X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskFlattener.v;h=60a205fa94a0e2797bd10c134d111765f1f9ae30;hp=65e06377035d27ee8313c43ac638dd9be4cf46e4;hb=b4857a6f575dffd5c9c9d5decbc21ff63a338270;hpb=94ad996f571e3c9fd622bc56d9b57118a7e5333a diff --git a/src/HaskFlattener.v b/src/HaskFlattener.v index 65e0637..60a205f 100644 --- a/src/HaskFlattener.v +++ b/src/HaskFlattener.v @@ -1,5 +1,5 @@ (*********************************************************************************************************************************) -(* HaskFlattener: *) +(* HaskFlattener: *) (* *) (* The Flattening Functor. *) (* *) @@ -18,31 +18,12 @@ Require Import HaskLiteralsAndTyCons. Require Import HaskStrongTypes. Require Import HaskProof. Require Import NaturalDeduction. -Require Import NaturalDeductionCategory. - -Require Import Algebras_ch4. -Require Import Categories_ch1_3. -Require Import Functors_ch1_4. -Require Import Isomorphisms_ch1_5. -Require Import ProductCategories_ch1_6_1. -Require Import OppositeCategories_ch1_6_2. -Require Import Enrichment_ch2_8. -Require Import Subcategories_ch7_1. -Require Import NaturalTransformations_ch7_4. -Require Import NaturalIsomorphisms_ch7_5. -Require Import BinoidalCategories. -Require Import PreMonoidalCategories. -Require Import MonoidalCategories_ch7_8. -Require Import Coherence_ch7_8. Require Import HaskStrongTypes. Require Import HaskStrong. Require Import HaskProof. Require Import HaskStrongToProof. Require Import HaskProofToStrong. -Require Import ProgrammingLanguage. -Require Import HaskProgrammingLanguage. -Require Import PCF. Open Scope nd_scope. @@ -56,67 +37,198 @@ Open Scope nd_scope. *) Section HaskFlattener. - (* this actually has nothing to do with categories; it shows that proofs [|-A]//[|-B] are one-to-one with []//[A|-B] *) - (* TODO Lemma hom_functor_full*) + Inductive TreeFlags {T:Type} : Tree T -> Type := + | tf_leaf_true : forall x, TreeFlags (T_Leaf x) + | tf_leaf_false : forall x, TreeFlags (T_Leaf x) + | tf_branch : forall b1 b2, TreeFlags b1 -> TreeFlags b2 -> TreeFlags (b1,,b2). - (* lemma: if a proof from no hypotheses contains no Brak's or Esc's, then the context contains no variables at level!=0 *) + Fixpoint mkFlags {T}(f:T -> bool)(t:Tree T) : TreeFlags t := + match t as T return TreeFlags T with + | T_Leaf x => if f x then tf_leaf_true x else tf_leaf_false x + | T_Branch b1 b2 => tf_branch _ _ (mkFlags f b1) (mkFlags f b2) + end. - (* In a tree of types, replace any type at level "lev" with None *) - Definition drop_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) := - mapTree (fun t => match t with - | Some (ttype @@ tlev) => if eqd_dec tlev lev then None else t - | _ => t - end) tt. - (* The opposite: replace any type which is NOT at level "lev" with None *) - Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) := - mapTree (fun t => match t with - | Some (ttype @@ tlev) => if eqd_dec tlev lev then Some ttype else None - | _ => None - end) tt. + (* take and drop are not exact inverses *) - (* In a tree of types, replace any type at depth "lev" or greater None *) - Definition drop_depth {Γ}(n:nat)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) := - mapTree (fun t => match t with - | Some (ttype @@ tlev) => if eqd_dec (length tlev) n then None else t - | _ => t - end) tt. + (* drop replaces each leaf where the flag is set with a [] *) + Fixpoint drop {T}{Σ:Tree ??T}(tf:TreeFlags Σ) : Tree ??T := + match tf with + | tf_leaf_true x => [] + | tf_leaf_false x => Σ + | tf_branch b1 b2 tb1 tb2 => (drop tb1),,(drop tb2) + end. + + (* take returns only those leaves for which the flag is set; all others are omitted entirely from the tree *) + Fixpoint take {T}{Σ:Tree T}(tf:TreeFlags Σ) : ??(Tree T) := + match tf with + | tf_leaf_true x => Some Σ + | tf_leaf_false x => None + | tf_branch b1 b2 tb1 tb2 => + match take tb1 with + | None => take tb2 + | Some b1' => match take tb2 with + | None => Some b1' + | Some b2' => Some (b1',,b2') + end + end + end. + + Definition maybeTree {T}(t:??(Tree ??T)) : Tree ??T := + match t with + | None => [] + | Some x => x + end. + + Definition setNone {T}(b:bool)(f:T -> bool) : ??T -> bool := + fun t => + match t with + | None => b + | Some x => f x + end. + + (* "Arrange" objects are parametric in the type of the leaves of the tree *) + Definition arrangeMap : + forall {T} (Σ₁ Σ₂:Tree ??T) {R} (f:T -> R), + Arrange Σ₁ Σ₂ -> + Arrange (mapOptionTree f Σ₁) (mapOptionTree f Σ₂). + intros. + induction X; simpl. + apply RCanL. + apply RCanR. + apply RuCanL. + apply RuCanR. + apply RAssoc. + apply RCossa. + apply RExch. + apply RWeak. + apply RCont. + apply RLeft; auto. + apply RRight; auto. + eapply RComp; [ apply IHX1 | apply IHX2 ]. + Defined. + + Definition bnot (b:bool) : bool := if b then false else true. + + Definition swapMiddle {T} (a b c d:Tree ??T) : + Arrange ((a,,b),,(c,,d)) ((a,,c),,(b,,d)). + eapply RComp. + apply RCossa. + eapply RComp. + eapply RLeft. + eapply RComp. + eapply RAssoc. + eapply RRight. + apply RExch. + eapply RComp. + eapply RLeft. + eapply RCossa. + eapply RAssoc. + Defined. + + Definition arrange : + forall {T} (Σ:Tree ??T) (f:T -> bool), + Arrange Σ (drop (mkFlags (setNone false f) Σ),,( (drop (mkFlags (setNone false (bnot ○ f)) Σ)))). + intros. + induction Σ. + simpl. + destruct a. + simpl. + destruct (f t); simpl. + apply RuCanL. + apply RuCanR. + simpl. + apply RuCanL. + simpl in *. + eapply RComp; [ idtac | apply swapMiddle ]. + eapply RComp. + eapply RLeft. + apply IHΣ2. + eapply RRight. + apply IHΣ1. + Defined. - (* delete from the tree any type which is NOT at level "lev" *) + Definition arrange' : + forall {T} (Σ:Tree ??T) (f:T -> bool), + Arrange (drop (mkFlags (setNone false f) Σ),,( (drop (mkFlags (setNone false (bnot ○ f)) Σ)))) Σ. + intros. + induction Σ. + simpl. + destruct a. + simpl. + destruct (f t); simpl. + apply RCanL. + apply RCanR. + simpl. + apply RCanL. + simpl in *. + eapply RComp; [ apply swapMiddle | idtac ]. + eapply RComp. + eapply RLeft. + apply IHΣ2. + eapply RRight. + apply IHΣ1. + Defined. + + Definition minus' n m := + match m with + | 0 => n + | _ => match n with + | 0 => 0 + | _ => n - m + end + end. + + Ltac eqd_dec_refl' := + match goal with + | [ |- context[@eqd_dec ?T ?V ?X ?X] ] => + destruct (@eqd_dec T V X X) as [eqd_dec1 | eqd_dec2]; + [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ] + end. - Fixpoint take_lev' {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : ??(Tree ??(HaskType Γ ★)) := + Fixpoint reduceTree {T}(unit:T)(merge:T -> T -> T)(tt:Tree ??T) : T := match tt with - | T_Leaf None => Some (T_Leaf None) (* FIXME: unsure of this; it ends up on both sides *) - | T_Branch b1 b2 => - let b1' := take_lev' lev b1 in - let b2' := take_lev' lev b2 in - match b1' with - | None => b2' - | Some b1'' => match b2' with - | None => Some b1'' - | Some b2'' => Some (b1'',,b2'') - end - end - | T_Leaf (Some (ttype@@tlev)) => - if eqd_dec tlev lev - then Some [ttype] - else None + | T_Leaf None => unit + | T_Leaf (Some x) => x + | T_Branch b1 b2 => merge (reduceTree unit merge b1) (reduceTree unit merge b2) end. - Context (ga' : forall TV, TV ★ -> Tree ??(RawHaskType TV ★) -> Tree ??(RawHaskType TV ★) -> RawHaskType TV ★). + Set Printing Width 130. - Definition ga : forall Γ, HaskTyVar Γ ★ -> Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> HaskType Γ ★ - := fun Γ ec ant suc => - fun TV ite => - ga' - TV - (ec TV ite) - (mapOptionTree (fun x => x TV ite) ant) - (mapOptionTree (fun x => x TV ite) suc). + 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 ★ }. - Implicit Arguments ga [ [Γ] ]. - Notation "a ~~~~> b" := (@ga _ _ a b) (at level 20). + 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). - Context (unit_type : forall TV, RawHaskType TV ★). + 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: @@ -124,49 +236,134 @@ Section HaskFlattener. * - 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}(depth:nat)(exp: RawHaskType TV κ) : RawHaskType TV κ := + 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 => flatten_rawtype depth (y v)) - | TApp _ _ x y => TApp (flatten_rawtype depth x) (flatten_rawtype depth y) - | TCon tc => TCon tc - | TCoerc _ t1 t2 t => TCoerc (flatten_rawtype depth t1) (flatten_rawtype depth t2) (flatten_rawtype depth t) - | TArrow => TArrow - | TCode v e => match depth with - | O => match v return RawHaskType TV ★ with - | TVar ★ ec => ga' TV ec [] [flatten_rawtype depth e] - | _ => unit_type TV - end - | (S depth') => TCode v (flatten_rawtype depth' e) - end - | TyFunApp tfc lt => TyFunApp tfc (flatten_rawtype_list _ depth lt) + | 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 flatten_rawtype_list {TV}(lk:list Kind)(depth:nat)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk := + 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 _ _ (flatten_rawtype depth t) (flatten_rawtype_list _ depth rest) + | 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). - Inductive AllT {T:Type}{P:T->Prop} : Tree ??T -> Prop := - | allt_none : AllT [] - | allt_some : forall t, P t -> AllT [t] - | allt_branch : forall b1 b2, AllT b1 -> AllT b2 -> AllT (b1,,b2) - . - Implicit Arguments AllT [[T]]. + Definition v2t {Γ}(ec:HaskTyVar Γ ECKind) := fun TV ite => TVar (ec TV ite). - Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ => Γ end. + 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 getSuc (j:Judg) : Tree ??(LeveledHaskType (getΓ j) ★) := - match j as J return Tree ??(LeveledHaskType (getΓ J) ★) with - Γ > _ > _ |- s => s - 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 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. + + (* In a tree of types, replace any type at depth "lev" or greater None *) + Definition mkDropFlags {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : TreeFlags tt := + mkFlags (setNone false (levelMatch lev)) tt. + + Definition drop_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) := + drop (mkDropFlags lev tt). - Definition getlev {Γ}{κ}(lht:LeveledHaskType Γ κ) := - match lht with t@@l => l end. + (* The opposite: replace any type which is NOT at level "lev" with None *) + Definition mkTakeFlags {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : TreeFlags tt := + mkFlags (setNone 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)) (drop (mkTakeFlags lev tt)). +(* + mapOptionTree (fun x => garrowfy_code_types (unlev x)) + (maybeTree (take tt (mkFlags ( + fun t => match t with + | Some (ttype @@ tlev) => if eqd_dec tlev lev then true else false + | _ => true + end + ) tt))). +*) + + Lemma drop_lev_lemma : forall Γ (lev:HaskLevel Γ) x, drop_lev lev [x @@ lev] = []. + intros; simpl. + Opaque eqd_dec. + unfold drop_lev. + simpl. + unfold mkDropFlags. + simpl. + Transparent eqd_dec. + eqd_dec_refl'. + auto. + Qed. + + Lemma drop_lev_lemma_s : forall Γ (lev:HaskLevel Γ) ec x, drop_lev (ec::lev) [x @@ (ec :: lev)] = []. + intros; simpl. + Opaque eqd_dec. + unfold drop_lev. + unfold mkDropFlags. + simpl. + Transparent eqd_dec. + eqd_dec_refl'. + auto. + Qed. + + Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [garrowfy_code_types x]. + intros; simpl. + Opaque eqd_dec. + unfold take_lev. + unfold mkTakeFlags. + simpl. + Transparent eqd_dec. + eqd_dec_refl'. + auto. + 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 (?E :: ?L) [ ?X @@ (?E :: ?L) ] ] ] => + rewrite (drop_lev_lemma_s G L E X) + | [ |- context[@drop_lev ?G ?N (?A,,?B)] ] => + change (@drop_lev G N (A,,B)) with ((@drop_lev G N A),,(@drop_lev G N B)) + | [ |- context[@drop_lev ?G ?N (T_Leaf None)] ] => + change (@drop_lev G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★))) + end. + + Ltac take_simplify := + match goal with + | [ |- 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. + + Axiom literal_types_unchanged : forall Γ l, garrowfy_code_types (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). (* 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 * picks nil *) + Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ => Γ end. + Definition getSuc (j:Judg) : Tree ??(LeveledHaskType (getΓ j) ★) := + match j as J return Tree ??(LeveledHaskType (getΓ J) ★) with Γ > _ > _ |- s => s end. Fixpoint getjlev {Γ}(tt:Tree ??(LeveledHaskType Γ ★)) : HaskLevel Γ := match tt with | T_Leaf None => nil @@ -178,161 +375,148 @@ Section HaskFlattener. end end. - (* an XJudg is a Judg for which the SUCCEDENT types all come from the same level, and that level is no deeper than "n" *) - (* even the empty succedent [] has a level... *) - Definition QJudg (n:nat)(j:Judg) := le (length (getjlev (getSuc j))) n. - -(* Definition qjudg2judg {n}(qj:QJudg n) : Judg := projT1 qj.*) - - Inductive QRule : nat -> Tree ??Judg -> Tree ??Judg -> Type := - mkQRule : forall n h c, - Rule h c -> - ITree _ (QJudg n) h -> - ITree _ (QJudg n) c -> - QRule n h c. - - Definition QND n := ND (QRule n). - - (* - * Any type "t" at a level with length "n" becomes (g () t) - * Any type "<[t]>" at a level with length "n-1" becomes (g () t) - *) - - Definition flatten_type {Γ}(n:nat)(ht:HaskType Γ ★) : HaskType Γ ★ := - fun TV ite => - flatten_rawtype n (ht TV ite). - - Definition minus' n m := - match m with - | 0 => n - | _ => n - m - end. - - (* to do: integrate this more cleanly with qjudg *) - Definition flatten_leveled_type {Γ}(n:nat)(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ := - match ht with - htt @@ htlev => - flatten_type (minus' n (length htlev)) htt @@ htlev - end. + Definition unlev' {Γ} (x:LeveledHaskType Γ ★) := + garrowfy_code_types (unlev x). - Definition flatten_qjudg_ok (n:nat)(j:Judg) : Judg := - match j with + (* "n" is the maximum depth remaining AFTER flattening *) + Definition flatten_judgment (j:Judg) := + match j as J return Judg with Γ > Δ > ant |- suc => - let ant' := mapOptionTree (flatten_leveled_type n) ant in - let suc' := mapOptionTree (flatten_leveled_type n) suc - in (Γ > Δ > ant' |- suc') + match getjlev suc with + | nil => Γ > Δ > mapOptionTree garrowfy_leveled_code_types ant + |- mapOptionTree garrowfy_leveled_code_types suc + + | (ec::lev') => Γ > Δ > mapOptionTree garrowfy_leveled_code_types (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] + end end. - Definition take_lev'' {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) := - match (take_lev' lev tt) with - | None => [] - | Some x => x - end. + 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] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply (@RLet Γ Δ [] ant y x lev) ]. + eapply nd_comp; [ apply nd_rlecnac | idtac ]. + apply nd_prod. + apply nd_id. + eapply nd_comp. + apply X. + eapply nd_rule. + eapply RArrange. + apply RuCanL. + Defined. - Definition flatten_qjudg : forall (n:nat), Judg -> Judg. - refine (fun {n}{j} => - match j as J return Judg with - Γ > Δ > ant |- suc => - match getjlev suc with - | nil => flatten_qjudg_ok n j - | (ec::lev') => if eqd_dec (length lev') n - then let ant_host := drop_depth (S n) ant in - let ant_guest := take_lev (ec::lev') ant in (* FIXME: I want take_lev''!!!!! *) - (Γ > Δ > ant_host |- [ga ec ant_guest (mapOptionTree unlev suc) @@ lev']) - else flatten_qjudg_ok n j - end - end). - Defined. + Definition postcompose' : ∀ Γ Δ ec lev a b c Σ, + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ] -> + ND Rule [] [ Γ > Δ > Σ,,[@ga_mk Γ ec b c @@ lev] |- [@ga_mk Γ ec a c @@ lev] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply (@RLet Γ Δ [@ga_mk _ ec b c @@lev] Σ (@ga_mk _ ec a c) (@ga_mk _ ec a b) lev) ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply X. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ]. + apply ga_comp. + Defined. - Axiom literal_types_unchanged : forall n Γ l, flatten_type n (literalType l) = literalType(Γ:=Γ) l. + Definition precompose' : ∀ Γ Δ ec lev a b c Σ, + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec b c @@ lev] ] -> + ND Rule [] [ Γ > Δ > Σ,,[@ga_mk Γ ec a b @@ lev] |- [@ga_mk Γ ec a c @@ lev] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply (@RLet Γ Δ [@ga_mk _ ec a b @@lev] Σ (@ga_mk _ ec a c) (@ga_mk _ ec b c) lev) ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply X. + apply ga_comp. + Defined. - Lemma drop_depth_lemma : forall Γ (lev:HaskLevel Γ) x, drop_depth (length lev) [x @@ lev] = []. - admit. - Defined. + Definition postcompose : ∀ Γ Δ ec lev a b c, + ND Rule [] [ Γ > Δ > [] |- [@ga_mk Γ ec a b @@ lev] ] -> + ND Rule [] [ Γ > Δ > [@ga_mk Γ ec b c @@ lev] |- [@ga_mk Γ ec a c @@ lev] ]. + intros. + eapply nd_comp. + apply postcompose'. + apply X. + apply nd_rule. + apply RArrange. + apply RCanL. + Defined. - Lemma drop_depth_lemma_s : forall Γ (lev:HaskLevel Γ) ec x, drop_depth (S (length lev)) [x @@ (ec :: lev)] = []. - admit. + Definition firstify : ∀ Γ Δ ec lev a b c Σ, + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ] -> + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) @@ lev] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply X. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanL ]. + apply ga_first. Defined. - Ltac drop_simplify := - match goal with - | [ |- context[@drop_depth ?G (length ?L) [ ?X @@ ?L ] ] ] => - rewrite (drop_depth_lemma G L X) - | [ |- context[@drop_depth ?G (S (length ?L)) [ ?X @@ (?E :: ?L) ] ] ] => - rewrite (drop_depth_lemma_s G L E X) - | [ |- context[@drop_depth ?G ?N (?A,,?B)] ] => - change (@drop_depth G N (A,,B)) with ((@drop_depth G N A),,(@drop_depth G N B)) - | [ |- context[@drop_depth ?G ?N (T_Leaf None)] ] => - change (@drop_depth G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★))) - end. - - Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [x]. - admit. + Definition secondify : ∀ Γ Δ ec lev a b c Σ, + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ] -> + ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) @@ lev] ]. + intros. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply X. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanL ]. + apply ga_second. Defined. - Ltac take_simplify := - match goal with - | [ |- 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. - - Class garrow := - { ga_id : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a a @@ lev] ] - ; ga_comp : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev],,[@ga Γ ec b c @@ lev] |- [@ga Γ ec a c @@ lev] ] - ; ga_cancelr : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,[]) a @@ lev] ] - ; ga_cancell : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ([],,a) a @@ lev] ] - ; ga_uncancelr : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,[]) @@ lev] ] - ; ga_uncancell : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a ([],,a) @@ lev] ] - ; ga_assoc : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ((a,,b),,c) (a,,(b,,c)) @@ lev] ] - ; ga_unassoc : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,(b,,c)) ((a,,b),,c) @@ lev] ] - ; ga_swap : ∀ Γ Δ ec lev a b , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,b) (b,,a) @@ lev] ] - ; ga_drop : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a [] @@ lev] ] - ; ga_copy : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,a) @@ lev] ] - ; ga_first : ∀ Γ Δ ec lev a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev] |- [@ga Γ ec (a,,x) (b,,x) @@ lev] ] - ; ga_second : ∀ Γ Δ ec lev a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev] |- [@ga Γ ec (x,,a) (x,,b) @@ lev] ] - ; ga_lit : ∀ Γ Δ ec lev lit , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec [] [literalType lit] @@ lev] ] - ; ga_curry : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [@ga Γ ec (a,,[b]) [c] @@ lev] |- [@ga Γ ec a [b ---> c] @@ lev] ] - ; ga_apply : ∀ Γ Δ ec lev a a' b c, ND Rule [] [Γ > Δ > - [@ga Γ ec a [b ---> c] @@ lev],,[@ga Γ ec a' [b] @@ lev] - |- - [@ga Γ ec (a,,a') [c] @@ lev] ] - }. + Lemma ga_unkappa : ∀ Γ Δ ec l 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] ]. + intros. + set (ga_comp Γ Δ ec l [] a b) as q. - Context (gar:garrow). + 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''. + eapply nd_comp. + Focus 2. + eapply nd_rule. + eapply RArrange. + apply RExch. - Definition boost : forall Γ Δ ant x y, - ND Rule [] [ Γ > Δ > x |- y ] -> - ND Rule [ Γ > Δ > ant |- x ] [ Γ > Δ > ant |- y ]. - admit. - Defined. - Definition postcompose : ∀ Γ Δ ec lev a b c, - ND Rule [] [ Γ > Δ > [] |- [@ga Γ ec a b @@ lev] ] -> - ND Rule [] [ Γ > Δ > [@ga Γ ec b c @@ lev] |- [@ga Γ ec a c @@ lev] ]. - admit. - Defined. - Definition precompose : ∀ Γ Δ lev a b c x, - ND Rule [] [ Γ > Δ > a |- x,,[b @@ lev] ] -> - ND Rule [] [ Γ > Δ > [b @@ lev] |- [c @@ lev] ] -> - ND Rule [] [ Γ > Δ > a,,x |- [c @@ lev] ]. - admit. - Defined. + eapply nd_comp. + Focus 2. + eapply nd_rule. + apply q''. - Set Printing Width 130. + idtac. + clear q'' q'. + eapply nd_comp. + apply nd_rlecnac. + apply nd_prod. + apply nd_id. + apply q. + Defined. +(* + 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' : forall Γ (Δ:CoercionEnv Γ) - (ec:HaskTyVar Γ ★) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2), - ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev] ]. + (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] ]. intros Γ Δ ec lev. refine (fix garrowfy ant1 ant2 (r:Arrange ant1 ant2): - ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev]] := + ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ nil]] := match r as R in Arrange A B return - ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ lev]] + ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ nil]] with | RCanL a => let case_RCanL := tt in ga_uncancell _ _ _ _ _ | RCanR a => let case_RCanR := tt in ga_uncancelr _ _ _ _ _ @@ -345,37 +529,45 @@ Section HaskFlattener. | 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 a b c r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (garrowfy _ _ r1) (garrowfy _ _ r2) + | 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. destruct case_RComp. - (* - set (ga_comp Γ Δ ec lev (``c) (``b) (``a)) as q. - eapply precompose. - Focus 2. - apply q. - refine ( _ ;; boost _ _ _ _ _ (ga_comp _ _ _ _ _ _ _)). - apply precompose. - Focus 2. - eapply ga_comp. - *) - admit. + 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 *. + 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')) ]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + apply nd_prod. + apply r2'. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RuCanL ]. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ]. + eapply nd_comp; [ idtac | eapply nd_rule; apply + (@RLet Γ Δ [@ga_mk _ (v2t ec) a' b' @@ _] [] (@ga_mk _ (v2t ec) a' c') (@ga_mk _ (v2t ec) b' c'))]. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + eapply nd_prod. + apply r1'. + apply ga_comp. Defined. Definition garrowfy_arrangement : forall Γ (Δ:CoercionEnv Γ) n - (ec:HaskTyVar Γ ★) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ, - (*ec :: lev = getjlev succ ->*) - (* H0 : left (Datatypes.length lev ≠ n) e = Peano_dec.eq_nat_dec (Datatypes.length lev) n *) + (ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ, ND Rule - [Γ > Δ > drop_depth n ant1 |- [@ga _ ec (take_lev (ec :: lev) ant1) (mapOptionTree unlev succ) @@ lev]] - [Γ > Δ > drop_depth n ant2 |- [@ga _ ec (take_lev (ec :: lev) ant2) (mapOptionTree unlev succ) @@ lev]]. + [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]]. intros. refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (garrowfy_arrangement' Γ Δ ec lev ant1 ant2 r)))). apply nd_rule. apply RArrange. refine ((fix garrowfy ant1 ant2 (r:Arrange ant1 ant2) := - match r as R in Arrange A B return Arrange (drop_depth _ A) (drop_depth _ B) with + 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 | RCanL a => let case_RCanL := tt in RCanL _ | RCanR a => let case_RCanR := tt in RCanR _ | RuCanL a => let case_RuCanL := tt in RuCanL _ @@ -392,9 +584,9 @@ Section HaskFlattener. Defined. Definition flatten_arrangement : - forall n Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2), - ND Rule (mapOptionTree (flatten_qjudg n) [Γ > Δ > ant1 |- succ]) - (mapOptionTree (flatten_qjudg n) [Γ > Δ > ant2 |- succ]). + 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]). intros. simpl. set (getjlev succ) as succ_lev. @@ -418,68 +610,193 @@ Section HaskFlattener. apply RRight; auto. eapply RComp; [ apply IHr1 | apply IHr2 ]. - set (Peano_dec.eq_nat_dec (Datatypes.length succ_lev) n) as lev_is_n. - assert (lev_is_n=Peano_dec.eq_nat_dec (Datatypes.length succ_lev) n). - reflexivity. - destruct lev_is_n. - rewrite <- e. - apply garrowfy_arrangement. + apply garrowfy_arrangement. apply r. - auto. - apply nd_rule. - apply RArrange. - induction r; simpl. - apply RCanL. - apply RCanR. - apply RuCanL. - apply RuCanR. - apply RAssoc. - apply RCossa. - apply RExch. - apply RWeak. - apply RCont. - apply RLeft; auto. - apply RRight; auto. - eapply RComp; [ apply IHr1 | apply IHr2 ]. Defined. - Lemma flatten_coercion : forall n Γ Δ σ τ (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)), - HaskCoercion Γ Δ (flatten_type n σ ∼∼∼ flatten_type n τ). - admit. + Definition ga_join Γ Δ Σ₁ Σ₂ a b ec : + ND Rule [] [Γ > Δ > Σ₁ |- [@ga_mk _ ec [] a @@ nil]] -> + ND Rule [] [Γ > Δ > Σ₂ |- [@ga_mk _ ec [] b @@ nil]] -> + ND Rule [] [Γ > Δ > Σ₁,,Σ₂ |- [@ga_mk _ ec [] (a,,b) @@ nil]]. + intro pfa. + intro pfb. + apply secondify with (c:=a) in pfb. + eapply nd_comp. + Focus 2. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. + eapply nd_comp; [ eapply nd_llecnac | idtac ]. + eapply nd_prod. + apply pfb. + clear pfb. + apply postcompose'. + eapply nd_comp. + apply pfa. + clear pfa. + apply boost. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ]. + apply precompose'. + apply ga_uncancelr. + apply nd_id. Defined. - Ltac eqd_dec_refl' := - match goal with - | [ |- context[@eqd_dec ?T ?V ?X ?X] ] => - destruct (@eqd_dec T V X X) as [eqd_dec1 | eqd_dec2]; - [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ] - end. - + 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]]. + intros. + unfold drop_lev. + set (@arrange' _ succ (levelMatch (ec::nil))) as q. + set (arrangeMap _ _ garrowfy_leveled_code_types q) as y. + eapply nd_comp. + Focus 2. + eapply nd_rule. + eapply RArrange. + apply y. + idtac. + clear y q. + simpl. + eapply nd_comp; [ apply nd_llecnac | idtac ]. + eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ]. + apply nd_prod. + Focus 2. + apply nd_id. + idtac. + induction succ; try destruct a; simpl. + unfold take_lev. + unfold mkTakeFlags. + unfold mkFlags. + unfold bnot. + simpl. + destruct l as [t' lev']. + destruct lev' as [|ec' lev']. + simpl. + apply ga_id. + unfold levelMatch. + set (@eqd_dec (HaskLevel Γ) (haskLevelEqDecidable Γ) (ec' :: lev') (ec :: nil)) as q. + destruct q. + inversion e; subst. + simpl. + apply nd_rule. + apply RVar. + simpl. + apply ga_id. + apply ga_id. + unfold take_lev. + simpl. + apply ga_join. + apply IHsucc1. + apply IHsucc2. + Defined. -(* - Lemma foog : forall Γ Δ, - ND Rule - ( [ Γ > Δ > Σ₁ |- a ],,[ Γ > Δ > Σ₂ |- b ] ) - [ Γ > Δ > Σ₁,,Σ₂ |- a,,b ] -*) + 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]]. + intros. + unfold drop_lev. + set (@arrange _ succ (levelMatch (ec::nil))) as q. + set (arrangeMap _ _ garrowfy_leveled_code_types q) as y. + eapply nd_comp. + eapply nd_rule. + eapply RArrange. + apply y. + idtac. + clear y q. + + induction succ. + destruct a. + destruct l. + simpl. + unfold mkDropFlags; simpl. + unfold take_lev. + unfold mkTakeFlags. + simpl. + destruct (General.list_eq_dec h0 (ec :: nil)). + simpl. + unfold garrowfy_leveled_code_types'. + rewrite e. + apply nd_id. + simpl. + apply nd_rule. + apply RArrange. + apply RLeft. + apply RWeak. + simpl. + apply nd_rule. + unfold take_lev. + simpl. + apply RArrange. + apply RLeft. + apply RWeak. + apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported"). + Defined. - Notation "` x" := (take_lev _ x) (at level 20). - Notation "`` x" := (mapOptionTree unlev x) (at level 20). - Notation "``` x" := (drop_depth _ x) (at level 20). + Lemma mapOptionTree_distributes + : forall T R (a b:Tree ??T) (f:T->R), + mapOptionTree f (a,,b) = (mapOptionTree f a),,(mapOptionTree f b). + reflexivity. + Qed. + + Axiom garrowfy_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). + + Axiom garrowfy_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)). + + Axiom garrowfy_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). + + 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 globals_do_not_have_code_types : forall (Γ:TypeEnv) (g:Global Γ) v, + garrowfy_code_types (g v) = g v. + + 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 flatten : - forall n {h}{c}, - QND (S n) h c -> - ND Rule (mapOptionTree (flatten_qjudg n) h) (mapOptionTree (flatten_qjudg n) c). + + Definition flatten_proof : + forall {h}{c}, + ND Rule 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 *. - inversion X. - subst. - refine (match X0 as R in Rule H C with + refine (match X 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 _ @@ -495,27 +812,61 @@ Section HaskFlattener. | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ lev => let case_RLet := tt in _ | RJoin Γ p lri m x q => let case_RJoin := tt in _ | RVoid _ _ => 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 _ + | RBrak Γ Δ t ec succ lev => let case_RBrak := tt in _ + | REsc Γ Δ t ec succ lev => let case_REsc := tt in _ | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _ | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _ - end); clear X X0 X1 X2 h c. + end); clear X h c. destruct case_RArrange. - apply (flatten_arrangement n Γ Δ a b x d). + apply (flatten_arrangement Γ Δ a b x d). destruct case_RBrak. - admit. + 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 (a)"). destruct case_REsc. - admit. + 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 (a)"). destruct case_RNote. simpl. destruct l; simpl. apply nd_rule; apply RNote; auto. - destruct (Peano_dec.eq_nat_dec (Datatypes.length l) n). - apply nd_rule; apply RNote; auto. apply nd_rule; apply RNote; auto. destruct case_RLit. @@ -523,52 +874,55 @@ Section HaskFlattener. destruct l0; simpl. rewrite literal_types_unchanged. apply nd_rule; apply RLit. - destruct (Peano_dec.eq_nat_dec (Datatypes.length l0) n); unfold mapTree; unfold mapOptionTree; simpl. unfold take_lev; simpl. - unfold drop_depth; simpl. - apply ga_lit. + unfold drop_lev; simpl. + unfold unlev'. + simpl. rewrite literal_types_unchanged. - apply nd_rule. - apply RLit. + apply ga_lit. destruct case_RVar. + Opaque flatten_judgment. simpl. - destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RVar | idtac ]. - destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RVar ]; simpl. - rewrite <- e. - clear e n. - repeat drop_simplify. + Transparent flatten_judgment. + idtac. + unfold flatten_judgment. + unfold getjlev. + destruct lev. + apply nd_rule. apply RVar. + repeat drop_simplify. + unfold unlev'. repeat take_simplify. - apply ga_id. + simpl. + apply ga_id. destruct case_RGlobal. simpl. - destruct l as [|ec lev]; simpl; [ apply nd_rule; apply RGlobal; auto | idtac ]. - destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RGlobal; auto ]; simpl. + rename l into g. + rename σ into l. + destruct l as [|ec lev]; simpl; [ apply nd_rule; rewrite globals_do_not_have_code_types; apply RGlobal; auto | idtac ]. apply (Prelude_error "found RGlobal at depth >0"). destruct case_RLam. + Opaque drop_lev. + Opaque take_lev. simpl. destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RLam; auto | idtac ]. - destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RLam; auto ]; simpl. - rewrite <- e. - clear e n. repeat drop_simplify. repeat take_simplify. eapply nd_comp. eapply nd_rule. eapply RArrange. + simpl. apply RCanR. apply boost. apply ga_curry. - + destruct case_RCast. simpl. destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RCast; auto | idtac ]. apply flatten_coercion; auto. - destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RCast; auto ]; simpl. apply (Prelude_error "RCast at level >0"). - apply flatten_coercion; auto. destruct case_RJoin. simpl. @@ -583,37 +937,85 @@ Section HaskFlattener. simpl. destruct lev as [|ec lev]. simpl. apply nd_rule. - replace (flatten_type n (tx ---> te)) with ((flatten_type n tx) ---> (flatten_type n te)). + replace (garrowfy_code_types (tx ---> te)) with ((garrowfy_code_types tx) ---> (garrowfy_code_types te)). apply RApp. - unfold flatten_type. - simpl. reflexivity. - destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n). - eapply nd_comp. - eapply nd_rule. - apply RJoin. - repeat drop_simplify. + repeat drop_simplify. repeat take_simplify. - apply boost. - apply ga_apply. - - replace (flatten_type (minus' n (length (ec::lev))) (tx ---> te)) - with ((flatten_type (minus' n (length (ec::lev))) tx) ---> (flatten_type (minus' n (length (ec::lev))) te)). - apply nd_rule. - apply RApp. - unfold flatten_type. - simpl. + 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 (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)). + apply (Prelude_error "FIXME: ga_apply"). reflexivity. - +(* + Notation "` x" := (take_lev _ x) (at level 20). + 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 "!<[@]>" := (garrowfy_leveled_code_types _) (at level 1). +*) destruct case_RLet. + apply (Prelude_error "FIXME: RLet"). +(* simpl. destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RLet; auto | idtac ]. destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RLet; auto ]; simpl. repeat drop_simplify. repeat take_simplify. - admit. (* FIXME *) + rename σ₁ into a. + rename σ₂ into b. + rewrite mapOptionTree_distributes. + rewrite mapOptionTree_distributes. + set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_lev (ec :: lev) Σ₁)) as A. + set (take_lev (ec :: lev) Σ₁) as A'. + set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_lev (ec :: lev) Σ₂)) as B. + set (take_lev (ec :: lev) Σ₂) as B'. + simpl. + + eapply nd_comp. + Focus 2. + eapply nd_rule. + eapply RLet. + + apply nd_prod. + + apply boost. + apply ga_second. + + eapply nd_comp. + Focus 2. + eapply boost. + apply ga_comp. + eapply nd_comp. + eapply nd_rule. + eapply RArrange. + eapply RCanR. + + eapply nd_comp. + Focus 2. + eapply nd_rule. + eapply RArrange. + eapply RExch. + idtac. + + eapply nd_comp. + apply nd_llecnac. + eapply nd_comp. + Focus 2. + eapply nd_rule. + apply RJoin. + apply nd_prod. + + eapply nd_rule. + eapply RVar. + + apply nd_id. +*) destruct case_RVoid. simpl. apply nd_rule. @@ -621,379 +1023,96 @@ Section HaskFlattener. destruct case_RAppT. simpl. destruct lev; simpl. - admit. - admit. + rewrite garrowfy_commutes_with_HaskTAll. + rewrite garrowfy_commutes_with_substT. + apply nd_rule. + apply RAppT. + apply Δ. + apply Δ. + apply (Prelude_error "ga_apply"). destruct case_RAbsT. simpl. destruct lev; simpl. - admit. - admit. + rewrite garrowfy_commutes_with_HaskTAll. + rewrite garrowfy_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'. + assert (a=q'). + unfold a. + unfold q'. + clear a q'. + induction Σ. + destruct a. + simpl. + rewrite garrowfy_commutes_with_weakLT. + reflexivity. + reflexivity. + simpl. + rewrite <- IHΣ1. + rewrite <- IHΣ2. + reflexivity. + rewrite H. + apply nd_id. + apply Δ. + apply Δ. + apply (Prelude_error "AbsT at depth>0"). destruct case_RAppCo. simpl. destruct lev; simpl. - admit. - admit. + unfold garrowfy_code_types. + simpl. + apply nd_rule. + apply RAppCo. + apply flatten_coercion. + apply γ. + apply (Prelude_error "AppCo at depth>0"). destruct case_RAbsCo. simpl. destruct lev; simpl. - admit. - admit. + unfold garrowfy_code_types. + simpl. + apply (Prelude_error "AbsCo not supported (FIXME)"). + apply (Prelude_error "AbsCo at depth>0"). destruct case_RLetRec. - simpl. - admit. + rename t into lev. + apply (Prelude_error "LetRec not supported (FIXME)"). destruct case_RCase. simpl. - admit. + apply (Prelude_error "Case not supported (FIXME)"). Defined. - Lemma flatten_qjudg_qjudg : forall {n}{j} (q:QJudg (S n) j), QJudg n (flatten_qjudg n j). - admit. - (* - intros. - destruct q. - destruct a. - unfold flatten_qjudg. - destruct j. - destruct (eqd_dec (Datatypes.length x) (S n)). - destruct x. - inversion e. - exists x; split. - simpl in e. - inversion e. - auto. - simpl in *. - apply allt_some. - simpl. - auto. - unfold QJudg. - exists x. - split; auto. - clear a. - Set Printing Implicit. - simpl. - set (length x) as x'. - assert (x'=length x). - reflexivity. - simpl in *. - rewrite <- H. - Unset Printing Implicit. - idtac. - omega. - simpl in *. - induction t0. - destruct a0; simpl in *. - apply allt_some. - inversion a. - subst. - destruct l0. - simpl. - auto. - apply allt_none. - simpl in *. - inversion a; subst. - apply allt_branch. - apply IHt0_1; auto. - apply IHt0_2; auto. - *) - 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 *) (* Instance FlatteningFunctor {Γ}{Δ}{ec} : Functor (JudgmentsL (PCF Γ Δ ec)) (TypesL (SystemFCa Γ Δ)) (obact) := { fmor := FlatteningFunctor_fmor }. - Admitted. Definition ReificationFunctor Γ Δ : Functor (JudgmentsL _ _ (PCF n Γ Δ)) SystemFCa' (mapOptionTree brakifyJudg). refine {| fmor := ReificationFunctor_fmor Γ Δ |}; unfold hom; unfold ob; simpl ; intros. - Admitted. Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME. refine {| plsmme_pl := PCF n Γ Δ |}. - admit. Defined. Definition SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME. refine {| plsmme_pl := SystemFCa n Γ Δ |}. - admit. Defined. Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n). - admit. Defined. (* 5.1.4 *) Definition PCF_SystemFCa_two_level n Γ Δ : TwoLevelLanguage (PCF_SMME n Γ Δ) (SystemFCa_SMME (S n) Γ Δ). - admit. Defined. *) (* ... and the retraction exists *) End HaskFlattener. - - - - - - - - -(* - - Old flattening code; ignore this - just to remind me how I used to do it - - (* - * Here it is, what you've all been waiting for! When reading this, - * it might help to have the definition for "Inductive ND" (see - * NaturalDeduction.v) handy as a cross-reference. - *) - Hint Constructors Rule_Flat. - Definition FlatteningFunctor_fmor - : forall h c, - (ND (PCFRule Γ Δ ec) h c) -> - ((obact h)====>(obact c)). - - set (@nil (HaskTyVar Γ ★)) as lev. - - unfold hom; unfold ob; unfold ehom; simpl; unfold pmon_I; unfold obact; intros. - - induction X; simpl. - - (* the proof from no hypotheses of no conclusions (nd_id0) becomes RVoid *) - apply nd_rule; apply (org_fc Γ Δ [] [(_,_)] (RVoid _ _)). apply Flat_RVoid. - - (* the proof from hypothesis X of conclusion X (nd_id1) becomes RVar *) - apply nd_rule; apply (org_fc _ _ [] [(_,_)] (RVar _ _ _ _)). apply Flat_RVar. - - (* the proof from hypothesis X of no conclusions (nd_weak) becomes RWeak;;RVoid *) - eapply nd_comp; - [ idtac - | eapply nd_rule - ; eapply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RWeak _))) - ; auto ]. - eapply nd_rule. - eapply (org_fc _ _ [] [(_,_)] (RVoid _ _)); auto. apply Flat_RVoid. - apply Flat_RArrange. - - (* the proof from hypothesis X of two identical conclusions X,,X (nd_copy) becomes RVar;;RJoin;;RCont *) - eapply nd_comp; [ idtac | eapply nd_rule; eapply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCont _))) ]. - eapply nd_comp; [ apply nd_llecnac | idtac ]. - set (snd_initial(SequentND:=pl_snd(ProgrammingLanguage:=SystemFCa Γ Δ)) - (mapOptionTree (guestJudgmentAsGArrowType) h @@@ lev)) as q. - eapply nd_comp. - eapply nd_prod. - apply q. - apply q. - apply nd_rule. - eapply (org_fc _ _ ([(_,_)],,[(_,_)]) [(_,_)] (RJoin _ _ _ _ _ _ )). - destruct h; simpl. - destruct o. - simpl. - apply Flat_RJoin. - apply Flat_RJoin. - apply Flat_RJoin. - apply Flat_RArrange. - - (* nd_prod becomes nd_llecnac;;nd_prod;;RJoin *) - eapply nd_comp. - apply (nd_llecnac ;; nd_prod IHX1 IHX2). - apply nd_rule. - eapply (org_fc _ _ ([(_,_)],,[(_,_)]) [(_,_)] (RJoin _ _ _ _ _ _ )). - apply (Flat_RJoin Γ Δ (mapOptionTree guestJudgmentAsGArrowType h1 @@@ nil) - (mapOptionTree guestJudgmentAsGArrowType h2 @@@ nil) - (mapOptionTree guestJudgmentAsGArrowType c1 @@@ nil) - (mapOptionTree guestJudgmentAsGArrowType c2 @@@ nil)). - - (* nd_comp becomes pl_subst (aka nd_cut) *) - eapply nd_comp. - apply (nd_llecnac ;; nd_prod IHX1 IHX2). - clear IHX1 IHX2 X1 X2. - apply (@snd_cut _ _ _ _ (pl_snd(ProgrammingLanguage:=SystemFCa Γ Δ))). - - (* nd_cancell becomes RVar;;RuCanL *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RuCanL _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - (* nd_cancelr becomes RVar;;RuCanR *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RuCanR _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - (* nd_llecnac becomes RVar;;RCanL *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCanL _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - (* nd_rlecnac becomes RVar;;RCanR *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCanR _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - (* nd_assoc becomes RVar;;RAssoc *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RAssoc _ _ _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - (* nd_cossa becomes RVar;;RCossa *) - eapply nd_comp; - [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCossa _ _ _))) ]. - apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))). - apply Flat_RArrange. - - destruct r as [r rp]. - rename h into h'. - rename c into c'. - rename r into r'. - - refine (match rp as R in @Rule_PCF _ _ _ H C _ - return - ND (OrgR Γ Δ) [] - [sequent (mapOptionTree guestJudgmentAsGArrowType H @@@ nil) - (mapOptionTree guestJudgmentAsGArrowType C @@@ nil)] - with - | PCF_RArrange h c r q => let case_RURule := tt in _ - | PCF_RLit lit => let case_RLit := tt in _ - | PCF_RNote Σ τ n => let case_RNote := tt in _ - | PCF_RVar σ => let case_RVar := tt in _ - | PCF_RLam Σ tx te => let case_RLam := tt in _ - | PCF_RApp Σ tx te p => let case_RApp := tt in _ - | PCF_RLet Σ σ₁ σ₂ p => let case_RLet := tt in _ - | PCF_RJoin b c d e => let case_RJoin := tt in _ - | PCF_RVoid => let case_RVoid := tt in _ - (*| PCF_RCase T κlen κ θ l x => let case_RCase := tt in _*) - (*| PCF_RLetRec Σ₁ τ₁ τ₂ lev => let case_RLetRec := tt in _*) - end); simpl in *. - clear rp h' c' r'. - - rewrite (unlev_lemma h (ec::nil)). - rewrite (unlev_lemma c (ec::nil)). - destruct case_RURule. - refine (match q as Q in Arrange H C - return - H=(h @@@ (ec :: nil)) -> - C=(c @@@ (ec :: nil)) -> - ND (OrgR Γ Δ) [] - [sequent - [ga_type (ga_rep (mapOptionTree unlev H)) (ga_rep r) @@ nil ] - [ga_type (ga_rep (mapOptionTree unlev C)) (ga_rep r) @@ nil ] ] - with - | RLeft a b c r => let case_RLeft := tt in _ - | RRight a b c r => let case_RRight := tt in _ - | RCanL b => let case_RCanL := tt in _ - | RCanR b => let case_RCanR := tt in _ - | RuCanL b => let case_RuCanL := tt in _ - | RuCanR b => let case_RuCanR := tt in _ - | RAssoc b c d => let case_RAssoc := tt in _ - | RCossa b c d => let case_RCossa := tt in _ - | RExch b c => let case_RExch := tt in _ - | RWeak b => let case_RWeak := tt in _ - | RCont b => let case_RCont := tt in _ - | RComp a b c f g => let case_RComp := tt in _ - end (refl_equal _) (refl_equal _)); - intros; simpl in *; - subst; - try rewrite <- unlev_lemma in *. - - destruct case_RCanL. - apply magic. - apply ga_uncancell. - - destruct case_RCanR. - apply magic. - apply ga_uncancelr. - - destruct case_RuCanL. - apply magic. - apply ga_cancell. - - destruct case_RuCanR. - apply magic. - apply ga_cancelr. - - destruct case_RAssoc. - apply magic. - apply ga_assoc. - - destruct case_RCossa. - apply magic. - apply ga_unassoc. - - destruct case_RExch. - apply magic. - apply ga_swap. - - destruct case_RWeak. - apply magic. - apply ga_drop. - - destruct case_RCont. - apply magic. - apply ga_copy. - - destruct case_RLeft. - apply magic. - (*apply ga_second.*) - admit. - - destruct case_RRight. - apply magic. - (*apply ga_first.*) - admit. - - destruct case_RComp. - apply magic. - (*apply ga_comp.*) - admit. - - destruct case_RLit. - apply ga_lit. - - (* hey cool, I figured out how to pass CoreNote's through... *) - destruct case_RNote. - eapply nd_comp. - eapply nd_rule. - eapply (org_fc _ _ [] [(_,_)] (RVar _ _ _ _)) . auto. - apply Flat_RVar. - apply nd_rule. - apply (org_fc _ _ [(_,_)] [(_,_)] (RNote _ _ _ _ _ n)). auto. - apply Flat_RNote. - - destruct case_RVar. - apply ga_id. - - (* - * class GArrow g (**) u => GArrowApply g (**) u (~>) where - * ga_applyl :: g (x**(x~>y) ) y - * ga_applyr :: g ( (x~>y)**x) y - * - * class GArrow g (**) u => GArrowCurry g (**) u (~>) where - * ga_curryl :: g (x**y) z -> g x (y~>z) - * ga_curryr :: g (x**y) z -> g y (x~>z) - *) - destruct case_RLam. - (* GArrowCurry.ga_curry *) - admit. - - destruct case_RApp. - (* GArrowApply.ga_apply *) - admit. - - destruct case_RLet. - admit. - - destruct case_RVoid. - apply ga_id. - - destruct case_RJoin. - (* this assumes we want effects to occur in syntactically-left-to-right order *) - admit. - Defined. -*) \ No newline at end of file +Implicit Arguments garrow [ ].