28 lines
1.2 KiB
Text
28 lines
1.2 KiB
Text
import data.sigma tools.tactic
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namespace sigma
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definition no_confusion_type {A : Type} {B : A → Type} (P : Type) (v₁ v₂ : sigma B) : Type :=
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rec_on v₁
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(λ (a₁ : A) (b₁ : B a₁), rec_on v₂
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(λ (a₂ : A) (b₂ : B a₂),
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(Π (eq₁ : a₁ = a₂), eq.rec_on eq₁ b₁ = b₂ → P) → P))
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definition no_confusion {A : Type} {B : A → Type} {P : Type} {v₁ v₂ : sigma B} : v₁ = v₂ → no_confusion_type P v₁ v₂ :=
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assume H₁₂ : v₁ = v₂,
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have aux : v₁ = v₁ → no_confusion_type P v₁ v₁, from
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assume H₁₁, rec_on v₁
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(λ (a₁ : A) (b₁ : B a₁) (h : Π (eq₁ : a₁ = a₁), eq.rec_on eq₁ b₁ = b₁ → P),
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h rfl rfl),
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eq.rec_on H₁₂ aux H₁₂
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theorem sigma.mk.inj_1 {A : Type} {B : A → Type} {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (Heq : dpair a₁ b₁ = dpair a₂ b₂) : a₁ = a₂ :=
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begin
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apply (no_confusion Heq), intros, assumption
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end
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theorem sigma.mk.inj_2 {A : Type} {B : A → Type} (a₁ a₂ : A) (b₁ : B a₁) (b₂ : B a₂) (Heq : dpair a₁ b₁ = dpair a₂ b₂) :
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eq.rec_on (sigma.mk.inj_1 Heq) b₁ = b₂ :=
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begin
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apply (no_confusion Heq), intros, eassumption
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end
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end sigma
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