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 Fixpoint count_arg_types {TV}{κ}(exp: RawHaskType TV κ) {struct exp} : nat :=
243 match exp as E in RawHaskType _ K return nat with
245 (match κ₁ as K1 return RawHaskType TV (κ₂ ⇛ K1) -> nat -> nat with
247 match κ₂ as K2 return RawHaskType TV (K2 ⇛ KindStar) -> nat -> nat with
248 | KindStar => fun x' =>
249 match x' return nat -> nat with
250 | TApp κ₁'' κ₂'' w'' x'' =>
251 match κ₂'' as K2'' return RawHaskType TV K2'' -> nat -> nat with
254 | TArrow => fun a b => S b
264 end) x (count_arg_types y)
268 Definition ite_unit : ∀ Γ, InstantiatedTypeEnv (fun _ => unit) Γ.
276 Definition take_arg_type {Γ}{κ}(ht:HaskType Γ κ) : (gt (count_arg_types (ht _ (ite_unit _))) 0) -> HaskType Γ κ :=
279 match take_arg_types (ht TV ite) with
280 | nil => Prelude_error "impossible"
284 (* From (t1->(t2->(t3-> ... t))), return t *)
285 (* this is a billion times uglier than it needs to be as a result of how primitive Coq's termiation checker is *)
286 Fixpoint drop_arg_types {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ :=
287 match exp as E in RawHaskType _ K return RawHaskType _ K with
290 (match κ₁ as K1 return RawHaskType TV (κ₂ ⇛ K1) -> (RawHaskType TV κ₂) -> ??(RawHaskType _ K1) with
292 match κ₂ as K2 return RawHaskType TV (K2 ⇛ KindStar) -> (RawHaskType TV K2) -> ??(RawHaskType _ KindStar) with
293 | KindStar => fun x' =>
294 match x' return (RawHaskType TV KindStar) -> ??(RawHaskType _ KindStar) with
295 | TApp κ₁'' κ₂'' w'' x'' =>
296 match κ₂'' as K2'' return RawHaskType TV K2'' -> (RawHaskType TV KindStar) -> ??(RawHaskType _ KindStar) with
299 | TArrow => fun _ b => Some b
300 | _ => fun _ b => None
302 | _ => fun _ b => None
306 | _ => fun _ _ => None
308 | _ => fun _ _ => None
309 end) x (drop_arg_types y)
320 (* yeah, things are kind of messy below this point *)
323 Definition unAddKindFromInstantiatedTypeEnv {Γ:TypeEnv}{κ:Kind}{TV:Kind->Type}(ite:InstantiatedTypeEnv TV (κ::Γ))
325 Definition addKindToCoercionEnv (Γ:TypeEnv)(Δ:CoercionEnv Γ)(κ:Kind) : CoercionEnv (κ::Γ) :=
326 map (fun f => (fun TV ite => f TV (unAddKindFromInstantiatedTypeEnv ite))) Δ.
327 Definition addKindToInstantiatedTypeEnv {Γ:TypeEnv}{TV:Kind->Type}(env:InstantiatedTypeEnv TV Γ)(κ:Kind)(tv:TV κ)
328 : InstantiatedTypeEnv TV (κ::Γ) := tv::::env.
329 Definition addKindToInstantiatedCoercionEnv {Γ:TypeEnv}{Δ}{TV:Kind->Type}{CV:Type}
330 (env:InstantiatedCoercionEnv TV CV Γ Δ)(κ:Kind)(tv:TV κ)
331 : InstantiatedCoercionEnv TV CV (κ::Γ) (addKindToCoercionEnv Γ Δ κ).
333 unfold InstantiatedCoercionEnv.
334 unfold addKindToCoercionEnv.
336 rewrite <- map_preserves_length.
339 Definition coercionEnvContainsCoercion {Γ}{Δ}{TV:Kind->Type}{CV:Type}(ite:InstantiatedTypeEnv TV Γ)
340 (ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)(ck:RawCoercionKind TV)
341 := @vec_In _ _ (cv,ck) (vec_zip ice (vec_map (fun f => f TV ite) (list2vec Δ))).
342 Definition addCoercionToCoercionEnv {Γ}(Δ:CoercionEnv Γ)(κ:HaskCoercionKind Γ) : CoercionEnv Γ :=
344 Definition addCoercionToInstantiatedCoercionEnv {Γ}{Δ}{κ}{TV CV}(ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)
345 : InstantiatedCoercionEnv TV CV Γ (addCoercionToCoercionEnv Δ κ).
347 unfold addCoercionToCoercionEnv; simpl.
348 unfold InstantiatedCoercionEnv; simpl.
349 apply vec_cons; auto.
351 (* the various "weak" functions turn a HaskXX-in-Γ into a HaskXX-in-(κ::Γ) *)
352 Definition weakITE {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (κ::Γ)) : InstantiatedTypeEnv TV Γ
354 Definition weakITE' {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (app κ Γ)) : InstantiatedTypeEnv TV Γ.
355 induction κ; auto. apply IHκ. inversion ite; subst. apply X0. Defined.
356 Definition weakCE {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (κ::Γ)
357 := map (fun x => (fun tv ite => x tv (weakITE ite))) Δ.
358 Definition weakV {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (κ::Γ) κv
359 := fun TV ite => (cv' TV (weakITE ite)).
360 Definition weakV' {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (app κ Γ) κv.
361 induction κ; auto. apply weakV; auto. Defined.
362 Definition weakT {Γ:TypeEnv}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (κ::Γ) κ₂
363 := fun TV ite => lt TV (weakITE ite).
364 Definition weakL {Γ}{κ}(lt:HaskLevel Γ) : HaskLevel (κ::Γ)
366 Definition weakT' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app κ Γ) κ₂.
367 induction κ; auto. apply weakT; auto. Defined.
368 Definition weakT'' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app Γ κ) κ₂.
369 unfold HaskType in *.
370 unfold InstantiatedTypeEnv in *.
372 apply ilist_chop in X.
376 Definition lamer {a}{b}{c}{κ}(lt:HaskType (app (app a b) c) κ) : HaskType (app a (app b c)) κ.
377 rewrite <- ass_app in lt.
380 Definition weakL' {Γ}{κ}(lev:HaskLevel Γ) : HaskLevel (app κ Γ).
381 induction κ; auto. apply weakL; auto. Defined.
382 Definition weakLT {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (κ::Γ) κ₂
383 := match lt with t @@ l => weakT t @@ weakL l end.
384 Definition weakLT' {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (app κ Γ) κ₂
385 := match lt with t @@ l => weakT' t @@ weakL' l end.
386 Definition weakCE' {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (app κ Γ).
387 induction κ; auto. apply weakCE; auto. Defined.
388 Definition weakICE {Γ:TypeEnv}{κ}{Δ:CoercionEnv Γ}{TV}{CV}(ice:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ))
389 : InstantiatedCoercionEnv TV CV Γ Δ.
391 unfold InstantiatedCoercionEnv; intros.
392 unfold InstantiatedCoercionEnv in ice.
393 unfold weakCE in ice.
395 rewrite <- map_preserves_length in ice.
398 Definition weakCK {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (κ::Γ).
399 unfold HaskCoercionKind in *.
401 apply hck; clear hck.
402 inversion X; subst; auto.
404 Definition weakCK' {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (app κ Γ).
409 Definition weakCK'' {Γ}{κ}(hck:list (HaskCoercionKind Γ)) : list (HaskCoercionKind (app κ Γ)) :=
411 Definition weakCV {Γ}{Δ}{κ}(cv':HaskCoVar Γ Δ) : HaskCoVar (κ::Γ) (weakCE Δ) :=
412 fun TV CV ite ice => (cv' TV CV (weakITE ite) (weakICE ice)).
413 Definition weakF {Γ:TypeEnv}{κ}{κ₂}(f:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV κ₂) :
414 forall TV (env:@InstantiatedTypeEnv TV (κ::Γ)), TV κ -> RawHaskType TV κ₂
415 := fun TV ite tv => (f TV (weakITE ite) tv).
417 Fixpoint caseType0 {Γ}(lk:list Kind) :
418 IList _ (HaskType Γ) lk ->
419 HaskType Γ (fold_right KindArrow ★ lk) ->
421 match lk as LK return
422 IList _ (HaskType Γ) LK ->
423 HaskType Γ (fold_right KindArrow ★ LK) ->
426 | nil => fun _ ht => ht
427 | k::lk' => fun tlist ht => caseType0 lk' (ilist_tail tlist) (fun TV env => TApp (ht TV env) (ilist_head tlist TV env))
430 Definition caseType {Γ}(tc:TyCon)(atypes:IList _ (HaskType Γ) (tyConKind tc)) : HaskType Γ ★ :=
431 caseType0 (tyConKind tc) atypes (fun TV env => TCon tc).
433 (* like a GHC DataCon, but using PHOAS representation for types and coercions *)
434 Record StrongAltCon {tc:TyCon} :=
436 ; sac_altcon : WeakAltCon
437 ; sac_numExTyVars : nat
438 ; sac_numCoerVars : nat
439 ; sac_numExprVars : nat
440 ; sac_ekinds : vec Kind sac_numExTyVars
441 ; sac_kinds := app (tyConKind tc) (vec2list sac_ekinds)
442 ; sac_Γ := fun Γ => app (vec2list sac_ekinds) Γ
443 ; sac_coercions : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskCoercionKind (sac_Γ Γ)) sac_numCoerVars
444 ; sac_types : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskType (sac_Γ Γ) ★) sac_numExprVars
445 ; sac_Δ := fun Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)) Δ => app (vec2list (sac_coercions Γ atypes)) Δ
447 Coercion sac_tc : StrongAltCon >-> TyCon.
448 Coercion sac_altcon : StrongAltCon >-> WeakAltCon.
451 Definition kindOfType {Γ}{κ}(ht:@HaskType Γ κ) : ???Kind := OK κ.
453 Axiom literal_tycons_are_of_ordinary_kind : forall lit, tyConKind (haskLiteralToTyCon lit) = nil.
455 Definition literalType (lit:HaskLiteral){Γ} : HaskType Γ ★.
456 set (fun TV (ite:InstantiatedTypeEnv TV Γ) => @TCon TV (haskLiteralToTyCon lit)) as z.
457 unfold tyConKind' in z.
458 rewrite literal_tycons_are_of_ordinary_kind in z.
463 Notation "a ∼∼∼ b" := (@mkHaskCoercionKind _ _ a b) (at level 18).
466 `{EQD_VV:EqDecidable VV}{Γ}
467 (ξ:VV -> LeveledHaskType Γ ★)
469 (vt:list (VV * HaskType Γ ★))
470 : VV -> LeveledHaskType Γ ★ :=
473 | (v,τ)::tl => fun v' => if eqd_dec v v' then τ @@ lev else (update_ξ ξ lev tl) v'
476 Lemma update_ξ_lemma0 `{EQD_VV:EqDecidable VV} : forall Γ ξ (lev:HaskLevel Γ)(varstypes:list (VV*_)) v,
477 not (In v (map (@fst _ _) varstypes)) ->
478 (update_ξ ξ lev varstypes) v = ξ v.
484 destruct (eqd_dec v0 v).
499 (***************************************************************************************************)
500 (* Well-Formedness of Types and Coercions *)
501 (* also represents production "S_n:κ" of Γ because these can only wind up in Γ via rule (Type) *)
502 Inductive TypeFunctionDecl (tfc:TyCon)(vk:vec Kind tfc) : Type :=
503 mkTFD : Kind -> TypeFunctionDecl tfc vk.
507 Context {TV:Kind->Type}.
510 (* local notations *)
511 Notation "ienv '⊢ᴛy' σ : κ" := (@WellKinded_RawHaskType TV _ ienv σ κ).
512 Notation "env ∋ cv : t1 ∼ t2 : Γ : t" := (@coercionEnvContainsCoercion Γ _ TV CV t env cv (@mkRawCoercionKind _ t1 t2))
513 (at level 20, t1 at level 99, t2 at level 99, t at level 99).
514 Reserved Notation "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite"
515 (at level 20, γ at level 99, b at level 99, Δ at level 99, ite at level 99, Γ at level 99).
517 (* Figure 8, lower half *)
518 Inductive WFCoercion:forall Γ (Δ:CoercionEnv Γ),
519 @InstantiatedTypeEnv TV Γ ->
520 @InstantiatedCoercionEnv TV CV Γ Δ ->
521 @RawHaskCoer TV CV -> @RawCoercionKind TV -> Prop :=
522 | CoTVar':∀ Γ Δ t e c σ τ,
523 (@coercionEnvContainsCoercion Γ _ TV CV t e c (@mkRawCoercionKind _ σ τ)) -> e⊢ᴄᴏ CoVar c : σ ∼ τ : Δ : Γ : t
524 | CoRefl :∀ Γ Δ t e τ κ, t ⊢ᴛy τ :κ -> e⊢ᴄᴏ CoType τ : τ ∼ τ : Δ :Γ: t
525 | 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
526 | 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
527 | 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
528 | 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
530 | SComp :∀ Γ Δ t e γ n S σ τ κ,
531 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
532 | 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),
533 ListWFCo Γ Δ t e γ (map TVar (vec2list σ₁)) (map TVar (vec2list σ₂)) ->
534 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₁)) (vec2list κ) ->
535 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₂)) (vec2list κ) ->
536 e⊢ᴄᴏ CoCFApp C γ : axd_σ _ _ _ ax σ₁ ∼ axd_τ _ _ _ ax σ₂ : Δ : Γ : t
538 | WFCoAll : forall Γ Δ κ (t:InstantiatedTypeEnv TV Γ) (e:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) γ σ τ ,
539 (∀ a, e ⊢ᴄᴏ ( γ a) : ( σ a) ∼ ( τ a) : _ : _ : (t + a : κ))
540 -> weakICE e ⊢ᴄᴏ (CoAll κ γ ) : (TAll κ σ ) ∼ (TAll κ τ ) : Δ : Γ : t
541 | Comp :forall Γ Δ t e γ₁ γ₂ σ₁ σ₂ τ₁ τ₂ κ,
542 (t ⊢ᴛy TApp σ₁ σ₂:κ)->
543 (e⊢ᴄᴏ γ₁:σ₁∼τ₁:Δ:Γ:t)->
544 (e⊢ᴄᴏ γ₂:σ₂∼τ₂:Δ:Γ:t) ->
545 e⊢ᴄᴏ (CoApp γ₁ γ₂) : (TApp σ₁ σ₂) ∼ (TApp τ₁ τ₂) : Δ:Γ:t
546 | CoInst :forall Γ Δ t e σ τ κ γ (v:∀ TV, InstantiatedTypeEnv TV Γ -> RawHaskType TV),
548 (e⊢ᴄᴏ γ:HaskTAll κ σ _ t ∼ HaskTAll κ τ _ t:Δ:Γ:t) ->
549 e⊢ᴄᴏ CoAppT γ (v TV t) : substT σ v TV t ∼substT τ v TV t : Δ : Γ : t
550 with ListWFCo : forall Γ (Δ:CoercionEnv Γ),
551 @InstantiatedTypeEnv TV Γ ->
552 InstantiatedCoercionEnv TV CV Γ Δ ->
553 list (RawHaskCoer TV CV) -> list (RawHaskType TV) -> list (RawHaskType TV) -> Prop :=
554 | LWFCo_nil : ∀ Γ Δ t e , ListWFCo Γ Δ t e nil nil nil
555 | LWFCo_cons : ∀ Γ Δ t e a b c la lb lc, (e⊢ᴄᴏ a : b∼c : Δ : Γ : t )->
556 ListWFCo Γ Δ t e la lb lc -> ListWFCo Γ Δ t e (a::la) (b::lb) (c::lc)
557 where "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" := (@WFCoercion Γ Δ ite ice γ (@mkRawCoercionKind _ a b)).
560 Definition WFCCo (Γ:TypeEnv)(Δ:CoercionEnv Γ)(γ:HaskCoercion Γ Δ)(a b:HaskType Γ) :=
561 forall {TV CV:Type}(env:@InstantiatedTypeEnv TV Γ)(cenv:InstantiatedCoercionEnv TV CV Γ Δ),
562 @WFCoercion _ _ Γ Δ env cenv (γ TV CV env cenv) (@mkRawCoercionKind _ (a TV env) (b TV env)).
563 Notation "Δ '⊢ᴄᴏ' γ : a '∼' b" := (@WFCCo _ Δ γ a b).
569 (* Decidable equality on PHOAS types *)
570 Fixpoint compareT (n:nat){κ₁}(t1:@RawHaskType (fun _ => nat) κ₁){κ₂}(t2:@RawHaskType (fun _ => nat) κ₂) : bool :=
572 | TVar _ x => match t2 with TVar _ x' => if eqd_dec x x' then true else false | _ => false end
573 | TAll _ y => match t2 with TAll _ y' => compareT (S n) (y n) (y' n) | _ => false end
574 | TApp _ _ x y => match t2 with TApp _ _ x' y' => if compareT n x x' then compareT n y y' else false | _ => false end
575 | TCon tc => match t2 with TCon tc' => if eqd_dec tc tc' then true else false | _ => false end
576 | TArrow => match t2 with TArrow => true | _ => false end
577 | TCode ec t => match t2 with TCode ec' t' => if compareT n ec ec' then compareT n t t' else false | _ => false end
578 | 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
579 | TyFunApp tfc kl k lt => match t2 with TyFunApp tfc' kl' k' lt' => eqd_dec tfc tfc' && compareTL n lt lt' | _ => false end
581 with compareTL (n:nat){κ₁}(t1:@RawHaskTypeList (fun _ => nat) κ₁){κ₂}(t2:@RawHaskTypeList (fun _ => nat) κ₂) : bool :=
583 | TyFunApp_nil => match t2 with TyFunApp_nil => true | _ => false end
584 | 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
587 Fixpoint count' (lk:list Kind)(n:nat) : IList _ (fun _ => nat) lk :=
588 match lk as LK return IList _ _ LK with
590 | h::t => n::::(count' t (S n))
593 Definition compareHT Γ κ (ht1 ht2:HaskType Γ κ) :=
594 compareT (length Γ) (ht1 (fun _ => nat) (count' Γ O)) (ht2 (fun _ => nat) (count' Γ O)).
598 * This is not provable in Coq's logic because the Coq function space
599 * is "too big" - although its only definable inhabitants are Coq
600 * functions, it is not provable in Coq that all function space
601 * inhabitants are definable (i.e. there are no "exotic" inhabitants).
602 * This is actually an important feature of Coq: it lets us reason
603 * about properties of non-computable (non-recursive) functions since
604 * any property proven to hold for the entire function space will hold
605 * even for those functions. However when representing binding
606 * structure using functions we would actually prefer the smaller
607 * function-space of *definable* functions only. These two axioms
609 Axiom compareHT_decides : forall Γ κ (ht1 ht2:HaskType Γ κ),
610 if compareHT Γ κ ht1 ht2
613 Axiom compareVars : forall Γ κ (htv1 htv2:HaskTyVar Γ κ),
614 if compareHT _ _ (haskTyVarToType htv1) (haskTyVarToType htv2)
618 (* using the axioms, we can now create an EqDecidable instance for HaskType, HaskTyVar, and HaskLevel *)
619 Instance haskTypeEqDecidable Γ κ : EqDecidable (HaskType Γ κ).
620 apply Build_EqDecidable.
622 set (compareHT_decides _ _ v1 v2) as z.
623 set (compareHT Γ κ v1 v2) as q.
624 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
629 Instance haskTyVarEqDecidable Γ κ : EqDecidable (HaskTyVar Γ κ).
630 apply Build_EqDecidable.
632 set (compareVars _ _ v1 v2) as z.
633 set (compareHT Γ κ (haskTyVarToType v1) (haskTyVarToType v2)) as q.
634 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
639 Instance haskLevelEqDecidable Γ : EqDecidable (HaskLevel Γ).
640 apply Build_EqDecidable.
642 unfold HaskLevel in *.
643 apply (eqd_dec v1 v2).
650 (* ToString instance for PHOAS types *)
651 Fixpoint typeToString' (needparens:bool)(n:nat){κ}(t:RawHaskType (fun _ => nat) κ) {struct t} : string :=
653 | TVar _ v => "tv" +++ toString v
654 | TCon tc => toString tc
655 | TCoerc _ t1 t2 t => "("+++typeToString' false n t1+++"~"
656 +++typeToString' false n t2+++")=>"
657 +++typeToString' needparens n t
660 | TApp _ _ TArrow t1 =>
662 then "("+++(typeToString' true n t1)+++"->"+++(typeToString' true n t2)+++")"
663 else (typeToString' true n t1)+++"->"+++(typeToString' true n t2)
666 then "("+++(typeToString' true n t1)+++" "+++(typeToString' false n t2)+++")"
667 else (typeToString' true n t1)+++" "+++(typeToString' false n t2)
670 | TAll k f => let alpha := "tv"+++ toString n
671 in "(forall "+++ alpha +++ ":"+++ toString k +++")"+++
672 typeToString' false (S n) (f n)
673 | TCode ec t => "<["+++(typeToString' true n t)+++"]>@"+++(typeToString' false n ec)
674 | TyFunApp tfc kl k lt => toString tfc+++ "_" +++ toString n+++" ["+++
675 (fold_left (fun x y => " \ "+++x+++y) (typeList2string false n lt) "")+++"]"
677 with typeList2string (needparens:bool)(n:nat){κ}(t:RawHaskTypeList κ) {struct t} : list string :=
679 | TyFunApp_nil => nil
680 | TyFunApp_cons κ kl rhk rhkl => (typeToString' needparens n rhk)::(typeList2string needparens n rhkl)
683 Definition typeToString {Γ}{κ}(ht:HaskType Γ κ) : string :=
684 typeToString' false (length Γ) (ht (fun _ => nat) (count' Γ O)).
686 Instance TypeToStringInstance {Γ} {κ} : ToString (HaskType Γ κ) :=
687 { toString := typeToString }.