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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

18
library/logic/eq.lean
View file

@ -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) :
@ -127,12 +127,12 @@ end
section section
variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type} {R : Type} variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type} {R : Type}
variables {a₁ a₂ : A} variables {a₁ a₂ : A}
{b₁ : B a₁} {b₂ : B a₂} {b₁ : B a₁} {b₂ : B a₂}
{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂}
{d₁ : D a₁ b₁ c₁} {d₂ : D a₂ b₂ c₂} {d₁ : D a₁ b₁ c₁} {d₂ : D a₂ b₂ c₂}
theorem congr_arg2_dep (f : Πa, B a → R) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) theorem congr_arg2_dep (f : Πa, B a → R) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂)
: f a₁ b₁ = f a₂ b₂ := : f a₁ b₁ = f a₂ b₂ :=
eq.rec_on H₁ eq.rec_on H₁
(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂), (λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂),
@ -152,7 +152,7 @@ section
-- for the moment the following theorem is commented out, because it takes long to prove -- for the moment the following theorem is commented out, because it takes long to prove
-- theorem congr_arg4_dep (f : Πa b c, D a b c → R) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) -- theorem congr_arg4_dep (f : Πa b c, D a b c → R) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂)
-- (H₃ : eq.rec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) -- (H₃ : eq.rec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂)
-- (H₄ : eq.rec_on (congr_arg3_dep D H₁ H₂ H₃) d₁ = d₂) : f a₁ b₁ c₁ d₁ = f a₂ b₂ c₂ d₂ := -- (H₄ : eq.rec_on (congr_arg3_dep D H₁ H₂ H₃) d₁ = d₂) : f a₁ b₁ c₁ d₁ = f a₂ b₂ c₂ d₂ :=
-- eq.rec_on H₁ -- eq.rec_on H₁
-- (λ b₂ H₂ c₂ H₃ d₂ (H₄ : _), -- (λ b₂ H₂ c₂ H₃ d₂ (H₄ : _),
@ -261,5 +261,5 @@ definition elim {A : Type} (H : subsingleton A) : ∀(a b : A), a = b :=
rec (fun p, p) H rec (fun p, p) H
end subsingleton end subsingleton
protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P := protected definition prop.subsingleton [instance] (P : Prop) : subsingleton P :=
subsingleton.intro (λa b, !proof_irrel) subsingleton.intro (λa b, !proof_irrel)