45 lines
1.9 KiB
Text
45 lines
1.9 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Leonardo de Moura, Jeremy Avigad
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import logic.classes.inhabited logic.core.eq
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open inhabited eq_ops
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inductive sigma {A : Type} (B : A → Type) : Type :=
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dpair : Πx : A, B x → sigma B
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notation `Σ` binders `,` r:(scoped P, sigma P) := r
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namespace sigma
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section
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parameters {A : Type} {B : A → Type}
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abbreviation dpr1 (p : Σ x, B x) : A := rec (λ a b, a) p
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abbreviation dpr2 (p : Σ x, B x) : B (dpr1 p) := rec (λ a b, b) p
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theorem dpr1_dpair (a : A) (b : B a) : dpr1 (dpair a b) = a := rfl
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theorem dpr2_dpair (a : A) (b : B a) : dpr2 (dpair a b) = b := rfl
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theorem destruct [protected] {P : sigma B → Prop} (p : sigma B) (H : ∀a b, P (dpair a b)) : P p :=
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rec H p
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theorem dpair_ext (p : sigma B) : dpair (dpr1 p) (dpr2 p) = p :=
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destruct p (take a b, rfl)
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theorem dpair_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
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dpair a₁ b₁ = dpair a₂ b₂ :=
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eq.rec_on H₁
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(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂),
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calc
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dpair a₁ b₁ = dpair a₁ (eq.rec_on H₁ b₁) : {(eq.rec_on_id H₁ b₁)⁻¹}
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... = dpair a₁ b₂ : {H₂})
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b₂ H₁ H₂
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theorem equal [protected] {p₁ p₂ : Σx : A, B x} :
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∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : eq.rec_on H₁ (dpr2 p₁) = (dpr2 p₂)), p₁ = p₂ :=
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destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))
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theorem is_inhabited [protected] [instance] (H₁ : inhabited A) (H₂ : inhabited (B (default A))) :
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inhabited (sigma B) :=
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inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (dpair (default A) b)))
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end
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end sigma
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