lean2/library/data/sigma.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
import logic.inhabited logic.eq
open inhabited eq.ops
inductive sigma {A : Type} (B : A → Type) : Type :=
dpair : Πx : A, B x → sigma B
notation `Σ` binders `,` r:(scoped P, sigma P) := r
namespace sigma
variables {A : Type} {B : A → Type}
--without reducible tag, slice.composition_functor in algebra.category.constructions fails
definition dpr1 [reducible] (p : Σ x, B x) : A := rec (λ a b, a) p
definition dpr2 [reducible] (p : Σ x, B x) : B (dpr1 p) := rec (λ a b, b) p
theorem dpr1_dpair (a : A) (b : B a) : dpr1 (dpair a b) = a := rfl
theorem dpr2_dpair (a : A) (b : B a) : dpr2 (dpair a b) = b := rfl
protected theorem destruct {P : sigma B → Prop} (p : sigma B) (H : ∀a b, P (dpair a b)) : P p :=
rec H p
theorem dpair_ext (p : sigma B) : dpair (dpr1 p) (dpr2 p) = p :=
destruct p (take a b, rfl)
theorem dpair_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
dpair a₁ b₁ = dpair a₂ b₂ :=
congr_arg2_dep dpair H₁ H₂
protected theorem equal {p₁ p₂ : Σx : A, B x} :
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : eq.rec_on H₁ (dpr2 p₁) = dpr2 p₂), p₁ = p₂ :=
destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))
protected definition is_inhabited [instance] (H₁ : inhabited A) (H₂ : inhabited (B (default A))) :
inhabited (sigma B) :=
inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (dpair (default A) b)))
variables {C : Πa, B a → Type} {D : Πa b, C a b → Type}
definition dtrip (a : A) (b : B a) (c : C a b) := dpair a (dpair b c)
definition dquad (a : A) (b : B a) (c : C a b) (d : D a b c) := dpair a (dpair b (dpair c d))
definition dpr1' (x : Σ a, B a) := dpr1 x
definition dpr2' (x : Σ a b, C a b) := dpr1 (dpr2 x)
definition dpr3 (x : Σ a b, C a b) := dpr2 (dpr2 x)
definition dpr3' (x : Σ a b c, D a b c) := dpr1 (dpr2 (dpr2 x))
definition dpr4 (x : Σ a b c, D a b c) := dpr2 (dpr2 (dpr2 x))
theorem dtrip_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂}
(H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) (H₃ : eq.rec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) :
dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
congr_arg3_dep dtrip H₁ H₂ H₃
theorem dtrip_eq_ndep {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) c₁ = c₂) :
dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
congr_arg3_ndep_dep dtrip H₁ H₂ H₃
theorem trip.equal_ndep {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : dpr2' p₁ = dpr2' p₂)
(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) (dpr3 p₁) = dpr3 p₂), p₁ = p₂ :=
destruct p₁ (take a₁ q₁, destruct q₁ (take b₁ c₁, destruct p₂ (take a₂ q₂, destruct q₂
(take b₂ c₂ H₁ H₂ H₃, dtrip_eq_ndep H₁ H₂ H₃))))
end sigma