lean2/library/logic/axioms/classical.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
-- logic.axioms.classical
-- ======================
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 iff_to_eq {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, iff_to_eq H)
(assume H, eq_to_iff H)
open relation
theorem iff_congruence [instance] (P : Prop → Prop) : congruence iff iff P :=
congruence.mk
(take (a b : Prop),
assume H : a ↔ b,
show P a ↔ P b, from eq_to_iff (iff_to_eq H ▸ eq.refl (P a)))