a618bd7d6c
Before this commit we were using overloading for concrete structures and type classes for abstract ones. This is the first of series of commits that implement this modification
62 lines
3 KiB
Text
62 lines
3 KiB
Text
/-
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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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Sigma types, aka dependent sum.
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-/
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import logic.cast
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open inhabited sigma.ops
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override eq.ops
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namespace sigma
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universe variables u v
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variables {A A' : Type.{u}} {B : A → Type.{v}} {B' : A' → Type.{v}}
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definition unpack {C : (Σa, B a) → Type} {u : Σa, B a} (H : C ⟨u.1 , u.2⟩) : C u :=
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destruct u (λx y H, H) H
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theorem dpair_heq {a : A} {a' : A'} {b : B a} {b' : B' a'}
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(HB : B == B') (Ha : a == a') (Hb : b == b') : ⟨a, b⟩ == ⟨a', b'⟩ :=
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hcongr_arg4 @mk (heq.type_eq Ha) HB Ha Hb
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protected theorem heq {p : Σa : A, B a} {p' : Σa' : A', B' a'} (HB : B == B') :
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∀(H₁ : p.1 == p'.1) (H₂ : p.2 == p'.2), p == p' :=
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destruct p (take a₁ b₁, destruct p' (take a₂ b₂ H₁ H₂, dpair_heq HB 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 ⟨default A, b⟩))
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theorem eq_rec_dpair_commute {C : Πa, B a → Type} {a a' : A} (H : a = a') (b : B a) (c : C a b) :
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eq.rec_on H ⟨b, c⟩ = ⟨eq.rec_on H b, eq.rec_on (dcongr_arg2 C H rfl) c⟩ :=
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eq.drec_on H (dpair_eq rfl (!eq.rec_on_id⁻¹))
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variables {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) := ⟨a, b, c⟩
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definition dquad (a : A) (b : B a) (c : C a b) (d : D a b c) := ⟨a, b, c, d⟩
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definition pr1' [reducible] (x : Σ a, B a) := x.1
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definition pr2' [reducible] (x : Σ a b, C a b) := x.2.1
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definition pr3 [reducible] (x : Σ a b, C a b) := x.2.2
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definition pr3' [reducible] (x : Σ a b c, D a b c) := x.2.2.1
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definition pr4 [reducible] (x : Σ a b c, D a b c) := x.2.2.2
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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 : cast (dcongr_arg2 C H₁ H₂) c₁ = c₂) :
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⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
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dcongr_arg3 dtrip H₁ H₂ H₃
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theorem ndtrip_eq {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 : cast (congr_arg2 C H₁ H₂) c₁ = c₂) : ⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
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hdcongr_arg3 dtrip H₁ (heq.of_eq H₂) H₃
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theorem ndtrip_equal {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
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∀(H₁ : pr1 p₁ = pr1 p₂) (H₂ : pr2' p₁ = pr2' p₂)
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(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) (pr3 p₁) = pr3 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, ndtrip_eq H₁ H₂ H₃))))
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end sigma
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