116 lines
4.5 KiB
Text
116 lines
4.5 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
|
|
using eq_proofs
|
|
|
|
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_elim {A B : Type} {C : Prop} {a : A} {b : B} (H1 : a == b) (H2 : ∀ (Hab : A = B), cast Hab a = b → C) : C :=
|
|
obtain w Hw, from H1, H2 w Hw
|
|
|
|
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) (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 : (cast_eq w a)⁻¹
|
|
... = b : Hw
|
|
|
|
theorem hrefl {A : Type} (a : A) : a == a :=
|
|
eq_to_heq (refl a)
|
|
|
|
theorem heqt_elim {a : Prop} (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 → Prop} (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, (cast_eq H a)⁻¹ ▸ H2,
|
|
obtain (Heq : A = B) (Hw : cast Heq a = b), from H1,
|
|
have Haux2 : P B (cast Heq a), from subst Heq Haux1 Heq,
|
|
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 type_eq {A B : Type} {a : A} {b : B} (H : a == b) : A = B :=
|
|
hsubst H (refl A)
|
|
|
|
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
|
|
assume 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 (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 (Hab ⬝ Hbc) a : hsymm (cast_heq (Hab ⬝ Hbc) a))
|
|
|
|
theorem dcongr2 {A : Type} {B : A → Type} (f : Πx, B x) {a b : A} (H : a = b) : f a == f b :=
|
|
have e1 : ∀ (H : B a = B a), cast H (f a) = f a, from
|
|
assume H, cast_eq H (f a),
|
|
have e2 : ∀ (H : B a = B b), cast H (f a) = f b, from
|
|
subst H e1,
|
|
have e3 : cast (congr2 B H) (f a) = f b, from
|
|
e2 (congr2 B H),
|
|
cast_eq_to_heq e3
|
|
|
|
theorem pi_eq {A : Type} {B B' : A → Type} (H : B = B') : (Π x, B x) = (Π x, B' x) :=
|
|
subst H (refl (Π x, B x))
|
|
|
|
theorem cast_app' {A : Type} {B B' : A → Type} (H : B = B') (f : Π x, B x) (a : A) : cast (pi_eq H) f a == f a :=
|
|
have H1 : ∀ (H : (Π x, B x) = (Π x, B x)), cast H f a == f a, from
|
|
assume H, eq_to_heq (congr1 (cast_eq H f) a),
|
|
have H2 : ∀ (H : (Π x, B x) = (Π x, B' x)), cast H f a == f a, from
|
|
subst H H1,
|
|
H2 (pi_eq H)
|
|
|
|
theorem cast_pull {A : Type} {B B' : A → Type} (H : B = B') (f : Π x, B x) (a : A) :
|
|
cast (pi_eq H) f a = cast (congr1 H a) (f a) :=
|
|
heq_to_eq (calc cast (pi_eq H) f a == f a : cast_app' H f a
|
|
... == cast (congr1 H a) (f a) : hsymm (cast_heq (congr1 H a) (f a)))
|
|
|
|
theorem hcongr1' {A : Type} {B B' : A → Type} {f : Π x, B x} {f' : Π x, B' x} (a : A) (H1 : f == f') (H2 : B = B')
|
|
: f a == f' a :=
|
|
heq_elim H1 (λ (Ht : (Π x, B x) = (Π x, B' x)) (Hw : cast Ht f = f'),
|
|
calc f a == cast (pi_eq H2) f a : hsymm (cast_app' H2 f a)
|
|
... = cast Ht f a : refl (cast Ht f a)
|
|
... = f' a : congr1 Hw a)
|