X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskFlattener.v;h=a5f42618b89ea2de829e7cb8e842ee82870c7510;hp=b46f4d22abd3ede84f4bc290ba5bc0b8073a8329;hb=bebffa435dbc5afd126f6972fbf220977455854d;hpb=2da83e6cfd727f142489859160b7d57bfa80a3be diff --git a/src/HaskFlattener.v b/src/HaskFlattener.v index b46f4d2..a5f4261 100644 --- a/src/HaskFlattener.v +++ b/src/HaskFlattener.v @@ -30,6 +30,8 @@ Require Import HaskStrongToProof. Require Import HaskProofToStrong. Require Import HaskWeakToStrong. +Require Import HaskSkolemizer. + Open Scope nd_scope. Set Printing Width 130. @@ -43,6 +45,9 @@ Set Printing Width 130. *) Section HaskFlattener. + Definition getlev {Γ}{κ}(lht:LeveledHaskType Γ κ) : HaskLevel Γ := + match lht with t @@ l => l end. + Definition arrange : forall {T} (Σ:Tree ??T) (f:T -> bool), Arrange Σ (dropT (mkFlags (liftBoolFunc false f) Σ),,( (dropT (mkFlags (liftBoolFunc false (bnot ○ f)) Σ)))). @@ -94,85 +99,7 @@ Section HaskFlattener. [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ] end. - Context (hetmet_flatten : WeakExprVar). - Context (hetmet_unflatten : WeakExprVar). - Context (hetmet_id : WeakExprVar). - Context {unitTy : forall TV, RawHaskType TV ★ }. - Context {prodTy : forall TV, RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }. - Context {gaTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }. - - Definition ga_mk_tree {Γ} (tr:Tree ??(HaskType Γ ★)) : HaskType Γ ★ := - fun TV ite => reduceTree (unitTy TV) (prodTy TV) (mapOptionTree (fun x => x TV ite) tr). - - Definition ga_mk {Γ}(ec:HaskType Γ ECKind )(ant suc:Tree ??(HaskType Γ ★)) : HaskType Γ ★ := - fun TV ite => gaTy TV (ec TV ite) (ga_mk_tree ant TV ite) (ga_mk_tree suc TV ite). - - Class garrow := - { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a @@ l] ] - ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a @@ l] ] - ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a @@ l] ] - ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) @@ l] ] - ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) @@ l] ] - ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ] - ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ] - ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) @@ l] ] - ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] @@ l] ] - ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) @@ l] ] - ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (a,,x) (b,,x) @@ l] ] - ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (x,,a) (x,,b) @@ l] ] - ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] @@ l] ] - ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] @@ l] ] - ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c @@ l] ] - ; ga_apply : ∀ Γ Δ ec l a a' b c, ND Rule [] - [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] @@ l] ] - ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule - [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ] - [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ] - }. - Context `(gar:garrow). - - Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20). - - (* - * The story: - * - code types <[t]>@c become garrows c () t - * - free variables of type t at a level lev deeper than the succedent become garrows c () t - * - free variables at the level of the succedent become - *) - Fixpoint garrowfy_raw_codetypes {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ := - match exp with - | TVar _ x => TVar x - | TAll _ y => TAll _ (fun v => garrowfy_raw_codetypes (y v)) - | TApp _ _ x y => TApp (garrowfy_raw_codetypes x) (garrowfy_raw_codetypes y) - | TCon tc => TCon tc - | TCoerc _ t1 t2 t => TCoerc (garrowfy_raw_codetypes t1) (garrowfy_raw_codetypes t2) - (garrowfy_raw_codetypes t) - | TArrow => TArrow - | TCode v e => gaTy TV v (unitTy TV) (garrowfy_raw_codetypes e) - | TyFunApp tfc kl k lt => TyFunApp tfc kl k (garrowfy_raw_codetypes_list _ lt) - end - with garrowfy_raw_codetypes_list {TV}(lk:list Kind)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk := - match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with - | TyFunApp_nil => TyFunApp_nil - | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (garrowfy_raw_codetypes t) (garrowfy_raw_codetypes_list _ rest) - end. - Definition garrowfy_code_types {Γ}{κ}(ht:HaskType Γ κ) : HaskType Γ κ := - fun TV ite => - garrowfy_raw_codetypes (ht TV ite). - - Definition v2t {Γ}(ec:HaskTyVar Γ ECKind) := fun TV ite => TVar (ec TV ite). - - Fixpoint garrowfy_leveled_code_types' {Γ}(ht:HaskType Γ ★)(lev:HaskLevel Γ) : HaskType Γ ★ := - match lev with - | nil => garrowfy_code_types ht - | ec::lev' => @ga_mk _ (v2t ec) [] [garrowfy_leveled_code_types' ht lev'] - end. - - Definition garrowfy_leveled_code_types {Γ}(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ := - match ht with - htt @@ lev => - garrowfy_leveled_code_types' htt lev @@ nil - end. + Definition v2t {Γ}(ec:HaskTyVar Γ ECKind) : HaskType Γ ECKind := fun TV ite => TVar (ec TV ite). Definition levelMatch {Γ}(lev:HaskLevel Γ) : LeveledHaskType Γ ★ -> bool := fun t => match t with ttype@@tlev => if eqd_dec tlev lev then true else false end. @@ -188,10 +115,10 @@ Section HaskFlattener. Definition mkTakeFlags {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : TreeFlags tt := mkFlags (liftBoolFunc true (bnot ○ levelMatch lev)) tt. - Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) := - mapOptionTree (fun x => garrowfy_code_types (unlev x)) (dropT (mkTakeFlags lev tt)). + Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) := + dropT (mkTakeFlags lev tt). (* - mapOptionTree (fun x => garrowfy_code_types (unlev x)) + mapOptionTree (fun x => flatten_type (unlev x)) (maybeTree (takeT tt (mkFlags ( fun t => match t with | Some (ttype @@ tlev) => if eqd_dec tlev lev then true else false @@ -229,7 +156,7 @@ Section HaskFlattener. auto. Qed. - Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [garrowfy_code_types x]. + Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [x @@ lev]. intros; simpl. Opaque eqd_dec. unfold take_lev. @@ -240,10 +167,39 @@ Section HaskFlattener. auto. Qed. + Lemma take_lemma' : forall Γ (lev:HaskLevel Γ) x, take_lev lev (x @@@ lev) = x @@@ lev. + intros. + induction x. + destruct a; simpl; try reflexivity. + apply take_lemma. + simpl. + rewrite <- IHx1 at 2. + rewrite <- IHx2 at 2. + reflexivity. + Qed. +(* + Lemma drop_lev_lemma' : forall Γ (lev:HaskLevel Γ) x, drop_lev lev (x @@@ lev) = []. + intros. + induction x. + destruct a; simpl; try reflexivity. + apply drop_lev_lemma. + simpl. + change (@drop_lev _ lev (x1 @@@ lev ,, x2 @@@ lev)) + with ((@drop_lev _ lev (x1 @@@ lev)) ,, (@drop_lev _ lev (x2 @@@ lev))). + simpl. + rewrite IHx1. + rewrite IHx2. + reflexivity. + Qed. +*) Ltac drop_simplify := match goal with | [ |- context[@drop_lev ?G ?L [ ?X @@ ?L ] ] ] => rewrite (drop_lev_lemma G L X) +(* + | [ |- context[@drop_lev ?G ?L [ ?X @@@ ?L ] ] ] => + rewrite (drop_lev_lemma' G L X) +*) | [ |- context[@drop_lev ?G (?E :: ?L) [ ?X @@ (?E :: ?L) ] ] ] => rewrite (drop_lev_lemma_s G L E X) | [ |- context[@drop_lev ?G ?N (?A,,?B)] ] => @@ -256,38 +212,101 @@ Section HaskFlattener. match goal with | [ |- context[@take_lev ?G ?L [ ?X @@ ?L ] ] ] => rewrite (take_lemma G L X) + | [ |- context[@take_lev ?G ?L [ ?X @@@ ?L ] ] ] => + rewrite (take_lemma' G L X) | [ |- context[@take_lev ?G ?N (?A,,?B)] ] => change (@take_lev G N (A,,B)) with ((@take_lev G N A),,(@take_lev G N B)) | [ |- context[@take_lev ?G ?N (T_Leaf None)] ] => change (@take_lev G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★))) end. + + (*******************************************************************************) + + + Context (hetmet_flatten : WeakExprVar). + Context (hetmet_unflatten : WeakExprVar). + Context (hetmet_id : WeakExprVar). + Context {unitTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ }. + Context {prodTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }. + Context {gaTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }. + + Definition ga_mk_tree' {TV}(ec:RawHaskType TV ECKind)(tr:Tree ??(RawHaskType TV ★)) : RawHaskType TV ★ := + reduceTree (unitTy TV ec) (prodTy TV ec) tr. + + Definition ga_mk_tree {Γ}(ec:HaskType Γ ECKind)(tr:Tree ??(HaskType Γ ★)) : HaskType Γ ★ := + fun TV ite => ga_mk_tree' (ec TV ite) (mapOptionTree (fun x => x TV ite) tr). + + Definition ga_mk_raw {TV}(ec:RawHaskType TV ECKind)(ant suc:Tree ??(RawHaskType TV ★)) : RawHaskType TV ★ := + gaTy TV ec + (ga_mk_tree' ec ant) + (ga_mk_tree' ec suc). + + Definition ga_mk {Γ}(ec:HaskType Γ ECKind)(ant suc:Tree ??(HaskType Γ ★)) : HaskType Γ ★ := + fun TV ite => gaTy TV (ec TV ite) (ga_mk_tree ec ant TV ite) (ga_mk_tree ec suc TV ite). + + (* + * The story: + * - code types <[t]>@c become garrows c () t + * - free variables of type t at a level lev deeper than the succedent become garrows c () t + * - free variables at the level of the succedent become + *) + Fixpoint flatten_rawtype {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ := + match exp with + | TVar _ x => TVar x + | TAll _ y => TAll _ (fun v => flatten_rawtype (y v)) + | TApp _ _ x y => TApp (flatten_rawtype x) (flatten_rawtype y) + | TCon tc => TCon tc + | TCoerc _ t1 t2 t => TCoerc (flatten_rawtype t1) (flatten_rawtype t2) (flatten_rawtype t) + | TArrow => TArrow + | TCode ec e => let e' := flatten_rawtype e + in ga_mk_raw ec (unleaves_ (take_arg_types e')) [drop_arg_types e'] + | TyFunApp tfc kl k lt => TyFunApp tfc kl k (flatten_rawtype_list _ lt) + end + with flatten_rawtype_list {TV}(lk:list Kind)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk := + match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with + | TyFunApp_nil => TyFunApp_nil + | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flatten_rawtype t) (flatten_rawtype_list _ rest) + end. + + Definition flatten_type {Γ}{κ}(ht:HaskType Γ κ) : HaskType Γ κ := + fun TV ite => flatten_rawtype (ht TV ite). + + Fixpoint levels_to_tcode {Γ}(ht:HaskType Γ ★)(lev:HaskLevel Γ) : HaskType Γ ★ := + match lev with + | nil => flatten_type ht + | ec::lev' => @ga_mk _ (v2t ec) [] [levels_to_tcode ht lev'] + end. + + Definition flatten_leveled_type {Γ}(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ := + levels_to_tcode (unlev ht) (getlev ht) @@ nil. + (* AXIOMS *) - Axiom literal_types_unchanged : forall Γ l, garrowfy_code_types (literalType l) = literalType(Γ:=Γ) l. + Axiom literal_types_unchanged : forall Γ l, flatten_type (literalType l) = literalType(Γ:=Γ) l. Axiom flatten_coercion : forall Γ Δ κ (σ τ:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)), - HaskCoercion Γ Δ (garrowfy_code_types σ ∼∼∼ garrowfy_code_types τ). + HaskCoercion Γ Δ (flatten_type σ ∼∼∼ flatten_type τ). - Axiom garrowfy_commutes_with_substT : + Axiom flatten_commutes_with_substT : forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★) (τ:HaskType Γ κ), - garrowfy_code_types (substT σ τ) = substT (fun TV ite v => garrowfy_raw_codetypes (σ TV ite v)) - (garrowfy_code_types τ). + flatten_type (substT σ τ) = substT (fun TV ite v => flatten_rawtype (σ TV ite v)) + (flatten_type τ). - Axiom garrowfy_commutes_with_HaskTAll : + Axiom flatten_commutes_with_HaskTAll : forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★), - garrowfy_code_types (HaskTAll κ σ) = HaskTAll κ (fun TV ite v => garrowfy_raw_codetypes (σ TV ite v)). + flatten_type (HaskTAll κ σ) = HaskTAll κ (fun TV ite v => flatten_rawtype (σ TV ite v)). - Axiom garrowfy_commutes_with_HaskTApp : + Axiom flatten_commutes_with_HaskTApp : forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★), - garrowfy_code_types (HaskTApp (weakF σ) (FreshHaskTyVar κ)) = - HaskTApp (weakF (fun TV ite v => garrowfy_raw_codetypes (σ TV ite v))) (FreshHaskTyVar κ). + flatten_type (HaskTApp (weakF σ) (FreshHaskTyVar κ)) = + HaskTApp (weakF (fun TV ite v => flatten_rawtype (σ TV ite v))) (FreshHaskTyVar κ). - Axiom garrowfy_commutes_with_weakLT : forall (Γ:TypeEnv) κ t, - garrowfy_leveled_code_types (weakLT(Γ:=Γ)(κ:=κ) t) = weakLT(Γ:=Γ)(κ:=κ) (garrowfy_leveled_code_types t). + Axiom flatten_commutes_with_weakLT : forall (Γ:TypeEnv) κ t, + flatten_leveled_type (weakLT(Γ:=Γ)(κ:=κ) t) = weakLT(Γ:=Γ)(κ:=κ) (flatten_leveled_type t). Axiom globals_do_not_have_code_types : forall (Γ:TypeEnv) (g:Global Γ) v, - garrowfy_code_types (g v) = g v. + flatten_type (g v) = g v. (* This tries to assign a single level to the entire succedent of a judgment. If the succedent has types from different * levels (should not happen) it just picks one; if the succedent has no non-None leaves (also should not happen) it @@ -306,25 +325,48 @@ Section HaskFlattener. end end. - Definition unlev' {Γ} (x:LeveledHaskType Γ ★) := - garrowfy_code_types (unlev x). - (* "n" is the maximum depth remaining AFTER flattening *) Definition flatten_judgment (j:Judg) := match j as J return Judg with Γ > Δ > ant |- suc => match getjlev suc with - | nil => Γ > Δ > mapOptionTree garrowfy_leveled_code_types ant - |- mapOptionTree garrowfy_leveled_code_types suc + | nil => Γ > Δ > mapOptionTree flatten_leveled_type ant + |- mapOptionTree flatten_leveled_type suc - | (ec::lev') => Γ > Δ > mapOptionTree garrowfy_leveled_code_types (drop_lev (ec::lev') ant) + | (ec::lev') => Γ > Δ > mapOptionTree flatten_leveled_type (drop_lev (ec::lev') ant) |- [ga_mk (v2t ec) - (take_lev (ec::lev') ant) - (mapOptionTree unlev' suc) (* we know the level of all of suc *) - @@ nil] + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec::lev') ant)) + (mapOptionTree (flatten_type ○ unlev) suc ) + @@ nil] (* we know the level of all of suc *) end end. + Class garrow := + { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a @@ l] ] + ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a @@ l] ] + ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a @@ l] ] + ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) @@ l] ] + ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) @@ l] ] + ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ] + ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ] + ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) @@ l] ] + ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] @@ l] ] + ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) @@ l] ] + ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (a,,x) (b,,x) @@ l] ] + ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (x,,a) (x,,b) @@ l] ] + ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] @@ l] ] + ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] @@ l] ] + ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c @@ l] ] + ; ga_apply : ∀ Γ Δ ec l a a' b c, + ND Rule [] [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] @@ l] ] + ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule + [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ] + [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ] + }. + Context `(gar:garrow). + + Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20). + Definition boost : forall Γ Δ ant x y {lev}, ND Rule [] [ Γ > Δ > [x@@lev] |- [y@@lev] ] -> ND Rule [ Γ > Δ > ant |- [x@@lev] ] [ Γ > Δ > ant |- [y@@lev] ]. @@ -420,15 +462,15 @@ Section HaskFlattener. apply ga_second. Defined. - Lemma ga_unkappa : ∀ Γ Δ ec l a b Σ, + Lemma ga_unkappa : ∀ Γ Δ ec l z a b Σ, ND Rule - [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ] - [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ]. + [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ] + [Γ > Δ > Σ,,[@ga_mk Γ ec z a @@ l] |- [@ga_mk Γ ec z b @@ l] ]. intros. - set (ga_comp Γ Δ ec l [] a b) as q. + set (ga_comp Γ Δ ec l z a b) as q. set (@RLet Γ Δ) as q'. - set (@RLet Γ Δ [@ga_mk _ ec [] a @@ l] Σ (@ga_mk _ ec [] b) (@ga_mk _ ec a b) l) as q''. + set (@RLet Γ Δ [@ga_mk _ ec z a @@ l] Σ (@ga_mk _ ec z b) (@ga_mk _ ec a b) l) as q''. eapply nd_comp. Focus 2. eapply nd_rule. @@ -449,23 +491,54 @@ Section HaskFlattener. apply q. Defined. + Lemma ga_unkappa' : ∀ Γ Δ ec l a b Σ x, + ND Rule + [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b @@ l] ] + [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b @@ l] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply ga_first. + + eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply postcompose. + apply ga_uncancell. + apply precompose. + Defined. + + Lemma ga_kappa' : ∀ Γ Δ ec l a b Σ x, + ND Rule + [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b @@ l] ] + [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b @@ l] ]. + apply (Prelude_error "ga_kappa not supported yet (BIG FIXME)"). + Defined. + (* useful for cutting down on the pretty-printed noise Notation "` x" := (take_lev _ x) (at level 20). Notation "`` x" := (mapOptionTree unlev x) (at level 20). Notation "``` x" := (drop_lev _ x) (at level 20). *) - Definition garrowfy_arrangement' : + Definition flatten_arrangement' : forall Γ (Δ:CoercionEnv Γ) (ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2), - ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ nil] ]. + ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2)) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil] ]. intros Γ Δ ec lev. - refine (fix garrowfy ant1 ant2 (r:Arrange ant1 ant2): - ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ nil]] := + refine (fix flatten ant1 ant2 (r:Arrange ant1 ant2): + ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2)) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil]] := match r as R in Arrange A B return - ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ nil]] + ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) B)) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) A)) @@ nil]] with + | RId a => let case_RId := tt in ga_id _ _ _ _ _ | RCanL a => let case_RCanL := tt in ga_uncancell _ _ _ _ _ | RCanR a => let case_RCanR := tt in ga_uncancelr _ _ _ _ _ | RuCanL a => let case_RuCanL := tt in ga_cancell _ _ _ _ _ @@ -475,15 +548,15 @@ Section HaskFlattener. | RExch a b => let case_RExch := tt in ga_swap _ _ _ _ _ _ | RWeak a => let case_RWeak := tt in ga_drop _ _ _ _ _ | RCont a => let case_RCont := tt in ga_copy _ _ _ _ _ - | RLeft a b c r' => let case_RLeft := tt in garrowfy _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _) - | RRight a b c r' => let case_RRight := tt in garrowfy _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _) - | RComp c b a r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (garrowfy _ _ r1) (garrowfy _ _ r2) - end); clear garrowfy; repeat take_simplify; repeat drop_simplify; intros. + | RLeft a b c r' => let case_RLeft := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _) + | RRight a b c r' => let case_RRight := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _) + | RComp c b a r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (flatten _ _ r1) (flatten _ _ r2) + end); clear flatten; repeat take_simplify; repeat drop_simplify; intros. destruct case_RComp. - set (take_lev (ec :: lev) a) as a' in *. - set (take_lev (ec :: lev) b) as b' in *. - set (take_lev (ec :: lev) c) as c' in *. + set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) a)) as a' in *. + set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) b)) as b' in *. + set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) c)) as c' in *. eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ]. eapply nd_comp; [ idtac | eapply nd_rule; apply (@RLet Γ Δ [] [] (@ga_mk _ (v2t ec) a' c') (@ga_mk _ (v2t ec) a' b')) ]. @@ -500,22 +573,27 @@ Section HaskFlattener. apply ga_comp. Defined. - Definition garrowfy_arrangement : + Definition flatten_arrangement : forall Γ (Δ:CoercionEnv Γ) n (ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ, ND Rule - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev n ant1) - |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant1) (mapOptionTree (unlev' ) succ) @@ nil]] - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev n ant2) - |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (mapOptionTree (unlev' ) succ) @@ nil]]. + [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant1) + |- [@ga_mk _ (v2t ec) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) + (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]] + [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant2) + |- [@ga_mk _ (v2t ec) + (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2)) + (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]]. intros. - refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (garrowfy_arrangement' Γ Δ ec lev ant1 ant2 r)))). + refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (flatten_arrangement' Γ Δ ec lev ant1 ant2 r)))). apply nd_rule. apply RArrange. - refine ((fix garrowfy ant1 ant2 (r:Arrange ant1 ant2) := + refine ((fix flatten ant1 ant2 (r:Arrange ant1 ant2) := match r as R in Arrange A B return - Arrange (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev _ A)) - (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev _ B)) with + Arrange (mapOptionTree (flatten_leveled_type ) (drop_lev _ A)) + (mapOptionTree (flatten_leveled_type ) (drop_lev _ B)) with + | RId a => let case_RId := tt in RId _ | RCanL a => let case_RCanL := tt in RCanL _ | RCanR a => let case_RCanR := tt in RCanR _ | RuCanL a => let case_RuCanL := tt in RuCanL _ @@ -525,13 +603,13 @@ Section HaskFlattener. | RExch a b => let case_RExch := tt in RExch _ _ | RWeak a => let case_RWeak := tt in RWeak _ | RCont a => let case_RCont := tt in RCont _ - | RLeft a b c r' => let case_RLeft := tt in RLeft _ (garrowfy _ _ r') - | RRight a b c r' => let case_RRight := tt in RRight _ (garrowfy _ _ r') - | RComp a b c r1 r2 => let case_RComp := tt in RComp (garrowfy _ _ r1) (garrowfy _ _ r2) - end) ant1 ant2 r); clear garrowfy; repeat take_simplify; repeat drop_simplify; intros. + | RLeft a b c r' => let case_RLeft := tt in RLeft _ (flatten _ _ r') + | RRight a b c r' => let case_RRight := tt in RRight _ (flatten _ _ r') + | RComp a b c r1 r2 => let case_RComp := tt in RComp (flatten _ _ r1) (flatten _ _ r2) + end) ant1 ant2 r); clear flatten; repeat take_simplify; repeat drop_simplify; intros. Defined. - Definition flatten_arrangement : + Definition flatten_arrangement'' : forall Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2), ND Rule (mapOptionTree (flatten_judgment ) [Γ > Δ > ant1 |- succ]) (mapOptionTree (flatten_judgment ) [Γ > Δ > ant2 |- succ]). @@ -545,6 +623,7 @@ Section HaskFlattener. apply nd_rule. apply RArrange. induction r; simpl. + apply RId. apply RCanL. apply RCanR. apply RuCanL. @@ -558,7 +637,7 @@ Section HaskFlattener. apply RRight; auto. eapply RComp; [ apply IHr1 | apply IHr2 ]. - apply garrowfy_arrangement. + apply flatten_arrangement. apply r. Defined. @@ -589,13 +668,13 @@ Section HaskFlattener. Definition arrange_brak : forall Γ Δ ec succ t, ND Rule - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: nil) succ),, - [(@ga_mk _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]] - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) succ |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]]. + [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ),, + [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [t @@ nil]] + [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]]. intros. unfold drop_lev. set (@arrange' _ succ (levelMatch (ec::nil))) as q. - set (arrangeMap _ _ garrowfy_leveled_code_types q) as y. + set (arrangeMap _ _ flatten_leveled_type q) as y. eapply nd_comp. Focus 2. eapply nd_rule. @@ -626,6 +705,12 @@ Section HaskFlattener. inversion e; subst. simpl. apply nd_rule. + unfold flatten_leveled_type. + simpl. + unfold flatten_type. + simpl. + unfold ga_mk. + simpl. apply RVar. simpl. apply ga_id. @@ -637,24 +722,116 @@ Section HaskFlattener. apply IHsucc2. Defined. + Definition arrange_empty_tree : forall {T}{A}(q:Tree A)(t:Tree ??T), + t = mapTree (fun _:A => None) q -> + Arrange t []. + intros T A q. + induction q; intros. + simpl in H. + rewrite H. + apply RId. + simpl in *. + destruct t; try destruct o; inversion H. + set (IHq1 _ H1) as x1. + set (IHq2 _ H2) as x2. + eapply RComp. + eapply RRight. + rewrite <- H1. + apply x1. + eapply RComp. + apply RCanL. + rewrite <- H2. + apply x2. + Defined. + +(* Definition unarrange_empty_tree : forall {T}{A}(t:Tree ??T)(q:Tree A), + t = mapTree (fun _:A => None) q -> + Arrange [] t. + Defined.*) + + Definition decide_tree_empty : forall {T:Type}(t:Tree ??T), + sum { q:Tree unit & t = mapTree (fun _ => None) q } unit. + intro T. + refine (fix foo t := + match t with + | T_Leaf x => _ + | T_Branch b1 b2 => let b1' := foo b1 in let b2' := foo b2 in _ + end). + intros. + destruct x. + right; apply tt. + left. + exists (T_Leaf tt). + auto. + destruct b1'. + destruct b2'. + destruct s. + destruct s0. + subst. + left. + exists (x,,x0). + reflexivity. + right; auto. + right; auto. + Defined. + Definition arrange_esc : forall Γ Δ ec succ t, ND Rule - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) succ |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]] - [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: nil) succ),, - [(@ga_mk _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]]. + [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]] + [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ),, + [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [t @@ nil]]. intros. - unfold drop_lev. set (@arrange _ succ (levelMatch (ec::nil))) as q. - set (arrangeMap _ _ garrowfy_leveled_code_types q) as y. + set (@drop_lev Γ (ec::nil) succ) as q'. + assert (@drop_lev Γ (ec::nil) succ=q') as H. + reflexivity. + unfold drop_lev in H. + unfold mkDropFlags in H. + rewrite H in q. + clear H. + set (arrangeMap _ _ flatten_leveled_type q) as y. eapply nd_comp. eapply nd_rule. eapply RArrange. apply y. - idtac. clear y q. + set (mapOptionTree flatten_leveled_type (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ))) as q. + destruct (decide_tree_empty q); [ idtac | apply (Prelude_error "escapifying open code not yet supported") ]. + destruct s. + + simpl. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ]. + set (fun z z' => @RLet Γ Δ z (mapOptionTree flatten_leveled_type q') t z' nil) as q''. + eapply nd_comp; [ idtac | eapply nd_rule; apply q'' ]. + clear q''. + eapply nd_comp; [ apply nd_rlecnac | idtac ]. + apply nd_prod. + apply nd_rule. + apply RArrange. + eapply RComp; [ idtac | apply RCanR ]. + apply RLeft. + apply (@arrange_empty_tree _ _ _ _ e). + + eapply nd_comp. + eapply nd_rule. + eapply (@RVar Γ Δ t nil). + apply nd_rule. + apply RArrange. + eapply RComp. + apply RuCanL. + apply RRight. + apply RWeak. +(* + eapply decide_tree_empty. + + simpl. + set (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ)) as escapified. + destruct (decide_tree_empty escapified). + induction succ. destruct a. + unfold drop_lev. destruct l. simpl. unfold mkDropFlags; simpl. @@ -663,7 +840,6 @@ Section HaskFlattener. simpl. destruct (General.list_eq_dec h0 (ec :: nil)). simpl. - unfold garrowfy_leveled_code_types'. rewrite e. apply nd_id. simpl. @@ -679,6 +855,7 @@ Section HaskFlattener. apply RLeft. apply RWeak. apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported"). +*) Defined. Lemma mapOptionTree_distributes @@ -687,43 +864,118 @@ Section HaskFlattener. reflexivity. Qed. - Definition decide_tree_empty : forall {T:Type}(t:Tree ??T), - sum { q:Tree unit & t = mapTree (fun _ => None) q } unit. - intro T. - refine (fix foo t := - match t with - | T_Leaf x => _ - | T_Branch b1 b2 => let b1' := foo b1 in let b2' := foo b2 in _ - end). + Lemma unlev_relev : forall {Γ}(t:Tree ??(HaskType Γ ★)) lev, mapOptionTree unlev (t @@@ lev) = t. intros. - destruct x. - right; apply tt. - left. - exists (T_Leaf tt). - auto. - destruct b1'. - destruct b2'. - destruct s. - destruct s0. - subst. - left. - exists (x,,x0). + induction t. + destruct a; reflexivity. + rewrite <- IHt1 at 2. + rewrite <- IHt2 at 2. reflexivity. - right; auto. - right; auto. + Qed. + + Lemma tree_of_nothing : forall Γ ec t a, + Arrange (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))) a. + intros. + induction t; try destruct o; try destruct a0. + simpl. + drop_simplify. + simpl. + apply RCanR. + simpl. + apply RCanR. + Opaque drop_lev. + simpl. + Transparent drop_lev. + drop_simplify. + simpl. + eapply RComp. + eapply RComp. + eapply RAssoc. + eapply RRight. + apply IHt1. + apply IHt2. + Defined. + + Lemma tree_of_nothing' : forall Γ ec t a, + Arrange a (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))). + intros. + induction t; try destruct o; try destruct a0. + simpl. + drop_simplify. + simpl. + apply RuCanR. + simpl. + apply RuCanR. + Opaque drop_lev. + simpl. + Transparent drop_lev. + drop_simplify. + simpl. + eapply RComp. + Focus 2. + eapply RComp. + Focus 2. + eapply RCossa. + Focus 2. + eapply RRight. + apply IHt1. + apply IHt2. Defined. + Lemma krunk : forall Γ (ec:HaskTyVar Γ ECKind) t, + flatten_type (<[ ec |- t ]>) + = @ga_mk Γ (v2t ec) + (mapOptionTree flatten_type (take_arg_types_as_tree t)) + [ flatten_type (drop_arg_types_as_tree t)]. + intros. + unfold flatten_type at 1. + simpl. + unfold ga_mk. + apply phoas_extensionality. + intros. + unfold v2t. + unfold ga_mk_raw. + unfold ga_mk_tree. + rewrite <- mapOptionTree_compose. + unfold take_arg_types_as_tree. + simpl. + replace (flatten_type (drop_arg_types_as_tree t) tv ite) + with (drop_arg_types (flatten_rawtype (t tv ite))). + replace (unleaves_ (take_arg_types (flatten_rawtype (t tv ite)))) + with ((mapOptionTree (fun x : HaskType Γ ★ => flatten_type x tv ite) + (unleaves_ + (take_trustme (count_arg_types (t (fun _ : Kind => unit) (ite_unit Γ))) + (fun TV : Kind → Type => take_arg_types ○ t TV))))). + reflexivity. + unfold flatten_type. + clear hetmet_flatten. + clear hetmet_unflatten. + clear hetmet_id. + clear gar. + set (t tv ite) as x. + admit. + admit. + Qed. + Definition flatten_proof : forall {h}{c}, - ND Rule h c -> - ND Rule (mapOptionTree (flatten_judgment ) h) (mapOptionTree (flatten_judgment ) c). + ND SRule h c -> + ND Rule (mapOptionTree (flatten_judgment ) h) (mapOptionTree (flatten_judgment ) c). intros. eapply nd_map'; [ idtac | apply X ]. clear h c X. intros. simpl in *. - refine (match X as R in Rule H C with + refine + (match X as R in SRule H C with + | SBrak Γ Δ t ec succ lev => let case_SBrak := tt in _ + | SEsc Γ Δ t ec succ lev => let case_SEsc := tt in _ + | SFlat h c r => let case_SFlat := tt in _ + end). + + destruct case_SFlat. + refine (match r as R in Rule H C with | RArrange Γ Δ a b x d => let case_RArrange := tt in _ | RNote Γ Δ Σ τ l n => let case_RNote := tt in _ | RLit Γ Δ l _ => let case_RLit := tt in _ @@ -746,49 +998,13 @@ Section HaskFlattener. end); clear X h c. destruct case_RArrange. - apply (flatten_arrangement Γ Δ a b x d). + apply (flatten_arrangement'' Γ Δ a b x d). destruct case_RBrak. - simpl. - destruct lev. - change ([garrowfy_code_types (<[ ec |- t ]>) @@ nil]) - with ([ga_mk (v2t ec) [] [garrowfy_code_types t] @@ nil]). - refine (ga_unkappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) _ - (mapOptionTree (garrowfy_leveled_code_types) (drop_lev (ec::nil) succ)) ;; _ ). - apply arrange_brak. - apply (Prelude_error "found Brak at depth >0 indicating 3-level code; only two-level code is currently supported"). + apply (Prelude_error "found unskolemized Brak rule; this shouldn't happen"). destruct case_REsc. - simpl. - destruct lev. - simpl. - change ([garrowfy_code_types (<[ ec |- t ]>) @@ nil]) - with ([ga_mk (v2t ec) [] [garrowfy_code_types t] @@ nil]). - eapply nd_comp; [ apply arrange_esc | idtac ]. - set (decide_tree_empty (take_lev (ec :: nil) succ)) as q'. - destruct q'. - destruct s. - rewrite e. - clear e. - - eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ]. - eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. - eapply nd_comp; [ apply nd_llecnac | idtac ]. - apply nd_prod; [ idtac | eapply boost ]. - induction x. - apply ga_id. - eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ]. - apply ga_join. - apply IHx1. - apply IHx2. - unfold unlev'. - simpl. - apply postcompose. - apply ga_drop. - - (* environment has non-empty leaves *) - apply (ga_kappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) _ _). - apply (Prelude_error "found Esc at depth >0 indicating 3-level code; only two-level code is currently supported"). + apply (Prelude_error "found unskolemized Esc rule; this shouldn't happen"). destruct case_RNote. simpl. @@ -799,11 +1015,12 @@ Section HaskFlattener. destruct case_RLit. simpl. destruct l0; simpl. + unfold flatten_leveled_type. + simpl. rewrite literal_types_unchanged. apply nd_rule; apply RLit. unfold take_lev; simpl. unfold drop_lev; simpl. - unfold unlev'. simpl. rewrite literal_types_unchanged. apply ga_lit. @@ -818,7 +1035,6 @@ Section HaskFlattener. destruct lev. apply nd_rule. apply RVar. repeat drop_simplify. - unfold unlev'. repeat take_simplify. simpl. apply ga_id. @@ -829,20 +1045,21 @@ Section HaskFlattener. rename σ into l. destruct l as [|ec lev]; simpl. destruct (eqd_dec (g:CoreVar) (hetmet_flatten:CoreVar)). - set (garrowfy_code_types (g wev)) as t. + set (flatten_type (g wev)) as t. set (RGlobal _ Δ nil (mkGlobal Γ t hetmet_id)) as q. simpl in q. apply nd_rule. apply q. apply INil. destruct (eqd_dec (g:CoreVar) (hetmet_unflatten:CoreVar)). - set (garrowfy_code_types (g wev)) as t. + set (flatten_type (g wev)) as t. set (RGlobal _ Δ nil (mkGlobal Γ t hetmet_id)) as q. simpl in q. apply nd_rule. apply q. apply INil. - apply nd_rule; rewrite globals_do_not_have_code_types. + unfold flatten_leveled_type. simpl. + apply nd_rule; rewrite globals_do_not_have_code_types. apply RGlobal. apply (Prelude_error "found RGlobal at depth >0; globals should never appear inside code brackets unless escaped"). @@ -859,11 +1076,13 @@ Section HaskFlattener. simpl. apply RCanR. apply boost. + simpl. apply ga_curry. destruct case_RCast. simpl. destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RCast; auto | idtac ]. + simpl. apply flatten_coercion; auto. apply (Prelude_error "RCast at level >0; casting inside of code brackets is currently not supported"). @@ -877,18 +1096,22 @@ Section HaskFlattener. simpl. destruct lev as [|ec lev]. simpl. apply nd_rule. - replace (garrowfy_code_types (tx ---> te)) with ((garrowfy_code_types tx) ---> (garrowfy_code_types te)). + unfold flatten_leveled_type at 4. + unfold flatten_leveled_type at 2. + simpl. + replace (flatten_type (tx ---> te)) + with (flatten_type tx ---> flatten_type te). apply RApp. reflexivity. repeat drop_simplify. repeat take_simplify. rewrite mapOptionTree_distributes. - set (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: lev) Σ₁)) as Σ₁'. - set (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: lev) Σ₂)) as Σ₂'. + set (mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: lev) Σ₁)) as Σ₁'. + set (mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: lev) Σ₂)) as Σ₂'. set (take_lev (ec :: lev) Σ₁) as Σ₁''. set (take_lev (ec :: lev) Σ₂) as Σ₂''. - replace (garrowfy_code_types (tx ---> te)) with ((garrowfy_code_types tx) ---> (garrowfy_code_types te)). + replace (flatten_type (tx ---> te)) with ((flatten_type tx) ---> (flatten_type te)). apply (Prelude_error "FIXME: ga_apply"). reflexivity. @@ -896,8 +1119,8 @@ Section HaskFlattener. Notation "` x" := (take_lev _ x). Notation "`` x" := (mapOptionTree unlev x) (at level 20). Notation "``` x" := ((drop_lev _ x)) (at level 20). - Notation "!<[]> x" := (garrowfy_code_types _ x) (at level 1). - Notation "!<[@]> x" := (mapOptionTree garrowfy_leveled_code_types x) (at level 1). + Notation "!<[]> x" := (flatten_type _ x) (at level 1). + Notation "!<[@]> x" := (mapOptionTree flatten_leveled_type x) (at level 1). *) destruct case_RLet. @@ -931,8 +1154,10 @@ Section HaskFlattener. destruct case_RAppT. simpl. destruct lev; simpl. - rewrite garrowfy_commutes_with_HaskTAll. - rewrite garrowfy_commutes_with_substT. + unfold flatten_leveled_type. + simpl. + rewrite flatten_commutes_with_HaskTAll. + rewrite flatten_commutes_with_substT. apply nd_rule. apply RAppT. apply Δ. @@ -941,12 +1166,15 @@ Section HaskFlattener. destruct case_RAbsT. simpl. destruct lev; simpl. - rewrite garrowfy_commutes_with_HaskTAll. - rewrite garrowfy_commutes_with_HaskTApp. + unfold flatten_leveled_type at 4. + unfold flatten_leveled_type at 2. + simpl. + rewrite flatten_commutes_with_HaskTAll. + rewrite flatten_commutes_with_HaskTApp. eapply nd_comp; [ idtac | eapply nd_rule; eapply RAbsT ]. simpl. - set (mapOptionTree (garrowfy_leveled_code_types ) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a. - set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (garrowfy_leveled_code_types ) Σ)) as q'. + set (mapOptionTree (flatten_leveled_type ) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a. + set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (flatten_leveled_type ) Σ)) as q'. assert (a=q'). unfold a. unfold q'. @@ -954,7 +1182,7 @@ Section HaskFlattener. induction Σ. destruct a. simpl. - rewrite garrowfy_commutes_with_weakLT. + rewrite flatten_commutes_with_weakLT. reflexivity. reflexivity. simpl. @@ -969,7 +1197,9 @@ Section HaskFlattener. destruct case_RAppCo. simpl. destruct lev; simpl. - unfold garrowfy_code_types. + unfold flatten_leveled_type at 4. + unfold flatten_leveled_type at 2. + unfold flatten_type. simpl. apply nd_rule. apply RAppCo. @@ -979,7 +1209,7 @@ Section HaskFlattener. destruct case_RAbsCo. simpl. destruct lev; simpl. - unfold garrowfy_code_types. + unfold flatten_type. simpl. apply (Prelude_error "AbsCo not supported (FIXME)"). apply (Prelude_error "found coercion abstraction at level >0; this is not supported"). @@ -992,8 +1222,111 @@ Section HaskFlattener. destruct case_RCase. simpl. apply (Prelude_error "Case not supported (BIG FIXME)"). + + destruct case_SBrak. + simpl. + destruct lev. + drop_simplify. + set (drop_lev (ec :: nil) (take_arg_types_as_tree t @@@ (ec :: nil))) as empty_tree. + take_simplify. + rewrite take_lemma'. + simpl. + rewrite mapOptionTree_compose. + rewrite mapOptionTree_compose. + rewrite unlev_relev. + rewrite <- mapOptionTree_compose. + unfold flatten_leveled_type at 4. + simpl. + rewrite krunk. + set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host. + set (mapOptionTree (flatten_type ○ unlev)(take_lev (ec :: nil) succ)) as succ_guest. + set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args. + unfold empty_tree. + eapply nd_comp; [ eapply nd_rule; eapply RArrange; apply tree_of_nothing | idtac ]. + refine (ga_unkappa' Γ Δ (v2t ec) nil _ _ _ _ ;; _). + unfold succ_host. + unfold succ_guest. + apply arrange_brak. + apply (Prelude_error "found Brak at depth >0 indicating 3-level code; only two-level code is currently supported"). + + destruct case_SEsc. + simpl. + destruct lev. + simpl. + unfold flatten_leveled_type at 2. + simpl. + rewrite krunk. + rewrite mapOptionTree_compose. + take_simplify. + drop_simplify. + simpl. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply tree_of_nothing' ]. + simpl. + rewrite take_lemma'. + rewrite unlev_relev. + rewrite <- mapOptionTree_compose. + eapply nd_comp; [ apply (arrange_esc _ _ ec) | idtac ]. + + set (decide_tree_empty (take_lev (ec :: nil) succ)) as q'. + destruct q'. + destruct s. + rewrite e. + clear e. + + set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host. + set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args. + + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ]. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod; [ idtac | eapply boost ]. + induction x. + apply ga_id. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ]. + simpl. + apply ga_join. + apply IHx1. + apply IHx2. + simpl. + apply postcompose. + + refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))). + apply ga_cancell. + apply firstify. + + induction x. + destruct a; simpl. + apply ga_id. + simpl. + refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))). + apply ga_cancell. + refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))). + eapply firstify. + apply IHx1. + apply secondify. + apply IHx2. + + (* environment has non-empty leaves *) + apply ga_kappa'. + + (* nesting too deep *) + apply (Prelude_error "found Esc at depth >0 indicating 3-level code; only two-level code is currently supported"). Defined. + Definition skolemize_and_flatten_proof : + forall {h}{c}, + ND Rule h c -> + ND Rule + (mapOptionTree (flatten_judgment ○ skolemize_judgment) h) + (mapOptionTree (flatten_judgment ○ skolemize_judgment) c). + intros. + rewrite mapOptionTree_compose. + rewrite mapOptionTree_compose. + apply flatten_proof. + apply skolemize_proof. + apply X. + Defined. + (* to do: establish some metric on judgments (max length of level of any succedent type, probably), show how to * calculate it, and show that the flattening procedure above drives it down by one *)