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added monoid_enriched_cat.v as the original pull request by Cyril, inconclusively modified enriched_cat.v (the version I'm working on)
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‎tests/enriched_cat.v‎

Lines changed: 84 additions & 10 deletions
Original file line numberDiff line numberDiff line change
@@ -3,11 +3,11 @@
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From HB Require Import structures.
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From Coq Require Import ssreflect ssrfun.
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HB.mixin Record isQuiver Obj := { hom : Obj -> Obj -> Type }.
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HB.mixin Record isQuiver (Obj: Type) : Type := { hom : Obj -> Obj -> Type }.
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HB.structure Definition Quiver := { Obj of isQuiver Obj }.
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HB.structure Definition Quiver : Type := { Obj of isQuiver Obj }.
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HB.mixin Record isMon A := {
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HB.mixin Record isMon (A: Type) : Type := {
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zero : A;
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add : A -> A -> A;
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addrA : associative add;
@@ -16,20 +16,94 @@ HB.mixin Record isMon A := {
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}.
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HB.structure
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Definition Monoid := { A of isMon A }.
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Definition Monoid : Type := { A of isMon A }.
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(**)
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HB.mixin Record hom_isMonT T of Quiver T :=
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HB.mixin Record hom_isMon T of Quiver T :=
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{ private : forall A B, isMon (@hom T A B) }.
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HB.structure
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Definition Monoid_enriched_quiverT :=
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{ Obj of isQuiver Obj & hom_isMonT Obj }.
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Definition Monoid_enriched_quiver :=
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{ Obj of isQuiver Obj & hom_isMon Obj }.
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(* unique projection from the axiom of Monoid_enriched_quiver *)
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HB.instance Definition _ (T : Monoid_enriched_quiver.type) (A B : T) : isMon (@hom T A B) :=
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@private T A B.
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HB.instance Definition _ := isQuiver.Build Type (fun A B => A -> B).
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(*********)
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HB.mixin Record hom_isMonX T of Quiver T : Type :=
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{ private : forall A B, isMon (@hom T A B) }.
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HB.structure
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Definition Monoid_enriched_quiverX :=
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{ Obj of isQuiver Obj & hom_isMonX Obj }.
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Record isQuiverS (Obj: Type) : Type := { homS : Obj -> Obj -> Type }.
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Structure QuiverS := { ObjS: Type; AxS: isQuiverS ObjS }.
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Definition hom_isMon_type T (X: isQuiverS T) (A B: T) : Type :=
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isMon (@homS T X A B).
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Record hom_isMonQ T (X: isQuiverS T) : Type :=
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hiMQ { privateQ : forall (A B: T), hom_isMon_type T X A B }.
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Definition my_hom_isMonQ T (X: isQuiverS T) (F: forall A B, hom_isMon_type T X A B) :
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hom_isMonQ T X := hiMQ T X F.
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Record Monoid_enriched_quiverQ := { ObjQ: Type; iQQ: isQuiverS ObjQ; hsM: hom_isMonQ ObjQ iQQ }.
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Record hom_wrapper T (X: isQuiverS T) (Str: Type -> Type) : Type :=
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{ privateW : forall (A B: T), Str (@homS T X A B) }.
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(**)
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Record hom_wrapperA T (Qv: Type -> Type) (hm: Qv T -> T -> T -> Type) (Str: Type -> Type) (x: Qv T) : Type :=
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{ privateWA : forall (A B: T), Str (hm x A B) }.
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Definition my_quiver (T: Type) : isQuiverS T.
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Admitted.
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Fail HB.structure
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Lemma my_quiver_mon (T: Type) : forall (A B: T), isMon (@homS T (my_quiver T) A B).
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Admitted.
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Definition my_hom_isMon (T: Type) : hom_isMonQ T (my_quiver T) :=
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my_hom_isMonQ T (my_quiver T) (my_quiver_mon T).
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Definition Mixin : Type := Type -> Type.
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(* write two versions of Monoid_enriched_quiver: one using hom_isMon
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(a mixin, hence a record), the other one using hom_isMon_type as naked
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field type. The former corresponds to the wrapped version, the latter
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to the intuitive, wrapper-less version. Now the latter should agree
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with the former... (broadly corresponding to 2?) *)
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Lemma quiver_ok (T: Type) (Str: Type -> Type) :
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forall (A B: T), @homS (my_quiver T) A B
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Class wrapper (T: Type) () (P: T -> T -> Prop) { prop: forall A B: T, P A B }.
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Definition wrapped_hom (T: Type) (F: isMon (@hom T A B)) := wrapper T (isMon (@hom T))
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Elpi Accumulate lp:{{
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pred wrapper-mixin o:mixinname, o:gref, o:mixinname.
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}}.
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HB.instance Definition homTypeMon (A B : Quiver.type) := isMon.Build (hom A B) (* ... *).
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(*********)
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HB.structure
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Definition Monoid_enriched_quiver :=
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{ Obj of isQuiver Obj &
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(forall A B : Obj, isMon (@hom (Quiver.clone Obj _) A B)) }.

‎tests/monoid_enriched_cat.v‎

Lines changed: 61 additions & 0 deletions
Original file line numberDiff line numberDiff line change
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From HB Require Import structures.
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From Coq Require Import ssreflect ssrfun.
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HB.mixin Record isQuiver Obj := { hom : Obj -> Obj -> Type }.
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HB.structure Definition Quiver := { Obj of isQuiver Obj }.
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HB.mixin Record isMon A := {
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zero : A;
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add : A -> A -> A;
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addrA : associative add;
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add0r : left_id zero add;
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addr0 : right_id zero add;
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}.
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HB.structure
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Definition Monoid := { A of isMon A }.
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Fail HB.structure
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Definition Monoid_enriched_quiver :=
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{ Obj of isQuiver Obj &
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(forall A B : Obj, isMon (@hom (Quiver.clone Obj _) A B)) }.
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(* Step 0: define a wrapper predicate in coq-elpi *)
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(* 5 lines of documentation + 1 line of elpi code in structure.v
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`pred wrapper-mixin o:mixinname, o:gref, o:mixinname`
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*)
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(* Step 1: add a wrapper attribute to declare wrappers,
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they should index:
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- the wrapped mixin (`isMon`)
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- the wrapper mixin (`hom_isMon`)
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- the new subject (`hom`)
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This attribute will add an entry in the `wrapper-mixin` database.
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As an addition substep, we should check that the wrapper has
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exactly one field, which is the wrapped mixin.
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*)
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#[wrapper]
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HB.mixin Record hom_isMon T of Quiver T :=
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{ private : forall A B, isMon (@hom T A B) }.
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(* Step 2: at structure declaration, export the main and only projection
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of each declared wrapper as an instance of the wrapped structure on
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its subject *)
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HB.structure
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Definition Monoid_enriched_quiver :=
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{ Obj of isQuiver Obj & hom_isMon Obj }.
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HB.instance Definition _ (T : Monoid_enriched_quiver.type) (A B : T) : isMon (@hom T A B) :=
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@private T A B.
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(* each instance of isMon should be tried as an instance of hom_isMon *)
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(* Step 3: for each instance of a wrapped mixin on a subject known
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to be wrapped, automatically produce an instance of the wrapper mixin too. *)
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HB.instance Definition _ := isQuiver.Build Type (fun A B => A -> B).
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Fail HB.instance Definition homTypeMon (A B : Quiver.type) := isMon.Build (hom A B) (* ... *).
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(* This last command should create a `Monoid_enriched_quiver`, in order to do so it should
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automatically instanciate the wrapper `hom_isMon`:
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HB.instance Definition _ := hom_isMon.Build Type homTypeMon.
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*)

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