(* Title: FOL/ex/Quantifiers_Int.thy ID: $Id: Quantifiers_Cla.thy,v 1.1 2007/07/22 20:01:30 wenzelm Exp $ 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
lemma
(∀x y. P(x, y)) --> (∀y x. P(x, y))
lemma
(∃x y. P(x, y)) --> (∃y x. P(x, y))
lemma
(∀x. P(x)) ∨ (∀x. Q(x)) --> (∀x. P(x) ∨ Q(x))
lemma
(∀x. P --> Q(x)) <-> P --> (∀x. Q(x))
lemma
(∀x. P(x) --> Q) <-> (∃x. P(x)) --> Q
lemma
(∃x. P(x) ∨ Q(x)) <-> (∃x. P(x)) ∨ (∃x. Q(x))
lemma
(∃x. P(x) ∧ Q(x)) --> (∃x. P(x)) ∧ (∃x. Q(x))
lemma
(∃y. ∀x. Q(x, y)) --> (∀x. ∃y. Q(x, y))
lemma
(∀x. Q(x)) --> (∃x. Q(x))
lemma
(∀x. P(x) --> Q(x)) ∧ (∃x. P(x)) --> (∃x. Q(x))
lemma
(P --> (∃x. Q(x))) ∧ P --> (∃x. Q(x))
lemma
(∀x. P(x) --> Q(f(x))) ∧ (∀x. Q(x) --> R(g(x))) ∧ P(d) --> R(g(f(d)))
lemma
(∀x. Q(x)) --> (∃x. Q(x))
lemma
(∀x y. P(x) --> Q(y)) <-> (∃x. P(x)) --> (∀y. Q(y))
lemma
(∃x y. P(x) ∧ Q(x, y)) <-> (∃x. P(x) ∧ (∃y. Q(x, y)))
lemma
(∃y. ∀x. P(x) --> Q(x, y)) --> (∀x. P(x) --> (∃y. Q(x, y)))