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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O b \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with kl lz
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> l
l \<parallel> z
z \<parallel> c
c \<parallel> q
[PROOF STEP]
obtain lzc where klzc:"k\<parallel>lzc" and lzcq:"lzc\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> z
z \<parallel> c
c \<parallel> q

goal (1 subgoal):
 1. (\<And>lzc. \<lbrakk>k \<parallel> lzc; lzc \<parallel> q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M5exist_var
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> z
z \<parallel> c
c \<parallel> q
\<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>lzc. \<lbrakk>k \<parallel> lzc; lzc \<parallel> q\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
k \<parallel> lzc
lzc \<parallel> q

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O b \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
obtain v' where qvp:"q\<parallel>v'"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<And>v'. q \<parallel> v' \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M3 meets_wd cq
[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
c \<parallel> q

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

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

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

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

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

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

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

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

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

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

goal (3 subgoals):
 1. p \<parallel> v' \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> v') \<and> (\<nexists>t. q \<parallel> t \<and> t \<parallel> v) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
 2. \<not> p \<parallel> v' \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> v') \<and> (\<nexists>t. q \<parallel> t \<and> t \<parallel> v) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
 3. \<not> p \<parallel> v' \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> v') \<and> (\<exists>t. q \<parallel> t \<and> t \<parallel> v) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with qvp kp klzc lzcq
[PROOF STATE]
proof (chain)
picking this:
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
p \<parallel> v'
[PROOF STEP]
have "(p,q) \<in> f^-1"
[PROOF STATE]
proof (prove)
using this:
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
p \<parallel> v'

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
using f
[PROOF STATE]
proof (prove)
using this:
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
p \<parallel> v'
((?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. (p, q) \<in> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> f\<inverse>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
 2. \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
 3. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[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> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
using x m
[PROOF STATE]
proof (prove)
using this:
p \<parallel> q
x = (p, q)
((?p, ?q) \<in> m) = ?p \<parallel> ?q

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
from lzcq cq ctp
[PROOF STATE]
proof (chain)
picking this:
lzc \<parallel> q
c \<parallel> q
c \<parallel> t'
[PROOF STEP]
have "lzc\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
lzc \<parallel> q
c \<parallel> q
c \<parallel> t'

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with pt tvp qvp kp klzc lzcq tpt
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t
t \<parallel> v'
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
t' \<parallel> t
lzc \<parallel> t'
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
t \<parallel> v'
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
t' \<parallel> t
lzc \<parallel> t'

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
t \<parallel> v'
q \<parallel> v'
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
t' \<parallel> t
lzc \<parallel> t'
((?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'. c \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[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> f\<inverse> \<union> d\<inverse>
[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> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> v' \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> v') \<and> (\<exists>t. q \<parallel> t \<and> t \<parallel> v) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with pv kp klzc lzcq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> v
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
q \<parallel> t
t \<parallel> v
[PROOF STEP]
have "(p,q) \<in> d^-1"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
q \<parallel> t
t \<parallel> v

goal (1 subgoal):
 1. (p, q) \<in> d\<inverse>
[PROOF STEP]
using d
[PROOF STATE]
proof (prove)
using this:
p \<parallel> v
k \<parallel> p
k \<parallel> lzc
lzc \<parallel> q
q \<parallel> t
t \<parallel> v
((?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\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> d\<inverse>

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed


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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> s O d\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
from \<open>(p,z) : s\<close>
[PROOF STATE]
proof (chain)
picking this:
(p, z) \<in> s
[PROOF STEP]
obtain k  u v  where kp:"k\<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> s

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> s O d\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
from kp kz kpz
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k \<parallel> z
k' \<parallel> z
[PROOF STEP]
have kpp:"k'\<parallel>p"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k \<parallel> z
k' \<parallel> z

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
using f
[PROOF STATE]
proof (prove)
using this:
q \<parallel> u'
k' \<parallel> p
k' \<parallel> l'
l' \<parallel> q
p \<parallel> u'
((?p, ?q) \<in> f) = (\<exists>k l u. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> k \<parallel> ?q \<and> ?q \<parallel> u)

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> f\<inverse>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> t) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with pt tup qup kpp kplp lpq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t
t \<parallel> u'
q \<parallel> u'
k' \<parallel> p
k' \<parallel> l'
l' \<parallel> q
l' \<parallel> t'
t' \<parallel> t
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
t \<parallel> u'
q \<parallel> u'
k' \<parallel> p
k' \<parallel> l'
l' \<parallel> q
l' \<parallel> t'
t' \<parallel> t

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> u' \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> u') \<and> (\<exists>t. q \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with pu kpp kplp lpq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u
k' \<parallel> p
k' \<parallel> l'
l' \<parallel> q
q \<parallel> t
t \<parallel> u
[PROOF STEP]
have "(p,q) \<in> d^-1"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
k' \<parallel> p
k' \<parallel> l'
l' \<parallel> q
q \<parallel> t
t \<parallel> u

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

goal (1 subgoal):
 1. (p, q) \<in> d\<inverse>
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
(p, q) \<in> d\<inverse>

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed