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

goal (1 subgoal):
 1. \<And>x. x \<in> m O f \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x \<in> m O f
[PROOF STEP]
obtain p q z where x:"x =(p,q)" and 1:"(p,z) \<in> m" and 2:"(z,q) \<in> f"
[PROOF STATE]
proof (prove)
using this:
x \<in> m O f

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

goal (1 subgoal):
 1. \<And>x. x \<in> m O f \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x = (p, q)
(p, z) \<in> m
(z, q) \<in> f
[PROOF STEP]
obtain k l u   where pz:"p\<parallel>z" and kq:"k\<parallel>q" and kl:"k\<parallel>l" and lz:"l\<parallel>z" and zu:"z\<parallel>u" and qu:"q\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
x = (p, q)
(p, z) \<in> m
(z, q) \<in> f

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)
[PROOF STEP]
using local.meets_atrans xor_distr_L[of ?A ?B ?C]
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q \<oplus> ((\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k \<parallel> t \<and> t \<parallel> p))
\<lbrakk>?p \<parallel> ?q; ?q \<parallel> ?r\<rbrakk> \<Longrightarrow> \<not> ?p \<parallel> ?r
k' \<parallel> q \<oplus> ((\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<oplus> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)) = (k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p)) \<or> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<or> \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p)

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

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using  s qu kpp pz zu
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
((?p, ?q) \<in> s) = (\<exists>k u v. k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)
q \<parallel> u
k' \<parallel> p
p \<parallel> z
z \<parallel> u

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

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

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

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

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

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

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

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

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

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

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

goal (2 subgoals):
 1. \<not> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from kq kl tq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
k \<parallel> l
t \<parallel> q
[PROOF STEP]
have "t\<parallel>l"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> l
t \<parallel> q

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

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

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

goal (2 subgoals):
 1. \<not> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from lz pz pz
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
p \<parallel> z
p \<parallel> z
[PROOF STEP]
have "l\<parallel>z"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
p \<parallel> z
p \<parallel> z

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov kpp qu pz zu
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> t
t \<parallel> q
t \<parallel> l
l \<parallel> z
((?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)
k' \<parallel> p
q \<parallel> u
p \<parallel> z
z \<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> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
(p, q) \<in> ov

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed


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

goal (1 subgoal):
 1. \<And>x. x \<in> m O ov\<inverse> \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x \<in> m O ov\<inverse>
[PROOF STEP]
obtain p q z where x:"x =(p,q)" and 1:"(p,z) \<in> m" and 2:"(z,q) \<in> ov^-1"
[PROOF STATE]
proof (prove)
using this:
x \<in> m O ov\<inverse>

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

goal (1 subgoal):
 1. \<And>x. x \<in> m O ov\<inverse> \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
x = (p, q)
(p, z) \<in> m
(z, q) \<in> ov\<inverse>
[PROOF STEP]
obtain k l c u v  where pz:"p\<parallel>z" and kq:"k\<parallel>q" and kl:"k\<parallel>l" and lz:"l\<parallel>z" and qu:"q\<parallel>u" and uv:"u\<parallel>v" and zv:"z\<parallel>v" and lc:"l\<parallel>c" and cu:"c\<parallel>u"
[PROOF STATE]
proof (prove)
using this:
x = (p, q)
(p, z) \<in> m
(z, q) \<in> ov\<inverse>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s kpp qu cu pc
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
((?p, ?q) \<in> s) = (\<exists>k u v. k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)
k' \<parallel> p
q \<parallel> u
c \<parallel> u
p \<parallel> c

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

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

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

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

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

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

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

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

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

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

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

goal (2 subgoals):
 1. \<not> k' \<parallel> q \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k' \<parallel> q \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from kq kl tq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
k \<parallel> l
t \<parallel> q
[PROOF STEP]
have "t\<parallel>l"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
k \<parallel> l
t \<parallel> q

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov kpp qu cu lc pc
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> t
t \<parallel> q
t \<parallel> l
((?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)
k' \<parallel> p
q \<parallel> u
c \<parallel> u
l \<parallel> c
p \<parallel> c

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

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> d
[PROOF STEP]
using d kq cu qu pc
[PROOF STATE]
proof (prove)
using this:
k \<parallel> t
t \<parallel> p
((?p, ?q) \<in> d) = (\<exists>k l u v. k \<parallel> l \<and> l \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)
k \<parallel> q
c \<parallel> u
q \<parallel> u
p \<parallel> c

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

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed


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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> ov O d \<Longrightarrow> x \<in> s \<union> ov \<union> d
[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 auto
[PROOF STATE]
proof (state)
this:
u \<parallel> u'

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

goal (1 subgoal):
 1. (\<And>uu. \<lbrakk>p \<parallel> uu; uu \<parallel> v'\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using \<open>u\<parallel>u'\<close>
[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 d \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from kp kpq
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> p
k' \<parallel> q
[PROOF STEP]
have "k\<parallel>q \<oplus> ((\<exists>t. k\<parallel>t \<and> t\<parallel>q) \<oplus> (\<exists>t. k'\<parallel>t \<and> t\<parallel>p))" (is "?A \<oplus> (?B \<oplus> ?C)")
[PROOF STATE]
proof (prove)
using this:
k \<parallel> p
k' \<parallel> q

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. (p, q) \<in> s
[PROOF STEP]
using s kp qvp puu uuvp
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
((?p, ?q) \<in> s) = (\<exists>k u v. k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> ?q \<and> ?q \<parallel> v)
k \<parallel> p
q \<parallel> v'
p \<parallel> uu
uu \<parallel> v'

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

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

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

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

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

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

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

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

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

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

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from cu pu puu
[PROOF STATE]
proof (chain)
picking this:
c \<parallel> u
p \<parallel> u
p \<parallel> uu
[PROOF STEP]
have "c\<parallel>uu"
[PROOF STATE]
proof (prove)
using this:
c \<parallel> u
p \<parallel> u
p \<parallel> uu

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

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

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

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from kpq tq kplp
[PROOF STATE]
proof (chain)
picking this:
k' \<parallel> q
t \<parallel> q
k' \<parallel> l'
[PROOF STEP]
have "t\<parallel>l'"
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> q
t \<parallel> q
k' \<parallel> l'

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

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

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

goal (2 subgoals):
 1. \<not> k \<parallel> q \<and> (\<exists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<nexists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
 2. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from lpz lz lc
[PROOF STATE]
proof (chain)
picking this:
l' \<parallel> z
l \<parallel> z
l \<parallel> c
[PROOF STEP]
have lpc:"l'\<parallel>c"
[PROOF STATE]
proof (prove)
using this:
l' \<parallel> z
l \<parallel> z
l \<parallel> c

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

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

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

goal (1 subgoal):
 1. (\<And>lc. \<lbrakk>t \<parallel> lc; lc \<parallel> uu\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using M5exist_var
[PROOF STATE]
proof (prove)
using this:
c \<parallel> uu
t \<parallel> l'
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>lc. \<lbrakk>t \<parallel> lc; lc \<parallel> uu\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
t \<parallel> lc
lc \<parallel> uu

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

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov kp kt tq puu uuvp qvp
[PROOF STATE]
proof (prove)
using this:
t \<parallel> lc
lc \<parallel> uu
((?p, ?q) \<in> ov) = (\<exists>k l u v t. (k \<parallel> ?p \<and> ?p \<parallel> u \<and> u \<parallel> v) \<and> (k \<parallel> l \<and> l \<parallel> ?q \<and> ?q \<parallel> v) \<and> l \<parallel> t \<and> t \<parallel> u)
k \<parallel> p
k \<parallel> t
t \<parallel> q
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'

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

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

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

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

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

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

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

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

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

goal (1 subgoal):
 1. \<not> k \<parallel> q \<and> (\<nexists>t. k \<parallel> t \<and> t \<parallel> q) \<and> (\<exists>t. k' \<parallel> t \<and> t \<parallel> p) \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
with puu uuvp qvp kpq
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
k' \<parallel> q
k' \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q) \<in> d"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> uu
uu \<parallel> v'
q \<parallel> v'
k' \<parallel> q
k' \<parallel> t
t \<parallel> p

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

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

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

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

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

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

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

goal:
No subgoals!
[PROOF STEP]
qed