src/HOL/ex/Coherent.thy
changeset 28323 8f12f7275637
child 32734 06c13b2e562e
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Coherent.thy	Mon Sep 22 22:59:11 2008 +0200
@@ -0,0 +1,97 @@
+(*  Title:      HOL/ex/Coherent
+    ID:         $Id$
+    Author:     Stefan Berghofer, TU Muenchen
+                Marc Bezem, Institutt for Informatikk, Universitetet i Bergen 
+*)
+
+header{* Coherent Logic Problems *}
+
+theory Coherent imports Main begin
+
+subsection{* Equivalence of two versions of Pappus' Axiom *}
+
+no_notation
+  comp (infixl "o" 55) and
+  rel_comp (infixr "O" 75)
+
+lemma p1p2:
+  assumes
+  "col a b c l \<and> col d e f m"
+  "col b f g n \<and> col c e g o"
+  "col b d h p \<and> col a e h q"
+  "col c d i r \<and> col a f i s"
+  "el n o \<Longrightarrow> goal"
+  "el p q \<Longrightarrow> goal"
+  "el s r \<Longrightarrow> goal"
+  "\<And>A. el A A \<Longrightarrow> pl g A \<Longrightarrow> pl h A \<Longrightarrow> pl i A \<Longrightarrow> goal"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl A D"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl B D"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl C D"
+  "\<And>A B. pl A B \<Longrightarrow> ep A A"
+  "\<And>A B. ep A B \<Longrightarrow> ep B A"
+  "\<And>A B C. ep A B \<Longrightarrow> ep B C \<Longrightarrow> ep A C"
+  "\<And>A B. pl A B \<Longrightarrow> el B B"
+  "\<And>A B. el A B \<Longrightarrow> el B A"
+  "\<And>A B C. el A B \<Longrightarrow> el B C \<Longrightarrow> el A C"
+  "\<And>A B C. ep A B \<Longrightarrow> pl B C \<Longrightarrow> pl A C"
+  "\<And>A B C. pl A B \<Longrightarrow> el B C \<Longrightarrow> 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 \<Longrightarrow> pl A C \<Longrightarrow> pl D B \<Longrightarrow> pl D C \<Longrightarrow> ep A D \<or> el B C"
+  "\<And>A B. ep A A \<Longrightarrow> ep B B \<Longrightarrow> \<exists>C. pl A C \<and> pl B C"
+  shows goal using assms
+  by coherent
+
+lemma p2p1:
+  assumes
+  "col a b c l \<and> col d e f m"
+  "col b f g n \<and> col c e g o"
+  "col b d h p \<and> col a e h q"
+  "col c d i r \<and> col a f i s"
+  "pl a m \<Longrightarrow> goal"
+  "pl b m \<Longrightarrow> goal"
+  "pl c m \<Longrightarrow> goal"
+  "pl d l \<Longrightarrow> goal"
+  "pl e l \<Longrightarrow> goal"
+  "pl f l \<Longrightarrow> goal"
+  "\<And>A. pl g A \<Longrightarrow> pl h A \<Longrightarrow> pl i A \<Longrightarrow> goal"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl A D"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl B D"
+  "\<And>A B C D. col A B C D \<Longrightarrow> pl C D"
+  "\<And>A B. pl A B \<Longrightarrow> ep A A"
+  "\<And>A B. ep A B \<Longrightarrow> ep B A"
+  "\<And>A B C. ep A B \<Longrightarrow> ep B C \<Longrightarrow> ep A C"
+  "\<And>A B. pl A B \<Longrightarrow> el B B"
+  "\<And>A B. el A B \<Longrightarrow> el B A"
+  "\<And>A B C. el A B \<Longrightarrow> el B C \<Longrightarrow> el A C"
+  "\<And>A B C. ep A B \<Longrightarrow> pl B C \<Longrightarrow> pl A C"
+  "\<And>A B C. pl A B \<Longrightarrow> el B C \<Longrightarrow> 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 \<Longrightarrow> pl C B \<Longrightarrow> pl D A \<Longrightarrow> pl D B \<Longrightarrow> ep C D \<or> el A B"
+  "\<And>A B C. ep A A \<Longrightarrow> ep B B \<Longrightarrow> \<exists>C. pl A C \<and> pl B C"
+  shows goal using assms
+  by coherent
+
+
+subsection {* Preservation of the Diamond Property under reflexive closure *}
+
+lemma diamond:
+  assumes
+  "reflexive_rewrite a b" "reflexive_rewrite a c"
+  "\<And>A. reflexive_rewrite b A \<Longrightarrow> reflexive_rewrite c A \<Longrightarrow> goal"
+  "\<And>A. equalish A A" 
+  "\<And>A B. equalish A B \<Longrightarrow> equalish B A"
+  "\<And>A B C. equalish A B \<Longrightarrow> reflexive_rewrite B C \<Longrightarrow> reflexive_rewrite A C"
+  "\<And>A B. equalish A B \<Longrightarrow> reflexive_rewrite A B"
+  "\<And>A B. rewrite A B \<Longrightarrow> reflexive_rewrite A B"
+  "\<And>A B. reflexive_rewrite A B \<Longrightarrow> equalish A B \<or> rewrite A B"
+  "\<And>A B C. rewrite A B \<Longrightarrow> rewrite A C \<Longrightarrow> \<exists>D. rewrite B D \<and> rewrite C D"
+  shows goal using assms
+  by coherent
+
+end