1 (*********************************************************************************************************************************)
2 (* HaskProofToStrong: convert HaskProof to HaskStrong *)
3 (*********************************************************************************************************************************)
5 Generalizable All Variables.
6 Require Import Preamble.
7 Require Import General.
8 Require Import NaturalDeduction.
9 Require Import Coq.Strings.String.
10 Require Import Coq.Lists.List.
11 Require Import Coq.Init.Specif.
12 Require Import HaskKinds.
13 Require Import HaskStrongTypes.
14 Require Import HaskStrong.
15 Require Import HaskProof.
17 Section HaskProofToStrong.
19 Context {VV:Type} {eqdec_vv:EqDecidable VV} {freshM:FreshMonad VV}.
21 Definition fresh := FMT_fresh freshM.
22 Definition FreshM := FMT freshM.
23 Definition FreshMon := FMT_Monad freshM.
24 Existing Instance FreshMon.
26 Definition ExprVarResolver Γ := VV -> LeveledHaskType Γ ★.
28 Definition judg2exprType (j:Judg) : Type :=
30 (Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
31 FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
34 Definition justOne Γ Δ ξ τ : ITree _ (fun t => Expr Γ Δ ξ t) [τ] -> Expr Γ Δ ξ τ.
39 Definition ileaf `(it:ITree X F [t]) : F t.
44 Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) lev l1 l2 q,
45 update_ξ ξ lev (app l1 l2) q = update_ξ (update_ξ ξ lev l2) lev l1 q.
55 Lemma quark {T} (l1:list T) l2 vf :
56 (In vf (app l1 l2)) <->
57 (In vf l1) \/ (In vf l2).
84 right; apply H0; auto.
85 right; apply H0; auto.
88 Lemma splitter {T} (l1:list T) l2 vf :
89 (In vf (app l1 l2) → False)
90 -> (In vf l1 → False) /\ (In vf l2 → False).
92 split; intros; apply H; rewrite quark.
98 : forall T Z {eqdt:EqDecidable T}(tl:Tree ??T)(vf:T) ξ (q:Z),
99 (In vf (leaves tl) -> False) ->
100 mapOptionTree (fun v' => if eqd_dec vf v' then q else ξ v') tl =
106 set (eqd_dec vf t) as x in *.
122 Lemma fresh_lemma'' Γ
123 : forall types ξ lev,
124 FreshM { varstypes : _
125 | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
126 /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
131 : forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars ->
132 FreshM { varstypes : _
133 | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ
134 /\ mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
135 /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
138 refine (bind vf = fresh (leaves vars) ; return _).
140 destruct vf as [ vf vf_pf ].
144 set (helper VV _ vars vf ξ (h@@lev) vf_pf) as q.
148 destruct (eqd_dec vf vf); [ idtac | set (n (refl_equal _)) as n'; inversion n' ]; auto.
164 intros vars Σ ξ lev pf; refine (bind x2 = IHtypes2 vars Σ ξ lev pf; _).
166 destruct x2 as [vt2 [pf21 [pf22 pfdist]]].
167 refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_ξ ξ lev
168 (leaves vt2)) _ _; return _).
174 clear IHtypes1 IHtypes2.
175 destruct x1 as [vt1 [pf11 pf12]].
176 exists (vt1,,vt2); split; auto.
178 set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
179 set (mapOptionTree_extensional _ _ q) as q'.
186 set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
187 set (mapOptionTree_extensional _ _ q) as q'.
191 rewrite <- mapOptionTree_compose.
192 rewrite <- mapOptionTree_compose.
193 rewrite <- mapOptionTree_compose in *.
198 rewrite <- mapOptionTree_compose.
204 Lemma fresh_lemma Γ ξ vars Σ Σ' lev
205 : Σ = mapOptionTree ξ vars ->
207 | mapOptionTree (update_ξ(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ
208 /\ mapOptionTree (update_ξ ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }.
210 set (fresh_lemma' Γ [Σ'] vars Σ ξ lev H) as q.
211 refine (q >>>= fun q' => return _).
214 destruct q' as [varstypes [pf1 [pf2 pfdist]]].
215 destruct varstypes; try destruct o; try destruct p; simpl in *.
216 destruct (eqd_dec v v); [ idtac | set (n (refl_equal _)) as n'; inversion n' ].
217 inversion pf2; subst.
219 destruct (eqd_dec v v); [ idtac | set (n (refl_equal _)) as n'; inversion n' ].
225 Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ : Type :=
226 forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ).
228 Definition urule2expr : forall Γ Δ h j t (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
229 ujudg2exprType Γ ξ Δ h t ->
230 ujudg2exprType Γ ξ Δ j t
233 refine (fix urule2expr h j t (r:@Arrange _ h j) ξ {struct r} :
234 ujudg2exprType Γ ξ Δ h t ->
235 ujudg2exprType Γ ξ Δ j t :=
236 match r as R in Arrange H C return
237 ujudg2exprType Γ ξ Δ H t ->
238 ujudg2exprType Γ ξ Δ C t
240 | RLeft h c ctx r => let case_RLeft := tt in (fun e => _) (urule2expr _ _ _ r)
241 | RRight h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ r)
242 | RId a => let case_RId := tt in _
243 | RCanL a => let case_RCanL := tt in _
244 | RCanR a => let case_RCanR := tt in _
245 | RuCanL a => let case_RuCanL := tt in _
246 | RuCanR a => let case_RuCanR := tt in _
247 | RAssoc a b c => let case_RAssoc := tt in _
248 | RCossa a b c => let case_RCossa := tt in _
249 | RExch a b => let case_RExch := tt in _
250 | RWeak a => let case_RWeak := tt in _
251 | RCont a => let case_RCont := tt in _
252 | RComp a b c f g => let case_RComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ f) (urule2expr _ _ _ g)
253 end); clear urule2expr; intros.
259 simpl; unfold ujudg2exprType; intros.
261 apply (X ([],,vars)).
262 simpl; rewrite <- H; auto.
265 simpl; unfold ujudg2exprType; intros.
267 apply (X (vars,,[])).
268 simpl; rewrite <- H; auto.
270 destruct case_RuCanL.
271 simpl; unfold ujudg2exprType; intros.
272 destruct vars; try destruct o; inversion H.
274 apply (X vars2); auto.
276 destruct case_RuCanR.
277 simpl; unfold ujudg2exprType; intros.
278 destruct vars; try destruct o; inversion H.
280 apply (X vars1); auto.
282 destruct case_RAssoc.
283 simpl; unfold ujudg2exprType; intros.
285 destruct vars; try destruct o; inversion H.
286 destruct vars1; try destruct o; inversion H.
287 apply (X (vars1_1,,(vars1_2,,vars2))).
290 destruct case_RCossa.
291 simpl; unfold ujudg2exprType; intros.
293 destruct vars; try destruct o; inversion H.
294 destruct vars2; try destruct o; inversion H.
295 apply (X ((vars1,,vars2_1),,vars2_2)).
299 simpl; unfold ujudg2exprType ; intros.
301 destruct vars; try destruct o; inversion H.
302 apply (X (vars2,,vars1)).
303 inversion H; subst; auto.
306 simpl; unfold ujudg2exprType; intros.
312 simpl; unfold ujudg2exprType ; intros.
314 apply (X (vars,,vars)).
320 intro vars; unfold ujudg2exprType; intro H.
321 destruct vars; try destruct o; inversion H.
322 apply (fun q => e ξ q vars2 H2).
326 unfold ujudg2exprType.
328 apply X with (vars:=vars1,,vars).
334 destruct case_RRight.
335 intro vars; unfold ujudg2exprType; intro H.
336 destruct vars; try destruct o; inversion H.
337 apply (fun q => e ξ q vars1 H1).
341 unfold ujudg2exprType.
343 apply X with (vars:=vars,,vars2).
355 Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' :
356 ITree (LeveledHaskType Γ ★)
357 (fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t)
358 (mapOptionTree (ξ' ○ (@fst _ _)) varstypes)
359 -> ELetRecBindings Γ Δ ξ' l varstypes.
362 destruct a; simpl in *.
365 apply ileaf in X. simpl in X.
368 destruct (eqd_dec (unlev (ξ' v)) τ).
372 destruct (eqd_dec h0 l).
375 apply (Prelude_error "level mismatch; should never happen").
376 apply (Prelude_error "letrec type mismatch; should never happen").
380 apply IHvarstypes1; inversion X; auto.
381 apply IHvarstypes2; inversion X; auto.
384 Definition unindex_tree {V}{F} : forall {t:Tree ??V}, ITree V F t -> Tree ??{ v:V & F v }.
385 refine (fix rec t it := match it as IT return Tree ??{ v:V & F v } with
386 | INone => T_Leaf None
387 | ILeaf x y => T_Leaf (Some _)
388 | IBranch _ _ b1 b2 => (rec _ b1),,(rec _ b2)
393 Definition fix_indexing X (F:X->Type)(J:X->Type)(t:Tree ??{ x:X & F x })
394 : ITree { x:X & F x } (fun x => J (projT1 x)) t
395 -> ITree X (fun x:X => J x) (mapOptionTree (@projT1 _ _) t).
397 induction it; simpl in *.
401 simpl; apply IBranch; auto.
404 Definition fix2 {X}{F} : Tree ??{ x:X & FreshM (F x) } -> Tree ??(FreshM { x:X & F x }).
405 refine (fix rec t := match t with
406 | T_Leaf None => T_Leaf None
407 | T_Leaf (Some x) => T_Leaf (Some _)
408 | T_Branch b1 b2 => T_Branch (rec b1) (rec b2)
410 destruct x as [x fx].
411 refine (bind fx' = fx ; return _).
417 Definition case_helper tc Γ Δ lev tbranches avars ξ :
418 forall pcb:{sac : StrongAltCon & ProofCaseBranch tc Γ Δ lev tbranches avars sac},
419 prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} ->
421 { scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac
422 & Expr (sac_Γ sac Γ) (sac_Δ sac Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev)) }) (projT1 pcb)).
427 destruct pcb as [sac pcb].
431 destruct s as [vars vars_pf].
433 refine (bind localvars = fresh_lemma' _ (unleaves (vec2list (sac_types sac _ avars))) vars
434 (mapOptionTree weakLT' (pcb_freevars pcb)) (weakLT' ○ ξ) (weakL' lev) _ ; _).
437 rewrite <- mapOptionTree_compose.
439 destruct localvars as [localvars [localvars_pf1 [localvars_pf2 localvars_dist ]]].
440 set (mapOptionTree (@fst _ _) localvars) as localvars'.
442 set (list2vec (leaves localvars')) as localvars''.
443 cut (length (leaves localvars') = sac_numExprVars sac). intro H''.
444 rewrite H'' in localvars''.
445 cut (distinct (vec2list localvars'')). intro H'''.
446 set (@Build_StrongCaseBranchWithVVs _ _ _ _ avars sac localvars'' H''') as scb.
448 refine (bind q = (f (scbwv_ξ scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _).
453 rewrite <- mapOptionTree_compose.
468 Definition gather_branch_variables
469 Γ Δ (ξ:VV -> LeveledHaskType Γ ★) tc avars tbranches lev (alts:Tree ?? {sac : StrongAltCon &
470 ProofCaseBranch tc Γ Δ lev tbranches avars sac})
473 mapOptionTreeAndFlatten (fun x => pcb_freevars(Γ:=Γ) (projT2 x)) alts = mapOptionTree ξ vars
474 -> ITree Judg judg2exprType (mapOptionTree (fun x => pcb_judg (projT2 x)) alts)
475 -> ITree _ (fun q => prod (judg2exprType (pcb_judg (projT2 q)))
476 { vars' : _ & pcb_freevars (projT2 q) = mapOptionTree ξ vars' })
482 destruct a; [ idtac | apply INone ].
484 apply ileaf in source.
486 destruct s as [sac pcb].
498 destruct vars; try destruct o; simpl in pf; inversion pf.
503 apply (IHalts1 vars1 H0 X); auto.
504 apply (IHalts2 vars2 H1 X0); auto.
509 Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j.
513 refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with
514 | RArrange a b c d e r => let case_RURule := tt in _
515 | RNote Γ Δ Σ τ l n => let case_RNote := tt in _
516 | RLit Γ Δ l _ => let case_RLit := tt in _
517 | RVar Γ Δ σ p => let case_RVar := tt in _
518 | RGlobal Γ Δ σ l wev => let case_RGlobal := tt in _
519 | RLam Γ Δ Σ tx te x => let case_RLam := tt in _
520 | RCast Γ Δ Σ σ τ γ x => let case_RCast := tt in _
521 | RAbsT Γ Δ Σ κ σ a => let case_RAbsT := tt in _
522 | RAppT Γ Δ Σ κ σ τ y => let case_RAppT := tt in _
523 | RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ l => let case_RAppCo := tt in _
524 | RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _
525 | RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
526 | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
527 | RJoin Γ p lri m x q => let case_RJoin := tt in _
528 | RVoid _ _ => let case_RVoid := tt in _
529 | RBrak Σ a b c n m => let case_RBrak := tt in _
530 | REsc Σ a b c n m => let case_REsc := tt in _
531 | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
532 | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
533 end); intro X_; try apply ileaf in X_; simpl in X_.
535 destruct case_RURule.
536 apply ILeaf. simpl. intros.
537 set (@urule2expr a b _ _ e r0 ξ) as q.
538 set (fun z => q z) as q'.
540 apply q' with (vars:=vars).
542 unfold ujudg2exprType.
544 apply X_ with (vars:=vars0).
549 apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
554 apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
559 apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
564 apply ILeaf; simpl; intros; refine (return ILeaf _ _).
568 apply ILeaf; simpl; intros; refine (return ILeaf _ _).
569 destruct vars; simpl in H; inversion H; destruct o. inversion H1. rewrite H2.
573 destruct case_RGlobal.
574 apply ILeaf; simpl; intros; refine (return ILeaf _ _).
580 refine (fresh_lemma _ ξ vars _ tx x H >>>= (fun pf => _)).
582 destruct pf as [ vnew [ pf1 pf2 ]].
583 set (update_ξ ξ x (((vnew, tx )) :: nil)) as ξ' in *.
584 refine (X_ ξ' (vars,,[vnew]) _ >>>= _).
594 apply ELam with (ev:=vnew).
601 apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
604 apply ileaf in X. simpl in X.
608 apply ILeaf; simpl; intros.
613 destruct vars; inversion H.
614 destruct o; inversion H3.
615 refine (X ξ vars1 H3 >>>= fun X' => X0 ξ vars2 H4 >>>= fun X0' => return _).
627 destruct vars. try destruct o; inversion H.
630 set (X1 ξ vars1 H5) as q1.
631 set (X2 ξ vars2 H6) as q2.
632 refine (q1 >>>= fun q1' => q2 >>>= fun q2' => return _).
639 apply (EApp _ _ _ _ _ _ q2' q1').
644 destruct vars; try destruct o; inversion H.
645 refine (fresh_lemma _ ξ vars1 _ σ₂ p H1 >>>= (fun pf => _)).
647 destruct pf as [ vnew [ pf1 pf2 ]].
648 set (update_ξ ξ p (((vnew, σ₂ )) :: nil)) as ξ' in *.
653 refine (X ξ vars2 _ >>>= fun X0' => _).
657 refine (X0 ξ' (vars1,,[vnew]) _ >>>= fun X1' => _).
671 apply ELet with (ev:=vnew)(tv:=σ₂).
676 apply ILeaf; simpl; intros.
681 apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
686 apply ILeaf; simpl; intros; refine (X_ (weakLT ○ ξ) vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
687 rewrite mapOptionTree_compose.
690 apply ileaf in X. simpl in *.
694 destruct case_RAppCo.
695 apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
701 destruct case_RAbsCo.
702 apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
707 destruct case_RLetRec.
708 apply ILeaf; simpl; intros.
709 refine (bind ξvars = fresh_lemma' _ y _ _ _ t H; _). apply FreshMon.
710 destruct ξvars as [ varstypes [ pf1[ pf2 pfdist]]].
711 refine (X_ ((update_ξ ξ t (leaves varstypes)))
712 (vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_. apply FreshMon.
718 inversion X; subst; clear X.
720 apply (@ELetRec _ _ _ _ _ _ _ varstypes).
722 apply (@letrec_helper Γ Δ t varstypes).
723 rewrite <- pf2 in X1.
724 rewrite mapOptionTree_compose.
730 apply ILeaf; simpl; intros.
737 (* body_freevars and alts_freevars are the types of variables in the body and alternatives (respectively) which are free
738 * from the viewpoint just outside the case block -- i.e. not bound by any of the branches *)
739 rename Σ into body_freevars_types.
740 rename vars into all_freevars.
741 rename X0 into body_expr.
742 rename X into alts_exprs.
744 destruct all_freevars; try destruct o; inversion H.
745 rename all_freevars2 into body_freevars.
746 rename all_freevars1 into alts_freevars.
748 set (gather_branch_variables _ _ _ _ _ _ _ _ _ H1 alts_exprs) as q.
749 set (itmap (fun pcb alt_expr => case_helper tc Γ Δ lev tbranches avars ξ pcb alt_expr) q) as alts_exprs'.
750 apply fix_indexing in alts_exprs'.
751 simpl in alts_exprs'.
752 apply unindex_tree in alts_exprs'.
753 simpl in alts_exprs'.
754 apply fix2 in alts_exprs'.
755 apply treeM in alts_exprs'.
757 refine ( alts_exprs' >>>= fun Y =>
759 >>>= fun X => return ILeaf _ (@ECase _ _ _ _ _ _ _ _ _ (ileaf X) Y)); auto.
765 Fixpoint closed2expr h j (pn:@SIND _ Rule h j) {struct pn} : ITree _ judg2exprType h -> ITree _ judg2exprType j :=
766 match pn in @SIND _ _ H J return ITree _ judg2exprType H -> ITree _ judg2exprType J with
767 | scnd_weak _ => let case_nil := tt in fun _ => INone _ _
768 | scnd_comp x h c cnd' r => let case_rule := tt in fun q => rule2expr _ _ r (closed2expr _ _ cnd' q)
769 | scnd_branch _ _ _ c1 c2 => let case_branch := tt in fun q => IBranch _ _ (closed2expr _ _ c1 q) (closed2expr _ _ c2 q)
772 Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
773 FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
775 induction Σ; intro ξ.
778 set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
779 refine (q >>>= fun q' => return _).
782 destruct q' as [varstypes [pf1 [pf2 distpf]]].
783 exists (mapOptionTree (@fst _ _) varstypes).
784 exists (update_ξ ξ l (leaves varstypes)).
789 refine (bind f1 = IHΣ1 ξ ; _).
791 destruct f1 as [vars1 [ξ1 pf1]].
792 refine (bind f2 = IHΣ2 ξ1 ; _).
794 destruct f2 as [vars2 [ξ2 pf22]].
796 exists (vars1,,vars2).
804 Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
805 {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] ->
806 FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
808 set (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) as cnd.
809 apply closed2expr in cnd.
813 refine (bind ξvars = manyFresh _ Σ ξ0; _).
815 destruct ξvars as [vars ξpf].
816 destruct ξpf as [ξ pf].
817 refine (cnd ξ vars _ >>>= fun it => _).
820 refine (return OK _).
826 End HaskProofToStrong.