re-arrange ProgrammingLanguage

diff --git a/src/ProgrammingLanguageArrow.v b/src/ProgrammingLanguageArrow.v
index 6bf0f24..f4a0d8e 100644 (file)
--- a/src/ProgrammingLanguageArrow.v
+++ b/src/ProgrammingLanguageArrow.v
@@ -31,38 +31,126 @@ Require Import ProgrammingLanguage.
 Require Import ProgrammingLanguageGeneralizedArrow.
 Require Import FreydCategories.
 
+Require Import GeneralizedArrow.
+
+Section ExtendFunctor.
+
+  Context `(F:Functor).
+  Context  (P:c1 -> Prop).
+
+  Definition domain_subcat := FullSubcategory c1 P.
+
+  Definition functor_restricts_to_full_subcat_on_domain_fobj (a:domain_subcat) : c2 :=
+    F (projT1 a).
+
+  Definition functor_restricts_to_full_subcat_on_domain_fmor (a b:domain_subcat)(f:a~~{domain_subcat}~~>b) :
+    (functor_restricts_to_full_subcat_on_domain_fobj a)~~{c2}~~>(functor_restricts_to_full_subcat_on_domain_fobj b) :=
+    F \ (projT1 f).
+
+  Lemma functor_restricts_to_full_subcat_on_domain : Functor domain_subcat c2 functor_restricts_to_full_subcat_on_domain_fobj.
+    refine {| fmor := functor_restricts_to_full_subcat_on_domain_fmor |};
+      unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
+    setoid_rewrite H; reflexivity.
+    setoid_rewrite fmor_preserves_id; reflexivity.
+    setoid_rewrite <- fmor_preserves_comp; reflexivity.
+    Defined.
+
+End ExtendFunctor.
+
+Section MonoidalSubCat.
+
+  (* a monoidal subcategory is a full subcategory, closed under tensor and containing the unit object *)
+  Class MonoidalSubCat {Ob}{Hom}{C:Category Ob Hom}{MFobj}{MF}{MI}(MC:MonoidalCat C MFobj MF MI) :=
+  { msc_P                   :  MC -> Prop
+  ; msc_closed_under_tensor :  forall o1 o2, msc_P o1 -> msc_P o2 -> msc_P (MC (pair_obj o1 o2))
+  ; msc_contains_unit       :  msc_P (mon_i MC)
+  ; msc_subcat              := FullSubcategory MC msc_P
+  }.
+  Local Coercion msc_subcat : MonoidalSubCat >-> SubCategory.
+
+  Context `(MSC:MonoidalSubCat).
+
+  (* any full subcategory of a monoidal category, , is itself monoidal *)
+  Definition mf_restricts_to_full_subcat_on_domain_fobj (a:MSC ×× MSC) : MSC.
+    destruct a.
+    destruct o.
+    destruct o0.
+    set (MC (pair_obj x x0)) as m'.
+    exists m'.
+    apply msc_closed_under_tensor; auto.
+    Defined.
+
+  Definition mf_restricts_to_full_subcat_on_domain_fmor
+    {a}{b}
+    (f:a~~{MSC ×× MSC}~~>b)
+    :
+    (mf_restricts_to_full_subcat_on_domain_fobj a)~~{MSC}~~>(mf_restricts_to_full_subcat_on_domain_fobj b).
+    destruct a as [[a1 a1pf] [a2 a2pf]].
+    destruct b as [[b1 b1pf] [b2 b2pf]].
+    destruct f as [[f1 f1pf] [f2 f2pf]].
+    simpl in *.
+    exists (MC \ (pair_mor (pair_obj a1 a2) (pair_obj b1 b2) f1 f2)); auto.
+    Defined.
+
+  Lemma mf_restricts_to_full_subcat_on_domain : Functor (MSC ×× MSC) MSC
+    mf_restricts_to_full_subcat_on_domain_fobj.
+    refine {| fmor := fun a b f => mf_restricts_to_full_subcat_on_domain_fmor f |};
+      unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
+    admit.
+    admit.
+    admit.
+    Defined.
+
+  Definition subcat_i : MSC.
+    exists (mon_i MC).
+    apply msc_contains_unit.
+    Defined.
+
+  Lemma full_subcat_is_monoidal : MonoidalCat MSC _ mf_restricts_to_full_subcat_on_domain subcat_i.
+    admit.
+    Defined.
+
+  Lemma inclusion_functor_monoidal : MonoidalFunctor full_subcat_is_monoidal MC (InclusionFunctor _ MSC).
+    admit.
+    Defined.
+
+End MonoidalSubCat.
+Coercion full_subcat_is_monoidal : MonoidalSubCat >-> MonoidalCat.
+
 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... *)
+
+  (* a host language: *)
+  Context (Host:ProgrammingLanguageSMME).
+
+  (* ... for which Types(L) is cartesian: *)
+  Context {MF}(center_of_TypesL:MonoidalCat (TypesL _ _ Host) (fun x => (fst_obj _ _ x),,(snd_obj _ _ x)) MF []).
 
-  Definition ArrowsAreGeneralizedArrows (Host:ProgrammingLanguageSMME)
-    {mf}{mn}{cc}{kehom}{CC}
-    (arrow:ArrowInProgrammingLanguage Host _ _ CC mf mn cc kehom) :  GeneralizedArrowInLanguage.
+  (* along with a monoidal subcategory of Z(Types(L)) *)
+  Context (VK:MonoidalSubCat center_of_TypesL).
 
-  Definition TwoLevelLanguage := Reification Guest Host (me_i Host).
+  (* a premonoidal category enriched in aforementioned full subcategory *)
+  Context {Kehom:center_of_TypesL -> center_of_TypesL -> VK}.
 
-  Context (GuestHost:TwoLevelLanguage).
+  Context {KE:@ECategory _ _ VK _ _ _ VK center_of_TypesL Kehom}.
 
-  Definition FlatObject (x:TypesL _ _ Host) :=
-    forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
+  Context (CC:CartesianCat center_of_TypesL).
 
-  Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject.
+  Context {kbo}{kbc}(K:@PreMonoidalCat _ _ KE kbo kbc (@one _ _ _ (car_terminal(CartesianCat:=CC)))).
 
-  Section Flattening.
+  (*
+  Definition K_enrichment : Enrichment.
+    refine {| enr_c := KE |}.
+    Defined.
+  Context (K_surjective:SurjectiveEnrichment K_enrichment).
+    *)
 
-    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_of_TypesL _ _ _ _ K.
 
-  End Flattening.
+  Definition ArrowsAreGeneralizedArrows (arrow:ArrowInProgrammingLanguage) : GeneralizedArrow K_enrichment Host.
+    refine {| ga_functor_monoidal := inclusion_functor_monoidal VK |}.
+    Defined.
 
 End GArrowInLanguage.

diff --git a/src/ProgrammingLanguageGeneralizedArrow.v b/src/ProgrammingLanguageGeneralizedArrow.v
index 620ca9f..bac2397 100644 (file)
--- a/src/ProgrammingLanguageGeneralizedArrow.v
+++ b/src/ProgrammingLanguageGeneralizedArrow.v
@@ -29,34 +29,20 @@ Require Import NaturalDeductionCategory.
 
 Require Import FreydCategories.
 
+Require Import Reification.
+Require Import GeneralizedArrow.
 Require Import GeneralizedArrowFromReification.
-Section GArrowInLanguage.
+Require Import ProgrammingLanguage.
 
-  Definition GeneralizedArrowInLanguage (Guest:ProgrammingLanguageSMME) (Host :ProgrammingLanguageSMME)
-    := GeneralizedArrow Guest Host.
-
-  Definition ArrowsAreGeneralizedArrows (Host:ProgrammingLanguageSMME)
-    {mf}{mn}{cc}{kehom}{CC}
-    (arrow:ArrowInProgrammingLanguage Host _ _ CC mf mn cc kehom) :  GeneralizedArrowInLanguage.
-
-  Definition TwoLevelLanguage := Reification Guest Host (me_i Host).
+Require Import ReificationsAndGeneralizedArrows.
+Require Import ReificationFromGeneralizedArrow.
 
-  Context (GuestHost:TwoLevelLanguage).
+Section ProgrammingLanguageGeneralizedArrow.
 
-  Definition FlatObject (x:TypesL _ _ Host) :=
-    forall y1 y2, not ((reification_r_obj GuestHost y1 y2)=x).
+  Context (Guest:ProgrammingLanguageSMME) (Host :ProgrammingLanguageSMME).
 
-  Definition FlatSubCategory := FullSubcategory (TypesL _ _ Host) FlatObject.
-
-  Section Flattening.
-
-    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 GeneralizedArrowInLanguage 
+    := GeneralizedArrow Guest Host.
 
-  End Flattening.
+End ProgrammingLanguageGeneralizedArrow.
 
-End GArrowInLanguage.

diff --git a/src/ProgrammingLanguageReification.v b/src/ProgrammingLanguageReification.v
index 589f748..ab8bb90 100644 (file)
--- a/src/ProgrammingLanguageReification.v
+++ b/src/ProgrammingLanguageReification.v
@@ -29,6 +29,9 @@ Require Import NaturalDeduction.
 Require Import NaturalDeductionCategory.
 Require Import ProgrammingLanguage.
 
+Definition TwoLevelLanguage (Guest Host:ProgrammingLanguageSMME)
+  := Reification Guest Host (me_i Host).
+
 Inductive NLevelLanguage : nat -> ProgrammingLanguageSMME -> Type :=
 | NLevelLanguage_zero : forall lang,    NLevelLanguage O lang
 | NLevelLanguage_succ : forall (L1 L2:ProgrammingLanguageSMME) n,
@@ -37,3 +40,4 @@ Inductive NLevelLanguage : nat -> ProgrammingLanguageSMME -> Type :=
 Definition OmegaLevelLanguage : Type :=
   { f : nat -> ProgrammingLanguageSMME
   & forall n, TwoLevelLanguage (f n) (f (S n)) }.
+