Require Import RepresentableStructure_ch7_2.
Require Import FunctorCategories_ch7_7.
-Require Import Enrichments.
Require Import NaturalDeduction.
-Require Import NaturalDeductionCategory.
Section Programming_Language.
Context {T : Type}. (* types of the language *)
- Context (Judg : Type).
- Context (sequent : Tree ??T -> Tree ??T -> Judg).
+ Definition PLJudg := (Tree ??T) * (Tree ??T).
+ Definition sequent := @pair (Tree ??T) (Tree ??T).
Notation "cs |= ss" := (sequent cs ss) : pl_scope.
- Context {Rule : Tree ??Judg -> Tree ??Judg -> Type}.
+ Context {Rule : Tree ??PLJudg -> Tree ??PLJudg -> Type}.
Notation "H /⋯⋯/ C" := (ND Rule H C) : pl_scope.
Open Scope pl_scope.
Class ProgrammingLanguage :=
- { pl_eqv0 : @ND_Relation Judg Rule
- ; pl_snd :> @SequentND Judg Rule _ sequent
- ; pl_cnd :> @ContextND Judg Rule T sequent pl_snd
- ; pl_eqv1 :> @SequentND_Relation Judg Rule _ sequent pl_snd pl_eqv0
- ; pl_eqv :> @ContextND_Relation Judg Rule _ sequent pl_snd pl_cnd pl_eqv0 pl_eqv1
+ { pl_eqv0 :> @ND_Relation PLJudg Rule
+ ; pl_snd :> @SequentND PLJudg Rule _ sequent
+ ; pl_cnd :> @ContextND PLJudg Rule T sequent pl_snd
+ ; pl_eqv1 :> @SequentND_Relation PLJudg Rule _ sequent pl_snd pl_eqv0
+ ; pl_eqv :> @ContextND_Relation PLJudg Rule _ sequent pl_snd pl_cnd pl_eqv0 pl_eqv1
}.
Notation "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2) : temporary_scope3.
-
- Section LanguageCategory.
-
- Context (PL:ProgrammingLanguage).
-
- (* category of judgments in a fixed type/coercion context *)
- Definition Judgments_cartesian := @Judgments_Category_CartesianCat _ Rule pl_eqv.
-
- Definition JudgmentsL := Judgments_cartesian.
-
- Definition identityProof t : [] ~~{JudgmentsL}~~> [t |= t].
- unfold hom; simpl.
- apply snd_initial.
- Defined.
-
- Definition cutProof a b c : [a |= b],,[b |= c] ~~{JudgmentsL}~~> [a |= c].
- unfold hom; simpl.
- apply snd_cut.
- Defined.
-
- Existing Instance pl_eqv.
-
- Definition TypesL : ECategory JudgmentsL (Tree ??T) (fun x y => [x|=y]).
- refine
- {| eid := identityProof
- ; ecomp := cutProof
- |}; intros.
- apply (mon_commutative(MonoidalCat:=JudgmentsL)).
- apply (mon_commutative(MonoidalCat:=JudgmentsL)).
- unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto.
- unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto.
- unfold identityProof; unfold cutProof; simpl; eapply cndr_inert. apply pl_eqv. auto. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- Defined.
-
- Instance Types_first c : EFunctor TypesL TypesL (fun x => x,,c ) :=
- { efunc := fun x y => cnd_expand_right(ContextND:=pl_cnd) x y c }.
- intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
- intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
- apply (cndr_inert pl_cnd); auto.
- intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (cnd_expand_right _ _ c) _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (a,,c |= b,,c)) _ (cnd_expand_right _ _ c)).
- setoid_rewrite (@ndr_comp_right_identity _ _ pl_eqv _ [a,, c |= b,, c]).
- setoid_rewrite (@ndr_comp_left_identity _ _ pl_eqv [b |= c0]).
- simpl; eapply cndr_inert. apply pl_eqv. auto. auto.
- Defined.
-
- Instance Types_second c : EFunctor TypesL TypesL (fun x => c,,x) :=
- { efunc := fun x y => ((@cnd_expand_left _ _ _ _ _ _ x y c)) }.
- intros; apply (mon_commutative(MonoidalCat:=JudgmentsL)).
- intros. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
- eapply cndr_inert; auto. apply pl_eqv.
- intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (cnd_expand_left _ _ c) _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (c,,a |= c,,b)) _ (cnd_expand_left _ _ c)).
- setoid_rewrite (@ndr_comp_right_identity _ _ pl_eqv _ [c,,a |= c,,b]).
- setoid_rewrite (@ndr_comp_left_identity _ _ pl_eqv [b |= c0]).
- simpl; eapply cndr_inert. apply pl_eqv. auto. auto.
- Defined.
-
- Definition Types_binoidal : EBinoidalCat TypesL.
- refine
- {| ebc_first := Types_first
- ; ebc_second := Types_second
- |}.
- Defined.
-
- Instance Types_assoc_iso a b c : Isomorphic(C:=TypesL) ((a,,b),,c) (a,,(b,,c)) :=
- { iso_forward := snd_initial _ ;; cnd_ant_cossa _ a b c
- ; iso_backward := snd_initial _ ;; cnd_ant_assoc _ a b c
- }.
- simpl; eapply cndr_inert. unfold identityProof; apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- simpl; eapply cndr_inert. unfold identityProof; apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- Defined.
-
- Instance Types_cancelr_iso a : Isomorphic(C:=TypesL) (a,,[]) a :=
- { iso_forward := snd_initial _ ;; cnd_ant_rlecnac _ a
- ; iso_backward := snd_initial _ ;; cnd_ant_cancelr _ a
- }.
- unfold eqv; unfold comp; simpl.
- eapply cndr_inert. apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- unfold eqv; unfold comp; simpl.
- eapply cndr_inert. apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- Defined.
-
- Instance Types_cancell_iso a : Isomorphic(C:=TypesL) ([],,a) a :=
- { iso_forward := snd_initial _ ;; cnd_ant_llecnac _ a
- ; iso_backward := snd_initial _ ;; cnd_ant_cancell _ a
- }.
- unfold eqv; unfold comp; simpl.
- eapply cndr_inert. apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- unfold eqv; unfold comp; simpl.
- eapply cndr_inert. apply pl_eqv. auto.
- apply ndpc_comp; auto.
- apply ndpc_comp; auto.
- auto.
- 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 }.
- intros; unfold eqv; simpl.
- admit.
- Defined.
-
- Instance Types_cancelr : Types_first [] <~~~> functor_id _ :=
- { ni_iso := Types_cancelr_iso }.
- intros; simpl.
- admit.
- Defined.
-
- 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.
-
- Instance Types_PreMonoidal : PreMonoidalCat Types_binoidal [] :=
- { pmon_assoc := Types_assoc
- ; pmon_cancell := Types_cancell
- ; pmon_cancelr := Types_cancelr
- ; pmon_assoc_rr := Types_assoc_rr
- ; pmon_assoc_ll := Types_assoc_ll
- }.
-(*
- apply Build_Pentagon.
- intros; simpl.
- eapply cndr_inert. apply pl_eqv.
- apply ndpc_comp.
- apply ndpc_comp.
- auto.
- apply ndpc_comp.
- apply ndpc_prod.
- apply ndpc_comp.
- apply ndpc_comp.
- auto.
- apply ndpc_comp.
- auto.
- auto.
- auto.
- auto.
- auto.
- auto.
- apply ndpc_comp.
- apply ndpc_comp.
- auto.
- apply ndpc_comp.
- auto.
- auto.
- auto.
- apply Build_Triangle; intros; simpl.
- eapply cndr_inert. apply pl_eqv.
- auto.
- apply ndpc_comp.
- apply ndpc_comp.
- auto.
- apply ndpc_comp.
- auto.
- auto.
- auto.
- eapply cndr_inert. apply pl_eqv. auto.
- auto.
-*)
-admit.
-admit.
- intros; simpl; reflexivity.
- intros; simpl; reflexivity.
- admit. (* assoc central *)
- admit. (* cancelr central *)
- admit. (* cancell central *)
- Defined.
-
- Definition TypesEnrichedInJudgments : Enrichment.
- refine
- {| enr_v_mon := Judgments_Category_monoidal _
- ; enr_c_pm := Types_PreMonoidal
- ; enr_c_bin := Types_binoidal
- |}.
- Defined.
-
- Structure HasProductTypes :=
- {
- }.
-
- (*
- Lemma CartesianEnrMonoidal (e:PreMonoidalEnrichment)
- `(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.
+ Coercion pl_eqv : ProgrammingLanguage >-> ContextND_Relation.
+ Coercion pl_cnd : ProgrammingLanguage >-> ContextND.
End Programming_Language.
-(*
-Structure ProgrammingLanguageSMME :=
-{ plsmme_t : Type
-; plsmme_judg : Type
-; plsmme_sequent : Tree ??plsmme_t -> Tree ??plsmme_t -> plsmme_judg
-; plsmme_rule : Tree ??plsmme_judg -> Tree ??plsmme_judg -> Type
-; plsmme_pl : @ProgrammingLanguage plsmme_t plsmme_judg plsmme_sequent plsmme_rule
-; plsmme_smme : SurjectiveMonicMonoidalEnrichment (TypesEnrichedInJudgments _ _ plsmme_pl)
-}.
-Coercion plsmme_pl : ProgrammingLanguageSMME >-> ProgrammingLanguage.
-Coercion plsmme_smme : ProgrammingLanguageSMME >-> SurjectiveMonicMonoidalEnrichment.
-*)
-Implicit Arguments ND [ Judgment ].
+