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 n => 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 | RCut Γ Δ Σ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _
235 | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _
236 | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := 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 AuCanL ].
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 AuCanR ].
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 AuCanR ].
349 apply (Prelude_error "found RCast at level >0").
358 set (check_hof tx) as hof_tx.
359 destruct hof_tx; [ apply (Prelude_error "attempt tp apply a higher-order function at depth>0") | idtac ].
365 eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
370 eapply take_unarrange.
372 eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
373 eapply nd_comp; [ apply nd_exch | idtac ].
374 eapply nd_rule; eapply SFlat; eapply RCut.
377 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RCut | idtac ].
378 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₃) as Σ₃''.
379 set (mapOptionTree drop_arg_types_as_tree Σ₃) as Σ₃'''.
380 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₁₂) as Σ₁₂''.
381 set (mapOptionTree drop_arg_types_as_tree Σ₁₂) as Σ₁₂'''.
382 destruct (decide_tree_empty (Σ₁₂'' @@@ (h::l)));
383 [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
384 destruct (eqd_dec Σ₁₂ Σ₁₂'''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
395 eapply arrangeCancelEmptyTree with (q:=x).
399 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
400 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AAssoc ].
401 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RCut ].
416 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RLeft | idtac ].
417 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
418 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
419 set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
420 set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
421 destruct (decide_tree_empty (Σ' @@@ (h::l)));
422 [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
423 destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
427 set (arrangeUnCancelEmptyTree _ _ e) as q.
428 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ARight; eapply q ].
429 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanL; eapply q ].
430 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
435 destruct case_RRight.
436 simpl; destruct l; [ apply nd_rule; apply SFlat; apply RRight | idtac ].
437 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
438 set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
439 set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
440 set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
441 destruct (decide_tree_empty (Σ' @@@ (h::l)));
442 [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
443 destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
447 set (arrangeUnCancelEmptyTree _ _ e) as q.
448 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ALeft; eapply q ].
449 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanR ].
450 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
451 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AExch ]. (* yuck *)
452 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
463 eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuCanL ].
470 destruct lev; [ apply nd_rule; apply SFlat; apply RAppT | idtac ].
471 apply (Prelude_error "RAppT at depth>0").
478 apply (@RAbsT Γ Δ Σ κ σ nil n).
479 apply (Prelude_error "RAbsT at depth>0").
481 destruct case_RAppCo.
483 destruct lev; [ apply nd_rule; apply SFlat; apply RAppCo | idtac ].
485 apply (Prelude_error "RAppCo at depth>0").
487 destruct case_RAbsCo.
489 destruct lev; [ apply nd_rule; apply SFlat; apply RAbsCo | idtac ].
490 apply (Prelude_error "RAbsCo at depth>0").
492 destruct case_RLetRec.
497 apply (@RLetRec Γ Δ lri x y nil).
498 destruct (decide_tree_empty (mapOptionTreeAndFlatten take_arg_types_as_tree y @@@ (h :: t)));
499 [ idtac | apply (Prelude_error "used LetRec on a set of bindings involving a function type") ].
500 destruct (eqd_dec y (mapOptionTree drop_arg_types_as_tree y));
501 [ idtac | apply (Prelude_error "used LetRec on a set of bindings involving a function type") ].
512 apply (arrangeCancelEmptyTree _ _ e).
524 destruct lev; [ idtac | apply (Prelude_error "case at depth >0") ]; simpl.
527 rewrite <- mapOptionTree_compose.
529 ((mapOptionTree (fun x => skolemize_judgment (@pcb_judg tc Γ Δ nil tbranches avars (fst x) (snd x))) alts) =
530 (mapOptionTree (fun x => (@pcb_judg tc Γ Δ nil tbranches avars (fst x) (snd x))) alts)).
533 set (@RCase Γ Δ nil tc Σ avars tbranches alts) as q.
537 Transparent take_arg_types_as_tree.