77d5657813
Motivation: this file defines basic things such as function composition. In the HoTT library, it is located in the init folder.
97 lines
3.6 KiB
Text
97 lines
3.6 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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Author: Leonardo de Moura
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Unordered pairs
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-/
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import data.prod logic.identities
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open prod prod.ops quot function
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private definition eqv {A : Type} (p₁ p₂ : A × A) : Prop :=
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p₁ = p₂ ∨ swap p₁ = p₂
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infix `~` := eqv -- this is "~"
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private theorem eqv.refl [refl] {A : Type} : ∀ p : A × A, p ~ p :=
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take p, or.inl rfl
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private theorem eqv.symm [symm] {A : Type} : ∀ p₁ p₂ : A × A, p₁ ~ p₂ → p₂ ~ p₁ :=
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take p₁ p₂ h, or.elim h
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(λ e, by rewrite e)
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(λ e, begin esimp [eqv], rewrite [-e, swap_swap], right, reflexivity end)
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private theorem eqv.trans [trans] {A : Type} : ∀ p₁ p₂ p₃ : A × A, p₁ ~ p₂ → p₂ ~ p₃ → p₁ ~ p₃ :=
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take p₁ p₂ p₃ h₁ h₂, or.elim h₁
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(λ e₁₂, or.elim h₂
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(λ e₂₃, by rewrite [e₁₂, e₂₃])
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(λ es₂₃, begin esimp [eqv], rewrite -es₂₃, right, congruence, assumption end))
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(λ es₁₂, or.elim h₂
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(λ e₂₃, begin esimp [eqv], rewrite es₁₂, right, assumption end)
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(λ es₂₃, begin esimp [eqv], rewrite [-es₁₂ at es₂₃, swap_swap at es₂₃], left, assumption end))
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private theorem is_equivalence (A : Type) : equivalence (@eqv A) :=
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mk_equivalence (@eqv A) (@eqv.refl A) (@eqv.symm A) (@eqv.trans A)
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definition uprod.setoid [instance] (A : Type) : setoid (A × A) :=
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setoid.mk (@eqv A) (is_equivalence A)
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definition uprod (A : Type) : Type :=
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quot (uprod.setoid A)
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namespace uprod
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definition mk {A : Type} (a b : A) : uprod A :=
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⟦(a, b)⟧
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theorem mk_eq_mk {A : Type} (a b : A) : mk a b = mk b a :=
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quot.sound (or.inr rfl)
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private definition mem_fn {A : Type} (a : A) : A × A → Prop
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| (a₁, a₂) := a = a₁ ∨ a = a₂
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private lemma mem_swap {A : Type} {a : A} : ∀ {p : A × A}, mem_fn a p ↔ mem_fn a (swap p)
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| (a₁, a₂) := iff.intro
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(λ l : a = a₁ ∨ a = a₂, or.swap l)
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(λ r : a = a₂ ∨ a = a₁, or.swap r)
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private lemma mem_coherent {A : Type} : ∀ {p₁ p₂ : A × A} (a : A), p₁ ~ p₂ → mem_fn a p₁ = mem_fn a p₂
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| (a₁, b₁) (a₂, b₂) a h :=
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or.elim h
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(λ e, by rewrite e)
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(λ es, begin rewrite -es, apply propext, rewrite (propext mem_swap) end)
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definition mem {A : Type} (a : A) (u : uprod A) : Prop :=
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quot.lift_on u (λ p, mem_fn a p) (λ p₁ p₂ e, mem_coherent a e)
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infix `∈` := mem
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theorem mem_mk_left {A : Type} (a b : A) : a ∈ mk a b :=
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or.inl rfl
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theorem mem_mk_right {A : Type} (a b : A) : b ∈ mk a b :=
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or.inr rfl
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theorem mem_or_mem_of_mem_mk {A : Type} {a b c : A} : c ∈ mk a b → c = a ∨ c = b :=
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λ h, h
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private definition map_fn {A B : Type} (f : A → B) : A × A → uprod B
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| (a₁, a₂) := mk (f a₁) (f a₂)
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private lemma map_coherent {A B : Type} (f : A → B) : ∀ {p₁ p₂ : A × A}, p₁ ~ p₂ → map_fn f p₁ = map_fn f p₂
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| (a₁, b₁) (a₂, b₂) h :=
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or.elim h
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(λ e, by rewrite e)
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(λ es, begin rewrite -es, apply quot.sound, right, reflexivity end)
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definition map {A B : Type} (f : A → B) (u : uprod A) : uprod B :=
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quot.lift_on u (λ p, map_fn f p) (λ p₁ p₂ c, map_coherent f c)
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theorem mem_map_mk_left {A B : Type} (f : A → B) (a₁ a₂ : A) : f a₁ ∈ map f (mk a₁ a₂) :=
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or.inl rfl
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theorem mem_map_mk_right {A B : Type} (f : A → B) (a₁ a₂ : A) : f a₂ ∈ map f (mk a₁ a₂) :=
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or.inr rfl
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theorem map_map {A B C : Type} (g : B → C) (f : A → B) (u : uprod A) : map g (map f u) = map (g ∘ f) u :=
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quot.induction_on u (λ p : A × A, begin cases p, reflexivity end)
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end uprod
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