(*********************************************************************************************************************************) (* HaskStrongTypes: representation of types and coercions for HaskStrong *) (*********************************************************************************************************************************) Generalizable All Variables. Require Import Preamble. Require Import Coq.Strings.String. Require Import Coq.Lists.List. Require Import General. Require Import HaskKinds. Require Import HaskCoreLiterals. Require Import HaskCoreTypes. (* FIXME *) Require Import HaskCoreVars. (* FIXME *) Require Import HaskWeakTypes. Require Import HaskWeakVars. (* FIXME *) Require Import HaskCoreToWeak. (* FIXME *) Variable dataConTyCon : CoreDataCon -> TyCon. Extract Inlined Constant dataConTyCon => "DataCon.dataConTyCon". Variable dataConExVars_ : CoreDataCon -> list CoreVar. Extract Inlined Constant dataConExVars_ => "DataCon.dataConExTyVars". Variable dataConEqTheta_ : CoreDataCon -> list PredType. Extract Inlined Constant dataConEqTheta_ => "DataCon.dataConEqTheta". Variable dataConOrigArgTys_: CoreDataCon -> list CoreType. Extract Inlined Constant dataConOrigArgTys_=>"DataCon.dataConOrigArgTys". (* FIXME: might be a better idea to panic here than simply drop things that look wrong *) Definition dataConExTyVars cdc := filter (map (fun x => match coreVarToWeakVar x with WTypeVar v => Some v | _ => None end) (dataConExVars_ cdc)). Opaque dataConExTyVars. Definition dataConCoerKinds cdc := filter (map (fun x => match x with EqPred t1 t2 => match ( coreTypeToWeakType t1 >>= fun t1' => coreTypeToWeakType t2 >>= fun t2' => OK (t1',t2')) with OK z => Some z | _ => None end | _ => None end) (dataConEqTheta_ cdc)). Opaque dataConCoerKinds. Definition dataConFieldTypes cdc := filter (map (fun x => match coreTypeToWeakType x with | OK z => Some z | _ => None end) (dataConOrigArgTys_ cdc)). Definition tyConNumKinds (tc:TyCon) := length (tyConTyVars tc). Coercion tyConNumKinds : TyCon >-> nat. Inductive DataCon : TyCon -> Type := mkDataCon : forall cdc:CoreDataCon, DataCon (dataConTyCon cdc). Definition dataConToCoreDataCon `(dc:DataCon tc) : CoreDataCon := match dc with mkDataCon cdc => cdc end. Coercion mkDataCon : CoreDataCon >-> DataCon. Coercion dataConToCoreDataCon : DataCon >-> CoreDataCon. (*Opaque DataCon.*) Definition compare_datacons : forall tc, forall dc1 dc2:DataCon tc, bool. intros. destruct dc1. destruct dc2. exact (if eqd_dec cdc cdc0 then true else false). Defined. Axiom trust_compare_datacons : forall tc dc1 dc2, if compare_datacons tc dc1 dc2 then dc1=dc2 else not (dc1=dc2). Instance DataConEqDecidable : forall tc, EqDecidable (@DataCon tc). intros. apply Build_EqDecidable. intros. set (trust_compare_datacons _ v1 v2) as x. set (compare_datacons tc v1 v2) as y. assert (y=compare_datacons tc v1 v2). reflexivity. destruct y; rewrite <- H in x. left; auto. right; auto. Defined. Definition tyConKind' tc := fold_right KindTypeFunction ★ (tyConKind tc). (* types prefixed with "Raw" are NOT binder-polymorphic; they have had their PHOAS parameter instantiated already *) Section Raw. (* TV is the PHOAS type which stands for type variables of System FC *) Context {TV:Kind -> Type}. (* Figure 7: ρ, σ, τ, ν *) Inductive RawHaskType : Kind -> Type := | TVar : ∀ κ, TV κ -> RawHaskType κ (* a *) | TCon : ∀ tc, RawHaskType (tyConKind' tc) (* T *) | TArrow : RawHaskType (★ ⇛★ ⇛★ ) (* (->) *) | TCoerc : ∀ κ, RawHaskType κ -> RawHaskType κ -> RawHaskType ★ -> RawHaskType ★ (* (+>) *) | TApp : ∀ κ₁ κ₂, RawHaskType (κ₂⇛κ₁) -> RawHaskType κ₂ -> RawHaskType κ₁ (* φ φ *) | TAll : ∀ κ, (TV κ -> RawHaskType ★) -> RawHaskType ★ (* ∀a:κ.φ *) | TCode : RawHaskType ★ -> RawHaskType ★ -> RawHaskType ★ (* from λ^α *) | TyFunApp : ∀ tf, RawHaskTypeList (fst (tyFunKind tf)) -> RawHaskType (snd (tyFunKind tf)) (* S_n *) with RawHaskTypeList : list Kind -> Type := | TyFunApp_nil : RawHaskTypeList nil | TyFunApp_cons : ∀ κ kl, RawHaskType κ -> RawHaskTypeList kl -> RawHaskTypeList (κ::kl). (* the "kind" of a coercion is a pair of types *) Inductive RawCoercionKind : Type := mkRawCoercionKind : ∀ κ, RawHaskType κ -> RawHaskType κ -> RawCoercionKind. Section CV. (* the PHOAS type which stands for coercion variables of System FC *) (* Figure 7: γ, δ *) Inductive RawHaskCoer {CV:Type} : RawCoercionKind -> Prop := (* FIXME *). (* | CoVar : CV -> RawHaskCoer (* g *) | CoType : RawHaskType -> RawHaskCoer (* τ *) | CoApp : RawHaskCoer -> RawHaskCoer -> RawHaskCoer (* γ γ *) | CoAppT : RawHaskCoer -> RawHaskType -> RawHaskCoer (* γ@v *) | CoCFApp : ∀ n, CoFunConst n -> vec RawHaskCoer n -> RawHaskCoer (* C γⁿ *) | CoTFApp : ∀ n, TyFunConst n -> vec RawHaskCoer n -> RawHaskCoer (* S_n γⁿ *) | CoAll : Kind -> (TV -> RawHaskCoer) -> RawHaskCoer (* ∀a:κ.γ *) | CoSym : RawHaskCoer -> RawHaskCoer (* sym *) | CoComp : RawHaskCoer -> RawHaskCoer -> RawHaskCoer (* ◯ *) | CoLeft : RawHaskCoer -> RawHaskCoer (* left *) | CoRight : RawHaskCoer -> RawHaskCoer (* right *). *) End CV. End Raw. Implicit Arguments TCon [ [TV] ]. Implicit Arguments TyFunApp [ [TV] ]. Implicit Arguments RawHaskType [ ]. Implicit Arguments RawHaskCoer [ ]. Implicit Arguments RawCoercionKind [ ]. Implicit Arguments TVar [ [TV] [κ] ]. Implicit Arguments TCoerc [ [TV] [κ] ]. Implicit Arguments TApp [ [TV] [κ₁] [κ₂] ]. Implicit Arguments TAll [ [TV] ]. Notation "t1 ---> t2" := (fun TV env => (TApp (TApp TArrow (t1 TV env)) (t2 TV env))). Notation "φ₁ ∼∼ φ₂ ⇒ φ₃" := (fun TV env => TCoerc (φ₁ TV env) (φ₂ TV env) (φ₃ TV env)). (* Kind and Coercion Environments *) (* * In System FC, the environment consists of three components, each of * whose well-formedness depends on all of those prior to it: * * 1. (TypeEnv) The list of free type variables and their kinds * 2. (CoercionEnv) The list of free coercion variables and the pair of types between which it witnesses coercibility * 3. (Tree ??CoreVar) The list of free value variables and the type of each one *) Definition TypeEnv := list Kind. Definition InstantiatedTypeEnv (TV:Kind->Type) (Γ:TypeEnv) := IList _ TV Γ. Definition HaskCoercionKind (Γ:TypeEnv) := ∀ TV, InstantiatedTypeEnv TV Γ -> @RawCoercionKind TV. Definition CoercionEnv (Γ:TypeEnv) := list (HaskCoercionKind Γ). Definition InstantiatedCoercionEnv (TV:Kind->Type) CV (Γ:TypeEnv)(Δ:CoercionEnv Γ):= vec CV (length Δ). (* A (HaskXX Γ) is an XX which is valid in environments of shape Γ; they are always PHOAS-uninstantiated *) Definition HaskTyVar (Γ:TypeEnv) κ := forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ. Definition HaskCoVar Γ Δ := forall TV CV (env:@InstantiatedTypeEnv TV Γ)(cenv:@InstantiatedCoercionEnv TV CV Γ Δ), CV. Definition HaskLevel (Γ:TypeEnv) := list (HaskTyVar Γ ★). Definition HaskType (Γ:TypeEnv) κ := ∀ TV, @InstantiatedTypeEnv TV Γ -> RawHaskType TV κ. Definition haskTyVarToType {Γ}{κ}(htv:HaskTyVar Γ κ) : HaskType Γ κ := fun TV ite => TVar (htv TV ite). Inductive HaskTypeOfSomeKind (Γ:TypeEnv) := haskTypeOfSomeKind : ∀ κ, HaskType Γ κ -> HaskTypeOfSomeKind Γ. Implicit Arguments haskTypeOfSomeKind [ [Γ] [κ] ]. Definition kindOfHaskTypeOfSomeKind {Γ}(htosk:HaskTypeOfSomeKind Γ) := match htosk with haskTypeOfSomeKind κ _ => κ end. Coercion kindOfHaskTypeOfSomeKind : HaskTypeOfSomeKind >-> Kind. Definition haskTypeOfSomeKindToHaskType {Γ}(htosk:HaskTypeOfSomeKind Γ) : HaskType Γ htosk := match htosk as H return HaskType Γ H with haskTypeOfSomeKind _ ht => ht end. Coercion haskTypeOfSomeKindToHaskType : HaskTypeOfSomeKind >-> HaskType. Definition HaskCoercion Γ Δ (hk:HaskCoercionKind Γ) := forall TV CV (ite:@InstantiatedTypeEnv TV Γ), @InstantiatedCoercionEnv TV CV Γ Δ -> @RawHaskCoer TV CV (hk TV ite). Inductive LeveledHaskType (Γ:TypeEnv) κ := mkLeveledHaskType : HaskType Γ κ -> HaskLevel Γ -> LeveledHaskType Γ κ. Definition FreshHaskTyVar {Γ}(κ:Kind) : HaskTyVar (κ::Γ) κ := fun TV env => ilist_head env. Definition HaskTAll {Γ}(κ:Kind)(σ:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV ★) : HaskType Γ ★ := fun TV env => TAll κ (σ TV env). Definition HaskTApp {Γ}{κ}(σ:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV ★) (cv:HaskTyVar Γ κ) : HaskType Γ ★ := fun TV env => σ TV env (cv TV env). Definition HaskBrak {Γ}(v:HaskTyVar Γ ★)(t:HaskType Γ ★) : HaskType Γ ★:= fun TV env => @TCode TV (TVar (v TV env)) (t TV env). Definition HaskTCon {Γ}(tc:TyCon) : HaskType Γ (fold_right KindTypeFunction ★ (tyConKind tc)) := fun TV ite => TCon tc. Definition HaskAppT {Γ}{κ₁}{κ₂}(t1:HaskType Γ (κ₂⇛κ₁))(t2:HaskType Γ κ₂) : HaskType Γ κ₁ := fun TV ite => TApp (t1 TV ite) (t2 TV ite). Definition mkHaskCoercionKind {Γ}{κ}(t1:HaskType Γ κ)(t2:HaskType Γ κ) : HaskCoercionKind Γ := fun TV ite => mkRawCoercionKind _ (t1 TV ite) (t2 TV ite). (* PHOAS substitution on types *) Definition substT {Γ}{κ₁}{κ₂}(exp:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ₁ -> RawHaskType TV κ₂)(v:@HaskType Γ κ₁) : @HaskType Γ κ₂ := fun TV env => (fix flattenT {κ} (exp: RawHaskType (fun k => RawHaskType TV k) κ) : RawHaskType TV κ := match exp with | TVar _ x => x | TAll _ y => TAll _ (fun v => flattenT _ (y (TVar v))) | TApp _ _ x y => TApp (flattenT _ x) (flattenT _ y) | TCon tc => TCon tc | TCoerc _ t1 t2 t => TCoerc (flattenT _ t1) (flattenT _ t2) (flattenT _ t) | TArrow => TArrow | TCode v e => TCode (flattenT _ v) (flattenT _ e) | TyFunApp tfc lt => TyFunApp tfc (flattenTyFunApp _ lt) end with flattenTyFunApp (lk:list Kind)(exp:@RawHaskTypeList (fun k => RawHaskType TV k) lk) : @RawHaskTypeList TV lk := match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with | TyFunApp_nil => TyFunApp_nil | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flattenT _ t) (flattenTyFunApp _ rest) end for flattenT) _ (exp (fun k => RawHaskType TV k) (ilmap (fun κ tv => TVar tv) env) (v TV env)). Notation "t @@ l" := (@mkLeveledHaskType _ _ t l) (at level 20). Notation "t @@@ l" := (mapOptionTree (fun t' => t' @@ l) t) (at level 20). Notation "'<[' a '|-' t ']>'" := (@HaskBrak _ a t). (* yeah, things are kind of messy below this point *) Definition unAddKindFromInstantiatedTypeEnv {Γ:TypeEnv}{κ:Kind}{TV:Kind->Type}(ite:InstantiatedTypeEnv TV (κ::Γ)) := ilist_tail ite. Definition addKindToCoercionEnv (Γ:TypeEnv)(Δ:CoercionEnv Γ)(κ:Kind) : CoercionEnv (κ::Γ) := map (fun f => (fun TV ite => f TV (unAddKindFromInstantiatedTypeEnv ite))) Δ. Definition addKindToInstantiatedTypeEnv {Γ:TypeEnv}{TV:Kind->Type}(env:InstantiatedTypeEnv TV Γ)(κ:Kind)(tv:TV κ) : InstantiatedTypeEnv TV (κ::Γ) := tv::::env. Definition addKindToInstantiatedCoercionEnv {Γ:TypeEnv}{Δ}{TV:Kind->Type}{CV:Type} (env:InstantiatedCoercionEnv TV CV Γ Δ)(κ:Kind)(tv:TV κ) : InstantiatedCoercionEnv TV CV (κ::Γ) (addKindToCoercionEnv Γ Δ κ). simpl. unfold InstantiatedCoercionEnv. unfold addKindToCoercionEnv. simpl. rewrite <- map_preserves_length. apply env. Defined. Definition coercionEnvContainsCoercion {Γ}{Δ}{TV:Kind->Type}{CV:Type}(ite:InstantiatedTypeEnv TV Γ) (ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)(ck:RawCoercionKind TV) := @vec_In _ _ (cv,ck) (vec_zip ice (vec_map (fun f => f TV ite) (list2vec Δ))). Definition addCoercionToCoercionEnv {Γ}(Δ:CoercionEnv Γ)(κ:HaskCoercionKind Γ) : CoercionEnv Γ := κ::Δ. Definition addCoercionToInstantiatedCoercionEnv {Γ}{Δ}{κ}{TV CV}(ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV) : InstantiatedCoercionEnv TV CV Γ (addCoercionToCoercionEnv Δ κ). simpl. unfold addCoercionToCoercionEnv; simpl. unfold InstantiatedCoercionEnv; simpl. apply vec_cons; auto. Defined. (* the various "weak" functions turn a HaskXX-in-Γ into a HaskXX-in-(κ::Γ) *) Definition weakITE {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (κ::Γ)) : InstantiatedTypeEnv TV Γ := ilist_tail ite. Definition weakITE' {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (app κ Γ)) : InstantiatedTypeEnv TV Γ. induction κ; auto. apply IHκ. inversion ite; subst. apply X0. Defined. Definition weakCE {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (κ::Γ) := map (fun x => (fun tv ite => x tv (weakITE ite))) Δ. Definition weakV {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (κ::Γ) κv := fun TV ite => (cv' TV (weakITE ite)). Definition weakV' {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (app κ Γ) κv. induction κ; auto. apply weakV; auto. Defined. Definition weakT {Γ:TypeEnv}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (κ::Γ) κ₂ := fun TV ite => lt TV (weakITE ite). Definition weakL {Γ}{κ}(lt:HaskLevel Γ) : HaskLevel (κ::Γ) := map weakV lt. Definition weakT' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app κ Γ) κ₂. induction κ; auto. apply weakT; auto. Defined. Definition weakT'' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app Γ κ) κ₂. unfold HaskType in *. unfold InstantiatedTypeEnv in *. intros. apply ilist_chop in X. apply lt. apply X. Defined. Definition lamer {a}{b}{c}{κ}(lt:HaskType (app (app a b) c) κ) : HaskType (app a (app b c)) κ. rewrite <- ass_app in lt. exact lt. Defined. Definition weakL' {Γ}{κ}(lev:HaskLevel Γ) : HaskLevel (app κ Γ). induction κ; auto. apply weakL; auto. Defined. Definition weakLT {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (κ::Γ) κ₂ := match lt with t @@ l => weakT t @@ weakL l end. Definition weakLT' {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (app κ Γ) κ₂ := match lt with t @@ l => weakT' t @@ weakL' l end. Definition weakCE' {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (app κ Γ). induction κ; auto. apply weakCE; auto. Defined. Definition weakICE {Γ:TypeEnv}{κ}{Δ:CoercionEnv Γ}{TV}{CV}(ice:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) : InstantiatedCoercionEnv TV CV Γ Δ. intros. unfold InstantiatedCoercionEnv; intros. unfold InstantiatedCoercionEnv in ice. unfold weakCE in ice. simpl in ice. rewrite <- map_preserves_length in ice. apply ice. Defined. Definition weakCK {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (κ::Γ). unfold HaskCoercionKind in *. intros. apply hck; clear hck. inversion X; subst; auto. Defined. Definition weakCK' {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (app κ Γ). induction κ; auto. apply weakCK. apply IHκ. Defined. Definition weakCK'' {Γ}{κ}(hck:list (HaskCoercionKind Γ)) : list (HaskCoercionKind (app κ Γ)) := match κ as K return list (HaskCoercionKind (app K Γ)) with | nil => hck | _ => map weakCK' hck end. Definition weakCV {Γ}{Δ}{κ}(cv':HaskCoVar Γ Δ) : HaskCoVar (κ::Γ) (weakCE Δ) := fun TV CV ite ice => (cv' TV CV (weakITE ite) (weakICE ice)). Definition weakF {Γ:TypeEnv}{κ}{κ₂}(f:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV κ₂) : forall TV (env:@InstantiatedTypeEnv TV (κ::Γ)), TV κ -> RawHaskType TV κ₂ := fun TV ite tv => (f TV (weakITE ite) tv). (* I keep mixing up left/right folds *) (* Eval compute in (fold_right (fun x y => "("+++x+++y+++")") "x" ("a"::"b"::"c"::nil)). *) (*Eval compute in (fold_left (fun x y => "("+++x+++y+++")") ("a"::"b"::"c"::nil) "x").*) Fixpoint caseType0 {Γ}(lk:list Kind) : IList _ (HaskType Γ) lk -> HaskType Γ (fold_right KindTypeFunction ★ lk) -> HaskType Γ ★ := match lk as LK return IList _ (HaskType Γ) LK -> HaskType Γ (fold_right KindTypeFunction ★ LK) -> HaskType Γ ★ with | nil => fun _ ht => ht | k::lk' => fun tlist ht => caseType0 lk' (ilist_tail tlist) (fun TV env => TApp (ht TV env) (ilist_head tlist TV env)) end. Definition caseType {Γ}(tc:TyCon)(atypes:IList _ (HaskType Γ) (tyConKind tc)) : HaskType Γ ★ := caseType0 (tyConKind tc) atypes (fun TV env => TCon tc). (* like a GHC DataCon, but using PHOAS representation for types and coercions *) Record StrongAltCon {tc:TyCon} := { sac_tc := tc ; sac_altcon : AltCon ; sac_numExTyVars : nat ; sac_numCoerVars : nat ; sac_numExprVars : nat ; sac_ekinds : vec Kind sac_numExTyVars ; sac_kinds := app (tyConKind tc) (vec2list sac_ekinds) ; sac_Γ := fun Γ => app (tyConKind tc) Γ ; sac_coercions : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskCoercionKind (sac_Γ Γ)) sac_numCoerVars ; sac_types : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskType (sac_Γ Γ) ★) sac_numExprVars ; sac_Δ := fun Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)) Δ => app (vec2list (sac_coercions Γ atypes)) Δ }. Coercion sac_tc : StrongAltCon >-> TyCon. Coercion sac_altcon : StrongAltCon >-> AltCon. Definition kindOfType {Γ}{κ}(ht:@HaskType Γ κ) : ???Kind := OK κ. Axiom literal_tycons_are_of_ordinary_kind : forall lit, tyConKind (haskLiteralToTyCon lit) = nil. Definition literalType (lit:HaskLiteral){Γ} : HaskType Γ ★. set (fun TV (ite:InstantiatedTypeEnv TV Γ) => @TCon TV (haskLiteralToTyCon lit)) as z. unfold tyConKind' in z. rewrite literal_tycons_are_of_ordinary_kind in z. unfold HaskType. apply z. Defined. Notation "a ∼∼∼ b" := (@mkHaskCoercionKind _ _ a b) (at level 18). Fixpoint update_ξ `{EQD_VV:EqDecidable VV}{Γ} (ξ:VV -> LeveledHaskType Γ ★) (vt:list (VV * LeveledHaskType Γ ★)) : VV -> LeveledHaskType Γ ★ := match vt with | nil => ξ | (v,τ)::tl => fun v' => if eqd_dec v v' then τ else (update_ξ ξ tl) v' end. (***************************************************************************************************) (* Well-Formedness of Types and Coercions *) (* also represents production "S_n:κ" of Γ because these can only wind up in Γ via rule (Type) *) Inductive TypeFunctionDecl (tfc:TyCon)(vk:vec Kind tfc) : Type := mkTFD : Kind -> TypeFunctionDecl tfc vk. (* Section WFCo. Context {TV:Kind->Type}. Context {CV:Type}. (* local notations *) Notation "ienv '⊢ᴛy' σ : κ" := (@WellKinded_RawHaskType TV _ ienv σ κ). Notation "env ∋ cv : t1 ∼ t2 : Γ : t" := (@coercionEnvContainsCoercion Γ _ TV CV t env cv (@mkRawCoercionKind _ t1 t2)) (at level 20, t1 at level 99, t2 at level 99, t at level 99). Reserved Notation "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" (at level 20, γ at level 99, b at level 99, Δ at level 99, ite at level 99, Γ at level 99). (* Figure 8, lower half *) Inductive WFCoercion:forall Γ (Δ:CoercionEnv Γ), @InstantiatedTypeEnv TV Γ -> @InstantiatedCoercionEnv TV CV Γ Δ -> @RawHaskCoer TV CV -> @RawCoercionKind TV -> Prop := | CoTVar':∀ Γ Δ t e c σ τ, (@coercionEnvContainsCoercion Γ _ TV CV t e c (@mkRawCoercionKind _ σ τ)) -> e⊢ᴄᴏ CoVar c : σ ∼ τ : Δ : Γ : t | CoRefl :∀ Γ Δ t e τ κ, t ⊢ᴛy τ :κ -> e⊢ᴄᴏ CoType τ : τ ∼ τ : Δ :Γ: t | Sym :∀ Γ Δ t e γ σ τ, (e⊢ᴄᴏ γ : σ ∼ τ : Δ : Γ:t) -> e⊢ᴄᴏ CoSym γ : τ ∼ σ : Δ :Γ: t | Trans :∀ Γ Δ t e γ₁ γ₂ σ₁ σ₂ σ₃,(e⊢ᴄᴏ γ₁:σ₁∼σ₂:Δ:Γ:t) -> (e⊢ᴄᴏ γ₂:σ₂∼σ₃:Δ:Γ:t) -> e⊢ᴄᴏ CoComp γ₁ γ₂: σ₁ ∼ σ₃ : Δ :Γ: t | Left :∀ Γ Δ t e γ σ₁ σ₂ τ₁ τ₂,(e⊢ᴄᴏ γ : TApp σ₁ σ₂ ∼ TApp τ₁ τ₂ :Δ:Γ:t ) -> e⊢ᴄᴏ CoLeft γ : σ₁ ∼ τ₁ : Δ :Γ: t | Right :∀ Γ Δ t e γ σ₁ σ₂ τ₁ τ₂,(e⊢ᴄᴏ γ : TApp σ₁ σ₂ ∼ TApp τ₁ τ₂ :Δ:Γ:t ) -> e⊢ᴄᴏ CoRight γ : σ₂ ∼ τ₂ : Δ :Γ: t (* | SComp :∀ Γ Δ t e γ n S σ τ κ, ListWFCo Γ Δ t e γ σ τ -> t ⊢ᴛy TyFunApp(n:=n) S σ : κ -> e⊢ᴄᴏ CoTFApp S γ : TyFunApp S σ∼TyFunApp S τ : Δ : Γ : t | CoAx :∀ Γ Δ t e n C κ γ, forall (σ₁:vec TV n) (σ₂:vec TV n), forall (ax:@AxiomDecl n C κ TV), ListWFCo Γ Δ t e γ (map TVar (vec2list σ₁)) (map TVar (vec2list σ₂)) -> ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₁)) (vec2list κ) -> ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₂)) (vec2list κ) -> e⊢ᴄᴏ CoCFApp C γ : axd_σ _ _ _ ax σ₁ ∼ axd_τ _ _ _ ax σ₂ : Δ : Γ : t *) | WFCoAll : forall Γ Δ κ (t:InstantiatedTypeEnv TV Γ) (e:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) γ σ τ , (∀ a, e ⊢ᴄᴏ ( γ a) : ( σ a) ∼ ( τ a) : _ : _ : (t + a : κ)) -> weakICE e ⊢ᴄᴏ (CoAll κ γ ) : (TAll κ σ ) ∼ (TAll κ τ ) : Δ : Γ : t | Comp :forall Γ Δ t e γ₁ γ₂ σ₁ σ₂ τ₁ τ₂ κ, (t ⊢ᴛy TApp σ₁ σ₂:κ)-> (e⊢ᴄᴏ γ₁:σ₁∼τ₁:Δ:Γ:t)-> (e⊢ᴄᴏ γ₂:σ₂∼τ₂:Δ:Γ:t) -> e⊢ᴄᴏ (CoApp γ₁ γ₂) : (TApp σ₁ σ₂) ∼ (TApp τ₁ τ₂) : Δ:Γ:t | CoInst :forall Γ Δ t e σ τ κ γ (v:∀ TV, InstantiatedTypeEnv TV Γ -> RawHaskType TV), t ⊢ᴛy v TV t : κ -> (e⊢ᴄᴏ γ:HaskTAll κ σ _ t ∼ HaskTAll κ τ _ t:Δ:Γ:t) -> e⊢ᴄᴏ CoAppT γ (v TV t) : substT σ v TV t ∼substT τ v TV t : Δ : Γ : t with ListWFCo : forall Γ (Δ:CoercionEnv Γ), @InstantiatedTypeEnv TV Γ -> InstantiatedCoercionEnv TV CV Γ Δ -> list (RawHaskCoer TV CV) -> list (RawHaskType TV) -> list (RawHaskType TV) -> Prop := | LWFCo_nil : ∀ Γ Δ t e , ListWFCo Γ Δ t e nil nil nil | LWFCo_cons : ∀ Γ Δ t e a b c la lb lc, (e⊢ᴄᴏ a : b∼c : Δ : Γ : t )-> ListWFCo Γ Δ t e la lb lc -> ListWFCo Γ Δ t e (a::la) (b::lb) (c::lc) where "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" := (@WFCoercion Γ Δ ite ice γ (@mkRawCoercionKind _ a b)). End WFCo. Definition WFCCo (Γ:TypeEnv)(Δ:CoercionEnv Γ)(γ:HaskCoercion Γ Δ)(a b:HaskType Γ) := forall {TV CV:Type}(env:@InstantiatedTypeEnv TV Γ)(cenv:InstantiatedCoercionEnv TV CV Γ Δ), @WFCoercion _ _ Γ Δ env cenv (γ TV CV env cenv) (@mkRawCoercionKind _ (a TV env) (b TV env)). Notation "Δ '⊢ᴄᴏ' γ : a '∼' b" := (@WFCCo _ Δ γ a b). *) (* Decidable equality on PHOAS types *) Fixpoint compareT (n:nat){κ₁}(t1:@RawHaskType (fun _ => nat) κ₁){κ₂}(t2:@RawHaskType (fun _ => nat) κ₂) : bool := match t1 with | TVar _ x => match t2 with TVar _ x' => if eqd_dec x x' then true else false | _ => false end | TAll _ y => match t2 with TAll _ y' => compareT (S n) (y n) (y' n) | _ => false end | TApp _ _ x y => match t2 with TApp _ _ x' y' => if compareT n x x' then compareT n y y' else false | _ => false end | TCon tc => match t2 with TCon tc' => if eqd_dec tc tc' then true else false | _ => false end | TArrow => match t2 with TArrow => true | _ => false end | TCode ec t => match t2 with TCode ec' t' => if compareT n ec ec' then compareT n t t' else false | _ => false end | TCoerc _ t1 t2 t => match t2 with TCoerc _ t1' t2' t' => compareT n t1 t1' && compareT n t2 t2' && compareT n t t' | _ =>false end | TyFunApp tfc lt => match t2 with TyFunApp tfc' lt' => eqd_dec tfc tfc' && compareTL n lt lt' | _ => false end end with compareTL (n:nat){κ₁}(t1:@RawHaskTypeList (fun _ => nat) κ₁){κ₂}(t2:@RawHaskTypeList (fun _ => nat) κ₂) : bool := match t1 with | TyFunApp_nil => match t2 with TyFunApp_nil => true | _ => false end | TyFunApp_cons κ kl t r => match t2 with | TyFunApp_cons κ' kl' t' r' => compareT n t t' && compareTL n r r' | _ => false end end. Fixpoint count' (lk:list Kind)(n:nat) : IList _ (fun _ => nat) lk := match lk as LK return IList _ _ LK with | nil => INil | h::t => n::::(count' t (S n)) end. Definition compareHT Γ κ (ht1 ht2:HaskType Γ κ) := compareT (length Γ) (ht1 (fun _ => nat) (count' Γ O)) (ht2 (fun _ => nat) (count' Γ O)). (* This is not provable in Coq's logic because the Coq function space * is "too big" - although its only definable inhabitants are Coq * functions, it is not provable in Coq that all function space * inhabitants are definable (i.e. there are no "exotic" inhabitants). * This is actually an important feature of Coq: it lets us reason * about properties of non-computable (non-recursive) functions since * any property proven to hold for the entire function space will hold * even for those functions. However when representing binding * structure using functions we would actually prefer the smaller * function-space of *definable* functions only. These two axioms * assert that. *) Axiom compareHT_decides : forall Γ κ (ht1 ht2:HaskType Γ κ), if compareHT Γ κ ht1 ht2 then ht1=ht2 else ht1≠ht2. Axiom compareVars : forall Γ κ (htv1 htv2:HaskTyVar Γ κ), if compareHT _ _ (haskTyVarToType htv1) (haskTyVarToType htv2) then htv1=htv2 else htv1≠htv2. (* using the axioms, we can now create an EqDecidable instance for HaskType, HaskTyVar, and HaskLevel *) Instance haskTypeEqDecidable Γ κ : EqDecidable (HaskType Γ κ). apply Build_EqDecidable. intros. set (compareHT_decides _ _ v1 v2) as z. set (compareHT Γ κ v1 v2) as q. destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *. left; auto. right; auto. Defined. Instance haskTyVarEqDecidable Γ κ : EqDecidable (HaskTyVar Γ κ). apply Build_EqDecidable. intros. set (compareVars _ _ v1 v2) as z. set (compareHT Γ κ (haskTyVarToType v1) (haskTyVarToType v2)) as q. destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *. left; auto. right; auto. Defined. Instance haskLevelEqDecidable Γ : EqDecidable (HaskLevel Γ). apply Build_EqDecidable. intros. unfold HaskLevel in *. apply (eqd_dec v1 v2). Defined. (* ToString instance for PHOAS types *) Fixpoint typeToString' (needparens:bool)(n:nat){κ}(t:RawHaskType (fun _ => nat) κ) {struct t} : string := match t with | TVar _ v => "tv" +++ v | TCon tc => toString tc | TCoerc _ t1 t2 t => "("+++typeToString' false n t1+++"~" +++typeToString' false n t2+++")=>" +++typeToString' needparens n t | TApp _ _ t1 t2 => match t1 with | TApp _ _ TArrow t1 => if needparens then "("+++(typeToString' true n t1)+++"->"+++(typeToString' true n t2)+++")" else (typeToString' true n t1)+++"->"+++(typeToString' true n t2) | _ => if needparens then "("+++(typeToString' true n t1)+++" "+++(typeToString' false n t2)+++")" else (typeToString' true n t1)+++" "+++(typeToString' false n t2) end | TArrow => "(->)" | TAll k f => let alpha := "tv"+++n in "(forall "+++ alpha +++ "{:}"+++ k +++")"+++ typeToString' false (S n) (f n) | TCode ec t => "<["+++(typeToString' true n ec)+++"]>@"+++(typeToString' false n t) | TyFunApp tfc lt => tfc+++"_"+++n+++" ["+++(fold_left (fun x y => " \ "+++x+++y) (typeList2string false n lt) "")+++"]" end with typeList2string (needparens:bool)(n:nat){κ}(t:RawHaskTypeList κ) {struct t} : list string := match t with | TyFunApp_nil => nil | TyFunApp_cons κ kl rhk rhkl => (typeToString' needparens n rhk)::(typeList2string needparens n rhkl) end. Definition typeToString {Γ}{κ}(ht:HaskType Γ κ) : string := typeToString' false (length Γ) (ht (fun _ => nat) (count' Γ O)). Instance TypeToStringInstance {Γ} {κ} : ToString (HaskType Γ κ) := { toString := typeToString }.