src/Sequents/LK/Quantifiers.thy

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+++ b/src/Sequents/LK/Quantifiers.thy	Mon Nov 20 23:47:10 2006 +0100
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+(*  Title:      LK/Quantifiers.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Classical sequent calculus: examples with quantifiers.
+*)
+
+theory Quantifiers
+imports LK
+begin
+
+lemma "|- (ALL x. P)  <->  P"
+  by fast
+
+lemma "|- (ALL x y. P(x,y))  <->  (ALL y x. P(x,y))"
+  by fast
+
+lemma "ALL u. P(u), ALL v. Q(v) |- ALL u v. P(u) & Q(v)"
+  by fast
+
+
+text "Permutation of existential quantifier."
+
+lemma "|- (EX x y. P(x,y)) <-> (EX y x. P(x,y))"
+  by fast
+
+lemma "|- (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"
+  by fast
+
+(*Converse is invalid*)
+lemma "|- (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x)|Q(x))"
+  by fast
+
+
+text "Pushing ALL into an implication."
+
+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
+
+lemma "|- (EX x. P)  <->  P"
+  by fast
+
+
+text "Distribution of EX over disjunction."
+
+lemma "|- (EX x. P(x) | Q(x)) <-> (EX x. P(x))  |  (EX x. Q(x))"
+  by fast
+
+(*Converse is invalid*)
+lemma "|- (EX x. P(x) & Q(x))  -->  (EX x. P(x))  &  (EX x. Q(x))"
+  by fast
+
+
+text "Harder examples: classical theorems."
+
+lemma "|- (EX x. P-->Q(x))  <->  (P --> (EX x. Q(x)))"
+  by fast
+
+lemma "|- (EX x. P(x)-->Q)  <->  (ALL x. P(x)) --> Q"
+  by fast
+
+lemma "|- (ALL x. P(x)) | Q  <->  (ALL x. P(x) | Q)"
+  by fast
+
+
+text "Basic test of quantifier reasoning"
+
+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 are invalid!"
+
+(*INVALID*)
+lemma "|- (ALL x. EX y. Q(x,y))  -->  (EX y. ALL x. Q(x,y))"
+  apply fast?
+  apply (rule _)
+  oops
+
+(*INVALID*)
+lemma "|- (EX x. Q(x))  -->  (ALL x. Q(x))"
+  apply fast?
+  apply (rule _)
+  oops
+
+(*INVALID*)
+lemma "|- P(?a) --> (ALL x. P(x))"
+  apply fast?
+  apply (rule _)
+  oops
+
+(*INVALID*)
+lemma "|- (P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))"
+  apply fast?
+  apply (rule _)
+  oops
+
+
+text "Back to things that are provable..."
+
+lemma "|- (ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))"
+  by fast
+
+(*An example of why exR should be delayed as long as possible*)
+lemma "|- (P--> (EX x. Q(x))) & P--> (EX x. Q(x))"
+  by fast
+
+
+text "Solving for a Var"
+
+lemma "|- (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"
+  by fast
+
+
+text "Principia Mathematica *11.53"
+
+lemma "|- (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"
+  by fast
+
+
+text "Principia Mathematica *11.55"
+
+lemma "|- (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))"
+  by fast
+
+
+text "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
+
+
+(*21 August 88: loaded in 45.7 secs*)
+(*18 September 2005: loaded in 0.114 secs*)
+
+end