1 (*********************************************************************************************************************************)
2 (* ProgrammingLanguageArrow *)
4 (* Arrows in ProgrammingLanguages. *)
6 (*********************************************************************************************************************************)
8 Generalizable All Variables.
9 Require Import Preamble.
10 Require Import General.
11 Require Import Categories_ch1_3.
12 Require Import InitialTerminal_ch2_2.
13 Require Import Functors_ch1_4.
14 Require Import Isomorphisms_ch1_5.
15 Require Import ProductCategories_ch1_6_1.
16 Require Import OppositeCategories_ch1_6_2.
17 Require Import Enrichment_ch2_8.
18 Require Import Subcategories_ch7_1.
19 Require Import NaturalTransformations_ch7_4.
20 Require Import NaturalIsomorphisms_ch7_5.
21 Require Import MonoidalCategories_ch7_8.
22 Require Import Coherence_ch7_8.
23 Require Import Enrichment_ch2_8.
24 Require Import RepresentableStructure_ch7_2.
25 Require Import FunctorCategories_ch7_7.
27 Require Import NaturalDeduction.
28 Require Import NaturalDeductionCategory.
30 Require Import ProgrammingLanguage.
31 Require Import ProgrammingLanguageGeneralizedArrow.
32 Require Import FreydCategories.
34 Require Import GeneralizedArrow.
36 Section ArrowInLanguage.
38 (* an Arrow In A Programming Language consists of... *)
40 (* a host language: *)
41 Context `(Host:ProgrammingLanguage).
43 (* ... for which Types(L) is cartesian: *)
44 Context {MF}(center_of_TypesL:MonoidalCat (TypesL _ _ Host) (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []).
46 (* along with a monoidal subcategory of Z(Types(L)) *)
47 Context (VK:MonoidalSubCat center_of_TypesL).
49 (* a premonoidal category enriched in aforementioned full subcategory *)
50 Context (Kehom:center_of_TypesL -> center_of_TypesL -> @ob _ _ VK).
52 Context (KE:@ECategory (@ob _ _ VK) (@hom _ _ VK) VK _ VK
53 (mon_i (full_subcat_is_monoidal VK)) (full_subcat_is_monoidal VK) center_of_TypesL Kehom).
55 Context {kbo:center_of_TypesL -> center_of_TypesL -> center_of_TypesL}.
57 Context (kbc:@BinoidalCat center_of_TypesL _ (Underlying KE) kbo).
59 Check (@PreMonoidalCat)
60 Definition one' := @one _ _ _ (car_terminal(CartesianCat:=CC))
61 Context (K:@PreMonoidalCat _ _ KE kbo kbc ).
62 Context (CC:CartesianCat center_of_TypesL).
65 Definition K_enrichment : Enrichment.
66 refine {| enr_c := KE |}.
68 Context (K_surjective:SurjectiveEnrichment K_enrichment).
71 Definition ArrowInProgrammingLanguage :=
72 @FreydCategory _ _ _ _ _ _ center_of_TypesL _ _ _ _ K.
74 Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
75 refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.