[STATEMENT]
lemma cdid:"d^-1 O d \<subseteq> e \<union> ov \<union> ov^-1 \<union> d \<union> d^-1 \<union> s \<union> s^-1 \<union> f \<union> f^-1 "
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. d\<inverse> O d \<subseteq> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
fix x::"'a\<times>'a"
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "x \<in> d^-1 O d"
[PROOF STATE]
proof (state)
this:
x \<in> d\<inverse> O d

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x \<in> d\<inverse> O d
[PROOF STEP]
obtain p q z where x:"x = (p,q)" and "(p,z) \<in> d^-1" and "(z, q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
x \<in> d\<inverse> O d

goal (1 subgoal):
 1. (\<And>p q z. \<lbrakk>x = (p, q); (p, z) \<in> d\<inverse>; (z, q) \<in> d\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x = (p, q)
(p, z) \<in> d\<inverse>
(z, q) \<in> d

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from \<open>(p,z) \<in> d^-1\<close>
[PROOF STATE]
proof (chain)
picking this:
(p, z) \<in> d\<inverse>
[PROOF STEP]
obtain k l u v where kp:"k\<parallel>p" and kl:"k\<parallel>l" and lz:"l\<parallel>z" and pv:"p\<parallel>v" and zu:"z\<parallel>u" and uv:"u\<parallel>v"
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> d\<inverse>

goal (1 subgoal):
 1. (\<And>k l v u. \<lbrakk>k \<parallel> p; k \<parallel> l; l \<parallel> z; p \<parallel> v; z \<parallel> u; u \<parallel> v\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> d\<inverse>
((?p, ?q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. (\<And>k l v u. \<lbrakk>k \<parallel> p; k \<parallel> l; l \<parallel> z; p \<parallel> v; z \<parallel> u; u \<parallel> v\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k \<parallel> p
k \<parallel> l
l \<parallel> z
p \<parallel> v
z \<parallel> u
u \<parallel> v

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from \<open>(z,q) \<in> d\<close>
[PROOF STATE]
proof (chain)
picking this:
(z, q) \<in> d
[PROOF STEP]
obtain k' l'  u' v'  where kpq:"k'\<parallel>q" and kplp:"k'\<parallel>l'" and lpz:"l'\<parallel>z"  and qvp:"q\<parallel>v'" and zup:"z\<parallel>u'" and upvp:"u'\<parallel>v'"
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> d

goal (1 subgoal):
 1. (\<And>k' l' v' u'. \<lbrakk>k' \<parallel> q; k' \<parallel> l'; l' \<parallel> z; q \<parallel> v'; z \<parallel> u'; u' \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> d
((?p, ?q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. (\<And>k' l' v' u'. \<lbrakk>k' \<parallel> q; k' \<parallel> l'; l' \<parallel> z; q \<parallel> v'; z \<parallel> u'; u' \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k' \<parallel> q
k' \<parallel> l'
l' \<parallel> z
q \<parallel> v'
z \<parallel> u'
u' \<parallel> v'

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from kp kpq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k' \<parallel> q
[PROOF STEP]
have "k\<parallel>q \<oplus> ((\<exists>t. k\<parallel>t \<and> t\<parallel>q) \<oplus> (\<exists>t. k'\<parallel>t \<and> t\<parallel>p))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k' \<parallel> q

goal (1 subgoal):
 1. k \<parallel> q \<oplus> ((\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k' \<parallel> q
\<lbrakk>?p \<parallel> ?q; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?p \<parallel> ?s \<oplus> ((\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?s) \<oplus> (\<exists>t. ?r \<parallel> t \<and> t \<parallel> ?q))

goal (1 subgoal):
 1. k \<parallel> q \<oplus> ((\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k \<parallel> q \<oplus> ((\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p))

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q \<oplus> ((\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q \<oplus> ((\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p))

goal (1 subgoal):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
[PROOF STATE]
proof (state)
this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O d \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus "x \<in> e \<union> ov \<union> ov^-1 \<union> d \<union> d^-1 \<union> s \<union> s^-1 \<union> f \<union> f^-1"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof (elim disjE)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "?A\<and>\<not>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
have kq:?A
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. k \<parallel> q
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
k \<parallel> q

goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from pv qvp
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v
q \<parallel> v'
[PROOF STEP]
have "p\<parallel>v' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>v') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>v))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'
\<lbrakk>?p \<parallel> ?q; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?p \<parallel> ?s \<oplus> ((\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?s) \<oplus> (\<exists>t. ?r \<parallel> t \<and> t \<parallel> ?q))

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (1 subgoal):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof (elim disjE)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "?A\<and>\<not>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?A
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. p \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<parallel> v'

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with kq kp qvp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
p \<parallel> v'
[PROOF STEP]
have "p = q"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
p \<parallel> v'

goal (1 subgoal):
 1. p = q
[PROOF STEP]
using M4
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
p \<parallel> v'
\<lbrakk>?p \<parallel> ?q; ?q \<parallel> ?s; ?p \<parallel> ?r; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?q = ?r

goal (1 subgoal):
 1. p = q
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
p = q

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p = q

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x e
[PROOF STATE]
proof (prove)
using this:
p = q
x = (p, q)
((?p, ?q) \<in> e) = (?p = ?q)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?B
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with kq kp qvp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x s
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
q \<parallel> v'
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
x = (p, q)
((?p, ?q) \<in> s) = (\<exists>k u v. k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>\<not>?B\<and>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with kq kp pv
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
k \<parallel> p
p \<parallel> v
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
p \<parallel> v
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x s
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> p
p \<parallel> v
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
x = (p, q)
((?p, ?q) \<in> s) = (\<exists>k u v. k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
\<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
have ?B
[PROOF STATE]
proof (prove)
using this:
\<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. \<exists>t. k \<parallel> t \<and> t \<parallel> q
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t. k \<parallel> t \<and> t \<parallel> q

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t. k \<parallel> t \<and> t \<parallel> q
[PROOF STEP]
obtain t where kt:"k\<parallel>t" and tq:"t\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
\<exists>t. k \<parallel> t \<and> t \<parallel> q

goal (1 subgoal):
 1. (\<And>t. \<lbrakk>k \<parallel> t; t \<parallel> q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
k \<parallel> t
t \<parallel> q

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from pv qvp
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v
q \<parallel> v'
[PROOF STEP]
have "p\<parallel>v' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>v') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>v))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'
\<lbrakk>?p \<parallel> ?q; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?p \<parallel> ?s \<oplus> ((\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?s) \<oplus> (\<exists>t. ?r \<parallel> t \<and> t \<parallel> ?q))

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (1 subgoal):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof (elim disjE)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "?A\<and>\<not>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?A
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. p \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<parallel> v'

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with qvp kp kt tq
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> v'
k \<parallel> p
k \<parallel> t
t \<parallel> q
p \<parallel> v'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
q \<parallel> v'
k \<parallel> p
k \<parallel> t
t \<parallel> q
p \<parallel> v'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x f
[PROOF STATE]
proof (prove)
using this:
q \<parallel> v'
k \<parallel> p
k \<parallel> t
t \<parallel> q
p \<parallel> v'
x = (p, q)
((?p, ?q) \<in> f) = (\<exists>k l u. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> k \<parallel> ?q \<and> ?q \<parallel> u)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?B
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
obtain t' where ptp:"p\<parallel>t'" and tpvp:"t'\<parallel>v'"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (1 subgoal):
 1. (\<And>t'. \<lbrakk>p \<parallel> t'; t' \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
p \<parallel> t'
t' \<parallel> v'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from tq kpq kplp
[PROOF STATE]
proof (chain)
picking this:
t \<parallel> q
k' \<parallel> q
k' \<parallel> l'
[PROOF STEP]
have "t\<parallel>l'"
[PROOF STATE]
proof (prove)
using this:
t \<parallel> q
k' \<parallel> q
k' \<parallel> l'

goal (1 subgoal):
 1. t \<parallel> l'
[PROOF STEP]
using M1
[PROOF STATE]
proof (prove)
using this:
t \<parallel> q
k' \<parallel> q
k' \<parallel> l'
\<lbrakk>?p \<parallel> ?q; ?p \<parallel> ?s; ?r \<parallel> ?q\<rbrakk> \<Longrightarrow> ?r \<parallel> ?s

goal (1 subgoal):
 1. t \<parallel> l'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t \<parallel> l'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
t \<parallel> l'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with ptp pv uv
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t'
p \<parallel> v
u \<parallel> v
t \<parallel> l'
[PROOF STEP]
have "u\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t'
p \<parallel> v
u \<parallel> v
t \<parallel> l'

goal (1 subgoal):
 1. u \<parallel> t'
[PROOF STEP]
using M1
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t'
p \<parallel> v
u \<parallel> v
t \<parallel> l'
\<lbrakk>?p \<parallel> ?q; ?p \<parallel> ?s; ?r \<parallel> ?q\<rbrakk> \<Longrightarrow> ?r \<parallel> ?s

goal (1 subgoal):
 1. u \<parallel> t'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
u \<parallel> t'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
u \<parallel> t'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with lpz zu \<open>t\<parallel>l'\<close>
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> z
z \<parallel> u
t \<parallel> l'
u \<parallel> t'
[PROOF STEP]
obtain lzu where "t\<parallel>lzu" and "lzu\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> z
z \<parallel> u
t \<parallel> l'
u \<parallel> t'

goal (1 subgoal):
 1. (\<And>lzu. \<lbrakk>t \<parallel> lzu; lzu \<parallel> t'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using  M5exist_var
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> z
z \<parallel> u
t \<parallel> l'
u \<parallel> t'
\<lbrakk>?x \<parallel> ?y; ?y \<parallel> ?z; ?z \<parallel> ?w\<rbrakk> \<Longrightarrow> \<exists>t. ?x \<parallel> t \<and> t \<parallel> ?w

goal (1 subgoal):
 1. (\<And>lzu. \<lbrakk>t \<parallel> lzu; lzu \<parallel> t'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
t \<parallel> lzu
lzu \<parallel> t'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
t \<parallel> l'
u \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
t \<parallel> l'
u \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x ov kt tq kp ptp tpvp qvp
[PROOF STATE]
proof (prove)
using this:
t \<parallel> l'
u \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'
x = (p, q)
((?p, ?q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> ?q \<and> ?q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)
k \<parallel> t
t \<parallel> q
k \<parallel> p
p \<parallel> t'
t' \<parallel> v'
q \<parallel> v'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>\<not>?B\<and>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with pv kp kt tq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v
k \<parallel> p
k \<parallel> t
t \<parallel> q
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
k \<parallel> p
k \<parallel> t
t \<parallel> q
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x d
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
k \<parallel> p
k \<parallel> t
t \<parallel> q
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
x = (p, q)
((?p, ?q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>\<not>?B\<and>?C"
[PROOF STATE]
proof (state)
this:
\<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. \<exists>t. k' \<parallel> t \<and> t \<parallel> p
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<exists>t. k' \<parallel> t \<and> t \<parallel> p

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t. k' \<parallel> t \<and> t \<parallel> p
[PROOF STEP]
obtain t where kpt:"k'\<parallel>t" and tp:"t\<parallel>p"
[PROOF STATE]
proof (prove)
using this:
\<exists>t. k' \<parallel> t \<and> t \<parallel> p

goal (1 subgoal):
 1. (\<And>t. \<lbrakk>k' \<parallel> t; t \<parallel> p\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
k' \<parallel> t
t \<parallel> p

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from pv qvp
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v
q \<parallel> v'
[PROOF STEP]
have "p\<parallel>v' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>v') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>v))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
q \<parallel> v'
\<lbrakk>?p \<parallel> ?q; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?p \<parallel> ?s \<oplus> ((\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?s) \<oplus> (\<exists>t. ?r \<parallel> t \<and> t \<parallel> ?q))

goal (1 subgoal):
 1. p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))
[PROOF STEP]
have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v))

goal (1 subgoal):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<or> \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof (elim disjE)
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "?A\<and>\<not>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?A
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. p \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
p \<parallel> v'

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with kpq kpt tp qvp
[PROOF STATE]
proof (chain)
picking this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> v'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> v'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x f
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> v'
x = (p, q)
((?p, ?q) \<in> f) = (\<exists>k l u. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> k \<parallel> ?q \<and> ?q \<parallel> u)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>?B\<and>\<not>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?B
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'
[PROOF STEP]
obtain t' where "p\<parallel>t'" and "t'\<parallel>v'"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v'

goal (1 subgoal):
 1. (\<And>t'. \<lbrakk>p \<parallel> t'; t' \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
p \<parallel> t'
t' \<parallel> v'

goal (2 subgoals):
 1. \<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with  kpq kpt tp qvp
[PROOF STATE]
proof (chain)
picking this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> t'
t' \<parallel> v'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> t'
t' \<parallel> v'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x d
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
q \<parallel> v'
p \<parallel> t'
t' \<parallel> v'
x = (p, q)
((?p, ?q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "\<not>?A\<and>\<not>?B\<and>?C"
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v)

goal (1 subgoal):
 1. \<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v
[PROOF STEP]
obtain t' where qtp:"q\<parallel>t'" and tpv:"t'\<parallel>v"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v

goal (1 subgoal):
 1. (\<And>t'. \<lbrakk>q \<parallel> t'; t' \<parallel> v\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
q \<parallel> t'
t' \<parallel> v

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
from tp kp kl
[PROOF STATE]
proof (chain)
picking this:
t \<parallel> p
k \<parallel> p
k \<parallel> l
[PROOF STEP]
have "t\<parallel>l"
[PROOF STATE]
proof (prove)
using this:
t \<parallel> p
k \<parallel> p
k \<parallel> l

goal (1 subgoal):
 1. t \<parallel> l
[PROOF STEP]
using M1
[PROOF STATE]
proof (prove)
using this:
t \<parallel> p
k \<parallel> p
k \<parallel> l
\<lbrakk>?p \<parallel> ?q; ?p \<parallel> ?s; ?r \<parallel> ?q\<rbrakk> \<Longrightarrow> ?r \<parallel> ?s

goal (1 subgoal):
 1. t \<parallel> l
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t \<parallel> l

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
t \<parallel> l

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with qtp qvp upvp
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> t'
q \<parallel> v'
u' \<parallel> v'
t \<parallel> l
[PROOF STEP]
have "u'\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
q \<parallel> t'
q \<parallel> v'
u' \<parallel> v'
t \<parallel> l

goal (1 subgoal):
 1. u' \<parallel> t'
[PROOF STEP]
using M1
[PROOF STATE]
proof (prove)
using this:
q \<parallel> t'
q \<parallel> v'
u' \<parallel> v'
t \<parallel> l
\<lbrakk>?p \<parallel> ?q; ?p \<parallel> ?s; ?r \<parallel> ?q\<rbrakk> \<Longrightarrow> ?r \<parallel> ?s

goal (1 subgoal):
 1. u' \<parallel> t'
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
u' \<parallel> t'

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
u' \<parallel> t'

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with lz zup \<open>t\<parallel>l\<close>
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
z \<parallel> u'
t \<parallel> l
u' \<parallel> t'
[PROOF STEP]
obtain lzu where "t\<parallel>lzu" and "lzu\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
z \<parallel> u'
t \<parallel> l
u' \<parallel> t'

goal (1 subgoal):
 1. (\<And>lzu. \<lbrakk>t \<parallel> lzu; lzu \<parallel> t'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using  M5exist_var
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
z \<parallel> u'
t \<parallel> l
u' \<parallel> t'
\<lbrakk>?x \<parallel> ?y; ?y \<parallel> ?z; ?z \<parallel> ?w\<rbrakk> \<Longrightarrow> \<exists>t. ?x \<parallel> t \<and> t \<parallel> ?w

goal (1 subgoal):
 1. (\<And>lzu. \<lbrakk>t \<parallel> lzu; lzu \<parallel> t'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
t \<parallel> lzu
lzu \<parallel> t'

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
t \<parallel> l
u' \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
t \<parallel> l
u' \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x ov kpt tp kpq qtp tpv pv
[PROOF STATE]
proof (prove)
using this:
t \<parallel> l
u' \<parallel> t'
t \<parallel> lzu
lzu \<parallel> t'
x = (p, q)
((?p, ?q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> ?q \<and> ?q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)
k' \<parallel> t
t \<parallel> p
k' \<parallel> q
q \<parallel> t'
t' \<parallel> v
p \<parallel> v

goal (1 subgoal):
 1. x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> p \<parallel> v' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> v') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> v) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
x \<in> e \<union> ov \<union> ov\<inverse> \<union> d \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse>

goal:
No subgoals!
[PROOF STEP]
qed