add commented-out definitions for analytic proofs and cut elimination

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

diff --git a/src/ProgrammingLanguageArrow.v b/src/ProgrammingLanguageArrow.v
index 6bf0f24..f6a3940 100644 (file)
--- a/src/ProgrammingLanguageArrow.v
+++ b/src/ProgrammingLanguageArrow.v
@@ -18,6 +18,9 @@ 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 PreMonoidalCenter.
 Require Import MonoidalCategories_ch7_8.
 Require Import Coherence_ch7_8.
 Require Import Enrichment_ch2_8.
@@ -31,38 +34,41 @@ Require Import ProgrammingLanguage.
 Require Import ProgrammingLanguageGeneralizedArrow.
 Require Import FreydCategories.
 
+Require Import GeneralizedArrow.
+
 Section ArrowInLanguage.
 
-  Section ArrowInLanguage.
-    Context {MF}{mn:MonoidalCat TypesL (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []} (CC:CartesianCat mn).
-    Context {Kehom}(K:@ECategory _ _ TypesL _ mn [] mn TypesL Kehom).
-    Context {bc:BinoidalCat (Underlying K) (@T_Branch _)}.
-    Context (pmc:@PreMonoidalCat _ _ _ _ bc (@one _ _ _ (car_terminal(CartesianCat:=CC)))).
-    Definition ArrowInProgrammingLanguage := @FreydCategory _ _ _ _ _ _ mn _ _ _ _ pmc.
-  End ArrowInLanguage.
+  (* an Arrow In A Programming Language consists of... *)
 
-  Definition ArrowsAreGeneralizedArrows (Host:ProgrammingLanguageSMME)
-    {mf}{mn}{cc}{kehom}{CC}
-    (arrow:ArrowInProgrammingLanguage Host _ _ CC mf mn cc kehom) :  GeneralizedArrowInLanguage.
+  (* a host language: *)
+  Context `(Host:ProgrammingLanguage).
 
-  Definition TwoLevelLanguage := Reification Guest Host (me_i Host).
+  (* ... for which Types(L) is cartesian: *)
+  Context (center_of_TypesL:MonoidalCat (TypesL_PreMonoidal Host)).
 
-  Context (GuestHost:TwoLevelLanguage).
+  (* along with a full subcategory of Z(Types(L)) *)
+  Context {P}(VK:FullSubcategory (Center (TypesL_PreMonoidal Host)) P).
 
-  Definition FlatObject (x:TypesL _ _ Host) :=
-    forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
+  Context (Pobj_unit   : P []).
+  Context (Pobj_closed : forall a b, P a → P b → P (bin_obj(BinoidalCat:=Center_is_PreMonoidal (TypesL_PreMonoidal Host)) a b)).
+  Definition VKM :=
+    PreMonoidalFullSubcategory_PreMonoidal (Center_is_PreMonoidal (TypesL_PreMonoidal Host)) VK Pobj_unit Pobj_closed.
 
-  Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject.
+  (* a premonoidal category enriched in aforementioned full subcategory *)
+  Context (Kehom:(Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host)) -> @ob _ _ VK).
+  Context (KE   :@ECategory (@ob _ _ VK) (@hom _ _ VK) VK _ VKM (pmon_I VKM) VKM (Center (TypesL_PreMonoidal Host)) Kehom).
+  Context {kbo  :(Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host)) -> (Center (TypesL_PreMonoidal Host))}.
+  Context (kbc  :@BinoidalCat (Center (TypesL_PreMonoidal Host)) _ (Underlying KE) kbo).
 
-  Section Flattening.
+  Definition one' := @one _ _ _ (car_terminal(CartesianCat:=CC))
+  Context (K :@PreMonoidalCat _ _ KE kbo kbc ).
+  Context (CC:CartesianCat (Center (TypesL_PreMonoidal Host))).
 
-    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.
-      admit.
-      Qed.
+  Definition ArrowInProgrammingLanguage :=
+    @FreydCategory _ _ _ _ _ _ (Center (TypesL_PreMonoidal Host)) _ _ _ _ K.
 
-  End Flattening.
+  Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
+    refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.
+    Defined.
 
 End GArrowInLanguage.