src/FOL/ex/Quantifiers_Cla.thy

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+++ b/src/FOL/ex/Quantifiers_Cla.thy	Sun Jul 22 22:01:30 2007 +0200
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+(*  Title:      FOL/ex/Quantifiers_Int.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+*)
+
+header {* First-Order Logic: quantifier examples (classical version) *}
+
+theory Quantifiers_Cla
+imports FOL
+begin
+
+lemma "(ALL x y. P(x,y))  -->  (ALL y x. P(x,y))"
+  by fast
+
+lemma "(EX x y. P(x,y)) --> (EX y x. P(x,y))"
+  by fast
+
+
+-- {* Converse is false *}
+lemma "(ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x) | Q(x))"
+  by fast
+
+lemma "(ALL x. P-->Q(x))  <->  (P--> (ALL x. Q(x)))"
+  by fast
+
+
+lemma "(ALL x. P(x)-->Q)  <->  ((EX x. P(x)) --> Q)"
+  by fast
+
+
+text {* Some harder ones *}
+
+lemma "(EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))"
+  by fast
+
+-- {* Converse is false *}
+lemma "(EX x. P(x)&Q(x)) --> (EX x. P(x))  &  (EX x. Q(x))"
+  by fast
+
+
+text {* Basic test of quantifier reasoning *}
+
+-- {* TRUE *}
+lemma "(EX y. ALL x. Q(x,y)) -->  (ALL x. EX y. Q(x,y))"
+  by fast
+
+lemma "(ALL x. Q(x))  -->  (EX x. Q(x))"
+  by fast
+
+
+text {* The following should fail, as they are false! *}
+
+lemma "(ALL x. EX y. Q(x,y))  -->  (EX y. ALL x. Q(x,y))"
+  apply fast?
+  oops
+
+lemma "(EX x. Q(x))  -->  (ALL x. Q(x))"
+  apply fast?
+  oops
+
+lemma "P(?a) --> (ALL x. P(x))"
+  apply fast?
+  oops
+
+lemma "(P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))"
+  apply fast?
+  oops
+
+
+text {* Back to things that are provable \dots *}
+
+lemma "(ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))"
+  by fast
+
+-- {* An example of why exI should be delayed as long as possible *}
+lemma "(P --> (EX x. Q(x))) & P --> (EX x. Q(x))"
+  by fast
+
+lemma "(ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"
+  by fast
+
+lemma "(ALL x. Q(x))  -->  (EX x. Q(x))"
+  by fast
+
+
+text {* Some slow ones *}
+
+-- {* Principia Mathematica *11.53 *}
+lemma "(ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"
+  by fast
+
+(*Principia Mathematica *11.55  *)
+lemma "(EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))"
+  by fast
+
+(*Principia Mathematica *11.61  *)
+lemma "(EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))"
+  by fast
+
+end