merge proof correction

[coq-hetmet.git] / src / ProgrammingLanguageFlattening2.v

diff --git a/src/ProgrammingLanguageFlattening2.v b/src/ProgrammingLanguageFlattening2.v
new file mode 100644 (file)
index 0000000..4438dd2
--- /dev/null
+++ b/src/ProgrammingLanguageFlattening2.v
@@ -0,0 +1,92 @@
+(*********************************************************************************************************************************)
+(* 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.
+
+