(*********************************************************************************************************************************) (* ProgrammingLanguageFlattening *) (*********************************************************************************************************************************) 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 Reification. Require Import NaturalDeduction. Require Import NaturalDeductionCategory. Require Import GeneralizedArrow. Require Import GeneralizedArrowFromReification. Require Import ProgrammingLanguage. Require Import ProgrammingLanguageReification. Require Import ReificationsAndGeneralizedArrows. Section Flattening. Context (Guest:ProgrammingLanguageSMME) (Host :ProgrammingLanguageSMME). Context (GuestHost:TwoLevelLanguage Guest Host). Definition FlatObject (x:TypesL _ _ Host) := forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x). Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject. 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. apply if_inv. set (@roundtrip_reification_to_reification _ Guest _ _ Host GuestHost) as q. unfold mf_f in *; simpl in *. apply (if_comp q). clear q. unfold me_mf; simpl. unfold mf_f; simpl. refine (if_respects _ (if_id _)). unfold FlatteningOfReification. unfold mf_f; simpl. eapply if_comp. Focus 2. eapply if_inv. apply (if_associativity (garrow_functor Guest Host GuestHost) F (retraction_retraction F)). eapply if_comp. eapply if_inv. apply if_right_identity. refine (if_respects (if_id _) _). apply if_inv. apply retraction_composes. Qed. *) End Flattening.