-- 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

definition cast {A B : Type} (H : A = B) (a : A) : B
:= eq_rec a H

theorem cast_refl {A : Type} (a : A) : cast (refl A) a = a
:= refl (cast (refl A) a)

theorem cast_proof_irrel {A B : Type} (H1 H2 : A = B) (a : A) : cast H1 a = cast H2 a
:= refl (cast H1 a)

theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a
:= calc cast H a = cast (refl A) a : cast_proof_irrel H (refl A) a
            ...  = a               : cast_refl a

definition heq {A B : Type} (a : A) (b : B) := ∃ H, cast H a = b

infixl `==`:50 := heq

theorem heq_type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B
:= obtain w Hw, from H, w

theorem eq_to_heq {A : Type} {a b : A} (H : a = b) : a == b
:= exists_intro (refl A) (trans (cast_refl a) H)

theorem heq_to_eq {A : Type} {a b : A} (H : a == b) : a = b
:= obtain (w : A = A) (Hw : cast w a = b), from H,
    calc a = cast w a : symm (cast_eq w a)
      ...  = b        : Hw

theorem hrefl {A : Type} (a : A) : a == a
:= eq_to_heq (refl a)

theorem heqt_elim {a : Bool} (H : a == true) : a
:= eqt_elim (heq_to_eq H)

opaque_hint (hiding cast)

theorem hsubst {A B : Type} {a : A} {b : B} {P : ∀ (T : Type), T → Bool} (H1 : a == b) (H2 : P A a) : P B b
:= have Haux1 : ∀ H : A = A, P A (cast H a), from
     assume H : A = A, subst (symm (cast_eq H a)) H2,
   obtain (Heq : A = B) (Hw : cast Heq a = b), from H1,
   have Haux2 : P B (cast Heq a), from subst Heq Haux1 Heq,
     subst Hw Haux2

theorem hsymm {A B : Type} {a : A} {b : B} (H : a == b) : b == a
:= hsubst H (hrefl a)

theorem htrans {A B C : Type} {a : A} {b : B} {c : C} (H1 : a == b) (H2 : b == c) : a == c
:= hsubst H2 H1

theorem htrans_left {A B : Type} {a : A} {b c : B} (H1 : a == b) (H2 : b = c) : a == c
:= htrans H1 (eq_to_heq H2)

theorem htrans_right {A C : Type} {a b : A} {c : C} (H1 : a = b) (H2 : b == c) : a == c
:= htrans (eq_to_heq H1) H2

calc_trans htrans
calc_trans htrans_left
calc_trans htrans_right

theorem cast_heq {A B : Type} (H : A = B) (a : A) : cast H a == a
:= have H1 : ∀ (H : A = A) (a : A), cast H a == a, from
     λ H a, eq_to_heq (cast_eq H a),
   subst H H1 H a

theorem cast_eq_to_heq {A B : Type} {a : A} {b : B} {H : A = B} (H1 : cast H a = b) : a == b
:= calc a  == cast H a : hsymm (cast_heq H a)
       ... =  b        : H1

theorem cast_trans {A B C : Type} (Hab : A = B) (Hbc : B = C) (a : A) : cast Hbc (cast Hab a) = cast (trans Hab Hbc) a
:= heq_to_eq (calc cast Hbc (cast Hab a)   == cast Hab a             : cast_heq Hbc (cast Hab a)
                                      ...  == a                      : cast_heq Hab a
                                      ...  == cast (trans Hab Hbc) a : hsymm (cast_heq (trans Hab Hbc) a))