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 | RCut Γ Δ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _
236 | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _
237 | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := tt in _
238 | RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := tt in _
239 | RVoid _ _ l => let case_RVoid := tt in _
240 | RBrak Γ Δ t ec succ lev => let case_RBrak := tt in _
241 | REsc Γ Δ t ec succ lev => let case_REsc := tt in _
242 | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
243 | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
246 destruct case_RArrange.
261 destruct lev; [ idtac | apply (Prelude_error "Brak with nesting depth >1") ].
267 destruct lev; [ idtac | apply (Prelude_error "Esc with nesting depth >1") ].
287 set (check_hof (@literalType l Γ)) as hof.
288 destruct hof; [ apply (Prelude_error "attempt to use a literal with higher-order type at depth>0") | idtac ].
293 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply AuCanL ].
301 apply nd_rule; apply SFlat; apply RVar.
302 set (check_hof σ) as hof.
303 destruct hof; [ apply (Prelude_error "attempt to use a variable with higher-order type at depth>0") | idtac ].
308 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
313 destruct case_RGlobal.
316 apply nd_rule; apply SFlat; apply RGlobal.
317 set (check_hof (l wev)) as hof.
318 destruct hof; [ apply (Prelude_error "attempt to use a global with higher-order type at depth>0") | idtac ].
323 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
351 apply (Prelude_error "found RCast at level >0").
360 set (check_hof tx) as hof_tx.
361 destruct hof_tx; [ apply (Prelude_error "attempt tp apply a higher-order function at depth>0") | idtac ].
367 eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
372 eapply take_unarrange.
374 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
375 eapply nd_rule; eapply SFlat; apply RWhere.
383 set (check_hof σ₁) as hof_tx.
384 destruct hof_tx; [ apply (Prelude_error "attempt to let-bind a higher-order function at depth>0") | idtac ].
390 eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR | eapply nd_id ].
392 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.
393 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
394 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply q ].
402 destruct case_RWhere.
408 set (check_hof σ₁) as hof_tx.
409 destruct hof_tx; [ apply (Prelude_error "attempt to let-bind a higher-order function at depth>0") | idtac ].
415 eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
416 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
417 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AAssoc ].
418 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RWhere ].
419 apply nd_prod; [ idtac | eapply nd_id ].
420 eapply nd_rule; apply SFlat; eapply RArrange.
427 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RCut | idtac ].
428 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₃) as Σ₃''.
429 set (mapOptionTree drop_arg_types_as_tree Σ₃) as Σ₃'''.
430 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₁₂) as Σ₁₂''.
431 set (mapOptionTree drop_arg_types_as_tree Σ₁₂) as Σ₁₂'''.
432 destruct (decide_tree_empty Σ₁₂''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
433 destruct (eqd_dec Σ₁₂ Σ₁₂'''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
436 eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
437 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
438 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RCut ].
440 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
446 eapply arrangeCancelEmptyTree with (q:=x).
448 admit. (* FIXME, but not serious *)
452 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RLeft | idtac ].
453 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
454 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
455 set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
456 set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
457 destruct (decide_tree_empty (Σ' @@@ (h::l)));
458 [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
459 destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
463 set (arrangeUnCancelEmptyTree _ _ e) as q.
464 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ARight; eapply q ].
465 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanL; eapply q ].
466 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
471 destruct case_RRight.
472 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RRight | idtac ].
473 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
474 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
475 set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
476 set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
477 destruct (decide_tree_empty (Σ' @@@ (h::l)));
478 [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
479 destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
483 set (arrangeUnCancelEmptyTree _ _ e) as q.
484 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ALeft; eapply q ].
485 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanR ].
486 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
487 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AExch ]. (* yuck *)
488 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
499 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuCanL ].
506 destruct lev; [ apply nd_rule; apply SFlat; apply RAppT | idtac ].
507 apply (Prelude_error "RAppT at depth>0").
511 destruct lev; simpl; [ apply nd_rule; apply SFlat; apply (@RAbsT _ _ _ _ _ nil) | idtac ].
512 apply (Prelude_error "RAbsT at depth>0").
514 destruct case_RAppCo.
516 destruct lev; [ apply nd_rule; apply SFlat; apply RAppCo | idtac ].
518 apply (Prelude_error "RAppCo at depth>0").
520 destruct case_RAbsCo.
522 destruct lev; [ apply nd_rule; apply SFlat; apply RAbsCo | idtac ].
523 apply (Prelude_error "RAbsCo at depth>0").
525 destruct case_RLetRec.
530 apply (@RLetRec Γ Δ lri x y nil).
531 apply (Prelude_error "RLetRec at depth>0").
535 apply (Prelude_error "CASE: BIG FIXME").
538 Transparent take_arg_types_as_tree.