src/Sequents/LK/quant.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Sequents/LK/quant.ML	Fri Feb 05 21:14:17 1999 +0100
@@ -0,0 +1,160 @@
+(*  Title:      LK/ex/quant
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Classical sequent calculus: examples with quantifiers.
+*)
+
+
+writeln"LK/ex/quant: Examples with quantifiers";
+
+goal LK.thy "|- (ALL x. P)  <->  P";
+by (fast_tac LK_pack 1);
+result(); 
+
+goal LK.thy "|- (ALL x y. P(x,y))  <->  (ALL y x. P(x,y))";
+by (fast_tac LK_pack 1);
+result(); 
+
+goal LK.thy "ALL u. P(u), ALL v. Q(v) |- ALL u v. P(u) & Q(v)";
+by (fast_tac LK_pack 1);
+result(); 
+
+writeln"Permutation of existential quantifier.";
+goal LK.thy "|- (EX x y. P(x,y)) <-> (EX y x. P(x,y))";
+by (fast_tac LK_pack 1);
+result(); 
+
+goal LK.thy "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+(*Converse is invalid*)
+goal LK.thy "|- (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x)|Q(x))";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+writeln"Pushing ALL into an implication.";
+goal LK.thy "|- (ALL x. P --> Q(x))  <->  (P --> (ALL x. Q(x)))";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+goal LK.thy "|- (ALL x. P(x)-->Q)  <->  ((EX x. P(x)) --> Q)";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+goal LK.thy "|- (EX x. P)  <->  P";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+writeln"Distribution of EX over disjunction.";
+goal LK.thy "|- (EX x. P(x) | Q(x)) <-> (EX x. P(x))  |  (EX x. Q(x))";
+by (fast_tac LK_pack 1);
+result(); 
+(*5 secs*)
+
+(*Converse is invalid*)
+goal LK.thy "|- (EX x. P(x) & Q(x))  -->  (EX x. P(x))  &  (EX x. Q(x))";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+writeln"Harder examples: classical theorems.";
+
+goal LK.thy "|- (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))";
+by (fast_tac LK_pack 1);
+result(); 
+(*3 secs*)
+
+
+goal LK.thy "|- (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q";
+by (fast_tac LK_pack 1);
+result(); 
+(*5 secs*)
+
+
+goal LK.thy "|- (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)";
+by (fast_tac LK_pack 1);
+result(); 
+
+
+writeln"Basic test of quantifier reasoning";
+goal LK.thy
+   "|- (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))";
+by (fast_tac LK_pack 1);
+result();  
+
+
+goal LK.thy "|- (ALL x. Q(x))  -->  (EX x. Q(x))";
+by (fast_tac LK_pack 1);
+result();  
+
+
+writeln"The following are invalid!";
+
+(*INVALID*)
+goal LK.thy "|- (ALL x. EX y. Q(x,y))  -->  (EX y. ALL x. Q(x,y))";
+by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected"; 
+(*Check that subgoals remain: proof failed.*)
+getgoal 1; 
+
+(*INVALID*)
+goal LK.thy "|- (EX x. Q(x))  -->  (ALL x. Q(x))";
+by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1; 
+
+goal LK.thy "|- P(?a) --> (ALL x. P(x))";
+by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected";
+(*Check that subgoals remain: proof failed.*)
+getgoal 1;  
+
+goal LK.thy "|- (P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))";
+by (fast_tac LK_pack 1) handle ERROR => writeln"Failed, as expected";
+getgoal 1;  
+
+
+writeln"Back to things that are provable...";
+
+goal LK.thy "|- (ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))";
+by (fast_tac LK_pack 1); 
+result();  
+
+(*An example of why exR should be delayed as long as possible*)
+goal LK.thy "|- (P--> (EX x. Q(x))) & P--> (EX x. Q(x))";
+by (fast_tac LK_pack 1);
+result();  
+
+writeln"Solving for a Var";
+goal LK.thy 
+   "|- (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)";
+by (fast_tac LK_pack 1);
+result();
+
+
+writeln"Principia Mathematica *11.53";
+goal LK.thy 
+    "|- (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))";
+by (fast_tac LK_pack 1);
+result();
+
+
+writeln"Principia Mathematica *11.55";
+goal LK.thy "|- (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))";
+by (fast_tac LK_pack 1);
+result();
+
+writeln"Principia Mathematica *11.61";
+goal LK.thy
+   "|- (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))";
+by (fast_tac LK_pack 1);
+result();
+
+writeln"Reached end of file.";
+
+(*21 August 88: loaded in 45.7 secs*)