1 (*********************************************************************************************************************************)
2 (* HaskStrongTypes: representation of types and coercions for HaskStrong *)
3 (*********************************************************************************************************************************)
5 Generalizable All Variables.
6 Require Import Preamble.
7 Require Import Coq.Strings.String.
8 Require Import Coq.Lists.List.
9 Require Import General.
10 Require Import HaskKinds.
11 Require Import HaskLiterals.
12 Require Import HaskTyCons.
13 Require Import HaskCoreTypes.
14 Require Import HaskCoreVars.
15 Require Import HaskWeakTypes.
16 Require Import HaskWeakVars.
17 Require Import HaskWeak.
18 Require Import HaskCoreToWeak.
20 Variable dataConTyCon : CoreDataCon -> TyCon. Extract Inlined Constant dataConTyCon => "DataCon.dataConTyCon".
21 Variable dataConExVars_ : CoreDataCon -> list CoreVar. Extract Inlined Constant dataConExVars_ => "DataCon.dataConExTyVars".
22 Variable dataConEqTheta_ : CoreDataCon -> list PredType. Extract Inlined Constant dataConEqTheta_ => "DataCon.dataConEqTheta".
23 Variable dataConOrigArgTys_: CoreDataCon -> list CoreType. Extract Inlined Constant dataConOrigArgTys_=>"DataCon.dataConOrigArgTys".
25 Definition dataConExTyVars cdc :=
26 filter (map (fun x => match coreVarToWeakVar x with WTypeVar v => Some v | _ => None end) (dataConExVars_ cdc)).
27 Opaque dataConExTyVars.
28 Definition dataConCoerKinds cdc :=
29 filter (map (fun x => match x with EqPred t1 t2 =>
31 coreTypeToWeakType t1 >>= fun t1' =>
32 coreTypeToWeakType t2 >>= fun t2' =>
38 end) (dataConEqTheta_ cdc)).
39 Opaque dataConCoerKinds.
40 Definition dataConFieldTypes cdc :=
41 filter (map (fun x => match coreTypeToWeakType x with
44 end) (dataConOrigArgTys_ cdc)).
46 Definition tyConNumKinds (tc:TyCon) := length (tyConTyVars tc).
47 Coercion tyConNumKinds : TyCon >-> nat.
49 Inductive DataCon : TyCon -> Type :=
50 mkDataCon : forall cdc:CoreDataCon, DataCon (dataConTyCon cdc).
51 Definition dataConToCoreDataCon `(dc:DataCon tc) : CoreDataCon := match dc with mkDataCon cdc => cdc end.
52 Coercion mkDataCon : CoreDataCon >-> DataCon.
53 Coercion dataConToCoreDataCon : DataCon >-> CoreDataCon.
56 Definition tyConKind' tc := fold_right KindArrow ★ (tyConKind tc).
58 (* types prefixed with "Raw" are NOT binder-polymorphic; they have had their PHOAS parameter instantiated already *)
61 (* TV is the PHOAS type which stands for type variables of System FC *)
62 Context {TV:Kind -> Type}.
64 (* Figure 7: ρ, σ, τ, ν *)
65 Inductive RawHaskType : Kind -> Type :=
66 | TVar : ∀ κ, TV κ -> RawHaskType κ (* a *)
67 | TCon : ∀ tc, RawHaskType (tyConKind' tc) (* T *)
68 | TArrow : RawHaskType (★ ⇛★ ⇛★ ) (* (->) *)
69 | TCoerc : ∀ κ, RawHaskType κ -> RawHaskType κ -> RawHaskType ★ -> RawHaskType ★ (* (+>) *)
70 | TApp : ∀ κ₁ κ₂, RawHaskType (κ₂⇛κ₁) -> RawHaskType κ₂ -> RawHaskType κ₁ (* φ φ *)
71 | TAll : ∀ κ, (TV κ -> RawHaskType ★) -> RawHaskType ★ (* ∀a:κ.φ *)
72 | TCode : RawHaskType ECKind -> RawHaskType ★ -> RawHaskType ★ (* from λ^α *)
73 | TyFunApp : forall (tf:TyFun) kl k, RawHaskTypeList kl -> RawHaskType k (* S_n *)
74 with RawHaskTypeList : list Kind -> Type :=
75 | TyFunApp_nil : RawHaskTypeList nil
76 | TyFunApp_cons : ∀ κ kl, RawHaskType κ -> RawHaskTypeList kl -> RawHaskTypeList (κ::kl).
78 (* the "kind" of a coercion is a pair of types *)
79 Inductive RawCoercionKind : Type :=
80 mkRawCoercionKind : ∀ κ, RawHaskType κ -> RawHaskType κ -> RawCoercionKind.
82 (* Figure 7: γ, δ; CV is the PHOAS type which stands for coercion variables of System FC *)
83 Inductive RawHaskCoer {CV:Type} : RawCoercionKind -> Prop := .
85 * This has been disabled until we manage to reconcile SystemFC's
86 * coercions with what GHC actually implements (they are not the
89 | CoVar : CV -> RawHaskCoer (* g *)
90 | CoType : RawHaskType -> RawHaskCoer (* τ *)
91 | CoApp : RawHaskCoer -> RawHaskCoer -> RawHaskCoer (* γ γ *)
92 | CoAppT : RawHaskCoer -> RawHaskType -> RawHaskCoer (* γ@v *)
93 | CoCFApp : ∀ n, CoFunConst n -> vec RawHaskCoer n -> RawHaskCoer (* C γⁿ *)
94 | CoTFApp : ∀ n, TyFunConst n -> vec RawHaskCoer n -> RawHaskCoer (* S_n γⁿ *)
95 | CoAll : Kind -> (TV -> RawHaskCoer) -> RawHaskCoer (* ∀a:κ.γ *)
96 | CoSym : RawHaskCoer -> RawHaskCoer (* sym *)
97 | CoComp : RawHaskCoer -> RawHaskCoer -> RawHaskCoer (* ◯ *)
98 | CoLeft : RawHaskCoer -> RawHaskCoer (* left *)
99 | CoRight : RawHaskCoer -> RawHaskCoer (* right *).
103 Implicit Arguments TCon [ [TV] ].
104 Implicit Arguments TyFunApp [ [TV] ].
105 Implicit Arguments RawHaskType [ ].
106 Implicit Arguments RawHaskCoer [ ].
107 Implicit Arguments RawCoercionKind [ ].
108 Implicit Arguments TVar [ [TV] [κ] ].
109 Implicit Arguments TCoerc [ [TV] [κ] ].
110 Implicit Arguments TApp [ [TV] [κ₁] [κ₂] ].
111 Implicit Arguments TAll [ [TV] ].
113 Notation "t1 ---> t2" := (fun TV env => (TApp (TApp TArrow (t1 TV env)) (t2 TV env))).
114 Notation "φ₁ ∼∼ φ₂ ⇒ φ₃" := (fun TV env => TCoerc (φ₁ TV env) (φ₂ TV env) (φ₃ TV env)).
116 (* Kind and Coercion Environments *)
118 * In System FC, the environment consists of three components, each of
119 * whose well-formedness depends on all of those prior to it:
121 * 1. (TypeEnv) The list of free type variables and their kinds
122 * 2. (CoercionEnv) The list of free coercion variables and the pair of types between which it witnesses coercibility
123 * 3. (Tree ??CoreVar) The list of free value variables and the type of each one
126 Definition TypeEnv := list Kind.
127 Definition InstantiatedTypeEnv (TV:Kind->Type) (Γ:TypeEnv) := IList _ TV Γ.
128 Definition HaskCoercionKind (Γ:TypeEnv) := ∀ TV, InstantiatedTypeEnv TV Γ -> @RawCoercionKind TV.
129 Definition CoercionEnv (Γ:TypeEnv) := list (HaskCoercionKind Γ).
130 Definition InstantiatedCoercionEnv (TV:Kind->Type) CV (Γ:TypeEnv)(Δ:CoercionEnv Γ):= vec CV (length Δ).
132 (* A (HaskXX Γ) is an XX which is valid in environments of shape Γ; they are always PHOAS-uninstantiated *)
133 Definition HaskTyVar (Γ:TypeEnv) κ := forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ.
134 Definition HaskCoVar Γ Δ := forall TV CV (env:@InstantiatedTypeEnv TV Γ)(cenv:@InstantiatedCoercionEnv TV CV Γ Δ), CV.
135 Definition HaskLevel (Γ:TypeEnv) := list (HaskTyVar Γ ECKind).
136 Definition HaskType (Γ:TypeEnv) κ := ∀ TV, @InstantiatedTypeEnv TV Γ -> RawHaskType TV κ.
137 Definition haskTyVarToType {Γ}{κ}(htv:HaskTyVar Γ κ) : HaskType Γ κ := fun TV ite => TVar (htv TV ite).
139 Inductive HaskTypeOfSomeKind (Γ:TypeEnv) :=
140 haskTypeOfSomeKind : ∀ κ, HaskType Γ κ -> HaskTypeOfSomeKind Γ.
141 Implicit Arguments haskTypeOfSomeKind [ [Γ] [κ] ].
142 Definition kindOfHaskTypeOfSomeKind {Γ}(htosk:HaskTypeOfSomeKind Γ) :=
144 haskTypeOfSomeKind κ _ => κ
146 Coercion kindOfHaskTypeOfSomeKind : HaskTypeOfSomeKind >-> Kind.
147 Definition haskTypeOfSomeKindToHaskType {Γ}(htosk:HaskTypeOfSomeKind Γ) : HaskType Γ htosk :=
148 match htosk as H return HaskType Γ H with
149 haskTypeOfSomeKind _ ht => ht
151 Coercion haskTypeOfSomeKindToHaskType : HaskTypeOfSomeKind >-> HaskType.
153 Definition HaskCoercion Γ Δ (hk:HaskCoercionKind Γ) := forall TV CV (ite:@InstantiatedTypeEnv TV Γ),
154 @InstantiatedCoercionEnv TV CV Γ Δ -> @RawHaskCoer TV CV (hk TV ite).
155 Inductive LeveledHaskType (Γ:TypeEnv) κ := mkLeveledHaskType : HaskType Γ κ -> HaskLevel Γ -> LeveledHaskType Γ κ.
157 Definition FreshHaskTyVar {Γ}(κ:Kind) : HaskTyVar (κ::Γ) κ := fun TV env => ilist_head env.
158 Definition HaskTAll {Γ}(κ:Kind)(σ:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV ★) : HaskType Γ ★
159 := fun TV env => TAll κ (σ TV env).
160 Definition HaskTApp {Γ}{κ}(σ:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV ★)
161 (cv:HaskTyVar Γ κ) : HaskType Γ ★
162 := fun TV env => σ TV env (cv TV env).
163 Definition HaskBrak {Γ}(v:HaskTyVar Γ ECKind)(t:HaskType Γ ★) : HaskType Γ ★:=
164 fun TV env => @TCode TV (TVar (v TV env)) (t TV env).
165 Definition HaskTCon {Γ}(tc:TyCon) : HaskType Γ (fold_right KindArrow ★ (tyConKind tc))
166 := fun TV ite => TCon tc.
167 Definition HaskAppT {Γ}{κ₁}{κ₂}(t1:HaskType Γ (κ₂⇛κ₁))(t2:HaskType Γ κ₂) : HaskType Γ κ₁ :=
168 fun TV ite => TApp (t1 TV ite) (t2 TV ite).
169 Definition mkHaskCoercionKind {Γ}{κ}(t1:HaskType Γ κ)(t2:HaskType Γ κ) : HaskCoercionKind Γ :=
170 fun TV ite => mkRawCoercionKind _ (t1 TV ite) (t2 TV ite).
173 Context {TV:Kind -> Type }.
174 Fixpoint flattenT {κ} (exp: RawHaskType (fun k => RawHaskType TV k) κ) : RawHaskType TV κ :=
177 | TAll _ y => TAll _ (fun v => flattenT (y (TVar v)))
178 | TApp _ _ x y => TApp (flattenT x) (flattenT y)
180 | TCoerc _ t1 t2 t => TCoerc (flattenT t1) (flattenT t2) (flattenT t)
182 | TCode v e => TCode (flattenT v) (flattenT e)
183 | TyFunApp tfc kl k lt => TyFunApp tfc kl k (flattenTyFunApp _ lt)
185 with flattenTyFunApp (lk:list Kind)(exp:@RawHaskTypeList (fun k => RawHaskType TV k) lk) : @RawHaskTypeList TV lk :=
186 match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with
187 | TyFunApp_nil => TyFunApp_nil
188 | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flattenT t) (flattenTyFunApp _ rest)
192 (* PHOAS substitution on types *)
193 Definition substT {Γ}{κ₁}{κ₂}(exp:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ₁ -> RawHaskType TV κ₂)(v:@HaskType Γ κ₁)
196 flattenT (exp (fun k => RawHaskType TV k) (ilmap (fun κ tv => TVar tv) env) (v TV env)).
198 Notation "t @@ l" := (@mkLeveledHaskType _ _ t l) (at level 20).
199 Notation "t @@@ l" := (mapOptionTree (fun t' => t' @@ l) t) (at level 20).
200 Notation "'<[' a '|-' t ']>'" := (@HaskBrak _ a t).
202 Definition unlev {Γ}{κ}(lht:LeveledHaskType Γ κ) :=
203 match lht with t@@l => t end.
205 Structure Global Γ :=
206 { glob_wv : WeakExprVar
207 ; glob_kinds : list Kind
208 ; glob_tf : IList _ (fun κ => HaskType Γ κ) glob_kinds -> HaskType Γ ★
210 Coercion glob_tf : Global >-> Funclass.
211 Coercion glob_wv : Global >-> WeakExprVar.
217 (* yeah, things are kind of messy below this point *)
220 Definition unAddKindFromInstantiatedTypeEnv {Γ:TypeEnv}{κ:Kind}{TV:Kind->Type}(ite:InstantiatedTypeEnv TV (κ::Γ))
222 Definition addKindToCoercionEnv (Γ:TypeEnv)(Δ:CoercionEnv Γ)(κ:Kind) : CoercionEnv (κ::Γ) :=
223 map (fun f => (fun TV ite => f TV (unAddKindFromInstantiatedTypeEnv ite))) Δ.
224 Definition addKindToInstantiatedTypeEnv {Γ:TypeEnv}{TV:Kind->Type}(env:InstantiatedTypeEnv TV Γ)(κ:Kind)(tv:TV κ)
225 : InstantiatedTypeEnv TV (κ::Γ) := tv::::env.
226 Definition addKindToInstantiatedCoercionEnv {Γ:TypeEnv}{Δ}{TV:Kind->Type}{CV:Type}
227 (env:InstantiatedCoercionEnv TV CV Γ Δ)(κ:Kind)(tv:TV κ)
228 : InstantiatedCoercionEnv TV CV (κ::Γ) (addKindToCoercionEnv Γ Δ κ).
230 unfold InstantiatedCoercionEnv.
231 unfold addKindToCoercionEnv.
233 rewrite <- map_preserves_length.
236 Definition coercionEnvContainsCoercion {Γ}{Δ}{TV:Kind->Type}{CV:Type}(ite:InstantiatedTypeEnv TV Γ)
237 (ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)(ck:RawCoercionKind TV)
238 := @vec_In _ _ (cv,ck) (vec_zip ice (vec_map (fun f => f TV ite) (list2vec Δ))).
239 Definition addCoercionToCoercionEnv {Γ}(Δ:CoercionEnv Γ)(κ:HaskCoercionKind Γ) : CoercionEnv Γ :=
241 Definition addCoercionToInstantiatedCoercionEnv {Γ}{Δ}{κ}{TV CV}(ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)
242 : InstantiatedCoercionEnv TV CV Γ (addCoercionToCoercionEnv Δ κ).
244 unfold addCoercionToCoercionEnv; simpl.
245 unfold InstantiatedCoercionEnv; simpl.
246 apply vec_cons; auto.
248 (* the various "weak" functions turn a HaskXX-in-Γ into a HaskXX-in-(κ::Γ) *)
249 Definition weakITE {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (κ::Γ)) : InstantiatedTypeEnv TV Γ
251 Definition weakITE' {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (app κ Γ)) : InstantiatedTypeEnv TV Γ.
252 induction κ; auto. apply IHκ. inversion ite; subst. apply X0. Defined.
253 Definition weakCE {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (κ::Γ)
254 := map (fun x => (fun tv ite => x tv (weakITE ite))) Δ.
255 Definition weakV {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (κ::Γ) κv
256 := fun TV ite => (cv' TV (weakITE ite)).
257 Definition weakV' {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (app κ Γ) κv.
258 induction κ; auto. apply weakV; auto. Defined.
259 Definition weakT {Γ:TypeEnv}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (κ::Γ) κ₂
260 := fun TV ite => lt TV (weakITE ite).
261 Definition weakL {Γ}{κ}(lt:HaskLevel Γ) : HaskLevel (κ::Γ)
263 Definition weakT' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app κ Γ) κ₂.
264 induction κ; auto. apply weakT; auto. Defined.
265 Definition weakT'' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app Γ κ) κ₂.
266 unfold HaskType in *.
267 unfold InstantiatedTypeEnv in *.
269 apply ilist_chop in X.
273 Definition lamer {a}{b}{c}{κ}(lt:HaskType (app (app a b) c) κ) : HaskType (app a (app b c)) κ.
274 rewrite <- ass_app in lt.
277 Definition weakL' {Γ}{κ}(lev:HaskLevel Γ) : HaskLevel (app κ Γ).
278 induction κ; auto. apply weakL; auto. Defined.
279 Definition weakLT {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (κ::Γ) κ₂
280 := match lt with t @@ l => weakT t @@ weakL l end.
281 Definition weakLT' {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (app κ Γ) κ₂
282 := match lt with t @@ l => weakT' t @@ weakL' l end.
283 Definition weakCE' {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (app κ Γ).
284 induction κ; auto. apply weakCE; auto. Defined.
285 Definition weakICE {Γ:TypeEnv}{κ}{Δ:CoercionEnv Γ}{TV}{CV}(ice:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ))
286 : InstantiatedCoercionEnv TV CV Γ Δ.
288 unfold InstantiatedCoercionEnv; intros.
289 unfold InstantiatedCoercionEnv in ice.
290 unfold weakCE in ice.
292 rewrite <- map_preserves_length in ice.
295 Definition weakCK {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (κ::Γ).
296 unfold HaskCoercionKind in *.
298 apply hck; clear hck.
299 inversion X; subst; auto.
301 Definition weakCK' {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (app κ Γ).
306 Definition weakCK'' {Γ}{κ}(hck:list (HaskCoercionKind Γ)) : list (HaskCoercionKind (app κ Γ)) :=
308 Definition weakCV {Γ}{Δ}{κ}(cv':HaskCoVar Γ Δ) : HaskCoVar (κ::Γ) (weakCE Δ) :=
309 fun TV CV ite ice => (cv' TV CV (weakITE ite) (weakICE ice)).
310 Definition weakF {Γ:TypeEnv}{κ}{κ₂}(f:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV κ₂) :
311 forall TV (env:@InstantiatedTypeEnv TV (κ::Γ)), TV κ -> RawHaskType TV κ₂
312 := fun TV ite tv => (f TV (weakITE ite) tv).
314 Fixpoint caseType0 {Γ}(lk:list Kind) :
315 IList _ (HaskType Γ) lk ->
316 HaskType Γ (fold_right KindArrow ★ lk) ->
318 match lk as LK return
319 IList _ (HaskType Γ) LK ->
320 HaskType Γ (fold_right KindArrow ★ LK) ->
323 | nil => fun _ ht => ht
324 | k::lk' => fun tlist ht => caseType0 lk' (ilist_tail tlist) (fun TV env => TApp (ht TV env) (ilist_head tlist TV env))
327 Definition caseType {Γ}(tc:TyCon)(atypes:IList _ (HaskType Γ) (tyConKind tc)) : HaskType Γ ★ :=
328 caseType0 (tyConKind tc) atypes (fun TV env => TCon tc).
330 (* like a GHC DataCon, but using PHOAS representation for types and coercions *)
331 Record StrongAltCon {tc:TyCon} :=
333 ; sac_altcon : WeakAltCon
334 ; sac_numExTyVars : nat
335 ; sac_numCoerVars : nat
336 ; sac_numExprVars : nat
337 ; sac_ekinds : vec Kind sac_numExTyVars
338 ; sac_kinds := app (tyConKind tc) (vec2list sac_ekinds)
339 ; sac_Γ := fun Γ => app (vec2list sac_ekinds) Γ
340 ; sac_coercions : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskCoercionKind (sac_Γ Γ)) sac_numCoerVars
341 ; sac_types : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskType (sac_Γ Γ) ★) sac_numExprVars
342 ; sac_Δ := fun Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)) Δ => app (vec2list (sac_coercions Γ atypes)) Δ
344 Coercion sac_tc : StrongAltCon >-> TyCon.
345 Coercion sac_altcon : StrongAltCon >-> WeakAltCon.
348 Definition kindOfType {Γ}{κ}(ht:@HaskType Γ κ) : ???Kind := OK κ.
350 Axiom literal_tycons_are_of_ordinary_kind : forall lit, tyConKind (haskLiteralToTyCon lit) = nil.
352 Definition literalType (lit:HaskLiteral){Γ} : HaskType Γ ★.
353 set (fun TV (ite:InstantiatedTypeEnv TV Γ) => @TCon TV (haskLiteralToTyCon lit)) as z.
354 unfold tyConKind' in z.
355 rewrite literal_tycons_are_of_ordinary_kind in z.
360 Notation "a ∼∼∼ b" := (@mkHaskCoercionKind _ _ a b) (at level 18).
363 `{EQD_VV:EqDecidable VV}{Γ}
364 (ξ:VV -> LeveledHaskType Γ ★)
366 (vt:list (VV * HaskType Γ ★))
367 : VV -> LeveledHaskType Γ ★ :=
370 | (v,τ)::tl => fun v' => if eqd_dec v v' then τ @@ lev else (update_ξ ξ lev tl) v'
373 Lemma update_ξ_lemma0 `{EQD_VV:EqDecidable VV} : forall Γ ξ (lev:HaskLevel Γ)(varstypes:list (VV*_)) v,
374 not (In v (map (@fst _ _) varstypes)) ->
375 (update_ξ ξ lev varstypes) v = ξ v.
381 destruct (eqd_dec v0 v).
396 (***************************************************************************************************)
397 (* Well-Formedness of Types and Coercions *)
398 (* also represents production "S_n:κ" of Γ because these can only wind up in Γ via rule (Type) *)
399 Inductive TypeFunctionDecl (tfc:TyCon)(vk:vec Kind tfc) : Type :=
400 mkTFD : Kind -> TypeFunctionDecl tfc vk.
404 Context {TV:Kind->Type}.
407 (* local notations *)
408 Notation "ienv '⊢ᴛy' σ : κ" := (@WellKinded_RawHaskType TV _ ienv σ κ).
409 Notation "env ∋ cv : t1 ∼ t2 : Γ : t" := (@coercionEnvContainsCoercion Γ _ TV CV t env cv (@mkRawCoercionKind _ t1 t2))
410 (at level 20, t1 at level 99, t2 at level 99, t at level 99).
411 Reserved Notation "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite"
412 (at level 20, γ at level 99, b at level 99, Δ at level 99, ite at level 99, Γ at level 99).
414 (* Figure 8, lower half *)
415 Inductive WFCoercion:forall Γ (Δ:CoercionEnv Γ),
416 @InstantiatedTypeEnv TV Γ ->
417 @InstantiatedCoercionEnv TV CV Γ Δ ->
418 @RawHaskCoer TV CV -> @RawCoercionKind TV -> Prop :=
419 | CoTVar':∀ Γ Δ t e c σ τ,
420 (@coercionEnvContainsCoercion Γ _ TV CV t e c (@mkRawCoercionKind _ σ τ)) -> e⊢ᴄᴏ CoVar c : σ ∼ τ : Δ : Γ : t
421 | CoRefl :∀ Γ Δ t e τ κ, t ⊢ᴛy τ :κ -> e⊢ᴄᴏ CoType τ : τ ∼ τ : Δ :Γ: t
422 | 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
423 | 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
424 | 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
425 | 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
427 | SComp :∀ Γ Δ t e γ n S σ τ κ,
428 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
429 | 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),
430 ListWFCo Γ Δ t e γ (map TVar (vec2list σ₁)) (map TVar (vec2list σ₂)) ->
431 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₁)) (vec2list κ) ->
432 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₂)) (vec2list κ) ->
433 e⊢ᴄᴏ CoCFApp C γ : axd_σ _ _ _ ax σ₁ ∼ axd_τ _ _ _ ax σ₂ : Δ : Γ : t
435 | WFCoAll : forall Γ Δ κ (t:InstantiatedTypeEnv TV Γ) (e:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) γ σ τ ,
436 (∀ a, e ⊢ᴄᴏ ( γ a) : ( σ a) ∼ ( τ a) : _ : _ : (t + a : κ))
437 -> weakICE e ⊢ᴄᴏ (CoAll κ γ ) : (TAll κ σ ) ∼ (TAll κ τ ) : Δ : Γ : t
438 | Comp :forall Γ Δ t e γ₁ γ₂ σ₁ σ₂ τ₁ τ₂ κ,
439 (t ⊢ᴛy TApp σ₁ σ₂:κ)->
440 (e⊢ᴄᴏ γ₁:σ₁∼τ₁:Δ:Γ:t)->
441 (e⊢ᴄᴏ γ₂:σ₂∼τ₂:Δ:Γ:t) ->
442 e⊢ᴄᴏ (CoApp γ₁ γ₂) : (TApp σ₁ σ₂) ∼ (TApp τ₁ τ₂) : Δ:Γ:t
443 | CoInst :forall Γ Δ t e σ τ κ γ (v:∀ TV, InstantiatedTypeEnv TV Γ -> RawHaskType TV),
445 (e⊢ᴄᴏ γ:HaskTAll κ σ _ t ∼ HaskTAll κ τ _ t:Δ:Γ:t) ->
446 e⊢ᴄᴏ CoAppT γ (v TV t) : substT σ v TV t ∼substT τ v TV t : Δ : Γ : t
447 with ListWFCo : forall Γ (Δ:CoercionEnv Γ),
448 @InstantiatedTypeEnv TV Γ ->
449 InstantiatedCoercionEnv TV CV Γ Δ ->
450 list (RawHaskCoer TV CV) -> list (RawHaskType TV) -> list (RawHaskType TV) -> Prop :=
451 | LWFCo_nil : ∀ Γ Δ t e , ListWFCo Γ Δ t e nil nil nil
452 | LWFCo_cons : ∀ Γ Δ t e a b c la lb lc, (e⊢ᴄᴏ a : b∼c : Δ : Γ : t )->
453 ListWFCo Γ Δ t e la lb lc -> ListWFCo Γ Δ t e (a::la) (b::lb) (c::lc)
454 where "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" := (@WFCoercion Γ Δ ite ice γ (@mkRawCoercionKind _ a b)).
457 Definition WFCCo (Γ:TypeEnv)(Δ:CoercionEnv Γ)(γ:HaskCoercion Γ Δ)(a b:HaskType Γ) :=
458 forall {TV CV:Type}(env:@InstantiatedTypeEnv TV Γ)(cenv:InstantiatedCoercionEnv TV CV Γ Δ),
459 @WFCoercion _ _ Γ Δ env cenv (γ TV CV env cenv) (@mkRawCoercionKind _ (a TV env) (b TV env)).
460 Notation "Δ '⊢ᴄᴏ' γ : a '∼' b" := (@WFCCo _ Δ γ a b).
466 (* Decidable equality on PHOAS types *)
467 Fixpoint compareT (n:nat){κ₁}(t1:@RawHaskType (fun _ => nat) κ₁){κ₂}(t2:@RawHaskType (fun _ => nat) κ₂) : bool :=
469 | TVar _ x => match t2 with TVar _ x' => if eqd_dec x x' then true else false | _ => false end
470 | TAll _ y => match t2 with TAll _ y' => compareT (S n) (y n) (y' n) | _ => false end
471 | TApp _ _ x y => match t2 with TApp _ _ x' y' => if compareT n x x' then compareT n y y' else false | _ => false end
472 | TCon tc => match t2 with TCon tc' => if eqd_dec tc tc' then true else false | _ => false end
473 | TArrow => match t2 with TArrow => true | _ => false end
474 | TCode ec t => match t2 with TCode ec' t' => if compareT n ec ec' then compareT n t t' else false | _ => false end
475 | 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
476 | TyFunApp tfc kl k lt => match t2 with TyFunApp tfc' kl' k' lt' => eqd_dec tfc tfc' && compareTL n lt lt' | _ => false end
478 with compareTL (n:nat){κ₁}(t1:@RawHaskTypeList (fun _ => nat) κ₁){κ₂}(t2:@RawHaskTypeList (fun _ => nat) κ₂) : bool :=
480 | TyFunApp_nil => match t2 with TyFunApp_nil => true | _ => false end
481 | 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
484 Fixpoint count' (lk:list Kind)(n:nat) : IList _ (fun _ => nat) lk :=
485 match lk as LK return IList _ _ LK with
487 | h::t => n::::(count' t (S n))
490 Definition compareHT Γ κ (ht1 ht2:HaskType Γ κ) :=
491 compareT (length Γ) (ht1 (fun _ => nat) (count' Γ O)) (ht2 (fun _ => nat) (count' Γ O)).
495 * This is not provable in Coq's logic because the Coq function space
496 * is "too big" - although its only definable inhabitants are Coq
497 * functions, it is not provable in Coq that all function space
498 * inhabitants are definable (i.e. there are no "exotic" inhabitants).
499 * This is actually an important feature of Coq: it lets us reason
500 * about properties of non-computable (non-recursive) functions since
501 * any property proven to hold for the entire function space will hold
502 * even for those functions. However when representing binding
503 * structure using functions we would actually prefer the smaller
504 * function-space of *definable* functions only. These two axioms
506 Axiom compareHT_decides : forall Γ κ (ht1 ht2:HaskType Γ κ),
507 if compareHT Γ κ ht1 ht2
510 Axiom compareVars : forall Γ κ (htv1 htv2:HaskTyVar Γ κ),
511 if compareHT _ _ (haskTyVarToType htv1) (haskTyVarToType htv2)
515 (* using the axioms, we can now create an EqDecidable instance for HaskType, HaskTyVar, and HaskLevel *)
516 Instance haskTypeEqDecidable Γ κ : EqDecidable (HaskType Γ κ).
517 apply Build_EqDecidable.
519 set (compareHT_decides _ _ v1 v2) as z.
520 set (compareHT Γ κ v1 v2) as q.
521 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
526 Instance haskTyVarEqDecidable Γ κ : EqDecidable (HaskTyVar Γ κ).
527 apply Build_EqDecidable.
529 set (compareVars _ _ v1 v2) as z.
530 set (compareHT Γ κ (haskTyVarToType v1) (haskTyVarToType v2)) as q.
531 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
536 Instance haskLevelEqDecidable Γ : EqDecidable (HaskLevel Γ).
537 apply Build_EqDecidable.
539 unfold HaskLevel in *.
540 apply (eqd_dec v1 v2).
547 (* ToString instance for PHOAS types *)
548 Fixpoint typeToString' (needparens:bool)(n:nat){κ}(t:RawHaskType (fun _ => nat) κ) {struct t} : string :=
550 | TVar _ v => "tv" +++ toString v
551 | TCon tc => toString tc
552 | TCoerc _ t1 t2 t => "("+++typeToString' false n t1+++"~"
553 +++typeToString' false n t2+++")=>"
554 +++typeToString' needparens n t
557 | TApp _ _ TArrow t1 =>
559 then "("+++(typeToString' true n t1)+++"->"+++(typeToString' true n t2)+++")"
560 else (typeToString' true n t1)+++"->"+++(typeToString' true n t2)
563 then "("+++(typeToString' true n t1)+++" "+++(typeToString' false n t2)+++")"
564 else (typeToString' true n t1)+++" "+++(typeToString' false n t2)
567 | TAll k f => let alpha := "tv"+++ toString n
568 in "(forall "+++ alpha +++ ":"+++ toString k +++")"+++
569 typeToString' false (S n) (f n)
570 | TCode ec t => "<["+++(typeToString' true n t)+++"]>@"+++(typeToString' false n ec)
571 | TyFunApp tfc kl k lt => toString tfc+++ "_" +++ toString n+++" ["+++
572 (fold_left (fun x y => " \ "+++x+++y) (typeList2string false n lt) "")+++"]"
574 with typeList2string (needparens:bool)(n:nat){κ}(t:RawHaskTypeList κ) {struct t} : list string :=
576 | TyFunApp_nil => nil
577 | TyFunApp_cons κ kl rhk rhkl => (typeToString' needparens n rhk)::(typeList2string needparens n rhkl)
580 Definition typeToString {Γ}{κ}(ht:HaskType Γ κ) : string :=
581 typeToString' false (length Γ) (ht (fun _ => nat) (count' Γ O)).
583 Instance TypeToStringInstance {Γ} {κ} : ToString (HaskType Γ κ) :=
584 { toString := typeToString }.