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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O s \<Longrightarrow> x \<in> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
from lz lpz lpq
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
l' \<parallel> z
l' \<parallel> q
[PROOF STEP]
have lq:"l\<parallel>q"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
l' \<parallel> z
l' \<parallel> q

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> d\<inverse> O s \<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> d\<inverse> O s \<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> d\<inverse> O s \<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 kl lq qvp kp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> l
l \<parallel> q
q \<parallel> v'
k \<parallel> p
p \<parallel> v'
[PROOF STEP]
have "(p,q) \<in> f^-1"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> q
q \<parallel> v'
k \<parallel> p
p \<parallel> v'

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
using f
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> q
q \<parallel> v'
k \<parallel> p
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 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> 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]
from pt pv uv
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t
p \<parallel> v
u \<parallel> v
[PROOF STEP]
have "u\<parallel>t"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
p \<parallel> v
u \<parallel> v

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

goal (1 subgoal):
 1. u \<parallel> 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]
with lz zu
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
z \<parallel> u
u \<parallel> t
[PROOF STEP]
obtain zu where "l\<parallel>zu" and "zu\<parallel>t"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
z \<parallel> u
u \<parallel> t

goal (1 subgoal):
 1. (\<And>zu. \<lbrakk>l \<parallel> zu; zu \<parallel> t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M5exist_var
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
z \<parallel> u
u \<parallel> t
\<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>zu. \<lbrakk>l \<parallel> zu; zu \<parallel> t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
l \<parallel> zu
zu \<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]
with kp pt tvp kl lq qvp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> l
l \<parallel> q
q \<parallel> v'
l \<parallel> zu
zu \<parallel> t
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> l
l \<parallel> q
q \<parallel> v'
l \<parallel> zu
zu \<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> t
t \<parallel> v'
k \<parallel> l
l \<parallel> q
q \<parallel> v'
l \<parallel> zu
zu \<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 (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 "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 kl lq kp pv
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> l
l \<parallel> q
k \<parallel> p
p \<parallel> v
q \<parallel> t
t \<parallel> v
[PROOF STEP]
have "(p,q)\<in>d^-1"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> l
l \<parallel> q
k \<parallel> p
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:
k \<parallel> l
l \<parallel> q
k \<parallel> p
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 csim:"s^-1 O m \<subseteq> ov \<union> f^-1 \<union> d^-1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. s\<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> s\<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> s\<inverse> O m \<Longrightarrow> x \<in> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
assume "x \<in> s^-1 O m"
[PROOF STATE]
proof (state)
this:
x \<in> s\<inverse> O m

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> s\<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> s\<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> s\<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> s\<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> s\<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 kp kz zq qvp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k \<parallel> z
z \<parallel> q
q \<parallel> v'
p \<parallel> v'
[PROOF STEP]
have "(p,q) \<in> f^-1"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k \<parallel> z
z \<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:
k \<parallel> p
k \<parallel> z
z \<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 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> 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]
from pt pv uv
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t
p \<parallel> v
u \<parallel> v
[PROOF STEP]
have "u\<parallel>t"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
p \<parallel> v
u \<parallel> v

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

goal (1 subgoal):
 1. u \<parallel> 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]
with kp pt tvp kz zq qvp zu
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> z
z \<parallel> q
q \<parallel> v'
z \<parallel> u
u \<parallel> t
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> z
z \<parallel> q
q \<parallel> v'
z \<parallel> u
u \<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> t
t \<parallel> v'
k \<parallel> z
z \<parallel> q
q \<parallel> v'
z \<parallel> u
u \<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 (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 "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 kp kz zq pv
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k \<parallel> z
z \<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:
k \<parallel> p
k \<parallel> z
z \<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:
k \<parallel> p
k \<parallel> z
z \<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 csiov:"s^-1 O ov \<subseteq> ov \<union> f^-1 \<union> d^-1"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. s\<inverse> O ov \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
proof
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>x. x \<in> s\<inverse> O ov \<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> s\<inverse> O ov \<Longrightarrow> x \<in> ov \<union> f\<inverse> \<union> d\<inverse>
[PROOF STEP]
assume "x \<in> s^-1 O ov"
[PROOF STATE]
proof (state)
this:
x \<in> s\<inverse> O ov

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> s\<inverse> O ov \<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> s\<inverse> O ov \<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> s\<inverse> O ov \<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 kp kplp lpq qvp klp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k' \<parallel> l'
l' \<parallel> q
q \<parallel> v'
k \<parallel> l'
p \<parallel> v'
[PROOF STEP]
have "(p,q) \<in> f^-1"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k' \<parallel> l'
l' \<parallel> q
q \<parallel> v'
k \<parallel> l'
p \<parallel> v'

goal (1 subgoal):
 1. (p, q) \<in> f\<inverse>
[PROOF STEP]
using f
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k' \<parallel> l'
l' \<parallel> q
q \<parallel> v'
k \<parallel> l'
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 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> 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]
from pt pv uv
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> t
p \<parallel> v
u \<parallel> v
[PROOF STEP]
have "u\<parallel>t"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> t
p \<parallel> v
u \<parallel> v

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

goal (1 subgoal):
 1. u \<parallel> 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]
moreover
[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]
from cup zup zu
[PROOF STATE]
proof (chain)
picking this:
c \<parallel> u'
z \<parallel> u'
z \<parallel> u
[PROOF STEP]
have cu:"c\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
c \<parallel> u'
z \<parallel> u'
z \<parallel> u

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

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

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

goal (1 subgoal):
 1. (\<And>cu. \<lbrakk>l' \<parallel> cu; cu \<parallel> t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using lpc M5exist_var
[PROOF STATE]
proof (prove)
using this:
u \<parallel> t
c \<parallel> u
l' \<parallel> c
\<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>cu. \<lbrakk>l' \<parallel> cu; cu \<parallel> t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
l' \<parallel> cu
cu \<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]
with kp pt tvp klp lpq qvp
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> l'
l' \<parallel> q
q \<parallel> v'
l' \<parallel> cu
cu \<parallel> t
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
p \<parallel> t
t \<parallel> v'
k \<parallel> l'
l' \<parallel> q
q \<parallel> v'
l' \<parallel> cu
cu \<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> t
t \<parallel> v'
k \<parallel> l'
l' \<parallel> q
q \<parallel> v'
l' \<parallel> cu
cu \<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 (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 "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 kp klp lpq pv
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k \<parallel> l'
l' \<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:
k \<parallel> p
k \<parallel> l'
l' \<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:
k \<parallel> p
k \<parallel> l'
l' \<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