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remove workaround for coq/coq#8994 #1807

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remove workaround for coq/coq#8994 #1807

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22 changes: 9 additions & 13 deletions theories/WildCat/Equiv.v
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Expand Up @@ -13,21 +13,25 @@ Declare Scope wc_iso_scope.
Class HasEquivs (A : Type) `{Is1Cat A} :=
{
CatEquiv' : A -> A -> Type where "a $<~> b" := (CatEquiv' a b);
CatIsEquiv' : forall a b, (a $-> b) -> Type;
CatIsEquiv : forall a b, (a $-> b) -> Type;
cate_fun' : forall a b, (a $<~> b) -> (a $-> b);
cate_isequiv' : forall a b (f : a $<~> b), CatIsEquiv' a b (cate_fun' a b f);
cate_buildequiv' : forall a b (f : a $-> b), CatIsEquiv' a b f -> CatEquiv' a b;
cate_buildequiv_fun' : forall a b (f : a $-> b) (fe : CatIsEquiv' a b f),
cate_isequiv : forall a b (f : a $<~> b), CatIsEquiv a b (cate_fun' a b f);
cate_buildequiv' : forall a b (f : a $-> b), CatIsEquiv a b f -> CatEquiv' a b;
cate_buildequiv_fun' : forall a b (f : a $-> b) (fe : CatIsEquiv a b f),
cate_fun' a b (cate_buildequiv' a b f fe) $== f;
cate_inv' : forall a b (f : a $<~> b), b $-> a;
cate_issect' : forall a b (f : a $<~> b),
cate_inv' _ _ f $o cate_fun' _ _ f $== Id a;
cate_isretr' : forall a b (f : a $<~> b),
cate_fun' _ _ f $o cate_inv' _ _ f $== Id b;
catie_adjointify' : forall a b (f : a $-> b) (g : b $-> a)
(r : f $o g $== Id b) (s : g $o f $== Id a), CatIsEquiv' a b f;
(r : f $o g $== Id b) (s : g $o f $== Id a), CatIsEquiv a b f;
}.

Existing Class CatIsEquiv.
Arguments CatIsEquiv {A _ _ _ _ _ a b} f.
Global Existing Instance cate_isequiv.

(** Since apparently a field of a record can't be the source of a coercion (Coq complains about the uniform inheritance condition, although as officially stated that condition appears to be satisfied), we redefine all the fields of [HasEquivs]. *)

Definition CatEquiv {A} `{HasEquivs A} (a b : A)
Expand All @@ -43,14 +47,6 @@ Definition cate_fun {A} `{HasEquivs A} {a b : A} (f : a $<~> b)

Coercion cate_fun : CatEquiv >-> Hom.

(* Being an equivalence should be a typeclass, but we have to redefine it to work around https://github.com/coq/coq/issues/8994 . *)
Class CatIsEquiv {A} `{HasEquivs A} {a b : A} (f : a $-> b)
:= catisequiv : CatIsEquiv' a b f.

Global Instance cate_isequiv {A} `{HasEquivs A} {a b : A} (f : a $<~> b)
: CatIsEquiv f
:= cate_isequiv' a b f.

Definition Build_CatEquiv {A} `{HasEquivs A} {a b : A}
(f : a $-> b) {fe : CatIsEquiv f}
: a $<~> b
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2 changes: 1 addition & 1 deletion theories/WildCat/FunctorCat.v
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Expand Up @@ -92,7 +92,7 @@ Proof.
- intros a; exact _.
- intros ?.
snrapply Build_NatEquiv.
+ intros a; exact (Build_CatEquiv (alpha a)).
+ intros a; by nrapply (Build_CatEquiv (alpha a)).
+ cbn. refine (is1natural_homotopic alpha _).
intros a; apply cate_buildequiv_fun.
- cbn; intros; apply cate_buildequiv_fun.
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4 changes: 2 additions & 2 deletions theories/WildCat/Induced.v
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Expand Up @@ -78,9 +78,9 @@ Section Induced_category.
Proof.
srapply Build_HasEquivs; intros a b; cbn.
+ exact (f a $<~> f b).
+ apply CatIsEquiv'.
+ apply CatIsEquiv.
+ apply cate_fun.
+ apply cate_isequiv'.
+ apply cate_isequiv.
+ apply cate_buildequiv'.
+ nrapply cate_buildequiv_fun'.
+ apply cate_inv'.
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2 changes: 1 addition & 1 deletion theories/WildCat/NatTrans.v
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Expand Up @@ -212,7 +212,7 @@ Definition Build_NatEquiv' {A B : Type} `{IsGraph A} `{HasEquivs B}
Proof.
snrapply Build_NatEquiv.
- intro a.
refine (Build_CatEquiv (alpha a)).
by refine (Build_CatEquiv (alpha a)).
- intros a a' f.
refine (cate_buildequiv_fun _ $@R _ $@ _ $@ (_ $@L cate_buildequiv_fun _)^$).
apply (isnat alpha).
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2 changes: 1 addition & 1 deletion theories/WildCat/Opposite.v
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Expand Up @@ -164,7 +164,7 @@ Proof.
- exact (b $<~> a).
- apply CatIsEquiv.
- apply cate_fun'.
- apply cate_isequiv'.
- apply cate_isequiv.
- apply cate_buildequiv'.
- rapply cate_buildequiv_fun'.
- apply cate_inv'.
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4 changes: 2 additions & 2 deletions theories/WildCat/Prod.v
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Expand Up @@ -81,8 +81,8 @@ Proof.
- split; [ exact (fst f) | exact (snd f) ].
- split; exact _.
- intros [fe1 fe2]; split.
+ exact (Build_CatEquiv (fst f)).
+ exact (Build_CatEquiv (snd f)).
+ by rapply (Build_CatEquiv (fst f)).
+ by rapply (Build_CatEquiv (snd f)).
- intros [fe1 fe2]; cbn; split; apply cate_buildequiv_fun.
- split; [ exact ((fst f)^-1$) | exact ((snd f)^-1$) ].
- split; apply cate_issect.
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