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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from \<open>(p,z) \<in> b\<close>
[PROOF STATE]
proof (chain)
picking this:
(p, z) \<in> b
[PROOF STEP]
obtain c where pc:"p\<parallel>c" and cz:"c\<parallel>z"
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> b

goal (1 subgoal):
 1. (\<And>c. \<lbrakk>p \<parallel> c; c \<parallel> z\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using b
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> b
((?p, ?q) \<in> b) = (\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?q)

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
obtain a where ap:"a\<parallel>p"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>a. a \<parallel> p \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M3 meets_wd pc
[PROOF STATE]
proof (prove)
using this:
\<I> ?p \<Longrightarrow> \<exists>q r. q \<parallel> ?p \<and> ?p \<parallel> r
?p \<parallel> ?q \<Longrightarrow> \<I> ?p \<and> \<I> ?q
p \<parallel> c

goal (1 subgoal):
 1. (\<And>a. a \<parallel> p \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
a \<parallel> p

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from \<open>(z,q) \<in> ov^-1\<close>
[PROOF STATE]
proof (chain)
picking this:
(z, q) \<in> ov\<inverse>
[PROOF STEP]
obtain k l u v w where "k\<parallel>l" and lz:"l\<parallel>z" and kq:"k\<parallel>q" and zv:"z\<parallel>v" and qu:"q\<parallel>u" and uv:"u\<parallel>v" and lw:"l\<parallel>w" and wu:"w\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> ov\<inverse>

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from cz lz lw
[PROOF STATE]
proof (chain)
picking this:
c \<parallel> z
l \<parallel> z
l \<parallel> w
[PROOF STEP]
have "c\<parallel>w"
[PROOF STATE]
proof (prove)
using this:
c \<parallel> z
l \<parallel> z
l \<parallel> w

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with pc wu
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> c
w \<parallel> u
c \<parallel> w
[PROOF STEP]
obtain cw where pcw:"p\<parallel>cw" and cwu:"cw\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
w \<parallel> u
c \<parallel> w

goal (1 subgoal):
 1. (\<And>cw. \<lbrakk>p \<parallel> cw; cw \<parallel> u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M5exist_var
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
w \<parallel> u
c \<parallel> w
\<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>cw. \<lbrakk>p \<parallel> cw; cw \<parallel> u\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> cw
cw \<parallel> u

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from ap kq
[PROOF STATE]
proof (chain)
picking this:
a \<parallel> p
k \<parallel> q
[PROOF STEP]
have "a\<parallel>q \<oplus> ((\<exists>t. a\<parallel>t \<and> t\<parallel>q) \<oplus> (\<exists>t. k\<parallel>t \<and> t\<parallel>p))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
a \<parallel> p
k \<parallel> q

goal (1 subgoal):
 1. a \<parallel> q \<oplus> ((\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k \<parallel> t \<and> t \<parallel> p))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
a \<parallel> p
k \<parallel> q
\<lbrakk>?p \<parallel> ?q; ?r \<parallel> ?s\<rbrakk> \<Longrightarrow> ?p \<parallel> ?s \<oplus> ((\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?s) \<oplus> (\<exists>t. ?r \<parallel> t \<and> t \<parallel> ?q))

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O ov\<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:
a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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. a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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. a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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:
a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p)

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

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

goal (3 subgoals):
 1. a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with ap qu pcw cwu
[PROOF STATE]
proof (chain)
picking this:
a \<parallel> p
q \<parallel> u
p \<parallel> cw
cw \<parallel> u
a \<parallel> q
[PROOF STEP]
have "(p,q) \<in> s"
[PROOF STATE]
proof (prove)
using this:
a \<parallel> p
q \<parallel> u
p \<parallel> cw
cw \<parallel> u
a \<parallel> q

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s
[PROOF STATE]
proof (prove)
using this:
a \<parallel> p
q \<parallel> u
p \<parallel> cw
cw \<parallel> u
a \<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. a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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:
a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d

goal (2 subgoals):
 1. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p)

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

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

goal (2 subgoals):
 1. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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. a \<parallel> t \<and> t \<parallel> q
[PROOF STEP]
obtain t where at:"a\<parallel>t" and tq:"t\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
\<exists>t. a \<parallel> t \<and> t \<parallel> q

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

goal (2 subgoals):
 1. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from pc  tq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> c
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>c))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
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> c))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
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> c))
[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> c))

goal (2 subgoals):
 1. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> c))
[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> c))

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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)
[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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)

goal (2 subgoals):
 1. \<not> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c) \<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> c) \<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> c)
[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> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c) \<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> c)
[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> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c)
[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> c)

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

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> c) \<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> c
[PROOF STEP]
obtain t' where "t\<parallel>t'" and "t'\<parallel>c"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c

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

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> c) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with pc pcw
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> c
p \<parallel> cw
t \<parallel> t'
t' \<parallel> c
[PROOF STEP]
have "t'\<parallel>cw"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
p \<parallel> cw
t \<parallel> t'
t' \<parallel> c

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

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

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> c) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with at tq ap pcw cwu qu \<open>t\<parallel>t'\<close>
[PROOF STATE]
proof (chain)
picking this:
a \<parallel> t
t \<parallel> q
a \<parallel> p
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
t \<parallel> t'
t' \<parallel> cw
[PROOF STEP]
have "(p,q)\<in>ov"
[PROOF STATE]
proof (prove)
using this:
a \<parallel> t
t \<parallel> q
a \<parallel> p
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
t \<parallel> t'
t' \<parallel> cw

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
a \<parallel> t
t \<parallel> q
a \<parallel> p
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
t \<parallel> t'
t' \<parallel> cw
((?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> c) \<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> c) \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<exists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d

goal (1 subgoal):
 1. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)

goal (1 subgoal):
 1. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)

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

goal (1 subgoal):
 1. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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. k \<parallel> t \<and> t \<parallel> p
[PROOF STEP]
obtain t where "k\<parallel>t" and "t\<parallel>p"
[PROOF STATE]
proof (prove)
using this:
\<exists>t. k \<parallel> t \<and> t \<parallel> p

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

goal (1 subgoal):
 1. \<not> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with kq pcw cwu  qu
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
k \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
k \<parallel> t
t \<parallel> p

goal (1 subgoal):
 1. (p, q) \<in> d
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
p \<parallel> cw
cw \<parallel> u
q \<parallel> u
k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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> a \<parallel> q \<and> (\<nexists>t. a \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<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 cbmi:"b O m^-1 \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. b O m\<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> b O m\<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> b O m\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
assume "x \<in> b O m^-1"
[PROOF STATE]
proof (state)
this:
x \<in> b O m\<inverse>

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O m\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from \<open>(p,z) \<in> b\<close>
[PROOF STATE]
proof (chain)
picking this:
(p, z) \<in> b
[PROOF STEP]
obtain c where pc:"p\<parallel>c" and cz:"c\<parallel>z"
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> b

goal (1 subgoal):
 1. (\<And>c. \<lbrakk>p \<parallel> c; c \<parallel> z\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using b
[PROOF STATE]
proof (prove)
using this:
(p, z) \<in> b
((?p, ?q) \<in> b) = (\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?q)

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O m\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
obtain k where kp:"k\<parallel>p"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>k. k \<parallel> p \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M3 meets_wd pc
[PROOF STATE]
proof (prove)
using this:
\<I> ?p \<Longrightarrow> \<exists>q r. q \<parallel> ?p \<and> ?p \<parallel> r
?p \<parallel> ?q \<Longrightarrow> \<I> ?p \<and> \<I> ?q
p \<parallel> c

goal (1 subgoal):
 1. (\<And>k. k \<parallel> p \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k \<parallel> p

goal (1 subgoal):
 1. \<And>x. x \<in> b O m\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from \<open>(z,q) \<in> m^-1\<close>
[PROOF STATE]
proof (chain)
picking this:
(z, q) \<in> m\<inverse>
[PROOF STEP]
have qz:"q\<parallel>z"
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> m\<inverse>

goal (1 subgoal):
 1. q \<parallel> z
[PROOF STEP]
using m
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> m\<inverse>
((?p, ?q) \<in> m) = ?p \<parallel> ?q

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O m\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
obtain k' where kpq:"k'\<parallel>q"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>k'. k' \<parallel> q \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M3 meets_wd qz
[PROOF STATE]
proof (prove)
using this:
\<I> ?p \<Longrightarrow> \<exists>q r. q \<parallel> ?p \<and> ?p \<parallel> r
?p \<parallel> ?q \<Longrightarrow> \<I> ?p \<and> \<I> ?q
q \<parallel> z

goal (1 subgoal):
 1. (\<And>k'. k' \<parallel> q \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k' \<parallel> q

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> b O m\<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:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<or> \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

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

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

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

goal (3 subgoals):
 1. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with kp pc cz qz
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<parallel> q
[PROOF STEP]
have "(p,q) \<in> s"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<parallel> q

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<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. k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 3. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d

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

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

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

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

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

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
from pc tq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> c
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>c))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
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> c))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
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> c))
[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> c))

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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> c))
[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> c))

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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)
[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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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> c) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. t \<parallel> t' \<and> t' \<parallel> c) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c) \<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> c) \<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> c)
[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> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c) \<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> c)
[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> c)

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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c) \<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> c)

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> c) \<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> c)
[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> c)

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

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> c) \<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> c
[PROOF STEP]
obtain t' where "t\<parallel>t'" and "t'\<parallel>c"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. t \<parallel> t' \<and> t' \<parallel> c

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

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> c) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with pc cz qz kt tq kp
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<parallel> t
t \<parallel> q
k \<parallel> p
t \<parallel> t'
t' \<parallel> c
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<parallel> t
t \<parallel> q
k \<parallel> p
t \<parallel> t'
t' \<parallel> c

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
p \<parallel> c
c \<parallel> z
q \<parallel> z
k \<parallel> t
t \<parallel> q
k \<parallel> p
t \<parallel> t'
t' \<parallel> c
((?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> c) \<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> c) \<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> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p)

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

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

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

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

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> s \<union> d
[PROOF STEP]
with kpq pc cz qz
[PROOF STATE]
proof (chain)
picking this:
k' \<parallel> q
p \<parallel> c
c \<parallel> z
q \<parallel> z
k' \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
p \<parallel> c
c \<parallel> z
q \<parallel> z
k' \<parallel> t
t \<parallel> p

goal (1 subgoal):
 1. (p, q) \<in> d
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
p \<parallel> c
c \<parallel> z
q \<parallel> z
k' \<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> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> 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