[STATEMENT]
lemma lateEarlyOutput:
  fixes P  :: pi
  and   a  :: name
  and   b  :: name
  and   P' :: pi

  assumes "P \<longmapsto>\<^sub>la[b] \<prec>\<^sub>l P'"

  shows "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'

goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
proof(nominal_induct rule: Late_Semantics.outputInduct)
[PROOF STATE]
proof (state)
goal (9 subgoals):
 1. \<And>a b P. a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P
 2. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 6. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 7. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 8. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 9. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Output a b P)
[PROOF STATE]
proof (state)
this:

goal (9 subgoals):
 1. \<And>a b P. a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P
 2. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 6. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 7. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 8. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 9. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P
[PROOF STEP]
by(rule Early_Semantics.Output)
[PROOF STATE]
proof (state)
this:
a{b}.P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P

goal (8 subgoals):
 1. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 5. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 6. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 7. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 8. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (8 subgoals):
 1. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 5. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 6. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 7. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 8. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Match P a b P' c)
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (8 subgoals):
 1. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 5. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 6. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 7. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 8. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (8 subgoals):
 1. \<And>P a b P' c. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 5. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 6. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 7. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 8. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. [c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by(rule Early_Semantics.Match)
[PROOF STATE]
proof (state)
this:
[c\<frown>c]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (7 subgoals):
 1. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 6. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 7. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (7 subgoals):
 1. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 6. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 7. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Mismatch P a b P' c d)
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
c \<noteq> d

goal (7 subgoals):
 1. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 6. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 7. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'" and "c \<noteq> d"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' &&& c \<noteq> d
[PROOF STEP]
by fact+
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
c \<noteq> d

goal (7 subgoals):
 1. \<And>P a b P' c d. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; c \<noteq> d\<rbrakk> \<Longrightarrow> [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 4. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 5. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 6. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 7. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
c \<noteq> d

goal (1 subgoal):
 1. [c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by(rule Early_Semantics.Mismatch)
[PROOF STATE]
proof (state)
this:
[c\<noteq>d]P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (6 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 5. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 6. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (6 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 5. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 6. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Sum1 P a b P' Q)
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (6 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 5. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 6. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (6 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 3. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 4. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 5. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 6. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by(rule Early_Semantics.Sum1)
[PROOF STATE]
proof (state)
this:
P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (5 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 4. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 5. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (5 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 4. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 5. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Sum2 Q a b Q' P)
[PROOF STATE]
proof (state)
this:
Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (5 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 4. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 5. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "Q \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e Q'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (5 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
 2. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 3. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 4. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 5. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (1 subgoal):
 1. P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
[PROOF STEP]
by(rule Early_Semantics.Sum2)
[PROOF STATE]
proof (state)
this:
P \<oplus> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (4 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 3. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 4. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (4 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 3. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 4. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Par1 P a b P' Q)
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (4 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 3. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 4. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (4 subgoals):
 1. \<And>P a b P' Q. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
 2. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 3. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 4. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q
[PROOF STEP]
by(rule Early_Semantics.Par1F)
[PROOF STATE]
proof (state)
this:
P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' \<parallel> Q

goal (3 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 2. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 3. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 2. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 3. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Par2 Q a b Q' P)
[PROOF STATE]
proof (state)
this:
Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (3 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 2. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 3. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "Q \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e Q'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (3 subgoals):
 1. \<And>Q a b Q' P. \<lbrakk>Q \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l Q'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
 2. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 3. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'

goal (1 subgoal):
 1. P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'
[PROOF STEP]
by(rule Early_Semantics.Par2F)
[PROOF STATE]
proof (state)
this:
P \<parallel> Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P \<parallel> Q'

goal (2 subgoals):
 1. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 2. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 2. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Res P a b P' x)
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
x \<noteq> a
x \<noteq> b
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (2 subgoals):
 1. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 2. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'" and "x \<noteq> a" and "x \<noteq> b"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' &&& x \<noteq> a &&& x \<noteq> b
[PROOF STEP]
by fact+
[PROOF STATE]
proof (state)
this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
x \<noteq> a
x \<noteq> b

goal (2 subgoals):
 1. \<And>P a b P' x. \<lbrakk>P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; x \<noteq> a; x \<noteq> b; P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
 2. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
x \<noteq> a
x \<noteq> b

goal (1 subgoal):
 1. <\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'
[PROOF STEP]
by(force intro: Early_Semantics.ResF)
[PROOF STATE]
proof (state)
this:
<\<nu>x>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e <\<nu>x>P'

goal (1 subgoal):
 1. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
case(Bang P a b P')
[PROOF STATE]
proof (state)
this:
P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'
P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
have "P \<parallel> !P \<longmapsto>\<^sub>ea[b] \<prec>\<^sub>e P'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by fact
[PROOF STATE]
proof (state)
this:
P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. \<And>P a b P'. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'; P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'\<rbrakk> \<Longrightarrow> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
P \<parallel> !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal (1 subgoal):
 1. !P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'
[PROOF STEP]
by(rule Early_Semantics.Bang)
[PROOF STATE]
proof (state)
this:
!P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'

goal:
No subgoals!
[PROOF STEP]
qed


[STATEMENT]
lemma cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_ext_none [simp]:
  "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. cast (object_ptr.Ext (Inr (Inr (Inr object_ext_ptr)))) = None
[PROOF STEP]
by(simp add: cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def)


[STATEMENT]
lemma "ECQm(\<^bold>\<not>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ECQm \<^bold>\<not>
[PROOF STEP]
nitpick
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. ECQm \<^bold>\<not>
[PROOF STEP]
oops