src/HOL/ex/Coherent.thy
author griff
Tue Apr 03 17:26:30 2012 +0900 (2012-04-03)
changeset 47433 07f4bf913230
parent 32734 06c13b2e562e
child 58889 5b7a9633cfa8
permissions -rw-r--r--
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
     1 (*  Title:      HOL/ex/Coherent.thy
     2     Author:     Stefan Berghofer, TU Muenchen
     3     Author:     Marc Bezem, Institutt for Informatikk, Universitetet i Bergen 
     4 *)
     5 
     6 header {* Coherent Logic Problems *}
     7 
     8 theory Coherent
     9 imports Main
    10 begin
    11 
    12 subsection {* Equivalence of two versions of Pappus' Axiom *}
    13 
    14 no_notation
    15   comp (infixl "o" 55) and
    16   relcomp (infixr "O" 75)
    17 
    18 lemma p1p2:
    19   assumes
    20   "col a b c l \<and> col d e f m"
    21   "col b f g n \<and> col c e g o"
    22   "col b d h p \<and> col a e h q"
    23   "col c d i r \<and> col a f i s"
    24   "el n o \<Longrightarrow> goal"
    25   "el p q \<Longrightarrow> goal"
    26   "el s r \<Longrightarrow> goal"
    27   "\<And>A. el A A \<Longrightarrow> pl g A \<Longrightarrow> pl h A \<Longrightarrow> pl i A \<Longrightarrow> goal"
    28   "\<And>A B C D. col A B C D \<Longrightarrow> pl A D"
    29   "\<And>A B C D. col A B C D \<Longrightarrow> pl B D"
    30   "\<And>A B C D. col A B C D \<Longrightarrow> pl C D"
    31   "\<And>A B. pl A B \<Longrightarrow> ep A A"
    32   "\<And>A B. ep A B \<Longrightarrow> ep B A"
    33   "\<And>A B C. ep A B \<Longrightarrow> ep B C \<Longrightarrow> ep A C"
    34   "\<And>A B. pl A B \<Longrightarrow> el B B"
    35   "\<And>A B. el A B \<Longrightarrow> el B A"
    36   "\<And>A B C. el A B \<Longrightarrow> el B C \<Longrightarrow> el A C"
    37   "\<And>A B C. ep A B \<Longrightarrow> pl B C \<Longrightarrow> pl A C"
    38   "\<And>A B C. pl A B \<Longrightarrow> el B C \<Longrightarrow> pl A C"
    39   "\<And>A B C D E F G H I J K L M N O P Q.
    40      col A B C D \<Longrightarrow> col E F G H \<Longrightarrow> col B G I J \<Longrightarrow> col C F I K \<Longrightarrow>
    41      col B E L M \<Longrightarrow> col A F L N \<Longrightarrow> col C E O P \<Longrightarrow> col A G O Q \<Longrightarrow>
    42      (\<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"
    43   "\<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"
    44   "\<And>A B. ep A A \<Longrightarrow> ep B B \<Longrightarrow> \<exists>C. pl A C \<and> pl B C"
    45   shows goal using assms
    46   by coherent
    47 
    48 lemma p2p1:
    49   assumes
    50   "col a b c l \<and> col d e f m"
    51   "col b f g n \<and> col c e g o"
    52   "col b d h p \<and> col a e h q"
    53   "col c d i r \<and> col a f i s"
    54   "pl a m \<Longrightarrow> goal"
    55   "pl b m \<Longrightarrow> goal"
    56   "pl c m \<Longrightarrow> goal"
    57   "pl d l \<Longrightarrow> goal"
    58   "pl e l \<Longrightarrow> goal"
    59   "pl f l \<Longrightarrow> goal"
    60   "\<And>A. pl g A \<Longrightarrow> pl h A \<Longrightarrow> pl i A \<Longrightarrow> goal"
    61   "\<And>A B C D. col A B C D \<Longrightarrow> pl A D"
    62   "\<And>A B C D. col A B C D \<Longrightarrow> pl B D"
    63   "\<And>A B C D. col A B C D \<Longrightarrow> pl C D"
    64   "\<And>A B. pl A B \<Longrightarrow> ep A A"
    65   "\<And>A B. ep A B \<Longrightarrow> ep B A"
    66   "\<And>A B C. ep A B \<Longrightarrow> ep B C \<Longrightarrow> ep A C"
    67   "\<And>A B. pl A B \<Longrightarrow> el B B"
    68   "\<And>A B. el A B \<Longrightarrow> el B A"
    69   "\<And>A B C. el A B \<Longrightarrow> el B C \<Longrightarrow> el A C"
    70   "\<And>A B C. ep A B \<Longrightarrow> pl B C \<Longrightarrow> pl A C"
    71   "\<And>A B C. pl A B \<Longrightarrow> el B C \<Longrightarrow> pl A C"
    72   "\<And>A B C D E F G H I J K L M N O P Q.
    73      col A B C J \<Longrightarrow> col D E F K \<Longrightarrow> col B F G L \<Longrightarrow> col C E G M \<Longrightarrow>
    74      col B D H N \<Longrightarrow> col A E H O \<Longrightarrow> col C D I P \<Longrightarrow> col A F I Q \<Longrightarrow>
    75      (\<exists> R. col G H I R) \<or> el L M \<or> el N O \<or> el P Q"
    76   "\<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"
    77   "\<And>A B C. ep A A \<Longrightarrow> ep B B \<Longrightarrow> \<exists>C. pl A C \<and> pl B C"
    78   shows goal using assms
    79   by coherent
    80 
    81 
    82 subsection {* Preservation of the Diamond Property under reflexive closure *}
    83 
    84 lemma diamond:
    85   assumes
    86   "reflexive_rewrite a b" "reflexive_rewrite a c"
    87   "\<And>A. reflexive_rewrite b A \<Longrightarrow> reflexive_rewrite c A \<Longrightarrow> goal"
    88   "\<And>A. equalish A A" 
    89   "\<And>A B. equalish A B \<Longrightarrow> equalish B A"
    90   "\<And>A B C. equalish A B \<Longrightarrow> reflexive_rewrite B C \<Longrightarrow> reflexive_rewrite A C"
    91   "\<And>A B. equalish A B \<Longrightarrow> reflexive_rewrite A B"
    92   "\<And>A B. rewrite A B \<Longrightarrow> reflexive_rewrite A B"
    93   "\<And>A B. reflexive_rewrite A B \<Longrightarrow> equalish A B \<or> rewrite A B"
    94   "\<And>A B C. rewrite A B \<Longrightarrow> rewrite A C \<Longrightarrow> \<exists>D. rewrite B D \<and> rewrite C D"
    95   shows goal using assms
    96   by coherent
    97 
    98 end