1 (*********************************************************************************************************************************)
4 (* Skolemizes the portion of a proof which uses judgments at level >0 *)
6 (*********************************************************************************************************************************)
8 Generalizable All Variables.
9 Require Import Preamble.
10 Require Import General.
11 Require Import NaturalDeduction.
12 Require Import NaturalDeductionContext.
13 Require Import Coq.Strings.String.
14 Require Import Coq.Lists.List.
16 Require Import HaskKinds.
17 Require Import HaskCoreTypes.
18 Require Import HaskCoreVars.
19 Require Import HaskWeakTypes.
20 Require Import HaskWeakVars.
21 Require Import HaskLiterals.
22 Require Import HaskTyCons.
23 Require Import HaskStrongTypes.
24 Require Import HaskProof.
25 Require Import NaturalDeduction.
27 Require Import HaskStrongTypes.
28 Require Import HaskStrong.
29 Require Import HaskProof.
30 Require Import HaskStrongToProof.
31 Require Import HaskProofToStrong.
32 Require Import HaskWeakToStrong.
35 Set Printing Width 130.
37 Section HaskSkolemizer.
40 Fixpoint debruijn2phoas {κ} (exp: RawHaskType (fun _ => nat) κ) : HaskType TV κ :=
43 | TAll _ y => TAll _ (fun v => debruijn2phoas (y (TVar v)))
44 | TApp _ _ x y => TApp (debruijn2phoas x) (debruijn2phoas y)
46 | TCoerc _ t1 t2 t => TCoerc (debruijn2phoas t1) (debruijn2phoas t2) (debruijn2phoas t)
48 | TCode v e => TCode (debruijn2phoas v) (debruijn2phoas e)
49 | TyFunApp tfc kl k lt => TyFunApp tfc kl k (debruijn2phoasyFunApp _ lt)
51 with debruijn2phoasyFunApp (lk:list Kind)(exp:@RawHaskTypeList (fun _ => nat) lk) : @HaskTypeList TV lk :=
52 match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with
53 | TyFunApp_nil => TyFunApp_nil
54 | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (debruijn2phoas t) (debruijn2phoasyFunApp _ rest)
57 Definition isNotBrakOrEsc {h}{c} (r:Rule h c) : Prop :=
59 | RBrak _ _ _ _ _ _ => False
60 | REsc _ _ _ _ _ _ => False
64 Fixpoint mkArrows {Γ}(lt:list (HaskType Γ ★))(t:HaskType Γ ★) : HaskType Γ ★ :=
67 | a::b => mkArrows b (a ---> t)
71 Fixpoint unleaves_ {Γ}(t:Tree ??(LeveledHaskType Γ ★))(l:list (HaskType Γ ★)) lev : Tree ??(LeveledHaskType Γ ★) :=
74 | a::b => unleaves_ (t,,[a @@ lev]) b lev
77 (* weak inverse of "leaves" *)
78 Fixpoint unleaves_ {A:Type}(l:list A) : Tree (option A) :=
82 | (a::b) => [a],,(unleaves_ b)
85 (* rules of skolemized proofs *)
86 Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ @ _ => Γ end.
88 Fixpoint take_trustme {Γ}
90 (l:forall TV, InstantiatedTypeEnv TV Γ -> list (RawHaskType TV ★))
91 : list (HaskType Γ ★) :=
95 | S n' => (fun TV ite => match l TV ite with
96 | nil => Prelude_error "impossible"
100 take_trustme n' (fun TV ite => match l TV ite with
101 | nil => Prelude_error "impossible"
106 Axiom phoas_extensionality : forall Γ Q (f g:forall TV, InstantiatedTypeEnv TV Γ -> Q TV),
107 (forall tv ite, f tv ite = g tv ite) -> f=g.
109 Definition take_arg_types_as_tree {Γ}(ht:HaskType Γ ★) : Tree ??(HaskType Γ ★ ) :=
112 (count_arg_types (ht _ (ite_unit _)))
113 (fun TV ite => take_arg_types (ht TV ite))).
115 Definition drop_arg_types_as_tree {Γ} (ht:HaskType Γ ★) : HaskType Γ ★ :=
116 fun TV ite => drop_arg_types (ht TV ite).
118 Implicit Arguments take_arg_types_as_tree [[Γ]].
119 Implicit Arguments drop_arg_types_as_tree [[Γ]].
121 Definition take_arrange : forall {Γ} (tx te:HaskType Γ ★) lev,
122 Arrange ([tx @@ lev],,take_arg_types_as_tree te @@@ lev)
123 (take_arg_types_as_tree (tx ---> te) @@@ lev).
125 destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
129 unfold take_arg_types_as_tree.
130 Opaque take_arg_types_as_tree.
132 destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
134 replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
136 apply phoas_extensionality.
138 apply (Prelude_error "should not be possible").
140 Transparent take_arg_types_as_tree.
142 Definition take_unarrange : forall {Γ} (tx te:HaskType Γ ★) lev,
143 Arrange (take_arg_types_as_tree (tx ---> te) @@@ lev)
144 ([tx @@ lev],,take_arg_types_as_tree te @@@ lev).
146 destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
150 unfold take_arg_types_as_tree.
151 Opaque take_arg_types_as_tree.
153 destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
155 replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
157 apply phoas_extensionality.
159 apply (Prelude_error "should not be possible").
161 Transparent take_arg_types_as_tree.
163 Lemma drop_works : forall {Γ}(t1 t2:HaskType Γ ★),
164 drop_arg_types_as_tree (t1 ---> t2) = (drop_arg_types_as_tree t2).
166 unfold drop_arg_types_as_tree.
171 Inductive SRule : Tree ??Judg -> Tree ??Judg -> Type :=
172 (* | SFlat : forall h c (r:Rule h c), isNotBrakOrEsc r -> SRule h c*)
173 | SFlat : forall h c, Rule h c -> SRule h c
174 | SBrak : forall Γ Δ t ec Σ l,
176 [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
177 [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
179 | SEsc : forall Γ Δ t ec Σ l,
181 [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
182 [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
185 Definition take_arg_types_as_tree' {Γ}(lt:LeveledHaskType Γ ★) :=
186 match lt with t @@ l => take_arg_types_as_tree t @@@ l end.
188 Definition drop_arg_types_as_tree' {Γ}(lt:LeveledHaskType Γ ★) :=
189 match lt with t @@ l => drop_arg_types_as_tree t @@ l end.
191 Definition skolemize_judgment (j:Judg) : Judg :=
193 | Γ > Δ > Σ₁ |- Σ₂ @ nil => j
194 | Γ > Δ > Σ₁ |- Σ₂ @ lev =>
195 Γ > Δ > Σ₁,,(mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂ @@@ lev) |- mapOptionTree drop_arg_types_as_tree Σ₂ @ lev
198 Definition check_hof : forall {Γ}(t:HaskType Γ ★),
201 (take_arg_types_as_tree t = [] /\ drop_arg_types_as_tree t = t).
203 destruct (eqd_dec (take_arg_types_as_tree t) []);
204 destruct (eqd_dec (drop_arg_types_as_tree t) t).
211 Opaque take_arg_types_as_tree.
212 Definition skolemize_proof :
215 ND SRule (mapOptionTree skolemize_judgment h) (mapOptionTree skolemize_judgment c).
217 eapply nd_map'; [ idtac | apply X ].
221 refine (match X as R in Rule H C with
222 | RArrange Γ Δ a b x l d => let case_RArrange := tt in _
223 | RNote Γ Δ Σ τ l n => let case_RNote := tt in _
224 | RLit Γ Δ l _ => let case_RLit := tt in _
225 | RVar Γ Δ σ lev => let case_RVar := tt in _
226 | RGlobal Γ Δ σ l wev => let case_RGlobal := tt in _
227 | RLam Γ Δ Σ tx te lev => let case_RLam := tt in _
228 | RCast Γ Δ Σ σ τ lev γ => let case_RCast := tt in _
229 | RAbsT Γ Δ Σ κ σ lev => let case_RAbsT := tt in _
230 | RAppT Γ Δ Σ κ σ τ lev => let case_RAppT := tt in _
231 | RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ lev => let case_RAppCo := tt in _
232 | RAbsCo Γ Δ Σ κ σ σ₁ σ₂ lev => let case_RAbsCo := tt in _
233 | RApp Γ Δ Σ₁ Σ₂ tx te lev => let case_RApp := tt in _
234 | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ lev => let case_RLet := tt in _
235 | RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := tt in _
236 | RJoin Γ p lri m x q l => let case_RJoin := tt in _
237 | RVoid _ _ l => let case_RVoid := tt in _
238 | RBrak Γ Δ t ec succ lev => let case_RBrak := tt in _
239 | REsc Γ Δ t ec succ lev => let case_REsc := tt in _
240 | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
241 | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
244 destruct case_RArrange.
259 destruct lev; [ idtac | apply (Prelude_error "Brak with nesting depth >1") ].
265 destruct lev; [ idtac | apply (Prelude_error "Esc with nesting depth >1") ].
285 set (check_hof (@literalType l Γ)) as hof.
286 destruct hof; [ apply (Prelude_error "attempt to use a literal with higher-order type at depth>0") | idtac ].
291 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply RuCanL ].
299 apply nd_rule; apply SFlat; apply RVar.
300 set (check_hof σ) as hof.
301 destruct hof; [ apply (Prelude_error "attempt to use a variable with higher-order type at depth>0") | idtac ].
306 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
311 destruct case_RGlobal.
314 apply nd_rule; apply SFlat; apply RGlobal.
315 set (check_hof (l wev)) as hof.
316 destruct hof; [ apply (Prelude_error "attempt to use a global with higher-order type at depth>0") | idtac ].
321 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
349 apply (Prelude_error "found RCast at level >0").
357 apply (Prelude_error "found RJoin at level >0").
366 set (check_hof tx) as hof_tx.
367 destruct hof_tx; [ apply (Prelude_error "attempt tp apply a higher-order function at depth>0") | idtac ].
373 eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
378 eapply take_unarrange.
380 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
381 eapply nd_rule; eapply SFlat; apply RWhere.
389 set (check_hof σ₁) as hof_tx.
390 destruct hof_tx; [ apply (Prelude_error "attempt to let-bind a higher-order function at depth>0") | idtac ].
396 eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR | eapply nd_id ].
398 set (@RLet Γ Δ Σ₁ (Σ₂,,(take_arg_types_as_tree σ₂ @@@ (h::lev))) σ₁ (drop_arg_types_as_tree σ₂) (h::lev)) as q.
399 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
400 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply q ].
408 destruct case_RWhere.
414 set (check_hof σ₁) as hof_tx.
415 destruct hof_tx; [ apply (Prelude_error "attempt to let-bind a higher-order function at depth>0") | idtac ].
421 eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
422 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RAssoc ].
423 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RLeft; eapply RAssoc ].
424 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RWhere ].
425 apply nd_prod; [ idtac | eapply nd_id ].
426 eapply nd_rule; apply SFlat; eapply RArrange.
438 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RuCanL ].
445 destruct lev; [ apply nd_rule; apply SFlat; apply RAppT | idtac ].
446 apply (Prelude_error "RAppT at depth>0").
450 destruct lev; simpl; [ apply nd_rule; apply SFlat; apply (@RAbsT _ _ _ _ _ nil) | idtac ].
451 apply (Prelude_error "RAbsT at depth>0").
453 destruct case_RAppCo.
455 destruct lev; [ apply nd_rule; apply SFlat; apply RAppCo | idtac ].
457 apply (Prelude_error "RAppCo at depth>0").
459 destruct case_RAbsCo.
461 destruct lev; [ apply nd_rule; apply SFlat; apply RAbsCo | idtac ].
462 apply (Prelude_error "RAbsCo at depth>0").
464 destruct case_RLetRec.
469 apply (@RLetRec Γ Δ lri x y nil).
470 apply (Prelude_error "RLetRec at depth>0").
474 apply (Prelude_error "CASE: BIG FIXME").
477 Transparent take_arg_types_as_tree.