(* Title: HOL/Examples/Coherent.thy Author: Stefan Berghofer, TU Muenchen Author: Marc Bezem, Institutt for Informatikk, Universitetet i Bergen
*)
section \<open>Coherent Logic Problems\<close>
theory Coherent imports Main begin
subsection \<open>Equivalence of two versions of Pappus' Axiom\<close>
no_notation comp (infixl\<open>o\<close> 55)
unbundle no relcomp_syntax
lemma p1p2: assumes"col a b c l \ col d e f m" and"col b f g n \ col c e g o" and"col b d h p \ col a e h q" and"col c d i r \ col a f i s" and"el n o \ goal" and"el p q \ goal" and"el s r \ goal" and"\A. el A A \ pl g A \ pl h A \ pl i A \ goal" and"\A B C D. col A B C D \ pl A D" and"\A B C D. col A B C D \ pl B D" and"\A B C D. col A B C D \ pl C D" and"\A B. pl A B \ ep A A" and"\A B. ep A B \ ep B A" and"\A B C. ep A B \ ep B C \ ep A C" and"\A B. pl A B \ el B B" and"\A B. el A B \ el B A" and"\A B C. el A B \ el B C \ el A C" and"\A B C. ep A B \ pl B C \ pl A C" and"\A B C. pl A B \ el B C \ pl A C" and"\A B C D E F G H I J K L M N O P Q.
col A B C D \<Longrightarrow> col E F G H \<Longrightarrow> col B G I J \<Longrightarrow> col C F I K \<Longrightarrow>
col B E L M \<Longrightarrow> col A F L N \<Longrightarrow> col C E O P \<Longrightarrow> col A G O Q \<Longrightarrow>
(\<exists> R. col I L O R) \<or> pl A H \<or> pl B H \<or> pl C H \<or> pl E D \<or> pl F D \<or> pl G D" and"\A B C D. pl A B \ pl A C \ pl D B \ pl D C \ ep A D \ el B C" and"\A B. ep A A \ ep B B \ \C. pl A C \ pl B C" shows goal using assms by coherent
lemma p2p1: assumes"col a b c l \ col d e f m" and"col b f g n \ col c e g o" and"col b d h p \ col a e h q" and"col c d i r \ col a f i s" and"pl a m \ goal" and"pl b m \ goal" and"pl c m \ goal" and"pl d l \ goal" and"pl e l \ goal" and"pl f l \ goal" and"\A. pl g A \ pl h A \ pl i A \ goal" and"\A B C D. col A B C D \ pl A D" and"\A B C D. col A B C D \ pl B D" and"\A B C D. col A B C D \ pl C D" and"\A B. pl A B \ ep A A" and"\A B. ep A B \ ep B A" and"\A B C. ep A B \ ep B C \ ep A C" and"\A B. pl A B \ el B B" and"\A B. el A B \ el B A" and"\A B C. el A B \ el B C \ el A C" and"\A B C. ep A B \ pl B C \ pl A C" and"\A B C. pl A B \ el B C \ pl A C" and"\A B C D E F G H I J K L M N O P Q.
col A B C J \<Longrightarrow> col D E F K \<Longrightarrow> col B F G L \<Longrightarrow> col C E G M \<Longrightarrow>
col B D H N \<Longrightarrow> col A E H O \<Longrightarrow> col C D I P \<Longrightarrow> col A F I Q \<Longrightarrow>
(\<exists> R. col G H I R) \<or> el L M \<or> el N O \<or> el P Q" and"\A B C D. pl C A \ pl C B \ pl D A \ pl D B \ ep C D \ el A B" and"\A B C. ep A A \ ep B B \ \C. pl A C \ pl B C" shows goal using assms by coherent
subsection \<open>Preservation of the Diamond Property under reflexive closure\<close>
lemma diamond: assumes"reflexive_rewrite a b""reflexive_rewrite a c" and"\A. reflexive_rewrite b A \ reflexive_rewrite c A \ goal" and"\A. equalish A A" and"\A B. equalish A B \ equalish B A" and"\A B C. equalish A B \ reflexive_rewrite B C \ reflexive_rewrite A C" and"\A B. equalish A B \ reflexive_rewrite A B" and"\A B. rewrite A B \ reflexive_rewrite A B" and"\A B. reflexive_rewrite A B \ equalish A B \ rewrite A B" and"\A B C. rewrite A B \ rewrite A C \ \D. rewrite B D \ rewrite C D" shows goal using assms by coherent
end
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