(* Title: LK/Quantifiers.ML ID: $Id: Quantifiers.thy,v 1.1 2006/11/20 22:47:12 wenzelm Exp $ 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
lemma
|- (∀x. P) <-> P
lemma
|- (∀x y. P(x, y)) <-> (∀y x. P(x, y))
lemma
∀u. P(u), ∀v. Q(v) |- ∀u v. P(u) ∧ Q(v)
lemma
|- (∃x y. P(x, y)) <-> (∃y x. P(x, y))
lemma
|- (∀x. P(x) ∧ Q(x)) <-> (∀x. P(x)) ∧ (∀x. Q(x))
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) <-> P
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
|- (∃x. P --> Q(x)) <-> P --> (∃x. Q(x))
lemma
|- (∃x. P(x) --> Q) <-> (∀x. P(x)) --> Q
lemma
|- (∀x. P(x)) ∨ Q <-> (∀x. P(x) ∨ Q)
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 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)))