split HaskProofCategory into two files

[coq-hetmet.git] / src / ProgrammingLanguageFlattening.v

diff --git a/src/ProgrammingLanguageFlattening.v b/src/ProgrammingLanguageFlattening.v
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+(*********************************************************************************************************************************)
+(* 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 ProgrammingLanguage.
+Require Import ProgrammingReification.
+
+Section Flattening.
+
+  Context (Guest:ProgrammingLanguageSMME) (Host :ProgrammingLanguageSMME).
+  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.
+
+    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.
+
+