Theory Quantifiers_Cla

(*  Title:      FOL/ex/Quantifiers_Cla.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

section ‹First-Order Logic: quantifier examples (classical version)›

theory Quantifiers_Cla
imports FOL
begin

lemma ‹(∀x y. P(x,y)) ⟶ (∀y x. P(x,y))›
  by fast

lemma ‹(∃x y. P(x,y)) ⟶ (∃y x. P(x,y))›
  by fast


text ‹Converse is false.›
lemma ‹(∀x. P(x)) ∨ (∀x. Q(x)) ⟶ (∀x. P(x) ∨ Q(x))›
  by fast

lemma ‹(∀x. P ⟶ Q(x)) ⟷ (P ⟶ (∀x. Q(x)))›
  by fast


lemma ‹(∀x. P(x) ⟶ Q) ⟷ ((∃x. P(x)) ⟶ Q)›
  by fast


text ‹Some harder ones.›

lemma ‹(∃x. P(x) ∨ Q(x)) ⟷ (∃x. P(x)) ∨ (∃x. Q(x))›
  by fast

― ‹Converse is false.›
lemma ‹(∃x. P(x) ∧ Q(x)) ⟶ (∃x. P(x)) ∧ (∃x. Q(x))›
  by fast


text ‹Basic test of quantifier reasoning.›

― ‹TRUE›
lemma ‹(∃y. ∀x. Q(x,y)) ⟶ (∀x. ∃y. Q(x,y))›
  by fast

lemma ‹(∀x. Q(x)) ⟶ (∃x. Q(x))›
  by fast


text ‹The following should fail, as they are false!›

lemma ‹(∀x. ∃y. Q(x,y)) ⟶ (∃y. ∀x. Q(x,y))›
  apply fast?
  oops

lemma ‹(∃x. Q(x)) ⟶ (∀x. Q(x))›
  apply fast?
  oops

schematic_goal ‹P(?a) ⟶ (∀x. P(x))›
  apply fast?
  oops

schematic_goal ‹(P(?a) ⟶ (∀x. Q(x))) ⟶ (∀x. P(x) ⟶ Q(x))›
  apply fast?
  oops


text ‹Back to things that are provable \dots›

lemma ‹(∀x. P(x) ⟶ Q(x)) ∧ (∃x. P(x)) ⟶ (∃x. Q(x))›
  by fast

text ‹An example of why ‹exI› should be delayed as long as possible.›
lemma ‹(P ⟶ (∃x. Q(x))) ∧ P ⟶ (∃x. Q(x))›
  by fast

schematic_goal ‹(∀x. P(x) ⟶ Q(f(x))) ∧ (∀x. Q(x) ⟶ R(g(x))) ∧ P(d) ⟶ R(?a)›
  by fast

lemma ‹(∀x. Q(x)) ⟶ (∃x. Q(x))›
  by fast


text ‹Some slow ones›

text ‹Principia Mathematica *11.53›
lemma ‹(∀x y. P(x) ⟶ Q(y)) ⟷ ((∃x. P(x)) ⟶ (∀y. Q(y)))›
  by fast

(*Principia Mathematica *11.55  *)
lemma ‹(∃x y. P(x) ∧ Q(x,y)) ⟷ (∃x. P(x) ∧ (∃y. Q(x,y)))›
  by fast

(*Principia Mathematica *11.61  *)
lemma ‹(∃y. ∀x. P(x) ⟶ Q(x,y)) ⟶ (∀x. P(x) ⟶ (∃y. Q(x,y)))›
  by fast

end