src/FOL/ex/Propositional_Int.thy

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+++ b/src/FOL/ex/Propositional_Int.thy	Sun Jul 22 22:01:30 2007 +0200
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+(*  Title:      FOL/ex/Propositional_Int.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+*)
+
+header {* First-Order Logic: propositional examples (intuitionistic version) *}
+
+theory Propositional_Int
+imports IFOL
+begin
+
+text {* commutative laws of @{text "&"} and @{text "|"} *}
+
+lemma "P & Q  -->  Q & P"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "P | Q  -->  Q | P"
+  by (tactic "IntPr.fast_tac 1")
+
+
+text {* associative laws of @{text "&"} and @{text "|"} *}
+lemma "(P & Q) & R  -->  P & (Q & R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P | Q) | R  -->  P | (Q | R)"
+  by (tactic "IntPr.fast_tac 1")
+
+
+text {* distributive laws of @{text "&"} and @{text "|"} *}
+lemma "(P & Q) | R  --> (P | R) & (Q | R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P | R) & (Q | R)  --> (P & Q) | R"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P | Q) & R  --> (P & R) | (Q & R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P & R) | (Q & R)  --> (P | Q) & R"
+  by (tactic "IntPr.fast_tac 1")
+
+
+text {* Laws involving implication *}
+
+lemma "(P-->R) & (Q-->R) <-> (P|Q --> R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P & Q --> R) <-> (P--> (Q-->R))"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P --> Q & R) <-> (P-->Q)  &  (P-->R)"
+  by (tactic "IntPr.fast_tac 1")
+
+
+text {* Propositions-as-types *}
+
+-- {* The combinator K *}
+lemma "P --> (Q --> P)"
+  by (tactic "IntPr.fast_tac 1")
+
+-- {* The combinator S *}
+lemma "(P-->Q-->R)  --> (P-->Q) --> (P-->R)"
+  by (tactic "IntPr.fast_tac 1")
+
+
+-- {* Converse is classical *}
+lemma "(P-->Q) | (P-->R)  -->  (P --> Q | R)"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma "(P-->Q)  -->  (~Q --> ~P)"
+  by (tactic "IntPr.fast_tac 1")
+
+
+text {* Schwichtenberg's examples (via T. Nipkow) *}
+
+lemma stab_imp: "(((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma stab_to_peirce:
+  "(((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q)  
+                              --> ((P --> Q) --> P) --> P"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma peirce_imp1: "(((Q --> R) --> Q) --> Q)  
+                --> (((P --> Q) --> R) --> P --> Q) --> P --> Q"
+  by (tactic "IntPr.fast_tac 1")
+  
+lemma peirce_imp2: "(((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma mints: "((((P --> Q) --> P) --> P) --> Q) --> Q"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma mints_solovev: "(P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma tatsuta: "(((P7 --> P1) --> P10) --> P4 --> P5)  
+  --> (((P8 --> P2) --> P9) --> P3 --> P10)  
+  --> (P1 --> P8) --> P6 --> P7  
+  --> (((P3 --> P2) --> P9) --> P4)  
+  --> (P1 --> P3) --> (((P6 --> P1) --> P2) --> P9) --> P5"
+  by (tactic "IntPr.fast_tac 1")
+
+lemma tatsuta1: "(((P8 --> P2) --> P9) --> P3 --> P10)  
+  --> (((P3 --> P2) --> P9) --> P4)  
+  --> (((P6 --> P1) --> P2) --> P9)  
+  --> (((P7 --> P1) --> P10) --> P4 --> P5)  
+  --> (P1 --> P3) --> (P1 --> P8) --> P6 --> P7 --> P5"
+  by (tactic "IntPr.fast_tac 1")
+
+end