refactor(library/logic/cast): move heq declaration to a separate module

heq is be needed for some automatically generated constructions.
So, we want it available with the least number of dependencies.
This commit is contained in:
Leonardo de Moura 2014-11-08 10:09:54 -08:00
parent c7992f2cac
commit ad2ecfb7a8
2 changed files with 53 additions and 42 deletions

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@ -1,7 +1,7 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved. -- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE. -- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura -- Author: Leonardo de Moura
import .eq .quantifiers import logic.eq logic.heq logic.quantifiers
open eq.ops open eq.ops
-- cast.lean -- cast.lean
@ -23,55 +23,17 @@ section
rfl rfl
end end
inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
refl : heq a a
infixl `==`:50 := heq
namespace heq namespace heq
universe variable u universe variable u
variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C} 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)
theorem type_eq (H : a == b) : A = B :=
subst H (eq.refl A)
theorem from_eq (H : a = a') : a == a' :=
eq.subst H (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 to_cast_eq (H : a == b) : cast (type_eq H) a = b := theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
drec_on H !cast_eq drec_on H !cast_eq
theorem to_eq (H : a == a') : a = a' :=
calc
a = cast (eq.refl A) a : cast_eq
... = a' : to_cast_eq H
theorem elim {D : Type} (H₁ : a == b) (H₂ : ∀ (Hab : A = B), cast Hab a = b → D) : D := theorem elim {D : Type} (H₁ : a == b) (H₂ : ∀ (Hab : A = B), cast Hab a = b → D) : D :=
H₂ (type_eq H₁) (to_cast_eq H₁) H₂ (type_eq H₁) (to_cast_eq H₁)
end heq end heq
calc_trans heq.trans
calc_trans heq.trans_left
calc_trans heq.trans_right
calc_symm heq.symm
section section
universe variables u v universe variables u v
variables {A A' B C : Type.{u}} {P P' : A → Type.{v}} {a a' : A} {b : B} variables {A A' B C : Type.{u}} {P P' : A → Type.{v}} {a a' : A} {b : B}
@ -91,9 +53,6 @@ section
theorem hproof_irrel {a b : Prop} (H : a = b) (H₁ : a) (H₂ : b) : H₁ == H₂ := theorem hproof_irrel {a b : Prop} (H : a = b) (H₁ : a) (H₂ : b) : H₁ == H₂ :=
eq_rec_to_heq (proof_irrel (cast H H₁) H₂) eq_rec_to_heq (proof_irrel (cast H H₁) H₂)
theorem heq.true_elim {a : Prop} (H : a == true) : a :=
eq_true_elim (heq.to_eq H)
--TODO: generalize to eq.rec. This is a special case of rec_on_compose in eq.lean --TODO: generalize to eq.rec. This is a special case of rec_on_compose in eq.lean
theorem cast_trans (Hab : A = B) (Hbc : B = C) (a : A) : theorem cast_trans (Hab : A = B) (Hbc : B = C) (a : A) :
cast Hbc (cast Hab a) = cast (Hab ⬝ Hbc) a := cast Hbc (cast Hab a) = cast (Hab ⬝ Hbc) a :=

52
library/logic/heq.lean Normal file
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@ -0,0 +1,52 @@
-- 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)
theorem type_eq (H : a == b) : A = B :=
subst H (eq.refl A)
theorem from_eq (H : a = a') : a == a' :=
eq.subst H (refl a)
theorem to_eq (H : a == a') : a = a' :=
have H₁ : ∀ (Ht : A = A), eq.rec_on Ht a = a, from
take Ht, eq.refl (eq.rec_on Ht a),
have H₂ : ∀ (Ht : A = A), eq.rec_on Ht a = a', from
heq.rec_on H H₁,
H₂ (type_eq H)
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