Theory Quantifiers_Int

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

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

theory Quantifiers_Int
imports IFOL
begin

lemma ‹(∀x y. P(x,y)) ⟶ (∀y x. P(x,y))›
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma ‹(∃x y. P(x,y)) ⟶ (∃y x. P(x,y))›
  by (tactic "IntPr.fast_tac \<^context> 1")


― ‹Converse is false›
lemma ‹(∀x. P(x)) ∨ (∀x. Q(x)) ⟶ (∀x. P(x) ∨ Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma ‹(∀x. P ⟶ Q(x)) ⟷ (P ⟶ (∀x. Q(x)))›
  by (tactic "IntPr.fast_tac \<^context> 1")


lemma ‹(∀x. P(x) ⟶ Q) ⟷ ((∃x. P(x)) ⟶ Q)›
  by (tactic "IntPr.fast_tac \<^context> 1")


text ‹Some harder ones›

lemma ‹(∃x. P(x) ∨ Q(x)) ⟷ (∃x. P(x)) ∨ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")

― ‹Converse is false›
lemma ‹(∃x. P(x) ∧ Q(x)) ⟶ (∃x. P(x)) ∧ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")


text ‹Basic test of quantifier reasoning›

― ‹TRUE›
lemma ‹(∃y. ∀x. Q(x,y)) ⟶ (∀x. ∃y. Q(x,y))›
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma ‹(∀x. Q(x)) ⟶ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")


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

lemma ‹(∀x. ∃y. Q(x,y)) ⟶ (∃y. ∀x. Q(x,y))›
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

lemma ‹(∃x. Q(x)) ⟶ (∀x. Q(x))›
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

schematic_goal ‹P(?a) ⟶ (∀x. P(x))›
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

schematic_goal ‹(P(?a) ⟶ (∀x. Q(x))) ⟶ (∀x. P(x) ⟶ Q(x))›
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops


text ‹Back to things that are provable \dots›

lemma ‹(∀x. P(x) ⟶ Q(x)) ∧ (∃x. P(x)) ⟶ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")

― ‹An example of why exI should be delayed as long as possible›
lemma ‹(P ⟶ (∃x. Q(x))) ∧ P ⟶ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")

schematic_goal ‹(∀x. P(x) ⟶ Q(f(x))) ∧ (∀x. Q(x) ⟶ R(g(x))) ∧ P(d) ⟶ R(?a)›
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma ‹(∀x. Q(x)) ⟶ (∃x. Q(x))›
  by (tactic "IntPr.fast_tac \<^context> 1")


text ‹Some slow ones›

― ‹Principia Mathematica *11.53›
lemma ‹(∀x y. P(x) ⟶ Q(y)) ⟷ ((∃x. P(x)) ⟶ (∀y. Q(y)))›
  by (tactic "IntPr.fast_tac \<^context> 1")

(*Principia Mathematica *11.55  *)
lemma ‹(∃x y. P(x) ∧ Q(x,y)) ⟷ (∃x. P(x) ∧ (∃y. Q(x,y)))›
  by (tactic "IntPr.fast_tac \<^context> 1")

(*Principia Mathematica *11.61  *)
lemma ‹(∃y. ∀x. P(x) ⟶ Q(x,y)) ⟶ (∀x. P(x) ⟶ (∃y. Q(x,y)))›
  by (tactic "IntPr.fast_tac \<^context> 1")

end