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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov\<inverse> O m \<Longrightarrow> x \<in> 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 zq
[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
z \<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> ov\<inverse> O m \<Longrightarrow> x \<in> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
from zu zq cu
[PROOF STATE]
proof (chain)
picking this:
z \<parallel> u
z \<parallel> q
c \<parallel> u
[PROOF STEP]
have cq:"c\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
z \<parallel> u
z \<parallel> q
c \<parallel> u

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov\<inverse> O m \<Longrightarrow> x \<in> 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> ov\<inverse> O m \<Longrightarrow> x \<in> 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> ov\<inverse> O m \<Longrightarrow> x \<in> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
thus "x \<in> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with lp lc cq qvp
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> p
l \<parallel> c
c \<parallel> q
q \<parallel> v'
p \<parallel> v'
[PROOF STEP]
have "(p,q) \<in> f^-1"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
l \<parallel> c
c \<parallel> q
q \<parallel> v'
p \<parallel> v'

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
using f
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
l \<parallel> c
c \<parallel> q
q \<parallel> v'
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> 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> 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> 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> 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> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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> 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 ptp:"p\<parallel>t" and "t\<parallel>v'"
[PROOF STATE]
proof (prove)
using this:
\<exists>t. p \<parallel> t \<and> t \<parallel> v'

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using lp lc cq qvp cu ov
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
t \<parallel> v'
u \<parallel> t
l \<parallel> p
l \<parallel> c
c \<parallel> q
q \<parallel> v'
c \<parallel> u
((?p, ?q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> ?q \<and> ?q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)

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

goal (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> 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> 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> 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> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> 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> 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> 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> 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> 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> 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> 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> 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 qt:"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> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
with lp lc cq pv
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> p
l \<parallel> c
c \<parallel> q
p \<parallel> v
q \<parallel> t
t \<parallel> v
[PROOF STEP]
have "(p,q) \<in> d^-1"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> p
l \<parallel> c
c \<parallel> q
p \<parallel> v
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:
l \<parallel> p
l \<parallel> c
c \<parallel> q
p \<parallel> v
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> 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> 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> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> 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> ov \<union> f\<inverse> \<union> d\<inverse>

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

goal:
No subgoals!
[PROOF STEP]
qed


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

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

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

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

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

goal (1 subgoal):
 1. (\<And>k u l t v. \<lbrakk>k \<parallel> p; p \<parallel> u; k \<parallel> l; l \<parallel> z; l \<parallel> t; t \<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
p \<parallel> u
k \<parallel> l
l \<parallel> z
l \<parallel> t
t \<parallel> u
u \<parallel> v
z \<parallel> v

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

goal (1 subgoal):
 1. (\<And>k' l' y u' v'. \<lbrakk>k' \<parallel> l'; k' \<parallel> z; l' \<parallel> q; l' \<parallel> y; y \<parallel> u'; z \<parallel> u'; u' \<parallel> v'; q \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
(z, q) \<in> ov
((?p, ?q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> ?q \<and> ?q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
from lz kplp kpz
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
k' \<parallel> l'
k' \<parallel> z
[PROOF STEP]
have llp:"l\<parallel>l'"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
k' \<parallel> l'
k' \<parallel> z

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
from puu lpq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> uu
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>uu))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> uu
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> uu))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> uu
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> uu))
[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> uu))

goal (1 subgoal):
 1. \<And>x. x \<in> ov O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu))
[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> uu))

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O ov \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov"
[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> uu) \<or> \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<or> \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu)

goal (1 subgoal):
 1. x \<in> b \<union> m \<union> ov
[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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu)

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu)
[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> uu)

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<nexists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> q
[PROOF STEP]
have "(p,q) \<in> m"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> q

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

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

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

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

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu)

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu)
[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> uu)

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t'. p \<parallel> t' \<and> t' \<parallel> q) \<and> (\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q
[PROOF STEP]
have "(p,q) \<in> b"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. p \<parallel> t' \<and> t' \<parallel> q

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

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

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

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

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> uu)
[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> uu)

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu
[PROOF STEP]
obtain t' where lptp:"l'\<parallel>t'" and "t'\<parallel>uu"
[PROOF STATE]
proof (prove)
using this:
\<exists>t'. l' \<parallel> t' \<and> t' \<parallel> uu

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
from kl llp lpq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> l
l \<parallel> l'
l' \<parallel> q
[PROOF STEP]
obtain ll where  kll:"k\<parallel>ll" and llq:"ll\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> l'
l' \<parallel> q

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
with lpq lptp
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> q
l' \<parallel> t'
k \<parallel> ll
ll \<parallel> q
[PROOF STEP]
have "ll\<parallel>t'"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
l' \<parallel> t'
k \<parallel> ll
ll \<parallel> q

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

goal (1 subgoal):
 1. ll \<parallel> t'
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
ll \<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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
with kp puu uuv kll llq qvp \<open>t'\<parallel>uu\<close>
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> uu
uu \<parallel> v'
k \<parallel> ll
ll \<parallel> q
q \<parallel> v'
t' \<parallel> uu
ll \<parallel> t'
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> uu
uu \<parallel> v'
k \<parallel> ll
ll \<parallel> q
q \<parallel> v'
t' \<parallel> uu
ll \<parallel> t'

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

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> uu) \<Longrightarrow> x \<in> b \<union> m \<union> ov

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

goal:
No subgoals!
[PROOF STEP]
qed


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

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

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

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

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

goal (1 subgoal):
 1. (\<And>k u l c v. \<lbrakk>k \<parallel> p; p \<parallel> u; k \<parallel> l; l \<parallel> z; l \<parallel> c; c \<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
p \<parallel> u
k \<parallel> l
l \<parallel> z
l \<parallel> c
c \<parallel> u
u \<parallel> v
z \<parallel> v

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
from pu lpq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> u
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>u))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
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> u))
[PROOF STEP]
using M2
[PROOF STATE]
proof (prove)
using this:
p \<parallel> u
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> u))
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
p \<parallel> q \<oplus> ((\<exists>t. p \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u))

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O f\<inverse> \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
thus "x \<in> b \<union> m \<union> ov"
[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> u) \<or> \<not> p \<parallel> q \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> u) \<or> \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u)

goal (1 subgoal):
 1. x \<in> b \<union> m \<union> ov
[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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<exists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 3. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u)

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

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

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

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

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

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

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

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

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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov

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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
 2. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u)

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> b
[PROOF STEP]
using b
[PROOF STATE]
proof (prove)
using this:
\<exists>t. p \<parallel> t \<and> t \<parallel> q
((?p, ?q) \<in> b) = (\<exists>t. ?p \<parallel> t \<and> t \<parallel> ?q)

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

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

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

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

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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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> u)
[PROOF STEP]
have ?C
[PROOF STATE]
proof (prove)
using this:
\<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u)

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

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

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

goal (1 subgoal):
 1. \<not> p \<parallel> q \<and> (\<nexists>t. p \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. l' \<parallel> t \<and> t \<parallel> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
from kl llp lpq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> l
l \<parallel> l'
l' \<parallel> q
[PROOF STEP]
obtain ll where  kll:"k\<parallel>ll" and llr:"ll\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> l'
l' \<parallel> q

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

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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
with lpq lptp
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> q
l' \<parallel> t
k \<parallel> ll
ll \<parallel> q
[PROOF STEP]
have "ll\<parallel>t"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> q
l' \<parallel> t
k \<parallel> ll
ll \<parallel> q

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

goal (1 subgoal):
 1. ll \<parallel> t
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
ll \<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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[PROOF STEP]
with kp pu uv kll llr qv \<open>t\<parallel>u\<close>
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> u
u \<parallel> v
k \<parallel> ll
ll \<parallel> q
q \<parallel> v
t \<parallel> u
ll \<parallel> t
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> u
u \<parallel> v
k \<parallel> ll
ll \<parallel> q
q \<parallel> v
t \<parallel> u
ll \<parallel> t

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> u
u \<parallel> v
k \<parallel> ll
ll \<parallel> q
q \<parallel> v
t \<parallel> u
ll \<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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov
[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
[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
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
x \<in> b \<union> m \<union> ov

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> u) \<Longrightarrow> x \<in> b \<union> m \<union> ov

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

goal:
No subgoals!
[PROOF STEP]
qed