59 lines
2.1 KiB
Text
59 lines
2.1 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.connectives.eq
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using inhabited
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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 := sigma_rec (λ a b, a) p
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abbreviation dpr2 {A : Type} {B : A → Type} (p : Σ x, B x) : B (dpr1 p) := sigma_rec (λ a b, b) p
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theorem dpr1_dpair (a : A) (b : B a) : dpr1 (dpair a b) = a := refl a
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theorem dpr2_dpair (a : A) (b : B a) : dpr2 (dpair a b) = b := refl b
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-- TODO: remove prefix when we can protect it
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theorem sigma_destruct {P : sigma B → Prop} (p : sigma B) (H : ∀a b, P (dpair a b)) : P p :=
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sigma_rec H p
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theorem dpair_ext (p : sigma B) : dpair (dpr1 p) (dpr2 p) = p :=
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sigma_destruct p (take a b, refl _)
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-- Note that we give the general statment explicitly, to help the unifier
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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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(show ∀(b2 : B a2) (H1 : a1 = a2) (H2 : eq_rec_on H1 b1 = b2), dpair a1 b1 = dpair a2 b2, from
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eq_rec
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(take (b2' : B a1),
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assume (H1' : a1 = a1),
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assume (H2' : eq_rec_on H1' b1 = b2'),
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show dpair a1 b1 = dpair a1 b2', from
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calc
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dpair a1 b1 = dpair a1 (eq_rec_on H1' b1) : {symm (eq_rec_on_id H1' b1)}
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... = dpair a1 b2' : {H2'}) H1)
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b2 H1 H2
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theorem sigma_eq {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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sigma_destruct p1 (take a1 b1, sigma_destruct p2 (take a2 b2 H1 H2, dpair_eq H1 H2))
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theorem sigma_inhabited [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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instance sigma_inhabited
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end sigma
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