(*********************************************************************************************************************************) (* ProgrammingLanguageArrow *) (* *) (* Arrows in ProgrammingLanguages. *) (* *) (*********************************************************************************************************************************) Generalizable All Variables. Require Import Preamble. Require Import General. Require Import Categories_ch1_3. Require Import InitialTerminal_ch2_2. Require Import Functors_ch1_4. Require Import Isomorphisms_ch1_5. Require Import ProductCategories_ch1_6_1. Require Import OppositeCategories_ch1_6_2. Require Import Enrichment_ch2_8. Require Import Subcategories_ch7_1. Require Import NaturalTransformations_ch7_4. Require Import NaturalIsomorphisms_ch7_5. Require Import MonoidalCategories_ch7_8. Require Import Coherence_ch7_8. Require Import Enrichment_ch2_8. Require Import RepresentableStructure_ch7_2. Require Import FunctorCategories_ch7_7. Require Import NaturalDeduction. Require Import NaturalDeductionCategory. Require Import ProgrammingLanguage. Require Import ProgrammingLanguageGeneralizedArrow. Require Import FreydCategories. Section ArrowInLanguage. Section ArrowInLanguage. Context {MF}{mn:MonoidalCat TypesL (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []} (CC:CartesianCat mn). Context {Kehom}(K:@ECategory _ _ TypesL _ mn [] mn TypesL Kehom). Context {bc:BinoidalCat (Underlying K) (@T_Branch _)}. Context (pmc:@PreMonoidalCat _ _ _ _ bc (@one _ _ _ (car_terminal(CartesianCat:=CC)))). Definition ArrowInProgrammingLanguage := @FreydCategory _ _ _ _ _ _ mn _ _ _ _ pmc. End ArrowInLanguage. Definition ArrowsAreGeneralizedArrows (Host:ProgrammingLanguageSMME) {mf}{mn}{cc}{kehom}{CC} (arrow:ArrowInProgrammingLanguage Host _ _ CC mf mn cc kehom) : GeneralizedArrowInLanguage. Definition TwoLevelLanguage := Reification Guest Host (me_i Host). Context (GuestHost:TwoLevelLanguage). Definition FlatObject (x:TypesL _ _ Host) := forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x). Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject. Section Flattening. Context (F:Retraction (TypesL _ _ Host) FlatSubCategory). Definition FlatteningOfReification := garrow_from_reification Guest Host GuestHost >>>> F. Lemma FlatteningIsNotDestructive : FlatteningOfReification >>>> retraction_retraction F >>>> HomFunctor _ (me_i Host) ~~~~ GuestHost. admit. Qed. End Flattening. End GArrowInLanguage.