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 NaturalDeduction.
Require Import NaturalDeductionCategory.
-Require Import FreydCategories.
-
-Require Import Reification.
-Require Import GeneralizedArrow.
-Require Import GeneralizedArrowFromReification.
-
Section Programming_Language.
Context {T : Type}. (* types of the language *)
apply pl_subst.
Defined.
+ Existing Instance pl_eqv.
Definition TypesL : ECategory JudgmentsL (Tree ??T) (fun x y => [x|=y]).
refine
{| eid := identityProof
; ecomp := cutProof
|}; intros.
- apply MonoidalCat_all_central.
- apply MonoidalCat_all_central.
+ apply (mon_commutative(MonoidalCat:=JudgmentsL)).
+ apply (mon_commutative(MonoidalCat:=JudgmentsL)).
unfold identityProof; unfold cutProof; simpl.
apply nd_cut_left_identity.
unfold identityProof; unfold cutProof; simpl.
Definition Types_first c : EFunctor TypesL TypesL (fun x => x,,c ).
refine {| efunc := fun x y => (@se_expand_right _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y) |}.
- intros; apply MonoidalCat_all_central.
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
apply se_reflexive_right.
intros. unfold ehom. unfold comp. simpl. unfold cutProof.
Definition Types_second c : EFunctor TypesL TypesL (fun x => c,,x).
eapply Build_EFunctor.
instantiate (1:=(fun x y => ((@se_expand_left _ _ _ _ _ _ _ (@pl_sequent_join PL) c x y)))).
- intros; apply MonoidalCat_all_central.
+ intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
apply se_reflexive_left.
intros. unfold ehom. unfold comp. simpl. unfold cutProof.
|}.
Defined.
- Definition Types_assoc a b : Types_second a >>>> Types_first b <~~~> Types_first b >>>> Types_second a.
+ Instance Types_assoc_iso a b c : Isomorphic(C:=TypesL) ((a,,b),,c) (a,,(b,,c)) :=
+ { iso_forward := nd_seq_reflexive _ ;; tsr_ant_cossa _ a b c
+ ; iso_backward := nd_seq_reflexive _ ;; tsr_ant_assoc _ a b c
+ }.
+ admit.
+ admit.
+ Defined.
+
+ Instance Types_assoc a b : Types_second a >>>> Types_first b <~~~> Types_first b >>>> Types_second a :=
+ { ni_iso := fun c => Types_assoc_iso a c b }.
admit.
Defined.
- Definition Types_cancelr : Types_first [] <~~~> functor_id _.
+ Instance Types_cancelr_iso a : Isomorphic(C:=TypesL) (a,,[]) a :=
+ { iso_forward := nd_seq_reflexive _ ;; tsr_ant_rlecnac _ a
+ ; iso_backward := nd_seq_reflexive _ ;; tsr_ant_cancelr _ a
+ }.
+ admit.
admit.
Defined.
- Definition Types_cancell : Types_second [] <~~~> functor_id _.
+ Instance Types_cancelr : Types_first [] <~~~> functor_id _ :=
+ { ni_iso := Types_cancelr_iso }.
admit.
Defined.
- Definition Types_assoc_ll a b : Types_second (a,,b) <~~~> Types_second b >>>> Types_second a.
+ Instance Types_cancell_iso a : Isomorphic(C:=TypesL) ([],,a) a :=
+ { iso_forward := nd_seq_reflexive _ ;; tsr_ant_llecnac _ a
+ ; iso_backward := nd_seq_reflexive _ ;; tsr_ant_cancell _ a
+ }.
+ admit.
admit.
Defined.
- Definition Types_assoc_rr a b : Types_first (a,,b) <~~~> Types_first a >>>> Types_first b.
+ Instance Types_cancell : Types_second [] <~~~> functor_id _ :=
+ { ni_iso := Types_cancell_iso }.
+ admit.
+ Defined.
+
+ Instance Types_assoc_ll a b : Types_second (a,,b) <~~~> Types_second b >>>> Types_second a :=
+ { ni_iso := fun c => Types_assoc_iso a b c }.
+ admit.
+ Defined.
+
+ Instance Types_assoc_rr a b : Types_first (a,,b) <~~~> Types_first a >>>> Types_first b :=
+ { ni_iso := fun c => iso_inv _ _ (Types_assoc_iso c a b) }.
admit.
Defined.
}.
admit. (* pentagon law *)
admit. (* triangle law *)
- admit. (* assoc_rr/assoc coherence *)
- admit. (* assoc_ll/assoc coherence *)
+ intros; simpl; reflexivity.
+ intros; simpl; reflexivity.
+ admit. (* assoc central *)
+ admit. (* cancelr central *)
+ admit. (* cancell central *)
Defined.
+ (*
Definition TypesEnrichedInJudgments : Enrichment.
refine {| enr_c := TypesL |}.
Defined.
{
}.
+ Lemma CartesianEnrMonoidal (e:Enrichment) `(C:CartesianCat(Ob:= _)(Hom:= _)(C:=Underlying (enr_c e))) : MonoidalEnrichment e.
+ admit.
+ Defined.
+
(* need to prove that if we have cartesian tuples we have cartesian contexts *)
Definition LanguagesWithProductsAreSMME : HasProductTypes -> SurjectiveMonicMonoidalEnrichment TypesEnrichedInJudgments.
admit.
Defined.
-
+ *)
End LanguageCategory.
-End Programming_Language.
+End Programming_Language.
+(*
Structure ProgrammingLanguageSMME :=
{ plsmme_t : Type
; plsmme_judg : Type
}.
Coercion plsmme_pl : ProgrammingLanguageSMME >-> ProgrammingLanguage.
Coercion plsmme_smme : ProgrammingLanguageSMME >-> SurjectiveMonicMonoidalEnrichment.
-
-Section ArrowInLanguage.
- Context (Host:ProgrammingLanguageSMME).
- Context `(CC:CartesianCat (me_mon Host)).
- Context `(K:@ECategory _ _ _ _ _ _ (@car_mn _ _ _ _ _ _ _ CC) C Kehom).
- Context `(pmc:PreMonoidalCat K bobj mobj (@one _ _ _ (cartesian_terminal C))).
- (* FIXME *)
- (*
- Definition ArrowInProgrammingLanguage :=
- @FreydCategory _ _ _ _ _ _ (@car_mn _ _ _ _ _ _ _ CC) _ _ _ _ pmc.
- *)
-End ArrowInLanguage.
-
-Section GArrowInLanguage.
- Context (Guest:ProgrammingLanguageSMME).
- Context (Host :ProgrammingLanguageSMME).
- Definition GeneralizedArrowInLanguage := GeneralizedArrow Guest Host.
-
- (* FIXME
- Definition ArrowsAreGeneralizedArrows : ArrowInProgrammingLanguage -> GeneralizedArrowInLanguage.
- *)
- Definition TwoLevelLanguage := Reification Guest Host (me_i Host).
-
- 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.
-
- Section Flattening.
-
- Context (F:Retraction (TypesL _ _ Host) FlatSubCategory).
- Definition FlatteningOfReification := garrow_from_reification Guest Host GuestHost >>>> F.
- Lemma FlatteningIsNotDestructive :
- FlatteningOfReification >>>> retraction_retraction F >>>> RepresentableFunctor _ (me_i Host) ~~~~ GuestHost.
- admit.
- Qed.
-
- End Flattening.
-
-End GArrowInLanguage.
-
-Inductive NLevelLanguage : nat -> ProgrammingLanguageSMME -> Type :=
-| NLevelLanguage_zero : forall lang, NLevelLanguage O lang
-| NLevelLanguage_succ : forall (L1 L2:ProgrammingLanguageSMME) n,
- TwoLevelLanguage L1 L2 -> NLevelLanguage n L1 -> NLevelLanguage (S n) L2.
-
-Definition OmegaLevelLanguage : Type :=
- { f : nat -> ProgrammingLanguageSMME
- & forall n, TwoLevelLanguage (f n) (f (S n)) }.
-
+*)
Implicit Arguments ND [ Judgment ].
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