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 Section ArrowInLanguage.
36 Section ArrowInLanguage.
37 Context {MF}{mn:MonoidalCat TypesL (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []} (CC:CartesianCat mn).
38 Context {Kehom}(K:@ECategory _ _ TypesL _ mn [] mn TypesL Kehom).
39 Context {bc:BinoidalCat (Underlying K) (@T_Branch _)}.
40 Context (pmc:@PreMonoidalCat _ _ _ _ bc (@one _ _ _ (car_terminal(CartesianCat:=CC)))).
41 Definition ArrowInProgrammingLanguage := @FreydCategory _ _ _ _ _ _ mn _ _ _ _ pmc.
44 Definition ArrowsAreGeneralizedArrows (Host:ProgrammingLanguageSMME)
45 {mf}{mn}{cc}{kehom}{CC}
46 (arrow:ArrowInProgrammingLanguage Host _ _ CC mf mn cc kehom) : GeneralizedArrowInLanguage.
48 Definition TwoLevelLanguage := Reification Guest Host (me_i Host).
50 Context (GuestHost:TwoLevelLanguage).
52 Definition FlatObject (x:TypesL _ _ Host) :=
53 forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
55 Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject.
59 Context (F:Retraction (TypesL _ _ Host) FlatSubCategory).
60 Definition FlatteningOfReification := garrow_from_reification Guest Host GuestHost >>>> F.
61 Lemma FlatteningIsNotDestructive :
62 FlatteningOfReification >>>> retraction_retraction F >>>> HomFunctor _ (me_i Host) ~~~~ GuestHost.