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

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

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

goal (1 subgoal):
 1. \<And>x. x \<in> m O d \<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> d
[PROOF STEP]
obtain k l u v  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 uv:"u\<parallel>v" and qv:"q\<parallel>v"
[PROOF STATE]
proof (prove)
using this:
x = (p, q)
(p, z) \<in> m
(z, q) \<in> d

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

goal (1 subgoal):
 1. \<And>x. x \<in> m O d \<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 d \<Longrightarrow> x \<in> s \<union> ov \<union> d
[PROOF STEP]
from pz zu uv
[PROOF STATE]
proof (chain)
picking this:
p \<parallel> z
z \<parallel> u
u \<parallel> v
[PROOF STEP]
obtain zu where pzu:"p\<parallel>zu" and zuv:"zu\<parallel>v"
[PROOF STATE]
proof (prove)
using this:
p \<parallel> z
z \<parallel> u
u \<parallel> v

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

goal (1 subgoal):
 1. \<And>x. x \<in> m O d \<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 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]
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 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 qv kpp pzu zuv
[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> v
k' \<parallel> p
p \<parallel> zu
zu \<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 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 pzu
[PROOF STATE]
proof (chain)
picking this:
l \<parallel> z
p \<parallel> z
p \<parallel> zu
[PROOF STEP]
have "l\<parallel>zu"
[PROOF STATE]
proof (prove)
using this:
l \<parallel> z
p \<parallel> z
p \<parallel> zu

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

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

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> zu
[PROOF STEP]
have "(p,q) \<in> ov"
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> t
t \<parallel> q
t \<parallel> l
l \<parallel> zu

goal (1 subgoal):
 1. (p, q) \<in> ov
[PROOF STEP]
using ov kpp qv pzu zuv
[PROOF STATE]
proof (prove)
using this:
k' \<parallel> t
t \<parallel> q
t \<parallel> l
l \<parallel> zu
((?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> v
p \<parallel> zu
zu \<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 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 pzu zuv qv
[PROOF STATE]
proof (chain)
picking this:
k \<parallel> q
p \<parallel> zu
zu \<parallel> v
q \<parallel> v
k \<parallel> t
t \<parallel> p
[PROOF STEP]
have "(p,q)\<in>d"
[PROOF STATE]
proof (prove)
using this:
k \<parallel> q
p \<parallel> zu
zu \<parallel> v
q \<parallel> v
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> zu
zu \<parallel> v
q \<parallel> v
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