lean2/library/logic/heq.lean

50 lines
1.5 KiB
Text

-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.eq
inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
refl : heq a a
infixl `==`:50 := heq
namespace heq
universe variable u
variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
rec_on H₁ H₂
theorem symm (H : a == b) : b == a :=
subst H (refl a)
definition type_eq (H : a == b) : A = B :=
heq.rec_on H (eq.refl A)
theorem from_eq (H : a = a') : a == a' :=
eq.subst H (refl a)
definition to_eq (H : a == a') : a = a' :=
have H₁ : ∀ (Ht : A = A), eq.rec_on Ht a = a, from
λ Ht, eq.refl (eq.rec_on Ht a),
heq.rec_on H H₁ (eq.refl A)
theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
subst H₂ H₁
theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
trans H₁ (from_eq H₂)
theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
trans (from_eq H₁) H₂
theorem true_elim {a : Prop} (H : a == true) : a :=
eq_true_elim (heq.to_eq H)
end heq
calc_trans heq.trans
calc_trans heq.trans_left
calc_trans heq.trans_right
calc_symm heq.symm