+(*********************************************************************************************************************************)
+(* 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 BinoidalCategories.
+Require Import PreMonoidalCategories.
+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 ProgrammingLanguage.
+Require Import ProgrammingLanguageReification.
+Require Import SectionRetract_ch2_4.
+Require Import GeneralizedArrowFromReification.
+Require Import Enrichments.
+Require Import ReificationsAndGeneralizedArrows.
+
+Section Flattening.
+
+ Context `(Guest:ProgrammingLanguage) `(Host :ProgrammingLanguage).
+ Context (GuestHost:TwoLevelLanguage Guest Host).
+
+ Definition FlatObject (x:TypesL Host) :=
+ forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
+
+ Instance FlatSubCategory : FullSubcategory (TypesL Host) FlatObject.
+
+ Context (F:RetractionOfCategories (TypesL Host) (FullSubCategoriesAreCategories FlatSubCategory)).
+
+ Definition FlatteningOfReification HostMonic HostMonoidal :=
+ (ga_functor
+ (@garrow_from_reification
+ (TypesEnrichedInJudgments Guest)
+ (TypesEnrichedInJudgments Host)
+ HostMonic HostMonoidal GuestHost))
+ >>>> F.
+
+ Lemma FlatteningIsNotDestructive HostMonic HostMonoidal :
+ FlatteningOfReification HostMonic HostMonoidal >>>> retraction_retraction F >>>> HomFunctor _ []
+ ≃ (reification_rstar GuestHost).
+ apply if_inv.
+ 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 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.
+ 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.
+
+