X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FProgrammingLanguage.v;h=83b435ab06ac2b44d545a7f2a5bb6c769d0a2078;hp=de7c7f0597ef6326645372d410a8978cbc33ffcd;hb=423b0bd3972c5bcbbd757cb715e13b5b9104a9a6;hpb=9444d329585e0dc3400a3bbb8155900f9ad62b92 diff --git a/src/ProgrammingLanguage.v b/src/ProgrammingLanguage.v index de7c7f0..83b435a 100644 --- a/src/ProgrammingLanguage.v +++ b/src/ProgrammingLanguage.v @@ -18,6 +18,8 @@ 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 MonoidalCategories_ch7_8. Require Import Coherence_ch7_8. Require Import Enrichment_ch2_8. @@ -25,23 +27,16 @@ Require Import RepresentableStructure_ch7_2. Require Import FunctorCategories_ch7_7. 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 *) - 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. @@ -50,192 +45,15 @@ Section Programming_Language. 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. + Coercion pl_eqv : ProgrammingLanguage >-> ContextND_Relation. + Coercion pl_cnd : ProgrammingLanguage >-> ContextND. - 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 := - { - }. - - (* 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. -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. - -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 ].