feat(library/hott) add: if precompositions with f are equivalences, then f is
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@ -2,7 +2,7 @@
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jeremy Avigad, Jakob von Raumer
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-- Ported from Coq HoTT
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import .path
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import .path .trunc
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open path function
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-- Equivalences
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@ -186,7 +186,8 @@ namespace IsEquiv
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definition cancel_L (Hg : IsEquiv g) (Hgf : IsEquiv (g ∘ f)) : (IsEquiv f) :=
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homotopic (comp_closed Hgf (inv_closed Hg)) (λa, sect Hg (f a))
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definition transport (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
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--Transporting is an equivalence
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definition transport [instance] (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
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IsEquiv_mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
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--Rewrite rules
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@ -205,6 +206,40 @@ namespace IsEquiv
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definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv Hf) x :=
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(moveR_V (p⁻¹))⁻¹
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definition contr (HA: Contr A) : (Contr B) :=
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Contr.Contr_mk (f (center HA)) (λb, moveR_M (contr HA (inv Hf b)))
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end
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--If pre- or post-composing with a function is always an equivalence,
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--then that function is also an equivalence. It's convenient to know
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--that we only need to assume the equivalence when the other type is
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--the domain or the codomain.
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section
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definition precomp (C : Type) (h : B → C) : A → C := h ∘ f
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definition inv_precomp (C D : Type) (Ceq : IsEquiv (precomp C))
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(Deq : IsEquiv (@precomp A B f D)) (k : C → D) (h : A → C) :
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k ∘ (inv Ceq) h ≈ (inv Deq) (k ∘ h) :=
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have eq1 : (inv Deq) (k ∘ h) ≈ k ∘ ((inv Ceq) h),
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from calc (inv Deq) (k ∘ h) ≈ (inv Deq) (k ∘ (precomp C ((inv Ceq) h))) : retr Ceq h
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... ≈ k ∘ ((inv Ceq) h) : !sect,
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eq1⁻¹
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definition isequiv_precompose (Aeq : IsEquiv (@precomp A B f A))
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(Beq : IsEquiv (@precomp A B f B)) : (IsEquiv f) :=
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let sect' : Sect ((inv Aeq) id) f := (λx,
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calc f (inv Aeq id x) ≈ (f ∘ (inv Aeq) id) x : idp
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... ≈ (inv Beq) (f ∘ id) x : apD10 (!inv_precomp)
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... ≈ (inv Beq) (@precomp A B f B id) x : idp
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... ≈ x : apD10 (sect Beq id))
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in
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let retr' : Sect f ((inv Aeq) id) := (λx,
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calc inv Aeq id (f x) ≈ @precomp A B f A ((inv Aeq) id) x : idp
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... ≈ x : apD10 (retr Aeq id)) in
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adjointify f ((inv Aeq) id) sect' retr'
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end
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end IsEquiv
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@ -212,7 +247,7 @@ end IsEquiv
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namespace Equiv
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variables {A B C : Type} (eqf : A ≃ B)
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theorem id : A ≃ A := Equiv_mk id IsEquiv.id_closed
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definition id : A ≃ A := Equiv_mk id IsEquiv.id_closed
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theorem compose (eqg: B ≃ C) : A ≃ C :=
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Equiv_mk ((equiv_fun eqg) ∘ (equiv_fun eqf))
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@ -235,4 +270,19 @@ namespace Equiv
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theorem transport (P : A → Type) {x y : A} {p : x ≈ y} : (P x) ≃ (P y) :=
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Equiv_mk (transport P p) (IsEquiv.transport P p)
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theorem contr_closed (HA: Contr A) : (Contr B) :=
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@IsEquiv.contr A B (equiv_fun eqf) (equiv_isequiv eqf) HA
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end Equiv
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namespace Equiv
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variables {A B : Type} {HA : Contr A} {HB : Contr B}
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--Each two contractible types are equivalent.
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definition contr_contr : A ≃ B :=
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Equiv_mk
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(λa, center HB)
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(IsEquiv.adjointify (λa, center HB) (λb, center HA)
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(contr HB) (contr HA))
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end Equiv
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