Require Import FunctorCategories_ch7_7.
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_eqv : @ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)
- ; pl_tsr :> @TreeStructuralRules Judg Rule T sequent
- ; pl_sc :> @SequentCalculus Judg Rule _ sequent
- ; pl_subst :> @CutRule Judg Rule _ sequent pl_eqv pl_sc
- ; pl_sequent_join :> @SequentExpansion Judg Rule T sequent pl_eqv pl_sc pl_subst
+ { 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 nd_seq_reflexive.
- Defined.
-
- Definition cutProof a b c : [a |= b],,[b |= c] ~~{JudgmentsL}~~> [a |= c].
- unfold hom; simpl.
- apply pl_subst.
- Defined.
-
- 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.
- unfold identityProof; unfold cutProof; simpl.
- apply nd_cut_left_identity.
- unfold identityProof; unfold cutProof; simpl.
- apply nd_cut_right_identity.
- unfold identityProof; unfold cutProof; simpl.
- symmetry.
- apply nd_cut_associativity.
- Defined.
-
- 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. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
- apply se_reflexive_right.
- intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (se_expand_right _ c) _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (a,,c |= b,,c)) _ (se_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]).
- apply se_cut_right.
- Defined.
-
- 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. unfold ehom. unfold hom. unfold identityProof. unfold eid. simpl. unfold identityProof.
- apply se_reflexive_left.
- intros. unfold ehom. unfold comp. simpl. unfold cutProof.
- rewrite <- (@ndr_prod_preserves_comp _ _ pl_eqv _ _ (se_expand_left _ c) _ _ (nd_id1 (b|=c0))
- _ (nd_id1 (c,,a |= c,,b)) _ (se_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]).
- apply se_cut_left.
- Defined.
-
- Definition Types_binoidal : BinoidalCat TypesL (@T_Branch _).
- refine
- {| bin_first := Types_first
- ; bin_second := Types_second
- |}.
- Defined.
-
- Definition Types_assoc a b : Types_second a >>>> Types_first b <~~~> Types_first b >>>> Types_second a.
- admit.
- Defined.
-
- Definition Types_cancelr : Types_first [] <~~~> functor_id _.
- admit.
- Defined.
-
- Definition Types_cancell : Types_second [] <~~~> functor_id _.
- admit.
- Defined.
-
- Definition Types_assoc_ll a b : Types_second (a,,b) <~~~> Types_second b >>>> Types_second a.
- admit.
- Defined.
-
- Definition Types_assoc_rr a b : Types_first (a,,b) <~~~> Types_first a >>>> Types_first 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
- }.
- admit. (* pentagon law *)
- admit. (* triangle law *)
- admit. (* assoc_rr/assoc coherence *)
- admit. (* assoc_ll/assoc coherence *)
- Defined.
-
- Definition TypesEnrichedInJudgments : Enrichment.
- refine {| enr_c := TypesL |}.
- Defined.
-
- Structure HasProductTypes :=
- {
- }.
-
- 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.
+ 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 ].