Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
Require Import Enrichment_ch2_8.
Require Import NaturalDeduction.
Require Import NaturalDeductionCategory.
Require Import GeneralizedArrow.
-Require Import GeneralizedArrowFromReification.
-Require Import ProgrammingLanguage.
+Require Import ProgrammingLanguageEnrichment.
Require Import ProgrammingLanguageReification.
+Require Import SectionRetract_ch2_4.
+Require Import GeneralizedArrowFromReification.
+Require Import Enrichments.
Require Import ReificationsAndGeneralizedArrows.
Section Flattening.
- Context (Guest:ProgrammingLanguageSMME) (Host :ProgrammingLanguageSMME).
+ Context `(Guest:ProgrammingLanguage) `(Host :ProgrammingLanguage).
Context (GuestHost:TwoLevelLanguage Guest Host).
- Definition FlatObject (x:TypesL _ _ Host) :=
+ Definition FlatObject (x:TypesL Host) :=
forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
- Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject.
+ Instance FlatSubCategory : FullSubcategory (TypesL Host) FlatObject.
- Context (F:Retraction (TypesL _ _ Host) FlatSubCategory).
+ Context (F:RetractionOfCategories (TypesL Host) (FullSubCategoriesAreCategories FlatSubCategory)).
- Definition FlatteningOfReification :=
- garrow_from_reification Guest Host GuestHost >>>> F.
+ Definition FlatteningOfReification HostMonic HostMonoidal :=
+ (ga_functor
+ (@garrow_from_reification
+ (TypesEnrichedInJudgments Guest)
+ (TypesEnrichedInJudgments Host)
+ HostMonic HostMonoidal GuestHost))
+ >>>> F.
-(*
- Lemma FlatteningIsNotDestructive :
- FlatteningOfReification >>>> retraction_retraction F >>>> HomFunctor _ (me_i Host) ≃ GuestHost.
+ Lemma FlatteningIsNotDestructive HostMonic HostMonoidal :
+ FlatteningOfReification HostMonic HostMonoidal >>>> retraction_retraction F >>>> HomFunctor _ []
+ ≃ (reification_rstar GuestHost).
apply if_inv.
- set (@roundtrip_reification_to_reification _ Guest _ _ Host GuestHost) as q.
- unfold mf_f in *; simpl in *.
- apply (if_comp q).
+ set (@roundtrip_reification_to_reification (TypesEnrichedInJudgments Guest) (TypesEnrichedInJudgments Host)
+ HostMonic HostMonoidal GuestHost) as q.
+ unfold mf_F in *; simpl in *.
+ eapply if_comp.
+ apply q.
clear q.
- unfold me_mf; simpl.
- unfold mf_f; simpl.
- refine (if_respects _ (if_id _)).
+ unfold mf_F; simpl.
+ unfold pmon_I.
+ apply (if_respects
+ (garrow_functor (TypesEnrichedInJudgments Guest) HostMonic HostMonoidal GuestHost)
+ (FlatteningOfReification HostMonic HostMonoidal >>>> retraction_retraction F)
+ (HomFunctor (TypesL Host) [])
+ (HomFunctor (TypesL Host) [])); [ idtac | apply (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 _) _).
+ unfold mf_F; simpl.
apply if_inv.
+ eapply if_comp.
+ apply (if_associativity (garrow_functor (TypesEnrichedInJudgments Guest) HostMonic HostMonoidal GuestHost) F
+ (retraction_retraction F)).
+ eapply if_comp; [ idtac | apply if_right_identity ].
+ apply (if_respects
+ (garrow_functor (TypesEnrichedInJudgments Guest) HostMonic HostMonoidal GuestHost)
+ (garrow_functor (TypesEnrichedInJudgments Guest) HostMonic HostMonoidal GuestHost)
+ (F >>>> retraction_retraction F)
+ (functor_id _)).
+ apply (if_id _).
apply retraction_composes.
Qed.
-*)
+
End Flattening.