/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Module: logic.axims.classical Author: Leonardo de Moura -/ import logic.quantifiers logic.cast algebra.relation open eq.ops axiom prop_complete (a : Prop) : a = true ∨ a = false theorem cases (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a := or.elim (prop_complete a) (assume Ht : a = true, Ht⁻¹ ▸ H1) (assume Hf : a = false, Hf⁻¹ ▸ H2) theorem cases_on (a : Prop) {P : Prop → Prop} (H1 : P true) (H2 : P false) : P a := cases P H1 H2 a -- this supercedes the em in decidable theorem em (a : Prop) : a ∨ ¬a := or.elim (prop_complete a) (assume Ht : a = true, or.inl (eq.true_elim Ht)) (assume Hf : a = false, or.inr (eq.false_elim Hf)) theorem prop_complete_swapped (a : Prop) : a = false ∨ a = true := cases (λ x, x = false ∨ x = true) (or.inr rfl) (or.inl rfl) a theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b := or.elim (prop_complete a) (assume Hat, or.elim (prop_complete b) (assume Hbt, Hat ⬝ Hbt⁻¹) (assume Hbf, false_elim (Hbf ▸ (Hab (eq.true_elim Hat))))) (assume Haf, or.elim (prop_complete b) (assume Hbt, false_elim (Haf ▸ (Hba (eq.true_elim Hbt)))) (assume Hbf, Haf ⬝ Hbf⁻¹)) theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b := iff.elim (assume H1 H2, propext H1 H2) H theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) := propext (assume H, eq.of_iff H) (assume H, iff.of_eq H) open relation theorem iff_congruence [instance] (P : Prop → Prop) : is_congruence iff iff P := is_congruence.mk (take (a b : Prop), assume H : a ↔ b, show P a ↔ P b, from iff.of_eq (eq.of_iff H ▸ eq.refl (P a)))