X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskProofToStrong.v;h=2ea0c4a1e4cfaea4311bcda04a7090166d87d53f;hp=19f2f62d8dd7a53105c6c4ebf185a7f42e586bdb;hb=171beb27508a340b24ab14837e72451d0b500805;hpb=164cdbf41ca206079b0dcfc18cd13625b286c38c diff --git a/src/HaskProofToStrong.v b/src/HaskProofToStrong.v index 19f2f62..2ea0c4a 100644 --- a/src/HaskProofToStrong.v +++ b/src/HaskProofToStrong.v @@ -6,6 +6,7 @@ Generalizable All Variables. Require Import Preamble. Require Import General. Require Import NaturalDeduction. +Require Import NaturalDeductionContext. Require Import Coq.Strings.String. Require Import Coq.Lists.List. Require Import Coq.Init.Specif. @@ -27,11 +28,11 @@ Section HaskProofToStrong. Definition judg2exprType (j:Judg) : Type := match j with - (Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars -> - FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ) + (Γ > Δ > Σ |- τ @ l) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars -> + FreshM (ITree _ (fun t => Expr Γ Δ ξ t l) τ) end. - Definition justOne Γ Δ ξ τ : ITree _ (fun t => Expr Γ Δ ξ t) [τ] -> Expr Γ Δ ξ τ. + Definition justOne Γ Δ ξ τ l : ITree _ (fun t => Expr Γ Δ ξ t l) [τ] -> Expr Γ Δ ξ τ l. intros. inversion X; auto. Defined. @@ -42,7 +43,7 @@ Section HaskProofToStrong. Defined. Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) lev l1 l2 q, - update_ξ ξ lev (app l1 l2) q = update_ξ (update_ξ ξ lev l2) lev l1 q. + update_xi ξ lev (app l1 l2) q = update_xi (update_xi ξ lev l2) lev l1 q. intros. induction l1. reflexivity. @@ -122,7 +123,7 @@ Section HaskProofToStrong. Lemma fresh_lemma'' Γ : forall types ξ lev, FreshM { varstypes : _ - | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev) + | mapOptionTree (update_xi(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev) /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }. admit. Defined. @@ -130,8 +131,8 @@ Section HaskProofToStrong. Lemma fresh_lemma' Γ : forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars -> FreshM { varstypes : _ - | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ - /\ mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev) + | mapOptionTree (update_xi(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ + /\ mapOptionTree (update_xi ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev) /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }. induction types. intros; destruct a. @@ -164,7 +165,7 @@ Section HaskProofToStrong. intros vars Σ ξ lev pf; refine (bind x2 = IHtypes2 vars Σ ξ lev pf; _). apply FreshMon. destruct x2 as [vt2 [pf21 [pf22 pfdist]]]. - refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_ξ ξ lev + refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_xi ξ lev (leaves vt2)) _ _; return _). apply FreshMon. simpl. @@ -204,8 +205,8 @@ Section HaskProofToStrong. Lemma fresh_lemma Γ ξ vars Σ Σ' lev : Σ = mapOptionTree ξ vars -> FreshM { vars' : _ - | mapOptionTree (update_ξ(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ - /\ mapOptionTree (update_ξ ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }. + | mapOptionTree (update_xi(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ + /\ mapOptionTree (update_xi ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }. intros. set (fresh_lemma' Γ [Σ'] vars Σ ξ lev H) as q. refine (q >>>= fun q' => return _). @@ -222,60 +223,64 @@ Section HaskProofToStrong. inversion pf2. Defined. - Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ : Type := - forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ). + Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ l : Type := + forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t l) τ). - Definition urule2expr : forall Γ Δ h j t (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★), - ujudg2exprType Γ ξ Δ h t -> - ujudg2exprType Γ ξ Δ j t + Definition urule2expr : forall Γ Δ h j t l (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★), + ujudg2exprType Γ ξ Δ h t l -> + ujudg2exprType Γ ξ Δ j t l . intros Γ Δ. - refine (fix urule2expr h j t (r:@Arrange _ h j) ξ {struct r} : - ujudg2exprType Γ ξ Δ h t -> - ujudg2exprType Γ ξ Δ j t := + refine (fix urule2expr h j t l (r:@Arrange _ h j) ξ {struct r} : + ujudg2exprType Γ ξ Δ h t l -> + ujudg2exprType Γ ξ Δ j t l := match r as R in Arrange H C return - ujudg2exprType Γ ξ Δ H t -> - ujudg2exprType Γ ξ Δ C t + ujudg2exprType Γ ξ Δ H t l -> + ujudg2exprType Γ ξ Δ C t l with - | RLeft h c ctx r => let case_RLeft := tt in (fun e => _) (urule2expr _ _ _ r) - | RRight h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ r) - | RCanL a => let case_RCanL := tt in _ - | RCanR a => let case_RCanR := tt in _ - | RuCanL a => let case_RuCanL := tt in _ - | RuCanR a => let case_RuCanR := tt in _ - | RAssoc a b c => let case_RAssoc := tt in _ - | RCossa a b c => let case_RCossa := tt in _ - | RExch a b => let case_RExch := tt in _ - | RWeak a => let case_RWeak := tt in _ - | RCont a => let case_RCont := tt in _ - | RComp a b c f g => let case_RComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ f) (urule2expr _ _ _ g) + | ALeft h c ctx r => let case_ALeft := tt in (fun e => _) (urule2expr _ _ _ _ r) + | ARight h c ctx r => let case_ARight := tt in (fun e => _) (urule2expr _ _ _ _ r) + | AId a => let case_AId := tt in _ + | ACanL a => let case_ACanL := tt in _ + | ACanR a => let case_ACanR := tt in _ + | AuCanL a => let case_AuCanL := tt in _ + | AuCanR a => let case_AuCanR := tt in _ + | AAssoc a b c => let case_AAssoc := tt in _ + | AuAssoc a b c => let case_AuAssoc := tt in _ + | AExch a b => let case_AExch := tt in _ + | AWeak a => let case_AWeak := tt in _ + | ACont a => let case_ACont := tt in _ + | AComp a b c f g => let case_AComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ _ f) (urule2expr _ _ _ _ g) end); clear urule2expr; intros. - destruct case_RCanL. + destruct case_AId. + apply X. + + destruct case_ACanL. simpl; unfold ujudg2exprType; intros. simpl in X. apply (X ([],,vars)). simpl; rewrite <- H; auto. - destruct case_RCanR. + destruct case_ACanR. simpl; unfold ujudg2exprType; intros. simpl in X. apply (X (vars,,[])). simpl; rewrite <- H; auto. - destruct case_RuCanL. + destruct case_AuCanL. simpl; unfold ujudg2exprType; intros. destruct vars; try destruct o; inversion H. simpl in X. apply (X vars2); auto. - destruct case_RuCanR. + destruct case_AuCanR. simpl; unfold ujudg2exprType; intros. destruct vars; try destruct o; inversion H. simpl in X. apply (X vars1); auto. - destruct case_RAssoc. + destruct case_AAssoc. simpl; unfold ujudg2exprType; intros. simpl in X. destruct vars; try destruct o; inversion H. @@ -283,7 +288,7 @@ Section HaskProofToStrong. apply (X (vars1_1,,(vars1_2,,vars2))). subst; auto. - destruct case_RCossa. + destruct case_AuAssoc. simpl; unfold ujudg2exprType; intros. simpl in X. destruct vars; try destruct o; inversion H. @@ -291,20 +296,20 @@ Section HaskProofToStrong. apply (X ((vars1,,vars2_1),,vars2_2)). subst; auto. - destruct case_RExch. + destruct case_AExch. simpl; unfold ujudg2exprType ; intros. simpl in X. destruct vars; try destruct o; inversion H. apply (X (vars2,,vars1)). inversion H; subst; auto. - destruct case_RWeak. + destruct case_AWeak. simpl; unfold ujudg2exprType; intros. simpl in X. apply (X []). auto. - destruct case_RCont. + destruct case_ACont. simpl; unfold ujudg2exprType ; intros. simpl in X. apply (X (vars,,vars)). @@ -312,7 +317,7 @@ Section HaskProofToStrong. rewrite <- H. auto. - destruct case_RLeft. + destruct case_ALeft. intro vars; unfold ujudg2exprType; intro H. destruct vars; try destruct o; inversion H. apply (fun q => e ξ q vars2 H2). @@ -327,7 +332,7 @@ Section HaskProofToStrong. simpl. reflexivity. - destruct case_RRight. + destruct case_ARight. intro vars; unfold ujudg2exprType; intro H. destruct vars; try destruct o; inversion H. apply (fun q => e ξ q vars1 H1). @@ -342,16 +347,16 @@ Section HaskProofToStrong. simpl. reflexivity. - destruct case_RComp. + destruct case_AComp. apply e2. apply e1. apply X. Defined. Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' : - ITree (LeveledHaskType Γ ★) - (fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t) - (mapOptionTree (ξ' ○ (@fst _ _)) varstypes) + ITree (HaskType Γ ★) + (fun t : HaskType Γ ★ => Expr Γ Δ ξ' t l) + (mapOptionTree (unlev ○ ξ' ○ (@fst _ _)) varstypes) -> ELetRecBindings Γ Δ ξ' l varstypes. intros. induction varstypes. @@ -367,6 +372,8 @@ Section HaskProofToStrong. simpl. destruct (eqd_dec h0 l). rewrite <- e0. + simpl in X. + subst. apply X. apply (Prelude_error "level mismatch; should never happen"). apply (Prelude_error "letrec type mismatch; should never happen"). @@ -415,7 +422,8 @@ Section HaskProofToStrong. prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} -> ((fun sac => FreshM { scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac - & Expr (sac_Γ sac Γ) (sac_Δ sac Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev)) }) (projT1 pcb)). + & Expr (sac_gamma sac Γ) (sac_delta sac Γ avars (weakCK'' Δ)) (scbwv_xi scb ξ lev) + (weakT' tbranches) (weakL' lev) }) (projT1 pcb)). intro pcb. intro X. simpl in X. @@ -441,10 +449,10 @@ Section HaskProofToStrong. cut (distinct (vec2list localvars'')). intro H'''. set (@Build_StrongCaseBranchWithVVs _ _ _ _ avars sac localvars'' H''') as scb. - refine (bind q = (f (scbwv_ξ scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _). + refine (bind q = (f (scbwv_xi scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _). apply FreshMon. simpl. - unfold scbwv_ξ. + unfold scbwv_xi. rewrite vars_pf. rewrite <- mapOptionTree_compose. clear localvars_pf1. @@ -501,13 +509,249 @@ Section HaskProofToStrong. Defined. + Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★), + FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }. + intros Γ Σ. + induction Σ; intro ξ. + destruct a. + destruct l as [τ l]. + set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q. + refine (q >>>= fun q' => return _). + apply FreshMon. + clear q. + destruct q' as [varstypes [pf1 [pf2 distpf]]]. + exists (mapOptionTree (@fst _ _) varstypes). + exists (update_xi ξ l (leaves varstypes)). + symmetry; auto. + refine (return _). + exists []. + exists ξ; auto. + refine (bind f1 = IHΣ1 ξ ; _). + apply FreshMon. + destruct f1 as [vars1 [ξ1 pf1]]. + refine (bind f2 = IHΣ2 ξ1 ; _). + apply FreshMon. + destruct f2 as [vars2 [ξ2 pf22]]. + refine (return _). + exists (vars1,,vars2). + exists ξ2. + simpl. + rewrite pf22. + rewrite pf1. + admit. (* freshness assumption *) + Defined. + + Definition rlet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p : + forall (X_ : ITree Judg judg2exprType + ([Γ > Δ > Σ₁ |- [σ₁] @ p],, [Γ > Δ > [σ₁ @@ p],, Σ₂ |- [σ₂] @ p])), + ITree Judg judg2exprType [Γ > Δ > Σ₁,, Σ₂ |- [σ₂] @ p]. + intros. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + + refine (fresh_lemma _ ξ _ _ σ₁ p H2 >>>= (fun pf => _)). + apply FreshMon. + + destruct pf as [ vnew [ pf1 pf2 ]]. + set (update_xi ξ p (((vnew, σ₁ )) :: nil)) as ξ' in *. + inversion X_. + apply ileaf in X. + apply ileaf in X0. + simpl in *. + + refine (X ξ vars1 _ >>>= fun X0' => _). + apply FreshMon. + simpl. + auto. + + refine (X0 ξ' ([vnew],,vars2) _ >>>= fun X1' => _). + apply FreshMon. + simpl. + rewrite pf2. + rewrite pf1. + reflexivity. + apply FreshMon. + + apply ILeaf. + apply ileaf in X1'. + apply ileaf in X0'. + simpl in *. + apply ELet with (ev:=vnew)(tv:=σ₁). + apply X0'. + apply X1'. + Defined. + + Definition vartree Γ Δ Σ lev ξ : + forall vars, Σ @@@ lev = mapOptionTree ξ vars -> + ITree (HaskType Γ ★) (fun t : HaskType Γ ★ => Expr Γ Δ ξ t lev) Σ. + induction Σ; intros. + destruct a. + intros; simpl in *. + apply ILeaf. + destruct vars; try destruct o; inversion H. + set (EVar Γ Δ ξ v) as q. + rewrite <- H1 in q. + apply q. + intros. + apply INone. + intros. + destruct vars; try destruct o; inversion H. + apply IBranch. + eapply IHΣ1. + apply H1. + eapply IHΣ2. + apply H2. + Defined. + + + Definition rdrop Γ Δ Σ₁ Σ₁₂ a lev : + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a,,Σ₁₂ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *. + intros. + set (X ξ vars H) as q. + simpl in q. + refine (q >>>= fun q' => return _). + apply FreshMon. + inversion q'. + apply X0. + Defined. + + Definition rdrop' Γ Δ Σ₁ Σ₁₂ a lev : + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂,,a @ lev] -> + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *. + intros. + set (X ξ vars H) as q. + simpl in q. + refine (q >>>= fun q' => return _). + apply FreshMon. + inversion q'. + auto. + Defined. + + Definition rdrop'' Γ Δ Σ₁ Σ₁₂ lev : + ITree Judg judg2exprType [Γ > Δ > [],,Σ₁ |- Σ₁₂ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + eapply X with (vars:=[],,vars). + rewrite H; reflexivity. + Defined. + + Definition rdrop''' Γ Δ a Σ₁ Σ₁₂ lev : + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > a,,Σ₁ |- Σ₁₂ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + eapply X with (vars:=vars2). + auto. + Defined. + + Definition rassoc Γ Δ Σ₁ a b c lev : + ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + destruct vars2; try destruct o; inversion H2. + apply X with (vars:=(vars1,,vars2_1),,vars2_2). + subst; reflexivity. + Defined. + + Definition rassoc' Γ Δ Σ₁ a b c lev : + ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + destruct vars1; try destruct o; inversion H1. + apply X with (vars:=vars1_1,,(vars1_2,,vars2)). + subst; reflexivity. + Defined. + + Definition swapr Γ Δ Σ₁ a b c lev : + ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > ((b,,a),,c) |- Σ₁ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + destruct vars1; try destruct o; inversion H1. + apply X with (vars:=(vars1_2,,vars1_1),,vars2). + subst; reflexivity. + Defined. + + Definition rdup Γ Δ Σ₁ a c lev : + ITree Judg judg2exprType [Γ > Δ > ((a,,a),,c) |- Σ₁ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > (a,,c) |- Σ₁ @ lev]. + intros. + apply ileaf in X. + apply ILeaf. + simpl in *; intros. + destruct vars; try destruct o; inversion H. + apply X with (vars:=(vars1,,vars1),,vars2). (* is this allowed? *) + subst; reflexivity. + Defined. + + (* holy cow this is ugly *) + Definition rcut Γ Δ Σ₃ lev Σ₁₂ : + forall Σ₁ Σ₂, + ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] -> + ITree Judg judg2exprType [Γ > Δ > Σ₁₂ @@@ lev,,Σ₂ |- [Σ₃] @ lev] -> + ITree Judg judg2exprType [Γ > Δ > Σ₁,,Σ₂ |- [Σ₃] @ lev]. + + induction Σ₁₂. + intros. + destruct a. + + eapply rlet. + apply IBranch. + apply X. + apply X0. + + simpl in X0. + apply rdrop'' in X0. + apply rdrop'''. + apply X0. + + intros. + simpl in X0. + apply rassoc in X0. + set (IHΣ₁₂1 _ _ (rdrop _ _ _ _ _ _ X) X0) as q. + set (IHΣ₁₂2 _ (Σ₁,,Σ₂) (rdrop' _ _ _ _ _ _ X)) as q'. + apply rassoc' in q. + apply swapr in q. + apply rassoc in q. + set (q' q) as q''. + apply rassoc' in q''. + apply rdup in q''. + apply q''. + Defined. Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j. intros h j r. refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with - | RArrange a b c d e r => let case_RURule := tt in _ + | RArrange a b c d e l r => let case_RURule := tt in _ | RNote Γ Δ Σ τ l n => let case_RNote := tt in _ | RLit Γ Δ l _ => let case_RLit := tt in _ | RVar Γ Δ σ p => let case_RVar := tt in _ @@ -520,8 +764,11 @@ Section HaskProofToStrong. | RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _ | RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _ | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _ - | RJoin Γ p lri m x q => let case_RJoin := tt in _ - | RVoid _ _ => let case_RVoid := tt in _ + | RCut Γ Δ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _ + | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _ + | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := tt in _ + | RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ p => let case_RWhere := tt in _ + | RVoid _ _ l => let case_RVoid := tt in _ | RBrak Σ a b c n m => let case_RBrak := tt in _ | REsc Σ a b c n m => let case_REsc := tt in _ | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _ @@ -530,12 +777,10 @@ Section HaskProofToStrong. destruct case_RURule. apply ILeaf. simpl. intros. - set (@urule2expr a b _ _ e r0 ξ) as q. - set (fun z => q z) as q'. - simpl in q'. - apply q' with (vars:=vars). - clear q' q. + set (@urule2expr a b _ _ e l r0 ξ) as q. unfold ujudg2exprType. + unfold ujudg2exprType in q. + apply q with (vars:=vars). intros. apply X_ with (vars:=vars0). auto. @@ -562,8 +807,11 @@ Section HaskProofToStrong. destruct case_RVar. apply ILeaf; simpl; intros; refine (return ILeaf _ _). - destruct vars; simpl in H; inversion H; destruct o. inversion H1. rewrite H2. - apply EVar. + destruct vars; simpl in H; inversion H; destruct o. inversion H1. + set (@EVar _ _ _ Δ ξ v) as q. + rewrite <- H2 in q. + simpl in q. + apply q. inversion H. destruct case_RGlobal. @@ -576,7 +824,7 @@ Section HaskProofToStrong. refine (fresh_lemma _ ξ vars _ tx x H >>>= (fun pf => _)). apply FreshMon. destruct pf as [ vnew [ pf1 pf2 ]]. - set (update_ξ ξ x (((vnew, tx )) :: nil)) as ξ' in *. + set (update_xi ξ x (((vnew, tx )) :: nil)) as ξ' in *. refine (X_ ξ' (vars,,[vnew]) _ >>>= _). apply FreshMon. simpl. @@ -600,19 +848,6 @@ Section HaskProofToStrong. apply ileaf in X. simpl in X. apply X. - destruct case_RJoin. - apply ILeaf; simpl; intros. - inversion X_. - apply ileaf in X. - apply ileaf in X0. - simpl in *. - destruct vars; inversion H. - destruct o; inversion H3. - refine (X ξ vars1 H3 >>>= fun X' => X0 ξ vars2 H4 >>>= fun X0' => return _). - apply FreshMon. - apply FreshMon. - apply IBranch; auto. - destruct case_RApp. apply ILeaf. inversion X_. @@ -632,41 +867,102 @@ Section HaskProofToStrong. apply ileaf in q1'. apply ileaf in q2'. simpl in *. - apply (EApp _ _ _ _ _ _ q2' q1'). + apply (EApp _ _ _ _ _ _ q1' q2'). destruct case_RLet. + eapply rlet. + apply X_. + + destruct case_RWhere. apply ILeaf. simpl in *; intros. - destruct vars; try destruct o; inversion H. - refine (fresh_lemma _ ξ vars1 _ σ₂ p H1 >>>= (fun pf => _)). + destruct vars; try destruct o; inversion H. + destruct vars2; try destruct o; inversion H2. + clear H2. + + assert ((Σ₁,,Σ₃) = mapOptionTree ξ (vars1,,vars2_2)) as H13; simpl; subst; auto. + refine (fresh_lemma _ ξ _ _ σ₁ p H13 >>>= (fun pf => _)). apply FreshMon. + destruct pf as [ vnew [ pf1 pf2 ]]. - set (update_ξ ξ p (((vnew, σ₂ )) :: nil)) as ξ' in *. + set (update_xi ξ p (((vnew, σ₁ )) :: nil)) as ξ' in *. inversion X_. apply ileaf in X. apply ileaf in X0. simpl in *. - refine (X ξ vars2 _ >>>= fun X0' => _). - apply FreshMon. - auto. - refine (X0 ξ' (vars1,,[vnew]) _ >>>= fun X1' => _). + refine (X ξ' (vars1,,([vnew],,vars2_2)) _ >>>= fun X0' => _). apply FreshMon. - rewrite H1. simpl. + inversion pf1. + rewrite H3. + rewrite H4. rewrite pf2. - rewrite pf1. - rewrite H1. reflexivity. - refine (return _). + refine (X0 ξ vars2_1 _ >>>= fun X1' => _). + apply FreshMon. + reflexivity. + apply FreshMon. + apply ILeaf. apply ileaf in X0'. apply ileaf in X1'. simpl in *. - apply ELet with (ev:=vnew)(tv:=σ₂). - apply X0'. + apply ELet with (ev:=vnew)(tv:=σ₁). apply X1'. + apply X0'. + + destruct case_RCut. + inversion X_. + subst. + clear X_. + induction Σ₃. + destruct a. + subst. + eapply rcut. + apply X. + apply X0. + + apply ILeaf. + simpl. + intros. + refine (return _). + apply INone. + set (IHΣ₃1 (rdrop _ _ _ _ _ _ X0)) as q1. + set (IHΣ₃2 (rdrop' _ _ _ _ _ _ X0)) as q2. + apply ileaf in q1. + apply ileaf in q2. + simpl in *. + apply ILeaf. + simpl. + intros. + refine (q1 _ _ H >>>= fun q1' => q2 _ _ H >>>= fun q2' => return _). + apply FreshMon. + apply FreshMon. + apply IBranch; auto. + + destruct case_RLeft. + apply ILeaf. + simpl; intros. + destruct vars; try destruct o; inversion H. + refine (X_ _ _ H2 >>>= fun X' => return _). + apply FreshMon. + apply IBranch. + eapply vartree. + apply H1. + apply X'. + + destruct case_RRight. + apply ILeaf. + simpl; intros. + destruct vars; try destruct o; inversion H. + refine (X_ _ _ H1 >>>= fun X' => return _). + apply FreshMon. + apply IBranch. + apply X'. + eapply vartree. + apply H2. destruct case_RVoid. apply ILeaf; simpl; intros. @@ -704,7 +1000,7 @@ Section HaskProofToStrong. apply ILeaf; simpl; intros. refine (bind ξvars = fresh_lemma' _ y _ _ _ t H; _). apply FreshMon. destruct ξvars as [ varstypes [ pf1[ pf2 pfdist]]]. - refine (X_ ((update_ξ ξ t (leaves varstypes))) + refine (X_ ((update_xi ξ t (leaves varstypes))) (vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_. apply FreshMon. simpl. rewrite pf2. @@ -716,11 +1012,19 @@ Section HaskProofToStrong. apply (@ELetRec _ _ _ _ _ _ _ varstypes). auto. apply (@letrec_helper Γ Δ t varstypes). - rewrite <- pf2 in X1. rewrite mapOptionTree_compose. + rewrite mapOptionTree_compose. + rewrite pf2. + replace ((mapOptionTree unlev (y @@@ t))) with y. + apply X0. + clear pf1 X0 X1 pfdist pf2 vars varstypes. + induction y; try destruct a; auto. + rewrite IHy1 at 1. + rewrite IHy2 at 1. + reflexivity. + apply ileaf in X1. + simpl in X1. apply X1. - apply ileaf in X0. - apply X0. destruct case_RCase. apply ILeaf; simpl; intros. @@ -765,41 +1069,9 @@ Section HaskProofToStrong. | scnd_branch _ _ _ c1 c2 => let case_branch := tt in fun q => IBranch _ _ (closed2expr _ _ c1 q) (closed2expr _ _ c2 q) end. - Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★), - FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }. - intros Γ Σ. - induction Σ; intro ξ. - destruct a. - destruct l as [τ l]. - set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q. - refine (q >>>= fun q' => return _). - apply FreshMon. - clear q. - destruct q' as [varstypes [pf1 [pf2 distpf]]]. - exists (mapOptionTree (@fst _ _) varstypes). - exists (update_ξ ξ l (leaves varstypes)). - symmetry; auto. - refine (return _). - exists []. - exists ξ; auto. - refine (bind f1 = IHΣ1 ξ ; _). - apply FreshMon. - destruct f1 as [vars1 [ξ1 pf1]]. - refine (bind f2 = IHΣ2 ξ1 ; _). - apply FreshMon. - destruct f2 as [vars2 [ξ2 pf22]]. - refine (return _). - exists (vars1,,vars2). - exists ξ2. - simpl. - rewrite pf22. - rewrite pf1. - admit. - Defined. - - Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★) - {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] -> - FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}). + Definition proof2expr Γ Δ τ l Σ (ξ0: VV -> LeveledHaskType Γ ★) + {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ] @ l] -> + FreshM (???{ ξ : _ & Expr Γ Δ ξ τ l}). intro pf. set (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) as cnd. apply closed2expr in cnd. @@ -815,7 +1087,9 @@ Section HaskProofToStrong. auto. refine (return OK _). exists ξ. - apply (ileaf it). + apply ileaf in it. + simpl in it. + apply it. apply INone. Defined.