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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 kl:"k\<parallel>l" and lp:"l\<parallel>p" 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>k l u v. \<lbrakk>k \<parallel> l; l \<parallel> p; 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>k l u v. \<lbrakk>k \<parallel> l; l \<parallel> p; 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:
k \<parallel> l
l \<parallel> p
k \<parallel> z
p \<parallel> u
u \<parallel> v
z \<parallel> v

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from zup zv uv
[PROOF STATE]
proof (chain)
picking this:
z \<parallel> u'
z \<parallel> v
u \<parallel> v
[PROOF STEP]
have "u\<parallel>u'"
[PROOF STATE]
proof (prove)
using this:
z \<parallel> u'
z \<parallel> v
u \<parallel> v

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with pu upvp
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u
u' \<parallel> v'
u \<parallel> u'
[PROOF STEP]
obtain uu where puu:"p\<parallel>uu" and uuvp:"uu\<parallel>v'"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
u' \<parallel> v'
u \<parallel> u'

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 ov \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov \<union> s \<union> d"
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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 ?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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lp puu uuvp qvp
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> p
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l \<parallel> q
[PROOF STEP]
have "(p,q) \<in> s"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l \<parallel> q

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l \<parallel> 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. (p, q) \<in> s
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> s

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> s

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

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

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov \<union> s \<union> d"
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> q

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

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> u
[PROOF STEP]
obtain t' where ttp:"t\<parallel>t'" and "t'\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> u

goal (1 subgoal):
 1. (\<And>t'. \<lbrakk>t \<parallel> t'; t' \<parallel> u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t \<parallel> t'
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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with pu puu
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u
p \<parallel> uu
t \<parallel> t'
t' \<parallel> u
[PROOF STEP]
have "t'\<parallel>uu"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
p \<parallel> uu
t \<parallel> t'
t' \<parallel> u

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

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

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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lp puu qvp uuvp lt tq  ttp
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> p
p \<parallel> uu
q \<parallel> v'
uu \<parallel> v'
l \<parallel> t
t \<parallel> q
t \<parallel> t'
t' \<parallel> uu
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> uu
q \<parallel> v'
uu \<parallel> v'
l \<parallel> t
t \<parallel> q
t \<parallel> t'
t' \<parallel> uu

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> uu
q \<parallel> v'
uu \<parallel> v'
l \<parallel> t
t \<parallel> q
t \<parallel> t'
t' \<parallel> uu
((?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. (p, q) \<in> ov
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> ov

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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> ov

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

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

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> m \<union> ov \<union> s \<union> d

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

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t. l' \<parallel> t \<and> t \<parallel> p
[PROOF STEP]
obtain t where "l'\<parallel>t" and "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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lpq puu uuvp qvp
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> q
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l' \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l' \<parallel> t
t \<parallel> p

goal (1 subgoal):
 1. (p, q) \<in> d
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
l' \<parallel> t
t \<parallel> p
((?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. (p, q) \<in> d
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> d

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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> d

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

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

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> m \<union> ov \<union> s \<union> d

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

goal:
No subgoals!
[PROOF STEP]
qed


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

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

goal (1 subgoal):
 1. (\<And>p q z. \<lbrakk>x = (p, q); (p, z) \<in> d; (z, q) \<in> f\<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> f\<inverse>

goal (1 subgoal):
 1. \<And>x. x \<in> d O f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 kl:"k\<parallel>l" and lp:"l\<parallel>p" 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>k l u v. \<lbrakk>k \<parallel> l; l \<parallel> p; 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>k l u v. \<lbrakk>k \<parallel> l; l \<parallel> p; 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:
k \<parallel> l
l \<parallel> p
k \<parallel> z
p \<parallel> u
u \<parallel> v
z \<parallel> v

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from zup zv uv
[PROOF STATE]
proof (chain)
picking this:
z \<parallel> u'
z \<parallel> v
u \<parallel> v
[PROOF STEP]
have uup:"u\<parallel>u'"
[PROOF STATE]
proof (prove)
using this:
z \<parallel> u'
z \<parallel> v
u \<parallel> v

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d O f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[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 f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov \<union> s \<union> d"
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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 ?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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lp pu uup qup
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l \<parallel> q
[PROOF STEP]
have "(p,q) \<in> s"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l \<parallel> q

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l \<parallel> 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. (p, q) \<in> s
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> s

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> s

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

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

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov \<union> s \<union> d"
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
p \<parallel> q

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

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
 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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> u
[PROOF STEP]
obtain t' where ttp:"t\<parallel>t'" and tpu:"t'\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> u

goal (1 subgoal):
 1. (\<And>t'. \<lbrakk>t \<parallel> t'; t' \<parallel> u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
t \<parallel> t'
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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lt tq lp pu uup qup
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
t \<parallel> t'
t' \<parallel> u
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
t \<parallel> t'
t' \<parallel> u

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
l \<parallel> t
t \<parallel> q
l \<parallel> p
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
t \<parallel> t'
t' \<parallel> u
((?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. (p, q) \<in> ov
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> ov

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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> ov

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

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

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> m \<union> ov \<union> s \<union> d

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

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d

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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t. l' \<parallel> t \<and> t \<parallel> p
[PROOF STEP]
obtain t where "l'\<parallel>t" and "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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with lpq pu uup qup
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> q
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l' \<parallel> t
t \<parallel> p

goal (1 subgoal):
 1. (p, q) \<in> d
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
p \<parallel> u
u \<parallel> u'
q \<parallel> u'
l' \<parallel> t
t \<parallel> p
((?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. (p, q) \<in> d
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> d

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> m \<union> ov \<union> s \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> d

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

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

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> m \<union> ov \<union> s \<union> d

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

goal:
No subgoals!
[PROOF STEP]
qed