comp_associative
[PROOF STATE]
proof (prove)
using this:
arc e
arc ?x \<Longrightarrow> ?x\<^sup>T * top * ?x = ?x\<^sup>T * ?x
?x * ?y * ?z = ?x * (?y * ?z)

goal (1 subgoal):
 1. F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * top * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T)

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * (top * (?G\<^sup>T \<sqinter> (?E\<^sup>T \<sqinter> w\<^sup>T)))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)))
[PROOF STEP]
by (simp add: conv_dist_inf inf_assoc inf_commute inf_left_commute)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)))

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)))

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * (?G * (?E\<^sup>T \<sqinter> w\<^sup>T))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))
[PROOF STEP]
by (metis covector_comp_inf_1 covector_top_closed covector_mult_closed inf_top_left)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T))

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> (w \<sqinter> ?E \<sqinter> ?G) * (?G \<sqinter> ?E) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T
[PROOF STEP]
by (metis covector_comp_inf_1 covector_top_closed mult_assoc)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> (w \<sqinter> ?E) * (?G\<^sup>T \<sqinter> ?G \<sqinter> ?E) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T
[PROOF STEP]
by (metis covector_comp_inf_1 covector_mult_closed inf.sup_monoid.add_assoc vector_top_closed)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) * (top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (?E\<^sup>T \<sqinter> ?G\<^sup>T \<sqinter> ?G \<sqinter> ?E) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T
[PROOF STEP]
by (metis covector_comp_inf_1 covector_mult_closed inf.sup_monoid.add_assoc vector_top_closed)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * ((top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (?E\<^sup>T \<sqinter> ?E \<sqinter> (?G\<^sup>T \<sqinter> ?G)) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T
[PROOF STEP]
by (simp add: inf_commute inf_left_commute)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (?E\<^sup>T \<sqinter> ?E \<sqinter> (-F * e\<^sup>T * top \<sqinter> ?G)) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T
[PROOF STEP]
by (simp add: assms(1) conv_complement conv_dist_comp mult_assoc)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (?E\<^sup>T \<sqinter> ?E \<sqinter> (-F * e\<^sup>T * ?G)) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (?E\<^sup>T \<sqinter> ?E \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (metis assms(3) arc_top_edge comp_associative)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F)) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> ?E \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (simp add: comp_associative conv_dist_comp conv_star_commute)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * (w\<^sup>\<star> * e * ?E \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> ?FF \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (metis assms(3) mult_assoc mult_1_right mult_left_isotone mult_right_isotone inf.sup_left_isotone inf.sup_right_isotone arc_expanded)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * ((w\<^sup>+ \<squnion> 1 \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (simp add: assms(2) cancel_separate_eq star_left_unfold_equal sup_monoid.add_commute)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?FF \<sqinter> w * ((w\<^sup>+ \<squnion> 1 \<squnion> w\<^sup>T\<^sup>+) \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
using star.circ_plus_one star_left_unfold_equal sup_assoc
[PROOF STATE]
proof (prove)
using this:
?x\<^sup>\<star> = (1::'a) \<squnion> ?x\<^sup>\<star>
(1::'a) \<squnion> ?x\<^sup>+ = ?x\<^sup>\<star>
?x \<squnion> ?y \<squnion> ?z = ?x \<squnion> (?y \<squnion> ?z)

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (?FF \<sqinter> w * (w\<^sup>+ \<sqinter> ?GG) * w\<^sup>T) \<squnion> (?FF \<sqinter> w * (1 \<sqinter> ?GG) * w\<^sup>T) \<squnion> (?FF \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> ?GG) * w\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by (simp add: inf_sup_distrib1 inf_sup_distrib2 semiring.distrib_left semiring.distrib_right)
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * ((w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w * (1 \<sqinter> ?GG) * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
using 9 10
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T = bot

goal (1 subgoal):
 1. F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<squnion> F * e\<^sup>T * e * F \<sqinter> w * (w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w * w\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * w\<^sup>T
[PROOF STEP]
by (metis inf.cobounded1 mult_1_right mult_left_isotone mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * ((1::'a) \<sqinter> - F * e\<^sup>T * e * - F) * w\<^sup>T \<le> w * w\<^sup>T

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> 1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. injective w
[PROOF STEP]
by (simp add: assms(2))
[PROOF STATE]
proof (state)
this:
injective w

goal (3 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
 2. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 3. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
[PROOF STEP]
show "(w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G) * top * (w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G)\<^sup>T \<le> 1"
[PROOF STATE]
proof (prove)
using this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "w\<^sup>T\<^sup>+ \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> ?G \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
using top_greatest inf.sup_left_isotone inf.sup_right_isotone mult_left_isotone
[PROOF STATE]
proof (prove)
using this:
?x \<le> top
?y \<le> ?x \<Longrightarrow> ?y \<sqinter> ?z \<le> ?x \<sqinter> ?z
?y \<le> ?x \<Longrightarrow> ?z \<sqinter> ?y \<le> ?z \<sqinter> ?x
?x \<le> ?y \<Longrightarrow> ?x * ?z \<le> ?y * ?z

goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w\<^sup>T\<^sup>+ \<sqinter> ?G \<sqinter> w\<^sup>T * ?F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top)
[PROOF STEP]
using comp_associative inf.sup_right_isotone mult_right_isotone top.extremum
[PROOF STATE]
proof (prove)
using this:
?x * ?y * ?z = ?x * (?y * ?z)
?y \<le> ?x \<Longrightarrow> ?z \<sqinter> ?y \<le> ?z \<sqinter> ?x
?x \<le> ?y \<Longrightarrow> ?z * ?x \<le> ?z * ?y
?a \<le> top

goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top)
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> ?F) \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top) = w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F
[PROOF STEP]
using assms(2) inf_assoc inf_commute inf_left_commute univalent_comp_left_dist_inf
[PROOF STATE]
proof (prove)
using this:
forest w
?x \<sqinter> ?y \<sqinter> ?z = ?x \<sqinter> (?y \<sqinter> ?z)
?x \<sqinter> ?y = ?y \<sqinter> ?x
?x \<sqinter> (?y \<sqinter> ?z) = ?y \<sqinter> (?x \<sqinter> ?z)
univalent ?x \<Longrightarrow> ?x * (?y \<sqinter> ?z) = ?x * ?y \<sqinter> ?x * ?z

goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top) = w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top) = w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> top * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top) = w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w\<^sup>T * (top * e * F) \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F \<le> w\<^sup>T * (top * e * F) \<sqinter> top * e * - F
[PROOF STEP]
using 6
[PROOF STATE]
proof (prove)
using this:
F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star> \<le> top * e * F

goal (1 subgoal):
 1. w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F \<le> w\<^sup>T * (top * e * F) \<sqinter> top * e * - F
[PROOF STEP]
by (metis inf.sup_monoid.add_commute inf.sup_right_isotone mult_right_isotone)
[PROOF STATE]
proof (state)
this:
w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F \<le> w\<^sup>T * (top * e * F) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>T * (w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top) \<sqinter> top * e * - F \<le> w\<^sup>T * (top * e * F) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> top * e * F \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>T * (top * e * F) \<sqinter> top * e * - F \<le> top * e * F \<sqinter> top * e * - F
[PROOF STEP]
by (metis comp_associative comp_inf_covector mult_left_isotone top.extremum)
[PROOF STATE]
proof (state)
this:
w\<^sup>T * (top * e * F) \<sqinter> top * e * - F \<le> top * e * F \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>T * (top * e * F) \<sqinter> top * e * - F \<le> top * e * F \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = bot"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * e * F \<sqinter> top * e * - F = bot
[PROOF STEP]
by (metis assms(3) conv_dist_comp conv_involutive conv_top inf_p mult_right_zero univalent_comp_left_dist_inf)
[PROOF STATE]
proof (state)
this:
top * e * F \<sqinter> top * e * - F = bot

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> bot
[PROOF STEP]
have 11: "w\<^sup>T\<^sup>+ \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot"
[PROOF STATE]
proof (prove)
using this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<le> bot

goal (1 subgoal):
 1. w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot
[PROOF STEP]
by (simp add: order.antisym)
[PROOF STATE]
proof (state)
this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence 12: "w\<^sup>+ \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot"
[PROOF STATE]
proof (prove)
using this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot

goal (1 subgoal):
 1. w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot
[PROOF STEP]
using assms(1) comp_associative conv_bot conv_complement conv_dist_comp conv_dist_inf conv_star_commute star.circ_plus_same
[PROOF STATE]
proof (prove)
using this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot
equivalence F
?x * ?y * ?z = ?x * (?y * ?z)
symmetric bot
(- ?x)\<^sup>T = - ?x\<^sup>T
(?x * ?y)\<^sup>T = ?y\<^sup>T * ?x\<^sup>T
(?x \<sqinter> ?y)\<^sup>T = ?x\<^sup>T \<sqinter> ?y\<^sup>T
?x\<^sup>\<star>\<^sup>T = ?x\<^sup>T\<^sup>\<star>
?x\<^sup>\<star> * ?x = ?x\<^sup>+

goal (1 subgoal):
 1. w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G)\<^sup>T * top * (w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G) = ((w \<sqinter> ?E \<sqinter> ?G)\<^sup>T \<sqinter> ?F\<^sup>T) * top * (?F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) = ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))
[PROOF STEP]
by (simp add: conv_dist_inf inf_commute inf_left_commute)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) = ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) = ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T * ?F * (?F \<sqinter> (w \<sqinter> ?E \<sqinter> ?G))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))
[PROOF STEP]
by (simp add: covector_inf_comp_3 vector_mult_closed)
[PROOF STATE]
proof (state)
this:
((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * top * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F))

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * (w \<sqinter> ?E \<sqinter> ?G)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)
[PROOF STEP]
using covector_comp_inf covector_inf_comp_3 vector_conv_covector vector_mult_closed
[PROOF STATE]
proof (prove)
using this:
covector ?z \<Longrightarrow> ?x * (?y \<sqinter> ?z) = ?x * ?y \<sqinter> ?z
vector ?x \<Longrightarrow> (?y \<sqinter> ?x\<^sup>T) * ?z = ?y * (?x \<sqinter> ?z)
vector ?v = covector (?v\<^sup>T)
vector ?y \<Longrightarrow> vector (?x * ?y)

goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top) * (F * e\<^sup>T * top \<sqinter> (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * (w \<sqinter> ?E) \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F
[PROOF STEP]
by (simp add: comp_associative comp_inf_covector)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F) = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> ?E \<sqinter> ?G)\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w \<sqinter> ?E \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F
[PROOF STEP]
by (simp add: comp_associative comp_inf_covector)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> top * e * - F = (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (?G\<^sup>T \<sqinter> (?E\<^sup>T \<sqinter> w\<^sup>T)) * (?F \<sqinter> ?F\<^sup>T) * w \<sqinter> ?E \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F
[PROOF STEP]
by (simp add: conv_dist_inf inf.left_commute inf.sup_monoid.add_commute)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F)\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?G\<^sup>T \<sqinter> (?E\<^sup>T \<sqinter> w\<^sup>T) * (?F \<sqinter> ?F\<^sup>T) * w \<sqinter> ?E \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F
[PROOF STEP]
by (metis (no_types) comp_associative conv_dist_comp conv_top vector_export_comp)
[PROOF STATE]
proof (state)
this:
((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
((top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T)) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?G\<^sup>T \<sqinter> ?E\<^sup>T \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w \<sqinter> ?E \<sqinter> ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F
[PROOF STEP]
by (metis (no_types) comp_associative inf_assoc conv_dist_comp conv_top vector_export_comp)
[PROOF STATE]
proof (state)
this:
(top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(top * e * - F)\<^sup>T \<sqinter> ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T) * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?E\<^sup>T \<sqinter> ?E \<sqinter> (?G\<^sup>T \<sqinter> ?G) \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (simp add: inf_assoc inf.left_commute inf.sup_monoid.add_commute)
[PROOF STATE]
proof (state)
this:
(top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(top * e * - F)\<^sup>T \<sqinter> (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> top * e * - F = (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * e * top \<sqinter> ?E \<sqinter> (?G\<^sup>T \<sqinter> ?G) \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (simp add: comp_associative conv_dist_comp conv_star_commute)
[PROOF STATE]
proof (state)
this:
(top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * e * ?E \<sqinter> (?G\<^sup>T \<sqinter> ?G) \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * e * top \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (?G\<^sup>T \<sqinter> ?G) \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<le> w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (metis assms(3) mult_assoc mult_1_right mult_left_isotone mult_right_isotone inf.sup_left_isotone arc_expanded)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<le> w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * e * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w \<le> w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (-F * e\<^sup>T * top \<sqinter> ?G) \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (simp add: assms(1) conv_complement conv_dist_comp mult_assoc)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> ((top * e * - F)\<^sup>T \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> -F * e\<^sup>T * ?G \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> (- F * e\<^sup>T * top \<sqinter> top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * (?F \<sqinter> ?F\<^sup>T) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w
[PROOF STEP]
by (metis assms(3) arc_top_edge mult_assoc)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * (top * e * - F) \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * (?F \<sqinter> top * e * F) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w
[PROOF STEP]
by (simp add: assms(1) conv_dist_comp mult_assoc)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> (F * e\<^sup>T * top)\<^sup>T) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * (?F * e * F) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w
[PROOF STEP]
by (metis comp_associative comp_inf_covector inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top \<sqinter> top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
by (metis assms(3) arc_top_edge mult_assoc)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * (F * e\<^sup>T * top * e * F) * w = w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w\<^sup>+ \<squnion> 1 \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
by (simp add: assms(2) cancel_separate_eq star_left_unfold_equal sup_monoid.add_commute)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * w\<^sup>T\<^sup>\<star> \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w\<^sup>+ \<squnion> 1 \<squnion> w\<^sup>T\<^sup>+) \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
using star.circ_plus_one star_left_unfold_equal sup_assoc
[PROOF STATE]
proof (prove)
using this:
?x\<^sup>\<star> = (1::'a) \<squnion> ?x\<^sup>\<star>
(1::'a) \<squnion> ?x\<^sup>+ = ?x\<^sup>\<star>
?x \<squnion> ?y \<squnion> ?z = ?x \<squnion> (?y \<squnion> ?z)

goal (1 subgoal):
 1. (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
by presburger
[PROOF STATE]
proof (state)
this:
(w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>\<star>) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w\<^sup>+ \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w) \<squnion> (1 \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w) \<squnion> (w\<^sup>T\<^sup>+ \<sqinter> -F * e\<^sup>T * e * -F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w
[PROOF STEP]
by (simp add: inf_sup_distrib2)
[PROOF STATE]
proof (state)
this:
(w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w\<^sup>+ \<squnion> (1::'a) \<squnion> w\<^sup>T\<^sup>+) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> 1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. coreflexive (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w)
[PROOF STEP]
using 11 12
[PROOF STATE]
proof (prove)
using this:
w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot
w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w = bot

goal (1 subgoal):
 1. coreflexive (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w)
[PROOF STEP]
by (simp add: inf.coboundedI1)
[PROOF STATE]
proof (state)
this:
coreflexive (w\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> (1::'a) \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w \<squnion> w\<^sup>T\<^sup>+ \<sqinter> - F * e\<^sup>T * e * - F \<sqinter> w\<^sup>T * F * e\<^sup>T * e * F * w)

goal (2 subgoals):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
 2. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
show "(w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G)\<^sup>T * top * (w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G) \<le> 1"
[PROOF STATE]
proof (prove)
using this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))

goal (1 subgoal):
 1. coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
coreflexive ((w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F)\<^sup>T * top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(w \<sqinter> -F) * (F \<sqinter> w\<^sup>T) \<le> w * w\<^sup>T \<sqinter> -F * F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) \<le> w * w\<^sup>T \<sqinter> - F * F
[PROOF STEP]
by (simp add: mult_isotone)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) \<le> w * w\<^sup>T \<sqinter> - F * F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) \<le> w * w\<^sup>T \<sqinter> - F * F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> 1 \<sqinter> -F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * w\<^sup>T \<sqinter> - F * F \<le> (1::'a) \<sqinter> - F
[PROOF STEP]
using assms(1,2) comp_inf.comp_isotone equivalence_comp_right_complement
[PROOF STATE]
proof (prove)
using this:
equivalence F
forest w
\<lbrakk>?x \<le> ?y; ?w \<le> ?z\<rbrakk> \<Longrightarrow> ?x \<sqinter> ?w \<le> ?y \<sqinter> ?z
equivalence ?x \<Longrightarrow> - ?x * ?x = - ?x

goal (1 subgoal):
 1. w * w\<^sup>T \<sqinter> - F * F \<le> (1::'a) \<sqinter> - F
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
w * w\<^sup>T \<sqinter> - F * F \<le> (1::'a) \<sqinter> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w * w\<^sup>T \<sqinter> - F * F \<le> (1::'a) \<sqinter> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = bot"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (1::'a) \<sqinter> - F = bot
[PROOF STEP]
using assms(1) bot_unique pp_isotone pseudo_complement_pp
[PROOF STATE]
proof (prove)
using this:
equivalence F
(?a \<le> bot) = (?a = bot)
?x \<le> ?y \<Longrightarrow> - - ?x \<le> - - ?y
(?x \<sqinter> ?y = bot) = (- - ?x \<le> - ?y)

goal (1 subgoal):
 1. (1::'a) \<sqinter> - F = bot
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(1::'a) \<sqinter> - F = bot

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) \<le> bot
[PROOF STEP]
have 13: "(w \<sqinter> -F) * (F \<sqinter> w\<^sup>T) = bot"
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) \<le> bot

goal (1 subgoal):
 1. (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) = bot
[PROOF STEP]
by (simp add: order.antisym)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) = bot

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "w \<sqinter> ?G \<le> F * (w \<sqinter> ?G)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F)
[PROOF STEP]
by (metis assms(1) mult_1_left mult_right_dist_sup sup.absorb_iff2)
[PROOF STATE]
proof (state)
this:
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (metis eq_refl le_supE star.circ_back_loop_fixpoint)
[PROOF STATE]
proof (state)
this:
F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
have 14: "w \<sqinter> ?G \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "w \<sqinter> top * e * F \<le> w * (e * F)\<^sup>T * e * F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w \<sqinter> top * e * F \<le> w * (e * F)\<^sup>T * e * F
[PROOF STEP]
by (metis (no_types) comp_inf.star_slide dedekind_2 inf_left_commute inf_top_right mult_assoc)
[PROOF STATE]
proof (state)
this:
w \<sqinter> top * e * F \<le> w * (e * F)\<^sup>T * e * F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w \<sqinter> top * e * F \<le> w * (e * F)\<^sup>T * e * F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w * (e * F)\<^sup>T * e * F \<le> F
[PROOF STEP]
using 3 assms(1)
[PROOF STATE]
proof (prove)
using this:
w * F * e\<^sup>T * e \<le> F
equivalence F

goal (1 subgoal):
 1. w * (e * F)\<^sup>T * e * F \<le> F
[PROOF STEP]
by (metis comp_associative conv_dist_comp mult_left_isotone preorder_idempotent)
[PROOF STATE]
proof (state)
this:
w * (e * F)\<^sup>T * e * F \<le> F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w \<sqinter> top * e * F \<le> F
[PROOF STEP]
have "w \<sqinter> -F \<le> -(top * e * F)"
[PROOF STATE]
proof (prove)
using this:
w \<sqinter> top * e * F \<le> F

goal (1 subgoal):
 1. w \<sqinter> - F \<le> - (top * e * F)
[PROOF STEP]
using order.trans p_shunting_swap pp_increasing
[PROOF STATE]
proof (prove)
using this:
w \<sqinter> top * e * F \<le> F
\<lbrakk>?a \<le> ?b; ?b \<le> ?c\<rbrakk> \<Longrightarrow> ?a \<le> ?c
(?x \<sqinter> ?y \<le> - ?z) = (?x \<sqinter> ?z \<le> - ?y)
?x \<le> - - ?x

goal (1 subgoal):
 1. w \<sqinter> - F \<le> - (top * e * F)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
w \<sqinter> - F \<le> - (top * e * F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
w \<sqinter> - F \<le> - (top * e * F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = ?G"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. - (top * e * F) = top * e * - F
[PROOF STEP]
by (metis assms(3) comp_mapping_complement conv_dist_comp conv_involutive conv_top)
[PROOF STATE]
proof (state)
this:
- (top * e * F) = top * e * - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
w \<sqinter> - F \<le> top * e * - F
[PROOF STEP]
have "(w \<sqinter> -F) * F * (w \<sqinter> ?G) = (w \<sqinter> -F \<sqinter> ?G) * F * (w \<sqinter> ?G)"
[PROOF STATE]
proof (prove)
using this:
w \<sqinter> - F \<le> top * e * - F

goal (1 subgoal):
 1. (w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) = (w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F)
[PROOF STEP]
by (simp add: inf.absorb1)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) = (w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) = (w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * F * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F) \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * F * w
[PROOF STEP]
by (simp add: comp_isotone)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F) \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * F * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * F * (w \<sqinter> top * e * - F) \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * F * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * forest_components (F \<sqinter> w) * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * F * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w
[PROOF STEP]
by (simp add: assms(5) mult_left_isotone mult_right_isotone)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * F * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * F * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w
[PROOF STEP]
by (simp add: mult_left_isotone mult_right_isotone star_isotone mult_assoc)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * forest_components (F \<sqinter> w) * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star>
[PROOF STEP]
by (simp add: comp_associative mult_right_isotone star.circ_plus_same star.left_plus_below_circ)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> * w \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> -F \<sqinter> ?G) * w\<^sup>\<star> \<squnion> (w \<sqinter> -F \<sqinter> ?G) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> = (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star>
[PROOF STEP]
by (metis comp_associative inf.sup_monoid.add_assoc mult_left_dist_sup star.circ_loop_fixpoint sup_commute)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> = (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>\<star> * w\<^sup>\<star> = (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * w\<^sup>\<star> \<squnion> (w \<sqinter> -F \<sqinter> ?G) * (F \<sqinter> w)\<^sup>T * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star> \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top
[PROOF STEP]
by (metis mult_assoc top_greatest mult_right_isotone sup_right_isotone)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star> \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T\<^sup>+ * w\<^sup>\<star> \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> -F \<sqinter> ?G) * w\<^sup>\<star> \<squnion> (w \<sqinter> -F) * (F \<sqinter> w)\<^sup>T * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top
[PROOF STEP]
using inf.cobounded1 mult_left_isotone sup_right_isotone
[PROOF STATE]
proof (prove)
using this:
?a \<sqinter> ?b \<le> ?a
?x \<le> ?y \<Longrightarrow> ?x * ?z \<le> ?y * ?z
?x \<le> ?y \<Longrightarrow> ?z \<squnion> ?x \<le> ?z \<squnion> ?y

goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F \<sqinter> top * e * - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> (w \<sqinter> ?G) * w\<^sup>\<star> \<squnion> (w \<sqinter> -F) * (F \<sqinter> w)\<^sup>T * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top
[PROOF STEP]
using inf.sup_monoid.add_assoc inf.sup_right_isotone mult_left_isotone sup_commute sup_right_isotone
[PROOF STATE]
proof (prove)
using this:
?a \<sqinter> ?b \<sqinter> ?c = ?a \<sqinter> (?b \<sqinter> ?c)
?y \<le> ?x \<Longrightarrow> ?z \<sqinter> ?y \<le> ?z \<sqinter> ?x
?x \<le> ?y \<Longrightarrow> ?x * ?z \<le> ?y * ?z
?x \<squnion> ?y = ?y \<squnion> ?x
?x \<le> ?y \<Longrightarrow> ?z \<squnion> ?x \<le> ?z \<squnion> ?y

goal (1 subgoal):
 1. (w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top \<le> (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (w \<sqinter> ?G) * w\<^sup>\<star> \<squnion> (w \<sqinter> -F) * (F \<sqinter> w\<^sup>T) * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top = (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top
[PROOF STEP]
by (simp add: assms(1) conv_dist_inf)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top = (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w)\<^sup>T * top = (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> 1 * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top \<le> (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
using 13
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) = bot

goal (1 subgoal):
 1. (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top \<le> (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top \<le> (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> top * e * - F) * w\<^sup>\<star> \<squnion> (w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) * top \<le> (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
using assms(1) mult_left_isotone
[PROOF STATE]
proof (prove)
using this:
equivalence F
?x \<le> ?y \<Longrightarrow> ?x * ?z \<le> ?y * ?z

goal (1 subgoal):
 1. (1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(1::'a) * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
have 15: "(w \<sqinter> -F) * F * (w \<sqinter> ?G) \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. (w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "(w \<sqinter> F) * F * (w \<sqinter> ?G) \<le> F * F * (w \<sqinter> ?G)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * F * (w \<sqinter> top * e * - F)
[PROOF STEP]
by (simp add: mult_left_isotone)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = F * (w \<sqinter> ?G)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * F * (w \<sqinter> top * e * - F) = F * (w \<sqinter> top * e * - F)
[PROOF STEP]
by (simp add: assms(1) preorder_idempotent)
[PROOF STATE]
proof (state)
this:
F * F * (w \<sqinter> top * e * - F) = F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
F * F * (w \<sqinter> top * e * - F) = F * (w \<sqinter> top * e * - F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (metis eq_refl le_supE star.circ_back_loop_fixpoint)
[PROOF STATE]
proof (state)
this:
F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
have "(w \<sqinter> F) * F * (w \<sqinter> ?G) \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. (w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "((w \<sqinter> F) \<squnion> (w \<sqinter> -F)) * F * (w \<sqinter> ?G) \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. (w \<sqinter> F \<squnion> w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
using 15
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
(w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. (w \<sqinter> F \<squnion> w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (simp add: semiring.distrib_right)
[PROOF STATE]
proof (state)
this:
(w \<sqinter> F \<squnion> w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "w * F * (w \<sqinter> ?G) \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
(w \<sqinter> F \<squnion> w \<sqinter> - F) * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (metis assms(4) maddux_3_11_pp)
[PROOF STATE]
proof (state)
this:
w * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "w * F * (w \<sqinter> ?G) * w\<^sup>\<star> \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
w * F * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w * F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (metis (full_types) comp_associative mult_left_isotone star.circ_transitive_equal)
[PROOF STATE]
proof (state)
this:
w * F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "w\<^sup>\<star> * (w \<sqinter> ?G) \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
w * F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w\<^sup>\<star> * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
using 14
[PROOF STATE]
proof (prove)
using this:
w * F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w\<^sup>\<star> * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (simp add: mult_assoc star_left_induct)
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence 16: "w\<^sup>+ \<sqinter> ?G \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
w\<^sup>\<star> * (w \<sqinter> top * e * - F) \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. w\<^sup>+ \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (simp add: covector_comp_inf covector_mult_closed star.circ_plus_same)
[PROOF STATE]
proof (state)
this:
w\<^sup>+ \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have 17: "e\<^sup>T * top * e\<^sup>T \<le> -F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T * top * e\<^sup>T \<le> - F
[PROOF STEP]
using assms(8) le_bot triple_schroeder_p
[PROOF STATE]
proof (prove)
using this:
e * F * e = bot
?a \<le> bot \<Longrightarrow> ?a = bot
(?x * ?y * ?z \<le> - ?w) = (?x\<^sup>T * ?w * ?z\<^sup>T \<le> - ?y)

goal (1 subgoal):
 1. e\<^sup>T * top * e\<^sup>T \<le> - F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top * e\<^sup>T \<le> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "(top * e)\<^sup>T * e\<^sup>T \<le> -F"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top * e\<^sup>T \<le> - F

goal (1 subgoal):
 1. (top * e)\<^sup>T * e\<^sup>T \<le> - F
[PROOF STEP]
by (simp add: conv_dist_comp)
[PROOF STATE]
proof (state)
this:
(top * e)\<^sup>T * e\<^sup>T \<le> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence 18: "e\<^sup>T \<le> ?G"
[PROOF STATE]
proof (prove)
using this:
(top * e)\<^sup>T * e\<^sup>T \<le> - F

goal (1 subgoal):
 1. e\<^sup>T \<le> top * e * - F
[PROOF STEP]
by (metis assms(3) shunt_mapping conv_dist_comp conv_involutive conv_top)
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> top * e * - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "e\<^sup>T \<le> -F"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. e\<^sup>T \<le> - F
[PROOF STEP]
using 17
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top * e\<^sup>T \<le> - F

goal (1 subgoal):
 1. e\<^sup>T \<le> - F
[PROOF STEP]
by (simp add: assms(3) arc_top_arc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> -1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. irreflexive (- F)
[PROOF STEP]
by (simp add: assms(1) p_antitone)
[PROOF STATE]
proof (state)
this:
irreflexive (- F)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
irreflexive (e\<^sup>T)
[PROOF STEP]
have "e\<^sup>T \<le> w\<^sup>\<star> \<sqinter> -1"
[PROOF STATE]
proof (prove)
using this:
irreflexive (e\<^sup>T)

goal (1 subgoal):
 1. e\<^sup>T \<le> w\<^sup>\<star> \<sqinter> - (1::'a)
[PROOF STEP]
using assms(9)
[PROOF STATE]
proof (prove)
using this:
irreflexive (e\<^sup>T)
e\<^sup>T \<le> w\<^sup>\<star>

goal (1 subgoal):
 1. e\<^sup>T \<le> w\<^sup>\<star> \<sqinter> - (1::'a)
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> w\<^sup>\<star> \<sqinter> - (1::'a)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> w\<^sup>\<star> \<sqinter> - (1::'a)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... \<le> w\<^sup>+"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. w\<^sup>\<star> \<sqinter> - (1::'a) \<le> w\<^sup>+
[PROOF STEP]
using shunting_var_p star_left_unfold_equal sup_commute
[PROOF STATE]
proof (prove)
using this:
(?x \<sqinter> - ?y \<le> ?z) = (?x \<le> ?z \<squnion> - - ?y)
(1::'a) \<squnion> ?x\<^sup>+ = ?x\<^sup>\<star>
?x \<squnion> ?y = ?y \<squnion> ?x

goal (1 subgoal):
 1. w\<^sup>\<star> \<sqinter> - (1::'a) \<le> w\<^sup>+
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
w\<^sup>\<star> \<sqinter> - (1::'a) \<le> w\<^sup>+

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
e\<^sup>T \<le> w\<^sup>+
[PROOF STEP]
have "e\<^sup>T \<le> w\<^sup>+ \<sqinter> ?G"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T \<le> w\<^sup>+

goal (1 subgoal):
 1. e\<^sup>T \<le> w\<^sup>+ \<sqinter> top * e * - F
[PROOF STEP]
using 18
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T \<le> w\<^sup>+
e\<^sup>T \<le> top * e * - F

goal (1 subgoal):
 1. e\<^sup>T \<le> w\<^sup>+ \<sqinter> top * e * - F
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> w\<^sup>+ \<sqinter> top * e * - F

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "e\<^sup>T \<le> F * (w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T \<le> w\<^sup>+ \<sqinter> top * e * - F

goal (1 subgoal):
 1. e\<^sup>T \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
using 16 order_trans
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T \<le> w\<^sup>+ \<sqinter> top * e * - F
w\<^sup>+ \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
\<lbrakk>?x \<le> ?y; ?y \<le> ?z\<rbrakk> \<Longrightarrow> ?x \<le> ?z

goal (1 subgoal):
 1. e\<^sup>T \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
e\<^sup>T \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = (F * w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> = (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (simp add: comp_associative comp_inf_covector)
[PROOF STATE]
proof (state)
this:
F * (w \<sqinter> top * e * - F) * w\<^sup>\<star> = (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
e\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
have "e\<^sup>T * top * e\<^sup>T \<le> (F * w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. e\<^sup>T * top * e\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (simp add: assms(3) arc_top_arc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top * e\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "e\<^sup>T * top * (e * top)\<^sup>T \<le> (F * w \<sqinter> ?G) * w\<^sup>\<star>"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top * e\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. e\<^sup>T * top * (e * top)\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>
[PROOF STEP]
by (metis conv_dist_comp conv_top vector_top_closed mult_assoc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top * (e * top)\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "e\<^sup>T * top \<le> (F * w \<sqinter> ?G) * w\<^sup>\<star> * e * top"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top * (e * top)\<^sup>T \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star>

goal (1 subgoal):
 1. e\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top
[PROOF STEP]
by (metis assms(3) shunt_bijective mult_assoc)
[PROOF STATE]
proof (state)
this:
e\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "(top * e)\<^sup>T * top \<le> (F * w \<sqinter> ?G) * w\<^sup>\<star> * e * top"
[PROOF STATE]
proof (prove)
using this:
e\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. (top * e)\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top
[PROOF STEP]
by (simp add: conv_dist_comp mult_assoc)
[PROOF STATE]
proof (state)
this:
(top * e)\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
hence "top \<le> top * e * (F * w \<sqinter> ?G) * w\<^sup>\<star> * e * top"
[PROOF STATE]
proof (prove)
using this:
(top * e)\<^sup>T * top \<le> (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. top \<le> top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top
[PROOF STEP]
by (metis assms(3) shunt_mapping conv_dist_comp conv_involutive conv_top mult_assoc)
[PROOF STATE]
proof (state)
this:
top \<le> top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top \<le> top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> ?G\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top = top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)
[PROOF STEP]
by (metis comp_associative comp_inf_vector_1)
[PROOF STATE]
proof (state)
this:
top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top = top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * e * (F * w \<sqinter> top * e * - F) * w\<^sup>\<star> * e * top = top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> ?G\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)
[PROOF STEP]
by (metis comp_inf_vector_1 inf_top.left_neutral)
[PROOF STATE]
proof (state)
this:
top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * e * F * w * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * (w \<sqinter> ?F) * (w\<^sup>\<star> * e * top \<sqinter> ?G\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)
[PROOF STEP]
by (simp add: assms(1) conv_dist_comp mult_assoc)
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> (top * e * F)\<^sup>T) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * (w \<sqinter> ?F) * (?E\<^sup>T \<sqinter> ?G\<^sup>T)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T)
[PROOF STEP]
by (simp add: comp_associative conv_dist_comp conv_star_commute)
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> F * e\<^sup>T * top) * (w\<^sup>\<star> * e * top \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T)

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * (w \<sqinter> ?F \<sqinter> ?G) * ?E\<^sup>T"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T
[PROOF STEP]
by (simp add: comp_associative comp_inf_vector_1)
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
also
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> F * e\<^sup>T * top) * ((top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T \<sqinter> (top * e * - F)\<^sup>T) = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
have "... = top * (w \<sqinter> ?F \<sqinter> ?G \<sqinter> ?E) * top"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * top
[PROOF STEP]
using comp_inf_vector_1 mult_assoc
[PROOF STATE]
proof (prove)
using this:
(?x \<sqinter> top * ?y) * ?z = ?x * (?z \<sqinter> (top * ?y)\<^sup>T)
?a * ?b * ?c = ?a * (?b * ?c)

goal (1 subgoal):
 1. top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * top
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F) * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T = top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
finally
[PROOF STATE]
proof (chain)
picking this:
top \<le> top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * top
[PROOF STEP]
show "top * (w \<sqinter> ?E \<sqinter> ?F \<sqinter> ?G) * top = top"
[PROOF STATE]
proof (prove)
using this:
top \<le> top * (w \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>) * top

goal (1 subgoal):
 1. times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))
[PROOF STEP]
by (simp add: inf_commute inf_left_commute top_le)
[PROOF STATE]
proof (state)
this:
times_top_class.total (top * (w \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star> \<sqinter> F * e\<^sup>T * top \<sqinter> top * e * - F))

goal:
No subgoals!
[PROOF STEP]
qed