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(*
Title: Allen's qualitative temporal calculus
Author: Fadoua Ghourabi (fadouaghourabi@gmail.com)
Affiliation: Ochanomizu University, Japan
*)
theory jointly_exhaustive
imports
allen
begin
section \<open>JE property\<close>
text \<open>The 13 time interval relations are jointly exhaustive. For any two intervals $x$ and $y$, we can find a basic relation $r$ such that $(x,y) \in r$.\<close>
lemma (in arelations) jointly_exhaustive:
assumes "\<I> p" "\<I> q"
shows "(p::'a,q::'a) \<in> b \<or> (p,q) \<in> m \<or> (p,q) \<in> ov \<or> (p,q) \<in> s \<or> (p,q) \<in> d \<or> (p,q) \<in> f^-1 \<or> (p,q) \<in> e \<or>
(p,q) \<in> f \<or> (p,q) \<in> s^-1 \<or> (p,q) \<in> d^-1 \<or> (p,q) \<in> ov^-1 \<or> (p,q) \<in> m^-1 \<or> (p,q) \<in> b^-1 " (is ?R)
proof -
obtain k k' u u'::'a where kp:"k\<parallel>p" and kpq:"k'\<parallel>q" and pu:"p\<parallel>u" and qup:"q\<parallel>u'" using M3 meets_wd assms(1,2) by fastforce
from kp kpq 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)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{ assume "?A\<and>\<not>?B\<and>\<not>?C" then have kq:?A by simp
from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'::'a. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{assume "(?A\<and>\<not>?B\<and>\<not>?C)" then have "?A" by simp
with kp kq qup have "p = q" using M4 by auto
thus ?thesis using e by auto}
next
{assume "\<not>?A\<and>?B\<and>\<not>?C" then have "?B" by simp
with kq kp qup show ?thesis using s by blast}
next
{assume "(\<not>?A\<and>\<not>?B\<and>?C)" then have "?C" by simp
then obtain t' where "q\<parallel>t'" and "t'\<parallel>u" by blast
with kq kp pu show ?thesis using s by blast }
qed}
next
{ assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
then obtain t where kt:"k\<parallel>t" and tq:"t\<parallel>q" by auto
from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
with kp qup kt tq show ?thesis using f by blast}
next
{assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
then obtain t' where ptp:"p\<parallel>t'" and tpup:"t'\<parallel>u'" by auto
from pu tq have "p\<parallel>q \<oplus> ((\<exists>t''. p\<parallel>t'' \<and> t''\<parallel>q) \<oplus> (\<exists>t''. t\<parallel>t'' \<and> t''\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
thus ?thesis using m by auto}
next
{assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
thus ?thesis using b by auto}
next
{ assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
then obtain g where "t\<parallel>g" and "g\<parallel>u" by auto
moreover with pu ptp have "g\<parallel>t'" using M1 by blast
ultimately show ?thesis using ov kt tq kp ptp tpup qup by blast}
qed}
next
{assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
then obtain t' where "q\<parallel>t'" and "t'\<parallel>u" by auto
with kp kt tq pu show ?thesis using d by blast}
qed}
next
{ assume "\<not>?A\<and>\<not>?B\<and>?C" then have ?C by simp
then obtain t where kpt:"k'\<parallel>t" and tp:"t\<parallel>p" by auto
from pu qup have "p\<parallel>u' \<oplus> ((\<exists>t'. p\<parallel>t' \<and> t'\<parallel>u') \<oplus> (\<exists>t'. q\<parallel>t' \<and> t'\<parallel>u))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
with qup kpt tp kpq show ?thesis using f by blast}
next
{assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
with qup kpt tp kpq show ?thesis using d by blast}
next
{assume "\<not>?A\<and>\<not>?B\<and>?C" then obtain t' where qt':"q\<parallel>t'" and tpc:"t'\<parallel>u" by auto
from qup tp have "q\<parallel>p \<oplus> ((\<exists>t''. q\<parallel>t'' \<and> t''\<parallel>p) \<oplus> (\<exists>t''. t\<parallel>t'' \<and> t''\<parallel>u'))" (is "?A \<oplus> (?B \<oplus> ?C)") using M2 by blast
then have "(?A\<and>\<not>?B\<and>\<not>?C) \<or> ((\<not>?A\<and>?B\<and>\<not>?C) \<or> (\<not>?A\<and>\<not>?B\<and>?C))" by (insert xor_distr_L[of ?A ?B ?C], auto simp:elimmeets)
thus ?thesis
proof (elim disjE)
{assume "?A\<and>\<not>?B\<and>\<not>?C" then have ?A by simp
thus ?thesis using m by auto}
next
{assume "\<not>?A\<and>?B\<and>\<not>?C" then have ?B by simp
thus ?thesis using b by auto}
next
{ assume "\<not>?A\<and>\<not>?B\<and>?C" then obtain g where tg:"t\<parallel>g" and "g\<parallel>u'" by auto
with qup qt' have "g\<parallel>t'" using M1 by blast
with qt' tpc pu kpq kpt tp tg show ?thesis using ov by blast}
qed}
qed}
qed
qed
lemma (in arelations) JE:
assumes "\<I> p" "\<I> q"
shows "(p::'a,q::'a) \<in> b \<union> m \<union> ov \<union> s \<union> d \<union> f^-1 \<union> e \<union> f \<union> s^-1 \<union> d^-1 \<union> ov^-1 \<union> m^-1 \<union> b^-1 "
using jointly_exhaustive UnCI assms(1,2) by blast
end
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