[STATEMENT]
lemma cddi:"d O d^-1 \<subseteq> b \<union> b^-1 \<union> m \<union> m^-1 \<union> e \<union> ov \<union> ov^-1 \<union> s \<union> s^-1 \<union> d \<union> d^-1 \<union> f \<union> f^-1" (is "d O d^-1 \<subseteq> ?R")
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. d O d\<inverse> \<subseteq> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>x. x \<in> d O d\<inverse> \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<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 O d\<inverse> \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
assume "x \<in> d O d^-1"
[PROOF STATE]
proof (state)
this:
x \<in> d O d\<inverse>

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

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

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

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

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

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

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

goal (1 subgoal):
 1. l \<parallel> q \<oplus> ((\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
l' \<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. l \<parallel> q \<oplus> ((\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
l \<parallel> q \<oplus> ((\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p))

goal (1 subgoal):
 1. \<And>x. x \<in> d O d\<inverse> \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> q \<oplus> ((\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<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:
l \<parallel> q \<oplus> ((\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p))

goal (1 subgoal):
 1. l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<or> \<not> l \<parallel> q \<and> (\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<or> \<not> l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<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:
l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<or> \<not> l \<parallel> q \<and> (\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<or> \<not> l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p)

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

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

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

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

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

goal (1 subgoal):
 1. p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
q \<parallel> u'
\<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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

goal (3 subgoals):
 1. l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> l \<parallel> q \<and> (\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

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

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

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

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

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

goal (3 subgoals):
 1. p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> u' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with lq lp qup
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> q
l \<parallel> p
q \<parallel> u'
p \<parallel> u'
[PROOF STEP]
have "p = q"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> q
l \<parallel> p
q \<parallel> u'
p \<parallel> u'

goal (1 subgoal):
 1. p = q
[PROOF STEP]
using M4
[PROOF STATE]
proof (prove)
using this:
l \<parallel> q
l \<parallel> p
q \<parallel> u'
p \<parallel> u'
\<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> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> u' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p = q

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

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

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

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

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

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

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x s
[PROOF STATE]
proof (prove)
using this:
l \<parallel> q
l \<parallel> p
q \<parallel> u'
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u'
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> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>

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

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

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

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

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

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x s
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
l \<parallel> q
l \<parallel> p
\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u
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> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
q \<parallel> u'
\<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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

goal (2 subgoals):
 1. \<not> l \<parallel> q \<and> (\<exists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

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

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

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

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

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

goal (3 subgoals):
 1. p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> u' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with qup lt tq lp
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> u'
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> u'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> u'

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. p \<parallel> q \<oplus> ((\<exists>t''. p \<parallel> t'' \<and> t'' \<parallel> q) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
t \<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. p \<parallel> q \<oplus> ((\<exists>t''. p \<parallel> t'' \<and> t'' \<parallel> q) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> q \<oplus> ((\<exists>t''. p \<parallel> t'' \<and> t'' \<parallel> q) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x ov lt tq lp ptp tpup qup
[PROOF STATE]
proof (prove)
using this:
t \<parallel> g
g \<parallel> u
g \<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)
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> t'
t' \<parallel> u'
q \<parallel> u'

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
q \<parallel> u'
\<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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

goal (1 subgoal):
 1. \<not> l \<parallel> q \<and> (\<nexists>t. l \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))
[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> u' \<oplus> ((\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<oplus> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u))

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

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

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

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

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

goal (3 subgoals):
 1. p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 2. \<not> p \<parallel> u' \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<nexists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
 3. \<not> p \<parallel> u' \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> u') \<and> (\<exists>t'. q \<parallel> t' \<and> t' \<parallel> u) \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with qup lpt tp lpq
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
l' \<parallel> q
p \<parallel> u'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
l' \<parallel> q
p \<parallel> u'

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x f
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
l' \<parallel> q
p \<parallel> u'
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> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>

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

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

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

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

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

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x d
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
l' \<parallel> q
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> u'
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> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>

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

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

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

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

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

goal (1 subgoal):
 1. q \<parallel> p \<oplus> ((\<exists>t''. q \<parallel> t'' \<and> t'' \<parallel> p) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u'))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
t \<parallel> p
\<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. q \<parallel> p \<oplus> ((\<exists>t''. q \<parallel> t'' \<and> t'' \<parallel> p) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u'))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
q \<parallel> p \<oplus> ((\<exists>t''. q \<parallel> t'' \<and> t'' \<parallel> p) \<oplus> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u'))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> q \<parallel> p \<and> (\<nexists>t''. q \<parallel> t'' \<and> t'' \<parallel> p) \<and> (\<exists>t''. t \<parallel> t'' \<and> t'' \<parallel> u') \<Longrightarrow> x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
with qt' tpc pu lpq lpt tp tg
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> t'
t' \<parallel> u
p \<parallel> u
l' \<parallel> q
l' \<parallel> t
t \<parallel> p
t \<parallel> g
g \<parallel> t'
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
q \<parallel> t'
t' \<parallel> u
p \<parallel> u
l' \<parallel> q
l' \<parallel> t
t \<parallel> p
t \<parallel> g
g \<parallel> t'

goal (1 subgoal):
 1. x \<in> b \<union> b\<inverse> \<union> m \<union> m\<inverse> \<union> e \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> d \<union> d\<inverse> \<union> f \<union> f\<inverse>
[PROOF STEP]
using x ov
[PROOF STATE]
proof (prove)
using this:
q \<parallel> t'
t' \<parallel> u
p \<parallel> u
l' \<parallel> q
l' \<parallel> t
t \<parallel> p
t \<parallel> g
g \<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)

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

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

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed