src/Sequents/LK/Propositional.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK/Propositional.thy	Mon Nov 20 23:47:10 2006 +0100
@@ -0,0 +1,160 @@
+(*  Title:      LK/Propositional.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+*)
+
+header {* Classical sequent calculus: examples with propositional connectives *}
+
+theory Propositional
+imports LK
+begin
+
+text "absorptive laws of & and | "
+
+lemma "|- P & P <-> P"
+  by fast_prop
+
+lemma "|- P | P <-> P"
+  by fast_prop
+
+
+text "commutative laws of & and | "
+
+lemma "|- P & Q  <->  Q & P"
+  by fast_prop
+
+lemma "|- P | Q  <->  Q | P"
+  by fast_prop
+
+
+text "associative laws of & and | "
+
+lemma "|- (P & Q) & R  <->  P & (Q & R)"
+  by fast_prop
+
+lemma "|- (P | Q) | R  <->  P | (Q | R)"
+  by fast_prop
+
+
+text "distributive laws of & and | "
+
+lemma "|- (P & Q) | R  <-> (P | R) & (Q | R)"
+  by fast_prop
+
+lemma "|- (P | Q) & R  <-> (P & R) | (Q & R)"
+  by fast_prop
+
+
+text "Laws involving implication"
+
+lemma "|- (P|Q --> R) <-> (P-->R) & (Q-->R)"
+  by fast_prop
+
+lemma "|- (P & Q --> R) <-> (P--> (Q-->R))"
+  by fast_prop
+
+lemma "|- (P --> Q & R) <-> (P-->Q)  &  (P-->R)"
+  by fast_prop
+
+
+text "Classical theorems"
+
+lemma "|- P|Q --> P| ~P&Q"
+  by fast_prop
+
+lemma "|- (P-->Q)&(~P-->R)  -->  (P&Q | R)"
+  by fast_prop
+
+lemma "|- P&Q | ~P&R  <->  (P-->Q)&(~P-->R)"
+  by fast_prop
+
+lemma "|- (P-->Q) | (P-->R)  <->  (P --> Q | R)"
+  by fast_prop
+
+
+(*If and only if*)
+
+lemma "|- (P<->Q) <-> (Q<->P)"
+  by fast_prop
+
+lemma "|- ~ (P <-> ~P)"
+  by fast_prop
+
+
+(*Sample problems from 
+  F. J. Pelletier, 
+  Seventy-Five Problems for Testing Automatic Theorem Provers,
+  J. Automated Reasoning 2 (1986), 191-216.
+  Errata, JAR 4 (1988), 236-236.
+*)
+
+(*1*)
+lemma "|- (P-->Q)  <->  (~Q --> ~P)"
+  by fast_prop
+
+(*2*)
+lemma "|- ~ ~ P  <->  P"
+  by fast_prop
+
+(*3*)
+lemma "|- ~(P-->Q) --> (Q-->P)"
+  by fast_prop
+
+(*4*)
+lemma "|- (~P-->Q)  <->  (~Q --> P)"
+  by fast_prop
+
+(*5*)
+lemma "|- ((P|Q)-->(P|R)) --> (P|(Q-->R))"
+  by fast_prop
+
+(*6*)
+lemma "|- P | ~ P"
+  by fast_prop
+
+(*7*)
+lemma "|- P | ~ ~ ~ P"
+  by fast_prop
+
+(*8.  Peirce's law*)
+lemma "|- ((P-->Q) --> P)  -->  P"
+  by fast_prop
+
+(*9*)
+lemma "|- ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
+  by fast_prop
+
+(*10*)
+lemma "Q-->R, R-->P&Q, P-->(Q|R) |- P<->Q"
+  by fast_prop
+
+(*11.  Proved in each direction (incorrectly, says Pelletier!!)  *)
+lemma "|- P<->P"
+  by fast_prop
+
+(*12.  "Dijkstra's law"*)
+lemma "|- ((P <-> Q) <-> R)  <->  (P <-> (Q <-> R))"
+  by fast_prop
+
+(*13.  Distributive law*)
+lemma "|- P | (Q & R)  <-> (P | Q) & (P | R)"
+  by fast_prop
+
+(*14*)
+lemma "|- (P <-> Q) <-> ((Q | ~P) & (~Q|P))"
+  by fast_prop
+
+(*15*)
+lemma "|- (P --> Q) <-> (~P | Q)"
+  by fast_prop
+
+(*16*)
+lemma "|- (P-->Q) | (Q-->P)"
+  by fast_prop
+
+(*17*)
+lemma "|- ((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S))"
+  by fast_prop
+
+end