71 lines
3.2 KiB
Text
71 lines
3.2 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, Floris van Doorn
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import logic.inhabited logic.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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definition dpr1 (p : Σ x, B x) : A := rec (λ a b, a) p
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definition 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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protected theorem destruct {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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congr_arg2_dep dpair H₁ H₂
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protected theorem equal {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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protected definition is_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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section trip_quad
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parameters {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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definition dtrip (a : A) (b : B a) (c : C a b) := dpair a (dpair b c)
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definition dquad (a : A) (b : B a) (c : C a b) (d : D a b c) := dpair a (dpair b (dpair c d))
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definition dpr1' (x : Σ a, B a) := dpr1 x
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definition dpr2' (x : Σ a b, C a b) := dpr1 (dpr2 x)
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definition dpr3 (x : Σ a b, C a b) := dpr2 (dpr2 x)
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definition dpr3' (x : Σ a b c, D a b c) := dpr1 (dpr2 (dpr2 x))
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definition dpr4 (x : Σ a b c, D a b c) := dpr2 (dpr2 (dpr2 x))
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theorem dtrip_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂}
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(H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) (H₃ : eq.rec_on (congr_arg2_dep C H₁ H₂) c₁ = c₂) :
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dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
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congr_arg3_dep dtrip H₁ H₂ H₃
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end trip_quad
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theorem dtrip_eq_ndep {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
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{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
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(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) c₁ = c₂) :
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dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
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congr_arg3_ndep_dep dtrip H₁ H₂ H₃
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theorem trip.equal_ndep {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
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∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : dpr2' p₁ = dpr2' p₂)
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(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) (dpr3 p₁) = dpr3 p₂), p₁ = p₂ :=
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destruct p₁ (take a₁ q₁, destruct q₁ (take b₁ c₁, destruct p₂ (take a₂ q₂, destruct q₂
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(take b₂ c₂ H₁ H₂ H₃, dtrip_eq_ndep H₁ H₂ H₃))))
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end sigma
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