276 lines
12 KiB
Text
276 lines
12 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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Module: init.equiv
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Author: Jeremy Avigad, Jakob von Raumer
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Ported from Coq HoTT
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-/
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prelude
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import .path .function
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open eq function
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/- Equivalences -/
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-- This is our definition of equivalence. In the HoTT-book it's called
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-- ihae (half-adjoint equivalence).
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structure is_equiv [class] {A B : Type} (f : A → B) :=
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(inv : B → A)
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(retr : (f ∘ inv) ∼ id)
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(sect : (inv ∘ f) ∼ id)
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(adj : Πx, retr (f x) = ap f (sect x))
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attribute is_equiv.inv [quasireducible]
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-- A more bundled version of equivalence
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structure equiv (A B : Type) :=
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(to_fun : A → B)
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(to_is_equiv : is_equiv to_fun)
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namespace is_equiv
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/- Some instances and closure properties of equivalences -/
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postfix `⁻¹` := inv
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--a second notation for the inverse, which is not overloaded
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postfix [parsing-only] `⁻¹ᵉ`:std.prec.max_plus := inv
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section
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variables {A B C : Type} (f : A → B) (g : B → C) {f' : A → B}
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-- The identity function is an equivalence.
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-- TODO: make A explicit
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definition is_equiv_id : (@is_equiv A A id) := is_equiv.mk id (λa, idp) (λa, idp) (λa, idp)
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-- The composition of two equivalences is, again, an equivalence.
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definition is_equiv_compose [Hf : is_equiv f] [Hg : is_equiv g] : is_equiv (g ∘ f) :=
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is_equiv.mk ((inv f) ∘ (inv g))
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(λc, ap g (retr f (g⁻¹ c)) ⬝ retr g c)
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(λa, ap (inv f) (sect g (f a)) ⬝ sect f a)
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(λa, (whisker_left _ (adj g (f a))) ⬝
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(ap_con g _ _)⁻¹ ⬝
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ap02 g ((ap_con_eq_con (retr f) (sect g (f a)))⁻¹ ⬝
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(ap_compose (inv f) f _ ◾ adj f a) ⬝
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(ap_con f _ _)⁻¹
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) ⬝
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(ap_compose f g _)⁻¹
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)
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-- Any function equal to an equivalence is an equivlance as well.
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definition is_equiv_eq_closed [Hf : is_equiv f] (Heq : f = f') : (is_equiv f') :=
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eq.rec_on Heq Hf
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-- Any function pointwise equal to an equivalence is an equivalence as well.
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definition homotopy_closed [Hf : is_equiv f] (Hty : f ∼ f') : (is_equiv f') :=
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let sect' := (λ b, (Hty (inv f b))⁻¹ ⬝ retr f b) in
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let retr' := (λ a, (ap (inv f) (Hty a))⁻¹ ⬝ sect f a) in
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let adj' := (λ (a : A),
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let ff'a := Hty a in
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let invf := inv f in
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let secta := sect f a in
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let retrfa := retr f (f a) in
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let retrf'a := retr f (f' a) in
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have eq1 : _ = _,
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from calc ap f secta ⬝ ff'a
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= retrfa ⬝ ff'a : by rewrite adj
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... = ap (f ∘ invf) ff'a ⬝ retrf'a : by rewrite ap_con_eq_con
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... = ap f (ap invf ff'a) ⬝ retrf'a : by rewrite ap_compose,
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have eq2 : _ = _,
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from calc retrf'a
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= (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : eq_inv_con_of_con_eq _ _ _ eq1⁻¹
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... = (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ Hty a) : ap_inv invf ff'a
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... = (ap f (ap invf ff'a))⁻¹ ⬝ (Hty (invf (f a)) ⬝ ap f' secta) : by rewrite ap_con_eq_con_ap
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... = ((ap f (ap invf ff'a))⁻¹ ⬝ Hty (invf (f a))) ⬝ ap f' secta : by rewrite con.assoc
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... = (ap f (ap invf ff'a)⁻¹ ⬝ Hty (invf (f a))) ⬝ ap f' secta : by rewrite ap_inv
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... = (Hty (invf (f' a)) ⬝ ap f' (ap invf ff'a)⁻¹) ⬝ ap f' secta : by rewrite ap_con_eq_con_ap
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... = (Hty (invf (f' a)) ⬝ (ap f' (ap invf ff'a))⁻¹) ⬝ ap f' secta : by rewrite ap_inv
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... = Hty (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : by rewrite con.assoc,
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have eq3 : _ = _,
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from calc (Hty (invf (f' a)))⁻¹ ⬝ retrf'a
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= (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : inv_con_eq_of_eq_con _ _ _ eq2
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... = (ap f' (ap invf ff'a)⁻¹) ⬝ ap f' secta : by rewrite ap_inv
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... = ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : by rewrite ap_con,
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eq3) in
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is_equiv.mk (inv f) sect' retr' adj'
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end
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context
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parameters {A B : Type} (f : A → B) (g : B → A)
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(ret : f ∘ g ∼ id) (sec : g ∘ f ∼ id)
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private definition adjointify_sect' : g ∘ f ∼ id :=
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(λx, ap g (ap f (inverse (sec x))) ⬝ ap g (ret (f x)) ⬝ sec x)
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private definition adjointify_adj' : Π (x : A), ret (f x) = ap f (adjointify_sect' x) :=
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(λ (a : A),
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let fgretrfa := ap f (ap g (ret (f a))) in
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let fgfinvsect := ap f (ap g (ap f (sec a)⁻¹)) in
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let fgfa := f (g (f a)) in
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let retrfa := ret (f a) in
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have eq1 : ap f (sec a) = _,
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from calc ap f (sec a)
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= idp ⬝ ap f (sec a) : by rewrite idp_con
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... = (ret (f a) ⬝ (ret (f a))⁻¹) ⬝ ap f (sec a) : by rewrite con.right_inv
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... = ((ret fgfa)⁻¹ ⬝ ap (f ∘ g) (ret (f a))) ⬝ ap f (sec a) : by rewrite con_ap_eq_con
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... = ((ret fgfa)⁻¹ ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite ap_compose
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... = (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite con.assoc,
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have eq2 : ap f (sec a) ⬝ idp = (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)),
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from !con_idp ⬝ eq1,
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have eq3 : idp = _,
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from calc idp
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= (ap f (sec a))⁻¹ ⬝ ((ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a))) : eq_inv_con_of_con_eq _ _ _ eq2
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... = ((ap f (sec a))⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite con.assoc'
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... = (ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ (fgretrfa ⬝ ap f (sec a)) : by rewrite ap_inv
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... = ((ap f (sec a)⁻¹ ⬝ (ret fgfa)⁻¹) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite con.assoc'
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... = ((retrfa⁻¹ ⬝ ap (f ∘ g) (ap f (sec a)⁻¹)) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite con_ap_eq_con
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... = ((retrfa⁻¹ ⬝ fgfinvsect) ⬝ fgretrfa) ⬝ ap f (sec a) : by rewrite ap_compose
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... = (retrfa⁻¹ ⬝ (fgfinvsect ⬝ fgretrfa)) ⬝ ap f (sec a) : by rewrite con.assoc'
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... = retrfa⁻¹ ⬝ ap f (ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ ap f (sec a) : by rewrite ap_con
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... = retrfa⁻¹ ⬝ (ap f (ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ ap f (sec a)) : by rewrite con.assoc'
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... = retrfa⁻¹ ⬝ ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a) : by rewrite -ap_con,
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have eq4 : ret (f a) = ap f ((ap g (ap f (sec a)⁻¹) ⬝ ap g (ret (f a))) ⬝ sec a),
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from eq_of_idp_eq_inv_con _ _ eq3,
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eq4)
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definition adjointify : is_equiv f :=
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is_equiv.mk g ret adjointify_sect' adjointify_adj'
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end
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section
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variables {A B C : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f] (g : B → C)
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include Hf
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--The inverse of an equivalence is, again, an equivalence.
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definition is_equiv_inv [instance] : (is_equiv f⁻¹) :=
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adjointify f⁻¹ f (sect f) (retr f)
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definition cancel_right (g : B → C) [Hgf : is_equiv (g ∘ f)] : (is_equiv g) :=
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have Hfinv [visible] : is_equiv f⁻¹, from is_equiv_inv f,
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@homotopy_closed _ _ _ _ (is_equiv_compose f⁻¹ (g ∘ f)) (λb, ap g (@retr _ _ f _ b))
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definition cancel_left (g : C → A) [Hgf : is_equiv (f ∘ g)] : (is_equiv g) :=
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have Hfinv [visible] : is_equiv f⁻¹, from is_equiv_inv f,
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@homotopy_closed _ _ _ _ (is_equiv_compose (f ∘ g) f⁻¹) (λa, sect f (g a))
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definition is_equiv_ap [instance] (x y : A) : is_equiv (ap f) :=
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adjointify (ap f)
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(λq, (inverse (sect f x)) ⬝ ap f⁻¹ q ⬝ sect f y)
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(λq, !ap_con
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⬝ whisker_right !ap_con _
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⬝ ((!ap_inv ⬝ inverse2 (adj f _)⁻¹)
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◾ (inverse (ap_compose f⁻¹ f _))
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◾ (adj f _)⁻¹)
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⬝ con_ap_con_eq_con_con (retr f) _ _
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⬝ whisker_right !con.left_inv _
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⬝ !idp_con)
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(λp, whisker_right (whisker_left _ (ap_compose f f⁻¹ _)⁻¹) _
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⬝ con_ap_con_eq_con_con (sect f) _ _
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⬝ whisker_right !con.left_inv _
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⬝ !idp_con)
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-- The function equiv_rect says that given an equivalence f : A → B,
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-- and a hypothesis from B, one may always assume that the hypothesis
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-- is in the image of e.
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-- In fibrational terms, if we have a fibration over B which has a section
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-- once pulled back along an equivalence f : A → B, then it has a section
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-- over all of B.
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definition equiv_rect (P : B → Type) :
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(Πx, P (f x)) → (Πy, P y) :=
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(λg y, eq.transport _ (retr f y) (g (f⁻¹ y)))
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definition equiv_rect_comp (P : B → Type)
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(df : Π (x : A), P (f x)) (x : A) : equiv_rect f P df (f x) = df x :=
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calc equiv_rect f P df (f x)
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= transport P (retr f (f x)) (df (f⁻¹ (f x))) : by esimp
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... = transport P (eq.ap f (sect f x)) (df (f⁻¹ (f x))) : by rewrite adj
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... = transport (P ∘ f) (sect f x) (df (f⁻¹ (f x))) : by rewrite -transport_compose
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... = df x : by rewrite (apD df (sect f x))
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end
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section
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variables {A B : Type} {f : A → B} [Hf : is_equiv f] {a : A} {b : B}
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include Hf
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--Rewrite rules
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definition eq_of_eq_inv (p : a = f⁻¹ b) : f a = b :=
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ap f p ⬝ retr f b
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definition eq_of_inv_eq (p : f⁻¹ b = a) : b = f a :=
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(eq_of_eq_inv p⁻¹)⁻¹
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definition inv_eq_of_eq (p : b = f a) : f⁻¹ b = a :=
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ap f⁻¹ p ⬝ sect f a
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definition eq_inv_of_eq (p : f a = b) : a = f⁻¹ b :=
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(inv_eq_of_eq p⁻¹)⁻¹
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end
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--Transporting is an equivalence
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definition is_equiv_tr [instance] {A : Type} (P : A → Type) {x y : A} (p : x = y) : (is_equiv (transport P p)) :=
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is_equiv.mk (transport P p⁻¹) (tr_inv_tr P p) (inv_tr_tr P p) (tr_inv_tr_lemma P p)
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end is_equiv
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open is_equiv
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namespace equiv
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namespace ops
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attribute to_fun [coercion]
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end ops
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open equiv.ops
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attribute to_is_equiv [instance]
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infix `≃`:25 := equiv
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variables {A B C : Type}
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protected definition MK [reducible] (f : A → B) (g : B → A)
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(retr : f ∘ g ∼ id) (sect : g ∘ f ∼ id) : A ≃ B :=
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equiv.mk f (adjointify f g retr sect)
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definition to_inv [reducible] (f : A ≃ B) : B → A :=
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f⁻¹
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protected definition refl : A ≃ A :=
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equiv.mk id !is_equiv_id
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protected definition symm (f : A ≃ B) : B ≃ A :=
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equiv.mk f⁻¹ !is_equiv_inv
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protected definition trans (f : A ≃ B) (g: B ≃ C) : A ≃ C :=
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equiv.mk (g ∘ f) !is_equiv_compose
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definition equiv_of_eq_fn_of_equiv (f : A ≃ B) (f' : A → B) (Heq : f = f') : A ≃ B :=
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equiv.mk f' (is_equiv_eq_closed f Heq)
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definition eq_equiv_fn_eq (f : A → B) [H : is_equiv f] (a b : A) : (a = b) ≃ (f a = f b) :=
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equiv.mk (ap f) !is_equiv_ap
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definition eq_equiv_fn_eq_of_equiv (f : A ≃ B) (a b : A) : (a = b) ≃ (f a = f b) :=
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equiv.mk (ap f) !is_equiv_ap
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definition equiv_ap (P : A → Type) {a b : A} (p : a = b) : (P a) ≃ (P b) :=
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equiv.mk (transport P p) !is_equiv_tr
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--we need this theorem for the funext_of_ua proof
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theorem inv_eq {A B : Type} (eqf eqg : A ≃ B) (p : eqf = eqg) : (to_fun eqf)⁻¹ = (to_fun eqg)⁻¹ :=
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eq.rec_on p idp
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-- calc enviroment
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-- Note: Calculating with substitutions needs univalence
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definition equiv_of_equiv_of_eq {A B C : Type} (p : A = B) (q : B ≃ C) : A ≃ C := p⁻¹ ▹ q
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definition equiv_of_eq_of_equiv {A B C : Type} (p : A ≃ B) (q : B = C) : A ≃ C := q ▹ p
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calc_trans equiv.trans
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calc_refl equiv.refl
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calc_symm equiv.symm
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calc_trans equiv_of_equiv_of_eq
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calc_trans equiv_of_eq_of_equiv
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end equiv
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