114 lines
4.5 KiB
Text
114 lines
4.5 KiB
Text
/-
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Copyright (c) 2015 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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Module: init.quot
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Author: Leonardo de Moura
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Quotient types.
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-/
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prelude
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import init.sigma init.setoid init.logic
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open sigma.ops setoid
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constant quot.{l} : Π {A : Type.{l}}, setoid A → Type.{l}
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-- Remark: if we do not use propext here, then we would need a quot.lift for propositions.
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constant propext {a b : Prop} : a ↔ b → a = b
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namespace quot
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constant mk : Π {A : Type} [s : setoid A], A → quot s
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notation `⟦`:max a `⟧`:0 := mk a
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constant sound : Π {A : Type} [s : setoid A] {a b : A}, a ≈ b → ⟦a⟧ = ⟦b⟧
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constant exact : Π {A : Type} [s : setoid A] {a b : A}, ⟦a⟧ = ⟦b⟧ → a ≈ b
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constant lift : Π {A B : Type} [s : setoid A] (f : A → B), (∀ a b, a ≈ b → f a = f b) → quot s → B
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constant ind : ∀ {A : Type} [s : setoid A] {B : quot s → Prop}, (∀ a, B ⟦a⟧) → ∀ q, B q
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init_quotient
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protected theorem lift_beta {A B : Type} [s : setoid A] (f : A → B) (c : ∀ a b, a ≈ b → f a = f b) (a : A) : lift f c ⟦a⟧ = f a :=
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rfl
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protected theorem ind_beta {A : Type} [s : setoid A] {B : quot s → Prop} (p : ∀ a, B ⟦a⟧) (a : A) : ind p ⟦a⟧ = p a :=
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rfl
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protected definition lift_on [reducible] {A B : Type} [s : setoid A] (q : quot s) (f : A → B) (c : ∀ a b, a ≈ b → f a = f b) : B :=
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lift f c q
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protected theorem induction_on {A : Type} [s : setoid A] {B : quot s → Prop} (q : quot s) (H : ∀ a, B ⟦a⟧) : B q :=
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ind H q
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section
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variable {A : Type}
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variable [s : setoid A]
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variable {B : quot s → Type}
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include s
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protected definition indep [reducible] (f : Π a, B ⟦a⟧) (a : A) : Σ q, B q :=
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⟨⟦a⟧, f a⟩
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protected lemma indep_coherent (f : Π a, B ⟦a⟧)
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(H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
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: ∀ a b, a ≈ b → indep f a = indep f b :=
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λa b e, sigma.equal (sound e) (H a b e)
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protected lemma lift_indep_pr1
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(f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
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(q : quot s) : (lift (indep f) (indep_coherent f H) q).1 = q :=
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ind (λ a, by esimp) q
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protected definition rec [reducible]
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(f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
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(q : quot s) : B q :=
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let p := lift (indep f) (indep_coherent f H) q in
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eq.rec_on (lift_indep_pr1 f H q) (p.2)
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protected definition rec_on [reducible]
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(q : quot s) (f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b) : B q :=
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rec f H q
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protected definition rec_on_subsingleton [reducible]
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[H : ∀ a, subsingleton (B ⟦a⟧)] (q : quot s) (f : Π a, B ⟦a⟧) : B q :=
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rec f (λ a b h, !subsingleton.elim) q
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end
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section
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variables {A B C : Type}
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variables [s₁ : setoid A] [s₂ : setoid B]
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include s₁ s₂
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protected definition lift₂ [reducible]
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(f : A → B → C)(c : ∀ a₁ a₂ b₁ b₂, a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂)
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(q₁ : quot s₁) (q₂ : quot s₂) : C :=
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lift
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(λ a₁, lift (λ a₂, f a₁ a₂) (λ a b H, c a₁ a a₁ b (setoid.refl a₁) H) q₂)
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(λ a b H, ind (λ a', proof c a a' b a' H (setoid.refl a') qed) q₂)
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q₁
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protected definition lift_on₂ [reducible]
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(q₁ : quot s₁) (q₂ : quot s₂) (f : A → B → C) (c : ∀ a₁ a₂ b₁ b₂, a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂) : C :=
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lift₂ f c q₁ q₂
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protected theorem ind₂ {C : quot s₁ → quot s₂ → Prop} (H : ∀ a b, C ⟦a⟧ ⟦b⟧) (q₁ : quot s₁) (q₂ : quot s₂) : C q₁ q₂ :=
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quot.ind (λ a₁, quot.ind (λ a₂, H a₁ a₂) q₂) q₁
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protected theorem induction_on₂
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{C : quot s₁ → quot s₂ → Prop} (q₁ : quot s₁) (q₂ : quot s₂) (H : ∀ a b, C ⟦a⟧ ⟦b⟧) : C q₁ q₂ :=
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quot.ind (λ a₁, quot.ind (λ a₂, H a₁ a₂) q₂) q₁
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end
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section
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variables {A B : Type}
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variables [s₁ : setoid A] [s₂ : setoid B]
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include s₁ s₂
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variable {C : quot s₁ → quot s₂ → Type}
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protected definition rec_on_subsingleton₂ [reducible]
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{C : quot s₁ → quot s₂ → Type₁} [H : ∀ a b, subsingleton (C ⟦a⟧ ⟦b⟧)]
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(q₁ : quot s₁) (q₂ : quot s₂) (f : Π a b, C ⟦a⟧ ⟦b⟧) : C q₁ q₂:=
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@quot.rec_on_subsingleton _ _ _
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(λ a, quot.ind _ _)
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q₁ (λ a, quot.rec_on_subsingleton q₂ (λ b, f a b))
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end
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end quot
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