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 Definition take_arg_types : forall {TV}{κ}(exp: RawHaskType TV κ), list (RawHaskType TV κ).
216 refine (fix take_arg_types {TV}{κ}(exp: RawHaskType TV κ) {struct exp} : list (RawHaskType TV κ) :=
217 match exp as E in RawHaskType _ K return κ=K -> list (RawHaskType _ K) with
220 ((fun q:list (RawHaskType TV κ₂) =>
221 match x as X in RawHaskType _ KX return κ₂ ⇛ κ₁ = KX -> list (RawHaskType _ _) with
222 | TApp κ₁' κ₂' x' y' =>
223 fun eqpf' => match x' in RawHaskType _ KX' return (κ₂' ⇛ κ₁') = KX' -> _ with
224 | TArrow => fun eqpf'' => _
229 (take_arg_types TV _ y))
232 subst; inversion eqpf''; subst.
236 (* From (t1->(t2->(t3-> ... t))), return t *)
237 (* this is a billion times uglier than it needs to be as a result of how primitive Coq's termiation checker is *)
238 Definition drop_arg_types : forall {TV}{κ}(exp: RawHaskType TV κ), RawHaskType TV κ.
239 refine (fix drop_arg_types {TV}{κ}(exp: RawHaskType TV κ) {struct exp} : RawHaskType TV κ :=
240 match exp as E in RawHaskType _ K return κ=K -> RawHaskType _ K with
243 ((fun q:RawHaskType TV κ₂ =>
244 match x as X in RawHaskType _ KX return κ₂ ⇛ κ₁ = KX -> RawHaskType _ _ with
245 | TApp κ₁' κ₂' x' y' =>
246 fun eqpf' => match x' in RawHaskType _ KX' return (κ₂' ⇛ κ₁') = KX' -> _ with
247 | TArrow => fun eqpf'' => _
248 | z => fun _ => TApp x y
250 | z => fun _ => TApp x y
252 (drop_arg_types TV _ y))
255 subst; inversion eqpf''; subst.
262 (* yeah, things are kind of messy below this point *)
265 Definition unAddKindFromInstantiatedTypeEnv {Γ:TypeEnv}{κ:Kind}{TV:Kind->Type}(ite:InstantiatedTypeEnv TV (κ::Γ))
267 Definition addKindToCoercionEnv (Γ:TypeEnv)(Δ:CoercionEnv Γ)(κ:Kind) : CoercionEnv (κ::Γ) :=
268 map (fun f => (fun TV ite => f TV (unAddKindFromInstantiatedTypeEnv ite))) Δ.
269 Definition addKindToInstantiatedTypeEnv {Γ:TypeEnv}{TV:Kind->Type}(env:InstantiatedTypeEnv TV Γ)(κ:Kind)(tv:TV κ)
270 : InstantiatedTypeEnv TV (κ::Γ) := tv::::env.
271 Definition addKindToInstantiatedCoercionEnv {Γ:TypeEnv}{Δ}{TV:Kind->Type}{CV:Type}
272 (env:InstantiatedCoercionEnv TV CV Γ Δ)(κ:Kind)(tv:TV κ)
273 : InstantiatedCoercionEnv TV CV (κ::Γ) (addKindToCoercionEnv Γ Δ κ).
275 unfold InstantiatedCoercionEnv.
276 unfold addKindToCoercionEnv.
278 rewrite <- map_preserves_length.
281 Definition coercionEnvContainsCoercion {Γ}{Δ}{TV:Kind->Type}{CV:Type}(ite:InstantiatedTypeEnv TV Γ)
282 (ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)(ck:RawCoercionKind TV)
283 := @vec_In _ _ (cv,ck) (vec_zip ice (vec_map (fun f => f TV ite) (list2vec Δ))).
284 Definition addCoercionToCoercionEnv {Γ}(Δ:CoercionEnv Γ)(κ:HaskCoercionKind Γ) : CoercionEnv Γ :=
286 Definition addCoercionToInstantiatedCoercionEnv {Γ}{Δ}{κ}{TV CV}(ice:InstantiatedCoercionEnv TV CV Γ Δ)(cv:CV)
287 : InstantiatedCoercionEnv TV CV Γ (addCoercionToCoercionEnv Δ κ).
289 unfold addCoercionToCoercionEnv; simpl.
290 unfold InstantiatedCoercionEnv; simpl.
291 apply vec_cons; auto.
293 (* the various "weak" functions turn a HaskXX-in-Γ into a HaskXX-in-(κ::Γ) *)
294 Definition weakITE {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (κ::Γ)) : InstantiatedTypeEnv TV Γ
296 Definition weakITE' {Γ:TypeEnv}{κ}{TV}(ite:InstantiatedTypeEnv TV (app κ Γ)) : InstantiatedTypeEnv TV Γ.
297 induction κ; auto. apply IHκ. inversion ite; subst. apply X0. Defined.
298 Definition weakCE {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (κ::Γ)
299 := map (fun x => (fun tv ite => x tv (weakITE ite))) Δ.
300 Definition weakV {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (κ::Γ) κv
301 := fun TV ite => (cv' TV (weakITE ite)).
302 Definition weakV' {Γ:TypeEnv}{κ}{κv}(cv':HaskTyVar Γ κv) : HaskTyVar (app κ Γ) κv.
303 induction κ; auto. apply weakV; auto. Defined.
304 Definition weakT {Γ:TypeEnv}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (κ::Γ) κ₂
305 := fun TV ite => lt TV (weakITE ite).
306 Definition weakL {Γ}{κ}(lt:HaskLevel Γ) : HaskLevel (κ::Γ)
308 Definition weakT' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app κ Γ) κ₂.
309 induction κ; auto. apply weakT; auto. Defined.
310 Definition weakT'' {Γ}{κ}{κ₂}(lt:HaskType Γ κ₂) : HaskType (app Γ κ) κ₂.
311 unfold HaskType in *.
312 unfold InstantiatedTypeEnv in *.
314 apply ilist_chop in X.
318 Definition lamer {a}{b}{c}{κ}(lt:HaskType (app (app a b) c) κ) : HaskType (app a (app b c)) κ.
319 rewrite <- ass_app in lt.
322 Definition weakL' {Γ}{κ}(lev:HaskLevel Γ) : HaskLevel (app κ Γ).
323 induction κ; auto. apply weakL; auto. Defined.
324 Definition weakLT {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (κ::Γ) κ₂
325 := match lt with t @@ l => weakT t @@ weakL l end.
326 Definition weakLT' {Γ}{κ}{κ₂}(lt:LeveledHaskType Γ κ₂) : LeveledHaskType (app κ Γ) κ₂
327 := match lt with t @@ l => weakT' t @@ weakL' l end.
328 Definition weakCE' {Γ:TypeEnv}{κ}(Δ:CoercionEnv Γ) : CoercionEnv (app κ Γ).
329 induction κ; auto. apply weakCE; auto. Defined.
330 Definition weakICE {Γ:TypeEnv}{κ}{Δ:CoercionEnv Γ}{TV}{CV}(ice:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ))
331 : InstantiatedCoercionEnv TV CV Γ Δ.
333 unfold InstantiatedCoercionEnv; intros.
334 unfold InstantiatedCoercionEnv in ice.
335 unfold weakCE in ice.
337 rewrite <- map_preserves_length in ice.
340 Definition weakCK {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (κ::Γ).
341 unfold HaskCoercionKind in *.
343 apply hck; clear hck.
344 inversion X; subst; auto.
346 Definition weakCK' {Γ}{κ}(hck:HaskCoercionKind Γ) : HaskCoercionKind (app κ Γ).
351 Definition weakCK'' {Γ}{κ}(hck:list (HaskCoercionKind Γ)) : list (HaskCoercionKind (app κ Γ)) :=
353 Definition weakCV {Γ}{Δ}{κ}(cv':HaskCoVar Γ Δ) : HaskCoVar (κ::Γ) (weakCE Δ) :=
354 fun TV CV ite ice => (cv' TV CV (weakITE ite) (weakICE ice)).
355 Definition weakF {Γ:TypeEnv}{κ}{κ₂}(f:forall TV (env:@InstantiatedTypeEnv TV Γ), TV κ -> RawHaskType TV κ₂) :
356 forall TV (env:@InstantiatedTypeEnv TV (κ::Γ)), TV κ -> RawHaskType TV κ₂
357 := fun TV ite tv => (f TV (weakITE ite) tv).
359 Fixpoint caseType0 {Γ}(lk:list Kind) :
360 IList _ (HaskType Γ) lk ->
361 HaskType Γ (fold_right KindArrow ★ lk) ->
363 match lk as LK return
364 IList _ (HaskType Γ) LK ->
365 HaskType Γ (fold_right KindArrow ★ LK) ->
368 | nil => fun _ ht => ht
369 | k::lk' => fun tlist ht => caseType0 lk' (ilist_tail tlist) (fun TV env => TApp (ht TV env) (ilist_head tlist TV env))
372 Definition caseType {Γ}(tc:TyCon)(atypes:IList _ (HaskType Γ) (tyConKind tc)) : HaskType Γ ★ :=
373 caseType0 (tyConKind tc) atypes (fun TV env => TCon tc).
375 (* like a GHC DataCon, but using PHOAS representation for types and coercions *)
376 Record StrongAltCon {tc:TyCon} :=
378 ; sac_altcon : WeakAltCon
379 ; sac_numExTyVars : nat
380 ; sac_numCoerVars : nat
381 ; sac_numExprVars : nat
382 ; sac_ekinds : vec Kind sac_numExTyVars
383 ; sac_kinds := app (tyConKind tc) (vec2list sac_ekinds)
384 ; sac_Γ := fun Γ => app (vec2list sac_ekinds) Γ
385 ; sac_coercions : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskCoercionKind (sac_Γ Γ)) sac_numCoerVars
386 ; sac_types : forall Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)), vec (HaskType (sac_Γ Γ) ★) sac_numExprVars
387 ; sac_Δ := fun Γ (atypes:IList _ (HaskType Γ) (tyConKind tc)) Δ => app (vec2list (sac_coercions Γ atypes)) Δ
389 Coercion sac_tc : StrongAltCon >-> TyCon.
390 Coercion sac_altcon : StrongAltCon >-> WeakAltCon.
393 Definition kindOfType {Γ}{κ}(ht:@HaskType Γ κ) : ???Kind := OK κ.
395 Axiom literal_tycons_are_of_ordinary_kind : forall lit, tyConKind (haskLiteralToTyCon lit) = nil.
397 Definition literalType (lit:HaskLiteral){Γ} : HaskType Γ ★.
398 set (fun TV (ite:InstantiatedTypeEnv TV Γ) => @TCon TV (haskLiteralToTyCon lit)) as z.
399 unfold tyConKind' in z.
400 rewrite literal_tycons_are_of_ordinary_kind in z.
405 Notation "a ∼∼∼ b" := (@mkHaskCoercionKind _ _ a b) (at level 18).
408 `{EQD_VV:EqDecidable VV}{Γ}
409 (ξ:VV -> LeveledHaskType Γ ★)
411 (vt:list (VV * HaskType Γ ★))
412 : VV -> LeveledHaskType Γ ★ :=
415 | (v,τ)::tl => fun v' => if eqd_dec v v' then τ @@ lev else (update_ξ ξ lev tl) v'
418 Lemma update_ξ_lemma0 `{EQD_VV:EqDecidable VV} : forall Γ ξ (lev:HaskLevel Γ)(varstypes:list (VV*_)) v,
419 not (In v (map (@fst _ _) varstypes)) ->
420 (update_ξ ξ lev varstypes) v = ξ v.
426 destruct (eqd_dec v0 v).
441 (***************************************************************************************************)
442 (* Well-Formedness of Types and Coercions *)
443 (* also represents production "S_n:κ" of Γ because these can only wind up in Γ via rule (Type) *)
444 Inductive TypeFunctionDecl (tfc:TyCon)(vk:vec Kind tfc) : Type :=
445 mkTFD : Kind -> TypeFunctionDecl tfc vk.
449 Context {TV:Kind->Type}.
452 (* local notations *)
453 Notation "ienv '⊢ᴛy' σ : κ" := (@WellKinded_RawHaskType TV _ ienv σ κ).
454 Notation "env ∋ cv : t1 ∼ t2 : Γ : t" := (@coercionEnvContainsCoercion Γ _ TV CV t env cv (@mkRawCoercionKind _ t1 t2))
455 (at level 20, t1 at level 99, t2 at level 99, t at level 99).
456 Reserved Notation "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite"
457 (at level 20, γ at level 99, b at level 99, Δ at level 99, ite at level 99, Γ at level 99).
459 (* Figure 8, lower half *)
460 Inductive WFCoercion:forall Γ (Δ:CoercionEnv Γ),
461 @InstantiatedTypeEnv TV Γ ->
462 @InstantiatedCoercionEnv TV CV Γ Δ ->
463 @RawHaskCoer TV CV -> @RawCoercionKind TV -> Prop :=
464 | CoTVar':∀ Γ Δ t e c σ τ,
465 (@coercionEnvContainsCoercion Γ _ TV CV t e c (@mkRawCoercionKind _ σ τ)) -> e⊢ᴄᴏ CoVar c : σ ∼ τ : Δ : Γ : t
466 | CoRefl :∀ Γ Δ t e τ κ, t ⊢ᴛy τ :κ -> e⊢ᴄᴏ CoType τ : τ ∼ τ : Δ :Γ: t
467 | 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
468 | 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
469 | 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
470 | 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
472 | SComp :∀ Γ Δ t e γ n S σ τ κ,
473 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
474 | 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),
475 ListWFCo Γ Δ t e γ (map TVar (vec2list σ₁)) (map TVar (vec2list σ₂)) ->
476 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₁)) (vec2list κ) ->
477 ListWellKinded_RawHaskType TV Γ t (map TVar (vec2list σ₂)) (vec2list κ) ->
478 e⊢ᴄᴏ CoCFApp C γ : axd_σ _ _ _ ax σ₁ ∼ axd_τ _ _ _ ax σ₂ : Δ : Γ : t
480 | WFCoAll : forall Γ Δ κ (t:InstantiatedTypeEnv TV Γ) (e:InstantiatedCoercionEnv TV CV (κ::Γ) (weakCE Δ)) γ σ τ ,
481 (∀ a, e ⊢ᴄᴏ ( γ a) : ( σ a) ∼ ( τ a) : _ : _ : (t + a : κ))
482 -> weakICE e ⊢ᴄᴏ (CoAll κ γ ) : (TAll κ σ ) ∼ (TAll κ τ ) : Δ : Γ : t
483 | Comp :forall Γ Δ t e γ₁ γ₂ σ₁ σ₂ τ₁ τ₂ κ,
484 (t ⊢ᴛy TApp σ₁ σ₂:κ)->
485 (e⊢ᴄᴏ γ₁:σ₁∼τ₁:Δ:Γ:t)->
486 (e⊢ᴄᴏ γ₂:σ₂∼τ₂:Δ:Γ:t) ->
487 e⊢ᴄᴏ (CoApp γ₁ γ₂) : (TApp σ₁ σ₂) ∼ (TApp τ₁ τ₂) : Δ:Γ:t
488 | CoInst :forall Γ Δ t e σ τ κ γ (v:∀ TV, InstantiatedTypeEnv TV Γ -> RawHaskType TV),
490 (e⊢ᴄᴏ γ:HaskTAll κ σ _ t ∼ HaskTAll κ τ _ t:Δ:Γ:t) ->
491 e⊢ᴄᴏ CoAppT γ (v TV t) : substT σ v TV t ∼substT τ v TV t : Δ : Γ : t
492 with ListWFCo : forall Γ (Δ:CoercionEnv Γ),
493 @InstantiatedTypeEnv TV Γ ->
494 InstantiatedCoercionEnv TV CV Γ Δ ->
495 list (RawHaskCoer TV CV) -> list (RawHaskType TV) -> list (RawHaskType TV) -> Prop :=
496 | LWFCo_nil : ∀ Γ Δ t e , ListWFCo Γ Δ t e nil nil nil
497 | LWFCo_cons : ∀ Γ Δ t e a b c la lb lc, (e⊢ᴄᴏ a : b∼c : Δ : Γ : t )->
498 ListWFCo Γ Δ t e la lb lc -> ListWFCo Γ Δ t e (a::la) (b::lb) (c::lc)
499 where "ice '⊢ᴄᴏ' γ : a '∼' b : Δ : Γ : ite" := (@WFCoercion Γ Δ ite ice γ (@mkRawCoercionKind _ a b)).
502 Definition WFCCo (Γ:TypeEnv)(Δ:CoercionEnv Γ)(γ:HaskCoercion Γ Δ)(a b:HaskType Γ) :=
503 forall {TV CV:Type}(env:@InstantiatedTypeEnv TV Γ)(cenv:InstantiatedCoercionEnv TV CV Γ Δ),
504 @WFCoercion _ _ Γ Δ env cenv (γ TV CV env cenv) (@mkRawCoercionKind _ (a TV env) (b TV env)).
505 Notation "Δ '⊢ᴄᴏ' γ : a '∼' b" := (@WFCCo _ Δ γ a b).
511 (* Decidable equality on PHOAS types *)
512 Fixpoint compareT (n:nat){κ₁}(t1:@RawHaskType (fun _ => nat) κ₁){κ₂}(t2:@RawHaskType (fun _ => nat) κ₂) : bool :=
514 | TVar _ x => match t2 with TVar _ x' => if eqd_dec x x' then true else false | _ => false end
515 | TAll _ y => match t2 with TAll _ y' => compareT (S n) (y n) (y' n) | _ => false end
516 | TApp _ _ x y => match t2 with TApp _ _ x' y' => if compareT n x x' then compareT n y y' else false | _ => false end
517 | TCon tc => match t2 with TCon tc' => if eqd_dec tc tc' then true else false | _ => false end
518 | TArrow => match t2 with TArrow => true | _ => false end
519 | TCode ec t => match t2 with TCode ec' t' => if compareT n ec ec' then compareT n t t' else false | _ => false end
520 | 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
521 | TyFunApp tfc kl k lt => match t2 with TyFunApp tfc' kl' k' lt' => eqd_dec tfc tfc' && compareTL n lt lt' | _ => false end
523 with compareTL (n:nat){κ₁}(t1:@RawHaskTypeList (fun _ => nat) κ₁){κ₂}(t2:@RawHaskTypeList (fun _ => nat) κ₂) : bool :=
525 | TyFunApp_nil => match t2 with TyFunApp_nil => true | _ => false end
526 | 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
529 Fixpoint count' (lk:list Kind)(n:nat) : IList _ (fun _ => nat) lk :=
530 match lk as LK return IList _ _ LK with
532 | h::t => n::::(count' t (S n))
535 Definition compareHT Γ κ (ht1 ht2:HaskType Γ κ) :=
536 compareT (length Γ) (ht1 (fun _ => nat) (count' Γ O)) (ht2 (fun _ => nat) (count' Γ O)).
540 * This is not provable in Coq's logic because the Coq function space
541 * is "too big" - although its only definable inhabitants are Coq
542 * functions, it is not provable in Coq that all function space
543 * inhabitants are definable (i.e. there are no "exotic" inhabitants).
544 * This is actually an important feature of Coq: it lets us reason
545 * about properties of non-computable (non-recursive) functions since
546 * any property proven to hold for the entire function space will hold
547 * even for those functions. However when representing binding
548 * structure using functions we would actually prefer the smaller
549 * function-space of *definable* functions only. These two axioms
551 Axiom compareHT_decides : forall Γ κ (ht1 ht2:HaskType Γ κ),
552 if compareHT Γ κ ht1 ht2
555 Axiom compareVars : forall Γ κ (htv1 htv2:HaskTyVar Γ κ),
556 if compareHT _ _ (haskTyVarToType htv1) (haskTyVarToType htv2)
560 (* using the axioms, we can now create an EqDecidable instance for HaskType, HaskTyVar, and HaskLevel *)
561 Instance haskTypeEqDecidable Γ κ : EqDecidable (HaskType Γ κ).
562 apply Build_EqDecidable.
564 set (compareHT_decides _ _ v1 v2) as z.
565 set (compareHT Γ κ v1 v2) as q.
566 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
571 Instance haskTyVarEqDecidable Γ κ : EqDecidable (HaskTyVar Γ κ).
572 apply Build_EqDecidable.
574 set (compareVars _ _ v1 v2) as z.
575 set (compareHT Γ κ (haskTyVarToType v1) (haskTyVarToType v2)) as q.
576 destruct q as [ ] _eqn; unfold q in *; rewrite Heqb in *.
581 Instance haskLevelEqDecidable Γ : EqDecidable (HaskLevel Γ).
582 apply Build_EqDecidable.
584 unfold HaskLevel in *.
585 apply (eqd_dec v1 v2).
592 (* ToString instance for PHOAS types *)
593 Fixpoint typeToString' (needparens:bool)(n:nat){κ}(t:RawHaskType (fun _ => nat) κ) {struct t} : string :=
595 | TVar _ v => "tv" +++ toString v
596 | TCon tc => toString tc
597 | TCoerc _ t1 t2 t => "("+++typeToString' false n t1+++"~"
598 +++typeToString' false n t2+++")=>"
599 +++typeToString' needparens n t
602 | TApp _ _ TArrow t1 =>
604 then "("+++(typeToString' true n t1)+++"->"+++(typeToString' true n t2)+++")"
605 else (typeToString' true n t1)+++"->"+++(typeToString' true n t2)
608 then "("+++(typeToString' true n t1)+++" "+++(typeToString' false n t2)+++")"
609 else (typeToString' true n t1)+++" "+++(typeToString' false n t2)
612 | TAll k f => let alpha := "tv"+++ toString n
613 in "(forall "+++ alpha +++ ":"+++ toString k +++")"+++
614 typeToString' false (S n) (f n)
615 | TCode ec t => "<["+++(typeToString' true n t)+++"]>@"+++(typeToString' false n ec)
616 | TyFunApp tfc kl k lt => toString tfc+++ "_" +++ toString n+++" ["+++
617 (fold_left (fun x y => " \ "+++x+++y) (typeList2string false n lt) "")+++"]"
619 with typeList2string (needparens:bool)(n:nat){κ}(t:RawHaskTypeList κ) {struct t} : list string :=
621 | TyFunApp_nil => nil
622 | TyFunApp_cons κ kl rhk rhkl => (typeToString' needparens n rhk)::(typeList2string needparens n rhkl)
625 Definition typeToString {Γ}{κ}(ht:HaskType Γ κ) : string :=
626 typeToString' false (length Γ) (ht (fun _ => nat) (count' Γ O)).
628 Instance TypeToStringInstance {Γ} {κ} : ToString (HaskType Γ κ) :=
629 { toString := typeToString }.