src/FOL/ex/quant.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/FOL/ex/quant.ML	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,129 @@
+(*  Title: 	FOL/ex/quant
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+First-Order Logic: quantifier examples (intuitionistic and classical)
+Needs declarations of the theory "thy" and the tactic "tac"
+*)
+
+writeln"File FOL/ex/quant.";
+
+goal thy "(ALL x y.P(x,y))  -->  (ALL y x.P(x,y))";
+by tac;
+result();  
+
+
+goal thy "(EX x y.P(x,y)) --> (EX y x.P(x,y))";
+by tac;
+result();  
+
+
+(*Converse is false*)
+goal thy "(ALL x.P(x)) | (ALL x.Q(x)) --> (ALL x. P(x) | Q(x))";
+by tac;
+result();  
+
+goal thy "(ALL x. P-->Q(x))  <->  (P--> (ALL x.Q(x)))";
+by tac;
+result();  
+
+
+goal thy "(ALL x.P(x)-->Q)  <->  ((EX x.P(x)) --> Q)";
+by tac;
+result();  
+
+
+writeln"Some harder ones";
+
+goal thy "(EX x. P(x) | Q(x)) <-> (EX x.P(x)) | (EX x.Q(x))";
+by tac;
+result();  
+(*6 secs*)
+
+(*Converse is false*)
+goal thy "(EX x. P(x)&Q(x)) --> (EX x.P(x))  &  (EX x.Q(x))";
+by tac;
+result();  
+
+
+writeln"Basic test of quantifier reasoning";
+(*TRUE*)
+goal thy "(EX y. ALL x. Q(x,y)) -->  (ALL x. EX y. Q(x,y))";
+by tac;  
+result();  
+
+
+goal thy "(ALL x. Q(x))  -->  (EX x. Q(x))";
+by tac;  
+result();  
+
+
+writeln"The following should fail, as they are false!";
+
+goal thy "(ALL x. EX y. Q(x,y))  -->  (EX y. ALL x. Q(x,y))";
+by tac handle ERROR => writeln"Failed, as expected";  
+(*Check that subgoals remain: proof failed.*)
+getgoal 1; 
+
+goal thy "(EX x. Q(x))  -->  (ALL x. Q(x))";
+by tac handle ERROR => writeln"Failed, as expected";  
+getgoal 1; 
+
+goal thy "P(?a) --> (ALL x.P(x))";
+by tac handle ERROR => writeln"Failed, as expected";
+(*Check that subgoals remain: proof failed.*)
+getgoal 1;  
+
+goal thy
+    "(P(?a) --> (ALL x.Q(x))) --> (ALL x. P(x) --> Q(x))";
+by tac handle ERROR => writeln"Failed, as expected";
+getgoal 1;  
+
+
+writeln"Back to things that are provable...";
+
+goal thy "(ALL x.P(x)-->Q(x)) & (EX x.P(x)) --> (EX x.Q(x))";
+by tac;  
+result();  
+
+
+(*An example of why exI should be delayed as long as possible*)
+goal thy "(P --> (EX x.Q(x))) & P --> (EX x.Q(x))";
+by tac;  
+result();  
+
+goal thy "(ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)";
+by tac; 
+(*Verify that no subgoals remain.*) 
+uresult();  
+
+
+goal thy "(ALL x. Q(x))  -->  (EX x. Q(x))";
+by tac;
+result();  
+
+
+writeln"Some slow ones";
+
+
+(*Principia Mathematica *11.53  *)
+goal thy "(ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))";
+by tac;
+result();  
+(*6 secs*)
+
+(*Principia Mathematica *11.55  *)
+goal thy "(EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))";
+by tac;
+result();  
+(*9 secs*)
+
+(*Principia Mathematica *11.61  *)
+goal thy "(EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))";
+by tac;
+result();  
+(*3 secs*)
+
+writeln"Reached end of file.";
+