(* Title: FOLP/ex/Quantifiers_Int.thy ID: $Id: Quantifiers_Int.thy,v 1.1 2008/03/26 21:38:57 wenzelm Exp $ 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" *) theory Quantifiers_Int imports IFOLP begin lemma "?p : (ALL x y. P(x,y)) --> (ALL y x. P(x,y))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (EX x y. P(x,y)) --> (EX y x. P(x,y))" by (tactic {* IntPr.fast_tac 1 *}) (*Converse is false*) lemma "?p : (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x) | Q(x))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (ALL x. P-->Q(x)) <-> (P--> (ALL x. Q(x)))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (ALL x. P(x)-->Q) <-> ((EX x. P(x)) --> Q)" by (tactic {* IntPr.fast_tac 1 *}) text "Some harder ones" lemma "?p : (EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) (*Converse is false*) lemma "?p : (EX x. P(x)&Q(x)) --> (EX x. P(x)) & (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) text "Basic test of quantifier reasoning" (*TRUE*) lemma "?p : (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (ALL x. Q(x)) --> (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) text "The following should fail, as they are false!" lemma "?p : (ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))" apply (tactic {* IntPr.fast_tac 1 *})? oops lemma "?p : (EX x. Q(x)) --> (ALL x. Q(x))" apply (tactic {* IntPr.fast_tac 1 *})? oops lemma "?p : P(?a) --> (ALL x. P(x))" apply (tactic {* IntPr.fast_tac 1 *})? oops lemma "?p : (P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))" apply (tactic {* IntPr.fast_tac 1 *})? oops text "Back to things that are provable..." lemma "?p : (ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) (*An example of why exI should be delayed as long as possible*) lemma "?p : (P --> (EX x. Q(x))) & P --> (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)" by (tactic {* IntPr.fast_tac 1 *}) lemma "?p : (ALL x. Q(x)) --> (EX x. Q(x))" by (tactic {* IntPr.fast_tac 1 *}) text "Some slow ones" (*Principia Mathematica *11.53 *) lemma "?p : (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))" by (tactic {* IntPr.fast_tac 1 *}) (*Principia Mathematica *11.55 *) lemma "?p : (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))" by (tactic {* IntPr.fast_tac 1 *}) (*Principia Mathematica *11.61 *) lemma "?p : (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))" by (tactic {* IntPr.fast_tac 1 *}) end
lemma
(ALL x y. P(x, y)) --> (ALL y x. P(x, y))
lemma
(EX x y. P(x, y)) --> (EX y x. P(x, y))
lemma
(ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x) | Q(x))
lemma
(ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))
lemma
(ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q
lemma
(EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))
lemma
(EX x. P(x) & Q(x)) --> (EX x. P(x)) & (EX x. Q(x))
lemma
(EX y. ALL x. Q(x, y)) --> (ALL x. EX y. Q(x, y))
lemma
(ALL x. Q(x)) --> (EX x. Q(x))
lemma
(ALL x. P(x) --> Q(x)) & (EX x. P(x)) --> (EX x. Q(x))
lemma
(P --> (EX x. Q(x))) & P --> (EX x. Q(x))
lemma
(ALL x. P(x) --> Q(f(x))) & (ALL x. Q(x) --> R(g(x))) & P(d) --> R(g(f(d)))
lemma
(ALL x. Q(x)) --> (EX x. Q(x))
lemma
(ALL x y. P(x) --> Q(y)) <-> (EX x. P(x)) --> (ALL y. Q(y))
lemma
(EX x y. P(x) & Q(x, y)) <-> (EX x. P(x) & (EX y. Q(x, y)))
lemma
(EX y. ALL x. P(x) --> Q(x, y)) --> (ALL x. P(x) --> (EX y. Q(x, y)))