lean2/library/logic/axioms/prop_complete.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.
Module: logic.axioms.classical
Author: Leonardo de Moura
-/
import logic.connectives logic.quantifiers logic.cast algebra.relation
open eq.ops
axiom prop_complete (a : Prop) : a = true a = false
definition eq_true_or_eq_false := prop_complete
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 (of_eq_true Ht))
(assume Hf : a = false, or.inr (not_of_eq_false Hf))
theorem eq_false_or_eq_true (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 (of_eq_true Hat)))))
(assume Haf, or.elim (prop_complete b)
(assume Hbt, false.elim (Haf ▸ (Hba (of_eq_true 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)))