Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import PreMonoidalCenter.
Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
Require Import Enrichment_ch2_8.
Require Import NaturalDeduction.
Require Import NaturalDeductionCategory.
-Require Import ProgrammingLanguage.
-Require Import ProgrammingLanguageGeneralizedArrow.
+Require Import ProgrammingLanguageCategory.
Require Import FreydCategories.
-
+Require Import Enrichments.
Require Import GeneralizedArrow.
-Section ExtendFunctor.
-
- Context `(F:Functor).
- Context (P:c1 -> Prop).
-
- Definition domain_subcat := FullSubcategory c1 P.
-
- Definition functor_restricts_to_full_subcat_on_domain_fobj (a:domain_subcat) : c2 :=
- F (projT1 a).
-
- Definition functor_restricts_to_full_subcat_on_domain_fmor (a b:domain_subcat)(f:a~~{domain_subcat}~~>b) :
- (functor_restricts_to_full_subcat_on_domain_fobj a)~~{c2}~~>(functor_restricts_to_full_subcat_on_domain_fobj b) :=
- F \ (projT1 f).
-
- Lemma functor_restricts_to_full_subcat_on_domain : Functor domain_subcat c2 functor_restricts_to_full_subcat_on_domain_fobj.
- refine {| fmor := functor_restricts_to_full_subcat_on_domain_fmor |};
- unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
- setoid_rewrite H; reflexivity.
- setoid_rewrite fmor_preserves_id; reflexivity.
- setoid_rewrite <- fmor_preserves_comp; reflexivity.
- Defined.
-
-End ExtendFunctor.
-
-Section MonoidalSubCat.
-
- (* a monoidal subcategory is a full subcategory, closed under tensor and containing the unit object *)
- Class MonoidalSubCat {Ob}{Hom}{C:Category Ob Hom}{MFobj}{MF}{MI}(MC:MonoidalCat C MFobj MF MI) :=
- { msc_P : MC -> Prop
- ; msc_closed_under_tensor : forall o1 o2, msc_P o1 -> msc_P o2 -> msc_P (MC (pair_obj o1 o2))
- ; msc_contains_unit : msc_P (mon_i MC)
- ; msc_subcat := FullSubcategory MC msc_P
- }.
- Local Coercion msc_subcat : MonoidalSubCat >-> SubCategory.
-
- Context `(MSC:MonoidalSubCat).
-
- (* any full subcategory of a monoidal category, , is itself monoidal *)
- Definition mf_restricts_to_full_subcat_on_domain_fobj (a:MSC ×× MSC) : MSC.
- destruct a.
- destruct o.
- destruct o0.
- set (MC (pair_obj x x0)) as m'.
- exists m'.
- apply msc_closed_under_tensor; auto.
- Defined.
-
- Definition mf_restricts_to_full_subcat_on_domain_fmor
- {a}{b}
- (f:a~~{MSC ×× MSC}~~>b)
- :
- (mf_restricts_to_full_subcat_on_domain_fobj a)~~{MSC}~~>(mf_restricts_to_full_subcat_on_domain_fobj b).
- destruct a as [[a1 a1pf] [a2 a2pf]].
- destruct b as [[b1 b1pf] [b2 b2pf]].
- destruct f as [[f1 f1pf] [f2 f2pf]].
- simpl in *.
- exists (MC \ (pair_mor (pair_obj a1 a2) (pair_obj b1 b2) f1 f2)); auto.
- Defined.
-
- Lemma mf_restricts_to_full_subcat_on_domain : Functor (MSC ×× MSC) MSC
- mf_restricts_to_full_subcat_on_domain_fobj.
- refine {| fmor := fun a b f => mf_restricts_to_full_subcat_on_domain_fmor f |};
- unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
- admit.
- admit.
- admit.
- Defined.
-
- Definition subcat_i : MSC.
- exists (mon_i MC).
- apply msc_contains_unit.
- Defined.
-
- Lemma full_subcat_is_monoidal : MonoidalCat MSC _ mf_restricts_to_full_subcat_on_domain subcat_i.
- admit.
- Defined.
-
- Lemma inclusion_functor_monoidal : MonoidalFunctor full_subcat_is_monoidal MC (InclusionFunctor _ MSC).
- admit.
- Defined.
-
-End MonoidalSubCat.
-Coercion full_subcat_is_monoidal : MonoidalSubCat >-> MonoidalCat.
-
-(*
Section ArrowInLanguage.
(* an Arrow In A Programming Language consists of... *)
(* a host language: *)
- Context `(Host:ProgrammingLanguage).
-
- (* ... for which Types(L) is cartesian: *)
- Context {MF}(center_of_TypesL:MonoidalCat (TypesL _ _ Host) (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []).
+ Context `(Host : ProgrammingLanguage).
- (* along with a monoidal subcategory of Z(Types(L)) *)
- Context (VK:MonoidalSubCat center_of_TypesL).
+ Context (Host_Monoidal : MonoidalCat (TypesL_PreMonoidal Host)).
+ Context (Host_Cartesian : CartesianCat Host_Monoidal).
- (* a premonoidal category enriched in aforementioned full subcategory *)
- Context (Kehom:center_of_TypesL -> center_of_TypesL -> @ob _ _ VK).
-Check (@ECategory).
- Context (KE:@ECategory (@ob _ _ VK) (@hom _ _ VK) VK _ VK (mon_i (full_subcat_is_monoidal VK)) (full_subcat_is_monoidal VK) center_of_TypesL Kehom).
+ Context
+ {P}
+ (Pobj_unit : P [])
+ (Pobj_closed : forall a b, P a → P b → P (bin_obj(BinoidalCat:=Center_is_PreMonoidal (TypesL_PreMonoidal Host)) a b)).
-Check (Underlying KE).
+ Context (VK : FullSubcategory Host_Cartesian P).
- Context {kbo:center_of_TypesL -> center_of_TypesL -> center_of_TypesL}.
+ Context ehom KE (K_bin:@EBinoidalCat _ _ VK _ _ _
+ (PreMonoidalFullSubcategory_PreMonoidal Host_Cartesian VK Pobj_unit Pobj_closed)
+ (TypesL Host) ehom KE (bin_obj(BinoidalCat:=Host_Monoidal))).
- Context (kbc:@BinoidalCat center_of_TypesL _ (Underlying KE) kbo).
+ Context (K_premonoidal:PreMonoidalCat K_bin (one(TerminalObject:=Host_Cartesian))).
- Check (@PreMonoidalCat)
- Definition one' := @one _ _ _ (car_terminal(CartesianCat:=CC))
- Context (K:@PreMonoidalCat _ _ KE kbo kbc ).
- Context (CC:CartesianCat center_of_TypesL).
+ Definition ArrowInProgrammingLanguage :=
+ @FreydCategory _ _ _ _ _ _ _ _ Host_Cartesian _ _ K_bin K_premonoidal.
- (*
Definition K_enrichment : Enrichment.
- refine {| enr_c := KE |}.
- Defined.
- Context (K_surjective:SurjectiveEnrichment K_enrichment).
- *)
-Check (@FreydCategory).
-
- Definition ArrowInProgrammingLanguage :=
- @FreydCategory _ _ _ _ _ _ center_of_TypesL _ _ _ _ K.
+ refine
+ {| enr_c_pm := K_premonoidal
+ ; enr_v_mon := MonoidalFullSubcategory_Monoidal Host_Cartesian _ _ VK
+ |}.
+ Defined.
- Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
- refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.
- Defined.
+ Instance ArrowsAreGeneralizedArrows : GeneralizedArrow K_enrichment (TypesEnrichedInJudgments Host) :=
+ { ga_functor_monoidal :=
+ PreMonoidalFullSubcategoryInclusionFunctor_PreMonoidal Host_Cartesian VK Pobj_unit Pobj_closed Host_Cartesian }.
-End GArrowInLanguage.
-*)
\ No newline at end of file
+End ArrowInLanguage.
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