(*********************************************************************************************************************************)
(* ProgrammingLanguage *)
(* *)
-(* Basic assumptions about programming languages . *)
+(* Basic assumptions about programming languages. *)
(* *)
(*********************************************************************************************************************************)
Require Import GeneralizedArrowFromReification.
Require Import ReificationFromGeneralizedArrow.
-(*
- * Everything in the rest of this section is just groundwork meant to
- * build up to the definition of the ProgrammingLanguage class, which
- * appears at the end of the section. References to "the instance"
- * mean instances of that class. Think of this section as being one
- * big Class { ... } definition, except that we declare most of the
- * stuff outside the curly brackets in order to take advantage of
- * Coq's section mechanism.
- *)
Section Programming_Language.
Context {T : Type}. (* types of the language *)
Context (Judg : Type).
Context (sequent : Tree ??T -> Tree ??T -> Judg).
Notation "cs |= ss" := (sequent cs ss) : pl_scope.
- (* Because of term irrelevance we need only store the *erased* (def
- * 4.4) trees; for this reason there is no Coq type directly
- * corresponding to productions $e$ and $x$ of 4.1.1, and TreeOT can
- * be used for productions $\Gamma$ and $\Sigma$ *)
-
- (* to do: sequent calculus equals natural deduction over sequents, theorem equals sequent with null antecedent, *)
Context {Rule : Tree ??Judg -> Tree ??Judg -> Type}.
Open Scope nd_scope.
Open Scope pl_scope.
- (*
- *
- * Note that from this abstract interface, the terms (expressions)
- * in the proof are not accessible at all; they don't need to be --
- * so long as we have access to the equivalence relation upon
- * proof-conclusions. Moreover, hiding the expressions actually
- * makes the encoding in CiC work out easier for two reasons:
- *
- * 1. Because the denotation function is provided a proof rather
- * than a term, it is a total function (the denotation function is
- * often undefined for ill-typed terms).
- *
- * 2. We can define arr_composition of proofs without having to know how
- * to compose expressions. The latter task is left up to the client
- * function which extracts an expression from a completed proof.
- *
- * This also means that we don't need an explicit proof obligation for 4.1.2.
- *)
Class ProgrammingLanguage :=
{ pl_eqv : @ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)
; pl_tsr :> @TreeStructuralRules Judg Rule T sequent
Defined.
End LanguageCategory.
+End Programming_Language.
- Structure ProgrammingLanguageSMME :=
- { plsmme_pl : ProgrammingLanguage
- ; 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)) }.
-
- Close Scope temporary_scope3.
- Close Scope pl_scope.
- Close Scope nd_scope.
- Close Scope pf_scope.
+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 Programming_Language.
+ 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 ].