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fix(library/hott) : convert to new path notations

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Convert definitions and proofs to new notations for inverse and cocatenation. Adapt to now right associative of concatenation.
This commit is contained in:
Jakob von Raumer 2014-10-22 23:25:12 -04:00 committed by Leonardo de Moura
parent a169f791df
commit abd5c574ad
1 changed files with 37 additions and 37 deletions
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74
library/hott/equiv.lean
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@ -70,21 +70,21 @@ namespace IsEquiv
-- The identity function is an equivalence. -- The identity function is an equivalence.
definition idIsEquiv [instance] : (@IsEquiv A A id) := IsEquiv_mk id (λa, idp) (λa, idp) (λa, idp) definition id_closed [instance] : (@IsEquiv A A id) := IsEquiv_mk id (λa, idp) (λa, idp) (λa, idp)
-- The composition of two equivalences is, again, an equivalence. -- The composition of two equivalences is, again, an equivalence.
definition comp_closed [instance] (Hf : IsEquiv f) (Hg : IsEquiv g) : (IsEquiv (g ∘ f)) := definition comp_closed [instance] (Hf : IsEquiv f) (Hg : IsEquiv g) : (IsEquiv (g ∘ f)) :=
IsEquiv_mk ((inv Hf) ∘ (inv Hg)) IsEquiv_mk ((inv Hf) ∘ (inv Hg))
(λc, ap g (retr Hf ((inv Hg) c)) @ retr Hg c) (λc, ap g (retr Hf ((inv Hg) c)) ⬝ retr Hg c)
(λa, ap (inv Hf) (sect Hg (f a)) @ sect Hf a) (λa, ap (inv Hf) (sect Hg (f a)) ⬝ sect Hf a)
(λa, (whiskerL _ (adj Hg (f a))) @ (λa, (whiskerL _ (adj Hg (f a))) ⬝
(ap_pp g _ _)^ @ (ap_pp g _ _)⁻¹ ⬝
ap02 g (concat_A1p (retr Hf) (sect Hg (f a))^ @ ap02 g (concat_A1p (retr Hf) (sect Hg (f a))⁻¹ ⬝
(ap_compose (inv Hf) f _ @@ adj Hf a) @ (ap_compose (inv Hf) f _ ◾ adj Hf a) ⬝
(ap_pp f _ _)^ (ap_pp f _ _)⁻¹
) @ ) ⬝
(ap_compose f g _)^ (ap_compose f g _)⁻¹
) )
-- Any function equal to an equivalence is an equivlance as well. -- Any function equal to an equivalence is an equivlance as well.
@ -93,34 +93,34 @@ namespace IsEquiv
-- Any function pointwise equal to an equivalence is an equivalence as well. -- Any function pointwise equal to an equivalence is an equivalence as well.
definition homotopic (Hf : IsEquiv f) (Heq : f ∼ f') : (IsEquiv f') := definition homotopic (Hf : IsEquiv f) (Heq : f ∼ f') : (IsEquiv f') :=
let sect' := (λ b, (Heq (inv Hf b))^ @ retr Hf b) in let sect' := (λ b, (Heq (inv Hf b))⁻¹ ⬝ retr Hf b) in
let retr' := (λ a, (ap (inv Hf) (Heq a))^ @ sect Hf a) in let retr' := (λ a, (ap (inv Hf) (Heq a))⁻¹ ⬝ sect Hf a) in
let adj' := (λ (a : A), let adj' := (λ (a : A),
let ff'a := Heq a in let ff'a := Heq a in
let invf := inv Hf in let invf := inv Hf in
let secta := sect Hf a in let secta := sect Hf a in
let retrfa := retr Hf (f a) in let retrfa := retr Hf (f a) in
let retrf'a := retr Hf (f' a) in let retrf'a := retr Hf (f' a) in
have eq1 : ap f secta @ ff'a ≈ ap f (ap invf ff'a) @ retr Hf (f' a), have eq1 : _ ≈ _,
from calc ap f secta @ ff'a from calc ap f secta ⬝ ff'a
≈ retrfa @ ff'a : (ap _ (adj Hf _ ))^ ≈ retrfa ⬝ ff'a : (ap _ (adj Hf _ ))⁻¹
... ≈ ap (f ∘ invf) ff'a @ retrf'a : !concat_A1p^ ... ≈ ap (f ∘ invf) ff'a ⬝ retrf'a : !concat_A1p⁻¹
... ≈ ap f (ap invf ff'a) @ retr Hf (f' a) : {ap_compose invf f ff'a}, ... ≈ ap f (ap invf ff'a) ⬝ retr Hf (f' a) : {ap_compose invf f ff'a},
have eq2 : retrf'a ≈ Heq (invf (f' a)) @ ((ap f' (ap invf ff'a))^ @ ap f' secta), have eq2 : _ ≈ _,
from calc retrf'a from calc retrf'a
≈ (ap f (ap invf ff'a))^ @ (ap f secta @ ff'a) : moveL_Vp _ _ _ (eq1^) ≈ (ap f (ap invf ff'a))⁻¹ ⬝ (ap f secta ⬝ ff'a) : moveL_Vp _ _ _ (eq1⁻¹)
... ≈ ap f (ap invf ff'a)^ @ (ap f secta @ Heq a) : {ap_V invf ff'a} ... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (ap f secta ⬝ Heq a) : {ap_V invf ff'a}
... ≈ ap f (ap invf ff'a)^ @ (Heq (invf (f a)) @ ap f' secta) : {!concat_Ap} ... ≈ ap f (ap invf ff'a)⁻¹ ⬝ (Heq (invf (f a)) ⬝ ap f' secta) : {!concat_Ap}
... ≈ ap f (ap invf ff'a)^ @ Heq (invf (f a)) @ ap f' secta : {!concat_pp_p^} ... ≈ (ap f (ap invf ff'a)⁻¹ ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!concat_pp_p⁻¹}
... ≈ ap f ((ap invf ff'a)^) @ Heq (invf (f a)) @ ap f' secta : {!ap_V^} ... ≈ (ap f ((ap invf ff'a)⁻¹) ⬝ Heq (invf (f a))) ⬝ ap f' secta : {!ap_V⁻¹}
... ≈ Heq (invf (f' a)) @ ap f' ((ap invf ff'a)^) @ ap f' secta : {!concat_Ap} ... ≈ (Heq (invf (f' a)) ⬝ ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!concat_Ap}
... ≈ Heq (invf (f' a)) @ (ap f' (ap invf ff'a))^ @ ap f' secta : {!ap_V} ... ≈ (Heq (invf (f' a)) ⬝ (ap f' (ap invf ff'a))⁻¹) ⬝ ap f' secta : {!ap_V}
... ≈ Heq (invf (f' a)) @ ((ap f' (ap invf ff'a))^ @ ap f' secta) : !concat_pp_p, ... ≈ Heq (invf (f' a)) ⬝ ((ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta) : !concat_pp_p,
have eq3 : (Heq (invf (f' a)))^ @ retr Hf (f' a) ≈ ap f' ((ap invf ff'a)^ @ secta), have eq3 : _ ≈ _,
from calc (Heq (invf (f' a)))^ @ retr Hf (f' a) from calc (Heq (invf (f' a)))⁻¹ ⬝ retr Hf (f' a)
≈ (ap f' (ap invf ff'a))^ @ ap f' secta : moveR_Vp _ _ _ eq2 ≈ (ap f' (ap invf ff'a))⁻¹ ⬝ ap f' secta : moveR_Vp _ _ _ eq2
... ≈ (ap f' ((ap invf ff'a)^)) @ ap f' secta : {!ap_V^} ... ≈ (ap f' ((ap invf ff'a)⁻¹)) ⬝ ap f' secta : {!ap_V⁻¹}
... ≈ ap f' ((ap invf ff'a)^ @ secta) : !ap_pp^, ... ≈ ap f' ((ap invf ff'a)⁻¹ ⬝ secta) : !ap_pp⁻¹,
eq3) in eq3) in
IsEquiv_mk (inv Hf) sect' retr' adj' IsEquiv_mk (inv Hf) sect' retr' adj'
@ -135,23 +135,23 @@ namespace IsEquiv
homotopic (comp_closed Hgf (inv_closed Hg)) (λa, sect Hg (f a)) homotopic (comp_closed Hgf (inv_closed Hg)) (λa, sect Hg (f a))
definition transport (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) := definition transport (P : A → Type) {x y : A} (p : x ≈ y) : (IsEquiv (transport P p)) :=
IsEquiv_mk (transport P (p^)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p) IsEquiv_mk (transport P (p⁻¹)) (transport_pV P p) (transport_Vp P p) (transport_pVp P p)
--Rewrite rules --Rewrite rules
section section
variables {Hf : IsEquiv f} variables {Hf : IsEquiv f}
definition moveR_M {x : A} {y : B} (p : x ≈ (inv Hf) y) : (f x ≈ y) := definition moveR_M {x : A} {y : B} (p : x ≈ (inv Hf) y) : (f x ≈ y) :=
ap f p @ retr Hf y (ap f p) ⬝ (retr Hf y)
definition moveL_M {x : A} {y : B} (p : (inv Hf) y ≈ x) : (y ≈ f x) := definition moveL_M {x : A} {y : B} (p : (inv Hf) y ≈ x) : (y ≈ f x) :=
(moveR_M (p^))^ (moveR_M (p⁻¹))⁻¹
definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv Hf) x ≈ y := definition moveR_V {x : B} {y : A} (p : x ≈ f y) : (inv Hf) x ≈ y :=
ap (inv Hf) p @ sect Hf y ap (inv Hf) p ⬝ sect Hf y
definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv Hf) x := definition moveL_V {x : B} {y : A} (p : f y ≈ x) : y ≈ (inv Hf) x :=
(moveR_V (p^))^ (moveR_V (p⁻¹))⁻¹
end end
@ -161,7 +161,7 @@ namespace Equiv
variables {A B C : Type} (eqf : A ≃ B) variables {A B C : Type} (eqf : A ≃ B)
theorem id : A ≃ A := Equiv_mk id IsEquiv.idIsEquiv theorem id : A ≃ A := Equiv_mk id IsEquiv.id_closed
theorem compose (eqg: B ≃ C) : A ≃ C := theorem compose (eqg: B ≃ C) : A ≃ C :=
Equiv_mk ((equiv_fun eqg) ∘ (equiv_fun eqf)) Equiv_mk ((equiv_fun eqg) ∘ (equiv_fun eqf))