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