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.
213 (* From (t1->(t2->(t3-> ... t))), return t1::t2::t3::...nil *)
214 (* this is a billion times uglier than it needs to be as a result of how primitive Coq's termiation checker is *)
215 Fixpoint take_arg_types {TV}{κ}(exp: RawHaskType TV κ) {struct exp} : list (RawHaskType TV κ) :=
216 match exp as E in RawHaskType _ K return list (RawHaskType _ K) with
218 (match κ₁ as K1 return RawHaskType TV (κ₂ ⇛ K1) -> list (RawHaskType TV κ₂) -> list (RawHaskType _ K1) with
220 match κ₂ as K2 return RawHaskType TV (K2 ⇛ KindStar) -> list (RawHaskType TV K2) -> list (RawHaskType _ KindStar) with
221 | KindStar => fun x' =>
222 match x' return list (RawHaskType TV KindStar) -> list (RawHaskType _ KindStar) with
223 | TApp κ₁'' κ₂'' w'' x'' =>
224 match κ₂'' as K2'' return RawHaskType TV K2'' -> list (RawHaskType TV KindStar) ->
225 list (RawHaskType _ KindStar) with
228 | TArrow => fun a b => a::b
229 | _ => fun _ _ => nil
231 | _ => fun _ _ => nil
235 | _ => fun _ _ => nil
237 | _ => fun _ _ => nil
238 end) x (take_arg_types y)
242 (* From (t1->(t2->(t3-> ... t))), return t *)
243 (* this is a billion times uglier than it needs to be as a result of how primitive Coq's termiation checker is *)
244 Fixpoint drop_arg_types {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ :=
245 match exp as E in RawHaskType _ K return RawHaskType _ K with
248 (match κ₁ as K1 return RawHaskType TV (κ₂ ⇛ K1) -> (RawHaskType TV κ₂) -> ??(RawHaskType _ K1) with
250 match κ₂ as K2 return RawHaskType TV (K2 ⇛ KindStar) -> (RawHaskType TV K2) -> ??(RawHaskType _ KindStar) with
251 | KindStar => fun x' =>
252 match x' return (RawHaskType TV KindStar) -> ??(RawHaskType _ KindStar) with
253 | TApp κ₁'' κ₂'' w'' x'' =>
254 match κ₂'' as K2'' return RawHaskType TV K2'' -> (RawHaskType TV KindStar) -> ??(RawHaskType _ KindStar) with
257 | TArrow => fun _ b => Some b
258 | _ => fun _ b => None
260 | _ => fun _ b => None
264 | _ => fun _ _ => None
266 | _ => fun _ _ => None
267 end) x (drop_arg_types y)
278 (* yeah, things are kind of messy below this point *)
281 Definition unAddKindFromInstantiatedTypeEnv {Γ:TypeEnv}{κ:Kind}{TV:Kind->Type}(ite:InstantiatedTypeEnv TV (κ::Γ))
283 Definition addKindToCoercionEnv (Γ:TypeEnv)(Δ:CoercionEnv Γ)(κ:Kind) : CoercionEnv (κ::Γ) :=
284 map (fun f => (fun TV ite => f TV (unAddKindFromInstantiatedTypeEnv ite))) Δ.
285 Definition addKindToInstantiatedTypeEnv {Γ:TypeEnv}{TV:Kind->Type}(env:InstantiatedTypeEnv TV Γ)(κ:Kind)(tv:TV κ)
286 : InstantiatedTypeEnv TV (κ::Γ) := tv::::env.
287 Definition addKindToInstantiatedCoercionEnv {Γ:TypeEnv}{Δ}{TV:Kind->Type}{CV:Type}
288 (env:InstantiatedCoercionEnv TV CV Γ Δ)(κ:Kind)(tv:TV κ)
289 : InstantiatedCoercionEnv TV CV (κ::Γ) (addKindToCoercionEnv Γ Δ κ).
291 unfold InstantiatedCoercionEnv.
292 unfold addKindToCoercionEnv.
294 rewrite <- map_preserves_length.
297 Definition coercionEnvContainsCoercion {Γ}{Δ}{TV:Kind->Type}{CV:Type}(ite:InstantiatedTypeEnv TV Γ)
298 (ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)(ck:RawCoercionKind TV)
299 := @vec_In _ _ (cv,ck) (vec_zip ice (vec_map (fun f => f TV ite) (list2vec Δ))).
300 Definition addCoercionToCoercionEnv {Γ}(Δ:CoercionEnv Γ)(κ:HaskCoercionKind Γ) : CoercionEnv Γ :=
302 Definition addCoercionToInstantiatedCoercionEnv {Γ}{Δ}{κ}{TV CV}(ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)
303 : InstantiatedCoercionEnv TV CV Γ (addCoercionToCoercionEnv Δ κ).
305 unfold addCoercionToCoercionEnv; simpl.
306 unfold InstantiatedCoercionEnv; simpl.
307 apply vec_cons; auto.
309 (* the various "weak" functions turn a HaskXX-in-Γ into a HaskXX-in-(κ::Γ) *)
310 Definition weakITE {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (κ::Γ)) : InstantiatedTypeEnv TV Γ
312 Definition weakITE' {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (app κ Γ)) : InstantiatedTypeEnv TV Γ.
313 induction κ; auto. apply IHκ. inversion ite; subst. apply X0. Defined.
314 Definition weakCE {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (κ::Γ)
315 := map (fun x => (fun tv ite => x tv (weakITE ite))) Δ.
316 Definition weakV {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (κ::Γ) κv
317 := fun TV ite => (cv' TV (weakITE ite)).
318 Definition weakV' {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (app κ Γ) κv.
319 induction κ; auto. apply weakV; auto. Defined.
320 Definition weakT {Γ:TypeEnv}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (κ::Γ) κ₂
321 := fun TV ite => lt TV (weakITE ite).
322 Definition weakL {Γ}{κ}(lt:HaskLevel Γ) : HaskLevel (κ::Γ)
324 Definition weakT' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app κ Γ) κ₂.
325 induction κ; auto. apply weakT; auto. Defined.
326 Definition weakT'' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app Γ κ) κ₂.
327 unfold HaskType in *.
328 unfold InstantiatedTypeEnv in *.
330 apply ilist_chop in X.
334 Definition lamer {a}{b}{c}{κ}(lt:HaskType (app (app a b) c) κ) : HaskType (app a (app b c)) κ.
335 rewrite <- ass_app in lt.
338 Definition weakL' {Γ}{κ}(lev:HaskLevel Γ) : HaskLevel (app κ Γ).
339 induction κ; auto. apply weakL; auto. Defined.
340 Definition weakLT {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (κ::Γ) κ₂
341 := match lt with t @@ l => weakT t @@ weakL l end.
342 Definition weakLT' {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (app κ Γ) κ₂
343 := match lt with t @@ l => weakT' t @@ weakL' l end.
344 Definition weakCE' {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (app κ Γ).
345 induction κ; auto. apply weakCE; auto. Defined.
346 Definition weakICE {Γ:TypeEnv}{κ}{Δ:CoercionEnv Γ}{TV}{CV}(ice:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ))
347 : InstantiatedCoercionEnv TV CV Γ Δ.
349 unfold InstantiatedCoercionEnv; intros.
350 unfold InstantiatedCoercionEnv in ice.
351 unfold weakCE in ice.
353 rewrite <- map_preserves_length in ice.
356 Definition weakCK {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (κ::Γ).
357 unfold HaskCoercionKind in *.
359 apply hck; clear hck.
360 inversion X; subst; auto.
362 Definition weakCK' {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (app κ Γ).
367 Definition weakCK'' {Γ}{κ}(hck:list (HaskCoercionKind Γ)) : list (HaskCoercionKind (app κ Γ)) :=
369 Definition weakCV {Γ}{Δ}{κ}(cv':HaskCoVar Γ Δ) : HaskCoVar (κ::Γ) (weakCE Δ) :=
370 fun TV CV ite ice => (cv' TV CV (weakITE ite) (weakICE ice)).
371 Definition weakF {Γ:TypeEnv}{κ}{κ₂}(f:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV κ₂) :
372 forall TV (env:@InstantiatedTypeEnv TV (κ::Γ)), TV κ -> RawHaskType TV κ₂
373 := fun TV ite tv => (f TV (weakITE ite) tv).
375 Fixpoint caseType0 {Γ}(lk:list Kind) :
376 IList _ (HaskType Γ) lk ->
377 HaskType Γ (fold_right KindArrow ★ lk) ->
379 match lk as LK return
380 IList _ (HaskType Γ) LK ->
381 HaskType Γ (fold_right KindArrow ★ LK) ->
384 | nil => fun _ ht => ht
385 | k::lk' => fun tlist ht => caseType0 lk' (ilist_tail tlist) (fun TV env => TApp (ht TV env) (ilist_head tlist TV env))
388 Definition caseType {Γ}(tc:TyCon)(atypes:IList _ (HaskType Γ) (tyConKind tc)) : HaskType Γ ★ :=
389 caseType0 (tyConKind tc) atypes (fun TV env => TCon tc).
391 (* like a GHC DataCon, but using PHOAS representation for types and coercions *)
392 Record StrongAltCon {tc:TyCon} :=
394 ; sac_altcon : WeakAltCon
395 ; sac_numExTyVars : nat
396 ; sac_numCoerVars : nat
397 ; sac_numExprVars : nat
398 ; sac_ekinds : vec Kind sac_numExTyVars
399 ; sac_kinds := app (tyConKind tc) (vec2list sac_ekinds)
400 ; sac_Γ := fun Γ => app (vec2list sac_ekinds) Γ
401 ; sac_coercions : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskCoercionKind (sac_Γ Γ)) sac_numCoerVars
402 ; sac_types : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskType (sac_Γ Γ) ★) sac_numExprVars
403 ; sac_Δ := fun Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)) Δ => app (vec2list (sac_coercions Γ atypes)) Δ
405 Coercion sac_tc : StrongAltCon >-> TyCon.
406 Coercion sac_altcon : StrongAltCon >-> WeakAltCon.
409 Definition kindOfType {Γ}{κ}(ht:@HaskType Γ κ) : ???Kind := OK κ.
411 Axiom literal_tycons_are_of_ordinary_kind : forall lit, tyConKind (haskLiteralToTyCon lit) = nil.
413 Definition literalType (lit:HaskLiteral){Γ} : HaskType Γ ★.
414 set (fun TV (ite:InstantiatedTypeEnv TV Γ) => @TCon TV (haskLiteralToTyCon lit)) as z.
415 unfold tyConKind' in z.
416 rewrite literal_tycons_are_of_ordinary_kind in z.
421 Notation "a ∼∼∼ b" := (@mkHaskCoercionKind _ _ a b) (at level 18).
424 `{EQD_VV:EqDecidable VV}{Γ}
425 (ξ:VV -> LeveledHaskType Γ ★)
427 (vt:list (VV * HaskType Γ ★))
428 : VV -> LeveledHaskType Γ ★ :=
431 | (v,τ)::tl => fun v' => if eqd_dec v v' then τ @@ lev else (update_ξ ξ lev tl) v'
434 Lemma update_ξ_lemma0 `{EQD_VV:EqDecidable VV} : forall Γ ξ (lev:HaskLevel Γ)(varstypes:list (VV*_)) v,
435 not (In v (map (@fst _ _) varstypes)) ->
436 (update_ξ ξ lev varstypes) v = ξ v.
442 destruct (eqd_dec v0 v).
457 (***************************************************************************************************)
458 (* Well-Formedness of Types and Coercions *)
459 (* also represents production "S_n:κ" of Γ because these can only wind up in Γ via rule (Type) *)
460 Inductive TypeFunctionDecl (tfc:TyCon)(vk:vec Kind tfc) : Type :=
461 mkTFD : Kind -> TypeFunctionDecl tfc vk.
465 Context {TV:Kind->Type}.
468 (* local notations *)
469 Notation "ienv '⊢ᴛy' σ : κ" := (@WellKinded_RawHaskType TV _ ienv σ κ).
470 Notation "env ∋ cv : t1 ∼ t2 : Γ : t" := (@coercionEnvContainsCoercion Γ _ TV CV t env cv (@mkRawCoercionKind _ t1 t2))
471 (at level 20, t1 at level 99, t2 at level 99, t at level 99).
472 Reserved Notation "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite"
473 (at level 20, γ at level 99, b at level 99, Δ at level 99, ite at level 99, Γ at level 99).
475 (* Figure 8, lower half *)
476 Inductive WFCoercion:forall Γ (Δ:CoercionEnv Γ),
477 @InstantiatedTypeEnv TV Γ ->
478 @InstantiatedCoercionEnv TV CV Γ Δ ->
479 @RawHaskCoer TV CV -> @RawCoercionKind TV -> Prop :=
480 | CoTVar':∀ Γ Δ t e c σ τ,
481 (@coercionEnvContainsCoercion Γ _ TV CV t e c (@mkRawCoercionKind _ σ τ)) -> e⊢ᴄᴏ CoVar c : σ ∼ τ : Δ : Γ : t
482 | CoRefl :∀ Γ Δ t e τ κ, t ⊢ᴛy τ :κ -> e⊢ᴄᴏ CoType τ : τ ∼ τ : Δ :Γ: t
483 | 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
484 | 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
485 | 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
486 | 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
488 | SComp :∀ Γ Δ t e γ n S σ τ κ,
489 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
490 | 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),
491 ListWFCo Γ Δ t e γ (map TVar (vec2list σ₁)) (map TVar (vec2list σ₂)) ->
492 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₁)) (vec2list κ) ->
493 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₂)) (vec2list κ) ->
494 e⊢ᴄᴏ CoCFApp C γ : axd_σ _ _ _ ax σ₁ ∼ axd_τ _ _ _ ax σ₂ : Δ : Γ : t
496 | WFCoAll : forall Γ Δ κ (t:InstantiatedTypeEnv TV Γ) (e:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) γ σ τ ,
497 (∀ a, e ⊢ᴄᴏ ( γ a) : ( σ a) ∼ ( τ a) : _ : _ : (t + a : κ))
498 -> weakICE e ⊢ᴄᴏ (CoAll κ γ ) : (TAll κ σ ) ∼ (TAll κ τ ) : Δ : Γ : t
499 | Comp :forall Γ Δ t e γ₁ γ₂ σ₁ σ₂ τ₁ τ₂ κ,
500 (t ⊢ᴛy TApp σ₁ σ₂:κ)->
501 (e⊢ᴄᴏ γ₁:σ₁∼τ₁:Δ:Γ:t)->
502 (e⊢ᴄᴏ γ₂:σ₂∼τ₂:Δ:Γ:t) ->
503 e⊢ᴄᴏ (CoApp γ₁ γ₂) : (TApp σ₁ σ₂) ∼ (TApp τ₁ τ₂) : Δ:Γ:t
504 | CoInst :forall Γ Δ t e σ τ κ γ (v:∀ TV, InstantiatedTypeEnv TV Γ -> RawHaskType TV),
506 (e⊢ᴄᴏ γ:HaskTAll κ σ _ t ∼ HaskTAll κ τ _ t:Δ:Γ:t) ->
507 e⊢ᴄᴏ CoAppT γ (v TV t) : substT σ v TV t ∼substT τ v TV t : Δ : Γ : t
508 with ListWFCo : forall Γ (Δ:CoercionEnv Γ),
509 @InstantiatedTypeEnv TV Γ ->
510 InstantiatedCoercionEnv TV CV Γ Δ ->
511 list (RawHaskCoer TV CV) -> list (RawHaskType TV) -> list (RawHaskType TV) -> Prop :=
512 | LWFCo_nil : ∀ Γ Δ t e , ListWFCo Γ Δ t e nil nil nil
513 | LWFCo_cons : ∀ Γ Δ t e a b c la lb lc, (e⊢ᴄᴏ a : b∼c : Δ : Γ : t )->
514 ListWFCo Γ Δ t e la lb lc -> ListWFCo Γ Δ t e (a::la) (b::lb) (c::lc)
515 where "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" := (@WFCoercion Γ Δ ite ice γ (@mkRawCoercionKind _ a b)).
518 Definition WFCCo (Γ:TypeEnv)(Δ:CoercionEnv Γ)(γ:HaskCoercion Γ Δ)(a b:HaskType Γ) :=
519 forall {TV CV:Type}(env:@InstantiatedTypeEnv TV Γ)(cenv:InstantiatedCoercionEnv TV CV Γ Δ),
520 @WFCoercion _ _ Γ Δ env cenv (γ TV CV env cenv) (@mkRawCoercionKind _ (a TV env) (b TV env)).
521 Notation "Δ '⊢ᴄᴏ' γ : a '∼' b" := (@WFCCo _ Δ γ a b).
527 (* Decidable equality on PHOAS types *)
528 Fixpoint compareT (n:nat){κ₁}(t1:@RawHaskType (fun _ => nat) κ₁){κ₂}(t2:@RawHaskType (fun _ => nat) κ₂) : bool :=
530 | TVar _ x => match t2 with TVar _ x' => if eqd_dec x x' then true else false | _ => false end
531 | TAll _ y => match t2 with TAll _ y' => compareT (S n) (y n) (y' n) | _ => false end
532 | TApp _ _ x y => match t2 with TApp _ _ x' y' => if compareT n x x' then compareT n y y' else false | _ => false end
533 | TCon tc => match t2 with TCon tc' => if eqd_dec tc tc' then true else false | _ => false end
534 | TArrow => match t2 with TArrow => true | _ => false end
535 | TCode ec t => match t2 with TCode ec' t' => if compareT n ec ec' then compareT n t t' else false | _ => false end
536 | 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
537 | TyFunApp tfc kl k lt => match t2 with TyFunApp tfc' kl' k' lt' => eqd_dec tfc tfc' && compareTL n lt lt' | _ => false end
539 with compareTL (n:nat){κ₁}(t1:@RawHaskTypeList (fun _ => nat) κ₁){κ₂}(t2:@RawHaskTypeList (fun _ => nat) κ₂) : bool :=
541 | TyFunApp_nil => match t2 with TyFunApp_nil => true | _ => false end
542 | 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
545 Fixpoint count' (lk:list Kind)(n:nat) : IList _ (fun _ => nat) lk :=
546 match lk as LK return IList _ _ LK with
548 | h::t => n::::(count' t (S n))
551 Definition compareHT Γ κ (ht1 ht2:HaskType Γ κ) :=
552 compareT (length Γ) (ht1 (fun _ => nat) (count' Γ O)) (ht2 (fun _ => nat) (count' Γ O)).
556 * This is not provable in Coq's logic because the Coq function space
557 * is "too big" - although its only definable inhabitants are Coq
558 * functions, it is not provable in Coq that all function space
559 * inhabitants are definable (i.e. there are no "exotic" inhabitants).
560 * This is actually an important feature of Coq: it lets us reason
561 * about properties of non-computable (non-recursive) functions since
562 * any property proven to hold for the entire function space will hold
563 * even for those functions. However when representing binding
564 * structure using functions we would actually prefer the smaller
565 * function-space of *definable* functions only. These two axioms
567 Axiom compareHT_decides : forall Γ κ (ht1 ht2:HaskType Γ κ),
568 if compareHT Γ κ ht1 ht2
571 Axiom compareVars : forall Γ κ (htv1 htv2:HaskTyVar Γ κ),
572 if compareHT _ _ (haskTyVarToType htv1) (haskTyVarToType htv2)
576 (* using the axioms, we can now create an EqDecidable instance for HaskType, HaskTyVar, and HaskLevel *)
577 Instance haskTypeEqDecidable Γ κ : EqDecidable (HaskType Γ κ).
578 apply Build_EqDecidable.
580 set (compareHT_decides _ _ v1 v2) as z.
581 set (compareHT Γ κ v1 v2) as q.
582 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
587 Instance haskTyVarEqDecidable Γ κ : EqDecidable (HaskTyVar Γ κ).
588 apply Build_EqDecidable.
590 set (compareVars _ _ v1 v2) as z.
591 set (compareHT Γ κ (haskTyVarToType v1) (haskTyVarToType v2)) as q.
592 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
597 Instance haskLevelEqDecidable Γ : EqDecidable (HaskLevel Γ).
598 apply Build_EqDecidable.
600 unfold HaskLevel in *.
601 apply (eqd_dec v1 v2).
608 (* ToString instance for PHOAS types *)
609 Fixpoint typeToString' (needparens:bool)(n:nat){κ}(t:RawHaskType (fun _ => nat) κ) {struct t} : string :=
611 | TVar _ v => "tv" +++ toString v
612 | TCon tc => toString tc
613 | TCoerc _ t1 t2 t => "("+++typeToString' false n t1+++"~"
614 +++typeToString' false n t2+++")=>"
615 +++typeToString' needparens n t
618 | TApp _ _ TArrow t1 =>
620 then "("+++(typeToString' true n t1)+++"->"+++(typeToString' true n t2)+++")"
621 else (typeToString' true n t1)+++"->"+++(typeToString' true n t2)
624 then "("+++(typeToString' true n t1)+++" "+++(typeToString' false n t2)+++")"
625 else (typeToString' true n t1)+++" "+++(typeToString' false n t2)
628 | TAll k f => let alpha := "tv"+++ toString n
629 in "(forall "+++ alpha +++ ":"+++ toString k +++")"+++
630 typeToString' false (S n) (f n)
631 | TCode ec t => "<["+++(typeToString' true n t)+++"]>@"+++(typeToString' false n ec)
632 | TyFunApp tfc kl k lt => toString tfc+++ "_" +++ toString n+++" ["+++
633 (fold_left (fun x y => " \ "+++x+++y) (typeList2string false n lt) "")+++"]"
635 with typeList2string (needparens:bool)(n:nat){κ}(t:RawHaskTypeList κ) {struct t} : list string :=
637 | TyFunApp_nil => nil
638 | TyFunApp_cons κ kl rhk rhkl => (typeToString' needparens n rhk)::(typeList2string needparens n rhkl)
641 Definition typeToString {Γ}{κ}(ht:HaskType Γ κ) : string :=
642 typeToString' false (length Γ) (ht (fun _ => nat) (count' Γ O)).
644 Instance TypeToStringInstance {Γ} {κ} : ToString (HaskType Γ κ) :=
645 { toString := typeToString }.