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refactor(library/logic): use new K-like reduction to simplify some proofs

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Leonardo de Moura 2014-10-10 14:52:21 -07:00
parent 235b8975d2
commit a41850227a
2 changed files with 10 additions and 11 deletions
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3
library/logic/cast.lean
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@ -16,8 +16,7 @@ theorem cast_proof_irrel {A B : Type} (H₁ H₂ : A = B) (a : A) : cast H₁ a
rfl rfl
theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a := theorem cast_eq {A : Type} (H : A = A) (a : A) : cast H a = a :=
calc cast H a = cast (eq.refl A) a : rfl rfl
... = a : rfl
inductive heq.{l} {A : Type.{l}} (a : A) : Π {B : Type.{l}}, B → Prop := inductive heq.{l} {A : Type.{l}} (a : A) : Π {B : Type.{l}}, B → Prop :=
refl : heq a a refl : heq a a

6
library/logic/eq.lean
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@ -26,7 +26,7 @@ section
variables {A : Type} variables {A : Type}
variables {a b c : A} variables {a b c : A}
theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) := theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) :=
!proof_irrel rfl
theorem irrel (H₁ H₂ : a = b) : H₁ = H₂ := theorem irrel (H₁ H₂ : a = b) : H₁ = H₂ :=
!proof_irrel !proof_irrel
@ -58,7 +58,7 @@ namespace eq
eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁ eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁
theorem rec_on_id {A : Type} {a : A} {B : Πa' : A, a = a' → Type} (H : a = a) (b : B a H) : rec_on H b = b := theorem rec_on_id {A : Type} {a : A} {B : Πa' : A, a = a' → Type} (H : a = a) (b : B a H) : rec_on H b = b :=
refl (rec_on rfl b) rfl
theorem rec_on_constant {A : Type} {a a' : A} {B : Type} (H : a = a') (b : B) : rec_on H b = b := theorem rec_on_constant {A : Type} {a a' : A} {B : Type} (H : a = a') (b : B) : rec_on H b = b :=
rec_on H (λ(H' : a = a), rec_on_id H' b) H rec_on H (λ(H' : a = a), rec_on_id H' b) H
@ -72,7 +72,7 @@ namespace eq
rec_on H (λ(H : a = a) (H' : f a = f a), rec_on_id H b ⬝ rec_on_id H' b⁻¹) H H' rec_on H (λ(H : a = a) (H' : f a = f a), rec_on_id H b ⬝ rec_on_id H' b⁻¹) H H'
theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b := theorem rec_id {A : Type} {a : A} {B : A → Type} (H : a = a) (b : B a) : rec b H = b :=
id_refl H⁻¹ ▸ refl (eq.rec b (refl a)) rfl
theorem rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H₁ : a = b) (H₂ : b = c) theorem rec_on_compose {A : Type} {a b c : A} {P : A → Type} (H₁ : a = b) (H₂ : b = c)
(u : P a) : (u : P a) :