Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import HaskKinds.
Require Import HaskCoreTypes.
-Require Import HaskLiteralsAndTyCons.
+Require Import HaskCoreVars.
+Require Import HaskWeakTypes.
+Require Import HaskWeakVars.
+Require Import HaskLiterals.
+Require Import HaskTyCons.
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.
+Require Import HaskWeakToStrong.
+
+Require Import HaskSkolemizer.
Open Scope nd_scope.
+Set Printing Width 130.
(*
* The flattening transformation. Currently only TWO-level languages are
*)
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*)
-
- (* lemma: if a proof from no hypotheses contains no Brak's or Esc's, then the context contains no variables at level!=0 *)
-
- 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
[ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ]
end.
- (* 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.
+ 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.
(* In a tree of types, replace any type at depth "lev" or greater None *)
- Definition drop_depth {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.
+ Definition mkDropFlags {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : TreeFlags tt :=
+ mkFlags (liftBoolFunc false (levelMatch lev)) tt.
+
+ Definition drop_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
+ dropT (mkDropFlags lev tt).
+
+ (* 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 (liftBoolFunc true (bnot ○ levelMatch lev)) tt.
+
+ Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
+ dropT (mkTakeFlags lev tt).
+(*
+ 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
+ | _ => true
+ end
+ ) tt))).
+
+ Definition maybeTree {T}(t:??(Tree ??T)) : Tree ??T :=
+ match t with
+ | None => []
+ | Some x => x
+ end.
+*)
- Lemma drop_depth_lemma : forall Γ (lev:HaskLevel Γ) x, drop_depth lev [x @@ lev] = [].
+ Lemma drop_lev_lemma : forall Γ (lev:HaskLevel Γ) x, drop_lev lev [x @@ lev] = [].
intros; simpl.
Opaque eqd_dec.
- unfold drop_depth.
+ unfold drop_lev.
+ simpl.
+ unfold mkDropFlags.
simpl.
Transparent eqd_dec.
eqd_dec_refl'.
auto.
Qed.
- Lemma drop_depth_lemma_s : forall Γ (lev:HaskLevel Γ) ec x, drop_depth (ec::lev) [x @@ (ec :: lev)] = [].
+ 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_depth.
+ 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] = [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.
+ unfold mkTakeFlags.
simpl.
Transparent eqd_dec.
eqd_dec_refl'.
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.
+
Ltac drop_simplify :=
match goal with
- | [ |- context[@drop_depth ?G ?L [ ?X @@ ?L ] ] ] =>
- rewrite (drop_depth_lemma G L X)
- | [ |- context[@drop_depth ?G (?E :: ?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 ★)))
+ | [ |- 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 ?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.
- Fixpoint reduceTree {T}(unit:T)(merge:T -> T -> T)(tt:Tree ??T) : T :=
- match tt with
- | T_Leaf None => unit
- | T_Leaf (Some x) => x
- | T_Branch b1 b2 => merge (reduceTree unit merge b1) (reduceTree unit merge b2)
- end.
- Set Printing Width 130.
+ (*******************************************************************************)
- Context {unitTy : forall TV, RawHaskType TV ★ }.
- Context {prodTy : forall TV, RawHaskType TV (★ ⇛ ★ ⇛ ★) }.
- Context {gaTy : forall TV, RawHaskType TV (★ ⇛ ★ ⇛ ★ ⇛ ★)}.
- Definition ga_tree := fun TV tr => reduceTree (unitTy TV) (fun t1 t2 => TApp (TApp (prodTy TV) t1) t2) tr.
- Definition ga' := fun TV ec ant' suc' => TApp (TApp (TApp (gaTy TV) ec) (ga_tree TV ant')) (ga_tree TV suc').
- Definition ga {Γ} : HaskType Γ ★ -> Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> HaskType Γ ★ :=
- fun ec ant suc =>
- fun TV ite =>
- let ant' := mapOptionTree (fun x => x TV ite) ant in
- let suc' := mapOptionTree (fun x => x TV ite) suc in
- ga' TV (ec TV ite) ant' suc'.
+ 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 ★ }.
- Class garrow :=
- { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a a @@ l] ]
- ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,[]) a @@ l] ]
- ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ([],,a) a @@ l] ]
- ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,[]) @@ l] ]
- ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a ([],,a) @@ l] ]
- ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ]
- ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ]
- ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,b) (b,,a) @@ l] ]
- ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a [] @@ l] ]
- ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,a) @@ l] ]
- ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l] |- [@ga Γ ec (a,,x) (b,,x) @@ l] ]
- ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l] |- [@ga Γ ec (x,,a) (x,,b) @@ l] ]
- ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec [] [literalType lit] @@ l] ]
- ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga Γ ec (a,,[b]) [c] @@ l] |- [@ga Γ ec a [b ---> c] @@ l] ]
- ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l],,[@ga Γ ec b c @@ l] |- [@ga Γ ec a c @@ l] ]
- ; ga_apply : ∀ Γ Δ ec l a a' b c, ND Rule []
- [Γ > Δ > [@ga Γ ec a [b ---> c] @@ l],,[@ga Γ ec a' [b] @@ l] |- [@ga Γ ec (a,,a') [c] @@ l] ]
- ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule
- [Γ > Δ > Σ,,[@ga Γ ec [] a @@ l] |- [@ga Γ ec [] b @@ l] ]
- [Γ > Δ > Σ |- [@ga Γ ec a b @@ l] ]
- }.
- Context `(gar:garrow).
+ 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).
- Notation "a ~~~~> b" := (@ga _ _ a b) (at level 20).
+ 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:
* - 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}(depth:nat)(exp: RawHaskType TV κ) : RawHaskType TV κ :=
+ Fixpoint flatten_rawtype {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ :=
match exp with
- | TVar _ x => TVar x
- | TAll _ y => TAll _ (fun v => garrowfy_raw_codetypes depth (y v))
- | TApp _ _ x y => TApp (garrowfy_raw_codetypes depth x) (garrowfy_raw_codetypes depth y)
- | TCon tc => TCon tc
- | TCoerc _ t1 t2 t => TCoerc (garrowfy_raw_codetypes depth t1) (garrowfy_raw_codetypes depth t2)
- (garrowfy_raw_codetypes depth t)
- | TArrow => TArrow
- | TCode v e => match depth with
- | O => ga' TV v [] [(*garrowfy_raw_codetypes depth*) e]
- | (S depth') => TCode v (garrowfy_raw_codetypes depth' e)
- end
- | TyFunApp tfc lt => TyFunApp tfc (garrowfy_raw_codetypes_list _ depth lt)
+ | 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 garrowfy_raw_codetypes_list {TV}(lk:list Kind)(depth:nat)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk :=
+ 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 _ _ (garrowfy_raw_codetypes depth t) (garrowfy_raw_codetypes_list _ depth rest)
+ | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flatten_rawtype t) (flatten_rawtype_list _ rest)
end.
- Definition garrowfy_code_types {Γ}{κ}(n:nat)(ht:HaskType Γ κ) : HaskType Γ κ :=
- fun TV ite =>
- garrowfy_raw_codetypes n (ht TV ite).
- Definition garrowfy_leveled_code_types {Γ}(n:nat)(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
- match ht with htt @@ htlev => garrowfy_code_types (minus' n (length htlev)) htt @@ htlev end.
-
- Axiom literal_types_unchanged : forall n Γ l, garrowfy_code_types n (literalType l) = literalType(Γ:=Γ) l.
-
- Axiom flatten_coercion : forall n Γ Δ κ (σ τ:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)),
- HaskCoercion Γ Δ (garrowfy_code_types n σ ∼∼∼ garrowfy_code_types n 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
- | T_Leaf (Some (_ @@ lev)) => lev
- | T_Branch b1 b2 =>
- match getjlev b1 with
- | nil => getjlev b2
- | lev => lev
- 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 v2t {Γ}(ec:HaskTyVar Γ ★) := fun TV ite => TVar (ec TV ite).
+ 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, 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 (flatten_type σ ∼∼∼ flatten_type τ).
+
+ Axiom flatten_commutes_with_substT :
+ forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★) (τ:HaskType Γ κ),
+ flatten_type (substT σ τ) = substT (fun TV ite v => flatten_rawtype (σ TV ite v))
+ (flatten_type τ).
+
+ Axiom flatten_commutes_with_HaskTAll :
+ forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
+ flatten_type (HaskTAll κ σ) = HaskTAll κ (fun TV ite v => flatten_rawtype (σ TV ite v)).
+
+ Axiom flatten_commutes_with_HaskTApp :
+ forall κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
+ flatten_type (HaskTApp (weakF σ) (FreshHaskTyVar κ)) =
+ HaskTApp (weakF (fun TV ite v => flatten_rawtype (σ TV ite v))) (FreshHaskTyVar κ).
+
+ 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,
+ flatten_type (g v) = g v.
(* "n" is the maximum depth remaining AFTER flattening *)
- Definition flatten_judgment (n:nat)(j:Judg) :=
+ Definition flatten_judgment (j:Judg) :=
match j as J return Judg with
- Γ > Δ > ant |- suc =>
- match (match getjlev suc with
- | nil => inl _ tt
- | (ec::lev') => if eqd_dec (length lev') n
- (* If the judgment's level is the deepest in the proof, flatten it by turning
- * all antecedent variables at this level into None's, garrowfying any other
- * antecedent variables (from other levels) at the same depth, and turning the
- * succedent into a garrow type *)
- then inr _ (Γ > Δ > mapOptionTree (garrowfy_leveled_code_types n) (drop_depth (ec::lev') ant)
- |- [ga (v2t ec) (take_lev (ec::lev') ant) (mapOptionTree unlev suc) @@ lev'])
- else inl _ tt
- end) with
-
- (* otherwise, just garrowfy all code types of the specified depth, throughout the judgment *)
- | inl tt => Γ > Δ > mapOptionTree (garrowfy_leveled_code_types n) ant |- mapOptionTree (garrowfy_leveled_code_types n) suc
- | inr r => r
- end
+ | Γ > Δ > ant |- suc @ nil => Γ > Δ > mapOptionTree flatten_leveled_type ant
+ |- mapOptionTree flatten_type suc @ nil
+ | Γ > Δ > ant |- suc @ (ec::lev') => Γ > Δ > mapOptionTree flatten_leveled_type (drop_lev (ec::lev') ant)
+ |- [ga_mk (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec::lev') ant))
+ (mapOptionTree flatten_type suc )
+ ] @ nil
end.
- 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.
+ 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).
- 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.
+ Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20).
- Definition seq : ∀ Γ Δ lev a b,
- ND Rule [] [ Γ > Δ > [] |- [a @@ lev] ] ->
- ND Rule [] [ Γ > Δ > [] |- [b @@ lev] ] ->
- ND Rule [] [ Γ > Δ > [] |- [a @@ lev],,[b @@ lev] ].
- admit.
- Defined.
+ 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 ACanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_id.
+ eapply nd_comp.
+ apply X.
+ eapply nd_rule.
+ eapply RArrange.
+ apply AuCanR.
+ Defined.
- Lemma ga_unkappa : ∀ Γ Δ ec l a b Σ,
+ Definition precompose Γ Δ ec : forall a x y z lev,
ND Rule
- [Γ > Δ > Σ |- [@ga Γ ec a b @@ l] ]
- [Γ > Δ > Σ,,[@ga Γ ec [] a @@ l] |- [@ga Γ ec [] b @@ l] ].
+ [ Γ > Δ > a |- [@ga_mk _ ec y z ]@lev ]
+ [ Γ > Δ > a,,[@ga_mk _ ec x y @@ lev] |- [@ga_mk _ ec x z ]@lev ].
intros.
- set (ga_comp Γ Δ ec l [] a b) as q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_id.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ apply ga_comp.
+ Defined.
- set (@RLet Γ Δ) as q'.
- set (@RLet Γ Δ [@ga _ ec [] a @@ l] Σ (@ga _ ec [] b) (@ga _ ec a b) l) as q''.
- eapply nd_comp.
- Focus 2.
- eapply nd_rule.
- eapply RArrange.
- apply RExch.
+ Definition precompose' Γ Δ ec : forall a b x y z lev,
+ ND Rule
+ [ Γ > Δ > a,,b |- [@ga_mk _ ec y z ]@lev ]
+ [ Γ > Δ > a,,([@ga_mk _ ec x y @@ lev],,b) |- [@ga_mk _ ec x z ]@lev ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ALeft; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuAssoc ].
+ apply precompose.
+ Defined.
- eapply nd_comp.
- Focus 2.
- eapply nd_rule.
- apply q''.
+ Definition postcompose_ Γ Δ ec : forall a x y z lev,
+ ND Rule
+ [ Γ > Δ > a |- [@ga_mk _ ec x y ]@lev ]
+ [ Γ > Δ > a,,[@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z ]@lev ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_id.
+ apply ga_comp.
+ Defined.
- idtac.
- clear q'' q'.
+ Definition postcompose Γ Δ ec : forall x y z lev,
+ ND Rule [] [ Γ > Δ > [] |- [@ga_mk _ ec x y ]@lev ] ->
+ ND Rule [] [ Γ > Δ > [@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z ]@lev ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
+ eapply nd_comp; [ idtac | eapply postcompose_ ].
+ apply X.
+ Defined.
+
+ Definition first_nd : ∀ Γ Δ ec lev a b c Σ,
+ ND Rule [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ]
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) ]@lev ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_id.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply ga_first.
+ Defined.
+
+ 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.
- apply nd_rlecnac.
+ apply X.
+ apply first_nd.
+ Defined.
+
+ Definition second_nd : ∀ Γ Δ ec lev a b c Σ,
+ ND Rule
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ]
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) ]@lev ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
- apply q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply ga_second.
Defined.
-(*
+ 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.
+ apply X.
+ apply second_nd.
+ 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 RArrange; eapply AExch ].
+ 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.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ apply precompose.
+ 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_depth _ x) (at level 20).
-*)
- Definition garrowfy_arrangement' :
+ Notation "``` x" := (drop_lev _ x) (at level 20).
+ *)
+ Definition flatten_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 _ (v2t 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) (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 _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev]] :=
+ 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 _ (v2t ec) (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ lev]]
+ 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
- | 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 _ _ _ _ _
- | RuCanR a => let case_RuCanR := tt in ga_cancelr _ _ _ _ _
- | RAssoc a b c => let case_RAssoc := tt in ga_assoc _ _ _ _ _ _ _
- | RCossa a b c => let case_RCossa := tt in ga_unassoc _ _ _ _ _ _ _
- | 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 a b c 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.
- refine ( _ ;; boost _ _ _ _ _ (ga_comp _ _ _ _ _ _ _)).
- apply seq.
+ | AId a => let case_AId := tt in ga_id _ _ _ _ _
+ | ACanL a => let case_ACanL := tt in ga_uncancell _ _ _ _ _
+ | ACanR a => let case_ACanR := tt in ga_uncancelr _ _ _ _ _
+ | AuCanL a => let case_AuCanL := tt in ga_cancell _ _ _ _ _
+ | AuCanR a => let case_AuCanR := tt in ga_cancelr _ _ _ _ _
+ | AAssoc a b c => let case_AAssoc := tt in ga_assoc _ _ _ _ _ _ _
+ | AuAssoc a b c => let case_AuAssoc := tt in ga_unassoc _ _ _ _ _ _ _
+ | AExch a b => let case_AExch := tt in ga_swap _ _ _ _ _ _
+ | AWeak a => let case_AWeak := tt in ga_drop _ _ _ _ _
+ | ACont a => let case_ACont := tt in ga_copy _ _ _ _ _
+ | ALeft a b c r' => let case_ALeft := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _)
+ | ARight a b c r' => let case_ARight := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _)
+ | AComp c b a r1 r2 => let case_AComp := tt in (fun r1' r2' => _) (flatten _ _ r1) (flatten _ _ r2)
+ end); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
+
+ destruct case_AComp.
+ 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 ACanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply
+ (@RLet Γ Δ [] [] (@ga_mk _ (v2t ec) a' b') (@ga_mk _ (v2t ec) a' c')) ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
apply r2'.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ eapply nd_prod.
apply r1'.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ apply ga_comp.
Defined.
- Definition garrowfy_arrangement :
+ Definition flatten_arrangement :
forall Γ (Δ:CoercionEnv Γ) n
- (ec:HaskTyVar Γ ★) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ,
+ (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 ((length lev))) (drop_depth n ant1)
- |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant1) (mapOptionTree unlev succ) @@ lev]]
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth n ant2)
- |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant2) (mapOptionTree unlev succ) @@ lev]].
+ [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 ) 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 ) 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 ((length lev))) (drop_depth _ A))
- (mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth _ 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 _
- | RuCanR a => let case_RuCanR := tt in RuCanR _
- | RAssoc a b c => let case_RAssoc := tt in RAssoc _ _ _
- | RCossa a b c => let case_RCossa := tt in RCossa _ _ _
- | 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.
+ Arrange (mapOptionTree (flatten_leveled_type ) (drop_lev _ A))
+ (mapOptionTree (flatten_leveled_type ) (drop_lev _ B)) with
+ | AId a => let case_AId := tt in AId _
+ | ACanL a => let case_ACanL := tt in ACanL _
+ | ACanR a => let case_ACanR := tt in ACanR _
+ | AuCanL a => let case_AuCanL := tt in AuCanL _
+ | AuCanR a => let case_AuCanR := tt in AuCanR _
+ | AAssoc a b c => let case_AAssoc := tt in AAssoc _ _ _
+ | AuAssoc a b c => let case_AuAssoc := tt in AuAssoc _ _ _
+ | AExch a b => let case_AExch := tt in AExch _ _
+ | AWeak a => let case_AWeak := tt in AWeak _
+ | ACont a => let case_ACont := tt in ACont _
+ | ALeft a b c r' => let case_ALeft := tt in ALeft _ (flatten _ _ r')
+ | ARight a b c r' => let case_ARight := tt in ARight _ (flatten _ _ r')
+ | AComp a b c r1 r2 => let case_AComp := tt in AComp (flatten _ _ r1) (flatten _ _ r2)
+ end) ant1 ant2 r); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
Defined.
- Definition flatten_arrangement :
- forall n Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2),
- ND Rule (mapOptionTree (flatten_judgment n) [Γ > Δ > ant1 |- succ])
- (mapOptionTree (flatten_judgment n) [Γ > Δ > ant2 |- succ]).
+ Definition flatten_arrangement'' :
+ forall Γ Δ ant1 ant2 succ l (r:Arrange ant1 ant2),
+ ND Rule (mapOptionTree (flatten_judgment ) [Γ > Δ > ant1 |- succ @ l])
+ (mapOptionTree (flatten_judgment ) [Γ > Δ > ant2 |- succ @ l]).
intros.
simpl.
- set (getjlev succ) as succ_lev.
- assert (succ_lev=getjlev succ).
- reflexivity.
-
- destruct succ_lev.
+ destruct l.
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 ].
-
- 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 AId.
+ apply ACanL.
+ apply ACanR.
+ apply AuCanL.
+ apply AuCanR.
+ apply AAssoc.
+ apply AuAssoc.
+ apply AExch. (* TO DO: check for all-leaf trees here *)
+ apply AWeak.
+ apply ACont.
+ apply ALeft; auto.
+ apply ARight; auto.
+ eapply AComp; [ apply IHr1 | apply IHr2 ].
+
+ apply flatten_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.
+ 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.
+ apply firstify with (c:=[]) in pfa.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ eapply nd_llecnac | idtac ].
+ apply nd_prod.
+ apply pfa.
+ clear pfa.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
+ eapply nd_comp; [ idtac | eapply postcompose_ ].
+ apply ga_uncancelr.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply precompose ].
+ apply pfb.
+ Defined.
+
Definition arrange_brak : forall Γ Δ ec succ t,
- ND Rule
- [Γ > Δ >
- mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec :: nil) succ),,
- [(@ga _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |-
- [(@ga _ (v2t ec) [] [t]) @@ nil]]
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types 0) succ |- [(@ga _ (v2t ec) [] [t]) @@ nil]].
- admit.
+ ND Rule
+ [Γ > Δ >
+ [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil],,
+ mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t]@nil]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t]@nil].
+
+ intros.
+ unfold drop_lev.
+ set (@arrangeUnPartition _ succ (levelMatch (ec::nil))) as q.
+ set (arrangeMap _ _ flatten_leveled_type q) as y.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ apply y.
+ idtac.
+ clear y q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ 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.
+ unfold flatten_leveled_type.
+ simpl.
+ unfold flatten_type.
+ simpl.
+ unfold ga_mk.
+ simpl.
+ apply RVar.
+ simpl.
+ apply ga_id.
+ apply ga_id.
+ unfold take_lev.
+ simpl.
+ apply ga_join.
+ apply IHsucc1.
+ apply IHsucc2.
Defined.
Definition arrange_esc : forall Γ Δ ec succ t,
- ND Rule
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types 0) succ |- [(@ga _ (v2t ec) [] [t]) @@ nil]]
- [Γ > Δ >
- mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec :: nil) succ),,
- [(@ga _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga _ (v2t ec) [] [t]) @@ nil]].
- admit.
- Defined.
+ ND Rule
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t]@nil]
+ [Γ > Δ >
+ [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil],,
+ mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t]@nil].
+ intros.
+ set (@arrangePartition _ succ (levelMatch (ec::nil))) as q.
+ 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.
+ 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 AExch ].
+ 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 RLet ].
+ clear q''.
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_rule.
+ apply RArrange.
+ eapply AComp; [ idtac | apply ACanR ].
+ apply ALeft.
+ apply (@arrangeCancelEmptyTree _ _ _ _ e).
+
+ eapply nd_comp.
+ eapply nd_rule.
+ eapply (@RVar Γ Δ t nil).
+ apply nd_rule.
+ apply RArrange.
+ eapply AComp.
+ apply AuCanR.
+ apply ALeft.
+ apply AWeak.
+(*
+ 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.
+ unfold take_lev.
+ unfold mkTakeFlags.
+ simpl.
+ destruct (General.list_eq_dec h0 (ec :: nil)).
+ simpl.
+ rewrite e.
+ apply nd_id.
+ simpl.
+ apply nd_rule.
+ apply RArrange.
+ apply ALeft.
+ apply AWeak.
+ simpl.
+ apply nd_rule.
+ unfold take_lev.
+ simpl.
+ apply RArrange.
+ apply ALeft.
+ apply AWeak.
+ apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported").
+*)
+ Defined.
- Lemma mapOptionTree_distributes
- : forall T R (a b:Tree ??T) (f:T->R),
- mapOptionTree f (a,,b) = (mapOptionTree f a),,(mapOptionTree f b).
+ Lemma unlev_relev : forall {Γ}(t:Tree ??(HaskType Γ ★)) lev, mapOptionTree unlev (t @@@ lev) = t.
+ intros.
+ induction t.
+ destruct a; reflexivity.
+ rewrite <- IHt1 at 2.
+ rewrite <- IHt2 at 2.
reflexivity.
Qed.
- Lemma garrowfy_commutes_with_substT :
- forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★) (τ:HaskType Γ κ),
- garrowfy_code_types n (substT σ τ) = substT (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v))
- (garrowfy_code_types n τ).
- admit.
- Qed.
+ Lemma tree_of_nothing : forall Γ ec t,
+ Arrange (mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))) [].
+ intros.
+ induction t; try destruct o; try destruct a.
+ simpl.
+ drop_simplify.
+ simpl.
+ apply AId.
+ simpl.
+ apply AId.
+ eapply AComp; [ idtac | apply ACanL ].
+ eapply AComp; [ idtac | eapply ALeft; apply IHt2 ].
+ Opaque drop_lev.
+ simpl.
+ Transparent drop_lev.
+ idtac.
+ drop_simplify.
+ apply ARight.
+ apply IHt1.
+ Defined.
- Lemma garrowfy_commutes_with_HaskTAll :
- forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
- garrowfy_code_types n (HaskTAll κ σ) = HaskTAll κ (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v)).
- admit.
- Qed.
+ Lemma tree_of_nothing' : forall Γ ec t,
+ Arrange [] (mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))).
+ intros.
+ induction t; try destruct o; try destruct a.
+ simpl.
+ drop_simplify.
+ simpl.
+ apply AId.
+ simpl.
+ apply AId.
+ eapply AComp; [ apply AuCanL | idtac ].
+ eapply AComp; [ eapply ARight; apply IHt1 | idtac ].
+ Opaque drop_lev.
+ simpl.
+ Transparent drop_lev.
+ idtac.
+ drop_simplify.
+ apply ALeft.
+ apply IHt2.
+ Defined.
- Lemma garrowfy_commutes_with_HaskTApp :
- forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
- garrowfy_code_types n (HaskTApp (weakF σ) (FreshHaskTyVar κ)) =
- HaskTApp (weakF (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v))) (FreshHaskTyVar κ).
+ 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.
- Lemma garrowfy_commutes_with_weakLT : forall (Γ:TypeEnv) κ n t,
- garrowfy_leveled_code_types n (weakLT(Γ:=Γ)(κ:=κ) t) = weakLT(Γ:=Γ)(κ:=κ) (garrowfy_leveled_code_types n t).
- admit.
+ Lemma drop_to_nothing : forall (Γ:TypeEnv) Σ (lev:HaskLevel Γ),
+ drop_lev lev (Σ @@@ lev) = mapTree (fun _ => None) (mapTree (fun _ => tt) Σ).
+ intros.
+ induction Σ.
+ destruct a; simpl.
+ drop_simplify.
+ auto.
+ drop_simplify.
+ auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
Qed.
- Definition flatten_proof :
- forall n {h}{c},
- ND Rule h c ->
- ND Rule (mapOptionTree (flatten_judgment n) h) (mapOptionTree (flatten_judgment n) c).
+ Definition flatten_skolemized_proof :
+ forall {h}{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
- | RArrange Γ Δ a b x d => let case_RArrange := tt in _
+ 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 l 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 _
| RVar Γ Δ σ lev => let case_RVar := tt in _
| RAbsCo Γ Δ Σ κ σ σ₁ σ₂ lev => let case_RAbsCo := tt in _
| RApp Γ Δ Σ₁ Σ₂ tx te lev => let case_RApp := tt in _
| 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 _
+ | RCut Γ Δ Σ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _
+ | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _
+ | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := tt in _
+ | RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := tt in _
+ | RVoid _ _ l => let case_RVoid := 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 _
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.
- simpl.
- destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n).
- destruct lev.
- simpl.
- simpl.
- destruct n.
- change ([garrowfy_code_types 0 (<[ ec |- t ]>) @@ nil])
- with ([ga (v2t ec) [] [t] @@ nil]).
- refine (ga_unkappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) [t]
- (mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec::nil) succ)) ;; _).
- apply arrange_brak.
- inversion e.
- apply (Prelude_error "found Brak at depth >0").
- apply (Prelude_error "found Brak at depth >0").
+ apply (Prelude_error "found unskolemized Brak rule; this shouldn't happen").
destruct case_REsc.
- simpl.
- destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n).
- destruct lev.
- simpl.
- destruct n.
- change ([garrowfy_code_types 0 (<[ ec |- t ]>) @@ nil])
- with ([ga (v2t ec) [] [t] @@ nil]).
- refine (_ ;; ga_kappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) [t]
- (mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec::nil) succ))).
- apply arrange_esc.
- inversion e.
- apply (Prelude_error "found Esc at depth >0").
- apply (Prelude_error "found Esc at depth >0").
+ apply (Prelude_error "found unskolemized Esc rule; this shouldn't happen").
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.
simpl.
destruct l0; simpl.
+ unfold flatten_leveled_type.
+ 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.
+ simpl.
rewrite literal_types_unchanged.
- apply nd_rule.
- apply RLit.
+ apply ga_lit.
destruct case_RVar.
Opaque flatten_judgment.
Transparent flatten_judgment.
idtac.
unfold flatten_judgment.
- unfold getjlev.
destruct lev.
apply nd_rule. apply RVar.
- destruct (eqd_dec (Datatypes.length lev) n).
-
repeat drop_simplify.
repeat take_simplify.
simpl.
apply ga_id.
- apply nd_rule.
- apply RVar.
-
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.
- apply (Prelude_error "found RGlobal at depth >0").
+ rename l into g.
+ rename σ into l.
+ destruct l as [|ec lev]; simpl.
+ destruct (eqd_dec (g:CoreVar) (hetmet_flatten:CoreVar)).
+ 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 (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.
+ 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").
destruct case_RLam.
- Opaque drop_depth.
+ 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 ACanR.
apply boost.
+ simpl.
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.
- destruct (getjlev x); destruct (getjlev q).
- apply nd_rule.
- apply RJoin.
- apply (Prelude_error "RJoin at depth >0").
- apply (Prelude_error "RJoin at depth >0").
- apply (Prelude_error "RJoin at depth >0").
+ apply flatten_coercion; auto.
+ apply (Prelude_error "RCast at level >0; casting inside of code brackets is currently not supported").
destruct case_RApp.
simpl.
- destruct lev as [|ec lev]. simpl. apply nd_rule.
- replace (garrowfy_code_types n (tx ---> te)) with ((garrowfy_code_types n tx) ---> (garrowfy_code_types n te)).
- apply RApp.
- unfold garrowfy_code_types.
+ destruct lev as [|ec lev].
+ unfold flatten_type at 1.
simpl.
- reflexivity.
+ apply nd_rule.
+ apply RApp.
- 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 (garrowfy_code_types (minus' n (length (ec::lev))) (tx ---> te))
- with ((garrowfy_code_types (minus' n (length (ec::lev))) tx) --->
- (garrowfy_code_types (minus' n (length (ec::lev))) te)).
- apply nd_rule.
- apply RApp.
- unfold garrowfy_code_types.
- simpl.
+ rewrite mapOptionTree_distributes.
+ 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 (flatten_type (tx ---> te)) with ((flatten_type tx) ---> (flatten_type te)).
+ apply (Prelude_error "FIXME: ga_apply").
reflexivity.
+
(*
- Notation "` x" := (take_lev _ x) (at level 20).
+ Notation "` x" := (take_lev _ x).
Notation "`` x" := (mapOptionTree unlev x) (at level 20).
- Notation "``` x" := ((drop_depth _ x)) (at level 20).
- Notation "!<[]> x" := (garrowfy_code_types _ x) (at level 1).
- Notation "!<[@]>" := (garrowfy_leveled_code_types _) (at level 1).
+ Notation "``` x" := ((drop_lev _ x)) (at level 20).
+ Notation "!<[]> x" := (flatten_type _ x) (at level 1).
+ Notation "!<[@]> x" := (mapOptionTree flatten_leveled_type x) (at level 1).
*)
+
destruct case_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.
- rename σ₁ into a.
- rename σ₂ into b.
- rewrite mapOptionTree_distributes.
- rewrite mapOptionTree_distributes.
- set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_depth (ec :: lev) Σ₁)) as A.
- set (take_lev (ec :: lev) Σ₁) as A'.
- set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_depth (ec :: lev) Σ₂)) as B.
- set (take_lev (ec :: lev) Σ₂) as B'.
simpl.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as Σ₁'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_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''.
+
eapply nd_comp.
- Focus 2.
- eapply nd_rule.
- eapply RLet.
+ eapply nd_prod; [ idtac | apply nd_id ].
+ eapply boost.
+ apply (ga_first _ _ _ _ _ _ Σ₂').
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
apply nd_prod.
+ apply nd_id.
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ACanL | idtac ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch (* okay *)].
+ apply precompose.
- apply boost.
- apply ga_second.
+ destruct case_RWhere.
+ simpl.
+ destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RWhere; auto | idtac ].
+ repeat take_simplify.
+ repeat drop_simplify.
+ simpl.
- eapply nd_comp.
- Focus 2.
- eapply boost.
- apply ga_comp.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as Σ₁'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₂)) as Σ₂'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₃)) as Σ₃'.
+ 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 (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₃)) as Σ₃''.
eapply nd_comp.
- eapply nd_rule.
- eapply RArrange.
- eapply RCanR.
-
+ eapply nd_prod; [ eapply nd_id | idtac ].
+ eapply (first_nd _ _ _ _ _ _ Σ₃').
eapply nd_comp.
- Focus 2.
- eapply nd_rule.
- eapply RArrange.
- eapply RExch.
- idtac.
+ eapply nd_prod; [ eapply nd_id | idtac ].
+ eapply (second_nd _ _ _ _ _ _ Σ₁').
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RWhere ].
+ apply nd_prod; [ idtac | apply nd_id ].
+ eapply nd_comp; [ idtac | eapply precompose' ].
+ apply nd_rule.
+ apply RArrange.
+ apply ALeft.
+ apply ACanL.
+
+ destruct case_RCut.
+ simpl.
+ destruct l as [|ec lev]; simpl.
+ apply nd_rule.
+ replace (mapOptionTree flatten_leveled_type (Σ₁₂ @@@ nil)) with (mapOptionTree flatten_type Σ₁₂ @@@ nil).
+ apply RCut.
+ induction Σ₁₂; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ₁₂1.
+ rewrite <- IHΣ₁₂2.
+ reflexivity.
+ simpl; repeat drop_simplify.
+ simpl; repeat take_simplify.
+ simpl.
+ set (drop_lev (ec :: lev) (Σ₁₂ @@@ (ec :: lev))) as x1.
+ rewrite take_lemma'.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ rewrite <- mapOptionTree_compose.
+ rewrite <- mapOptionTree_compose.
+ rewrite <- mapOptionTree_compose.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ apply nd_prod.
+ apply nd_id.
eapply nd_comp.
- apply nd_llecnac.
- eapply nd_comp.
- Focus 2.
eapply nd_rule.
- apply RJoin.
+ eapply RArrange.
+ eapply ALeft.
+ eapply ARight.
+ unfold x1.
+ rewrite drop_to_nothing.
+ apply arrangeCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ₁₂)).
+ admit. (* OK *)
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ALeft; eapply ACanL | idtac ].
+ set (mapOptionTree flatten_type Σ₁₂) as a.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as b.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₂)) as c.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₂)) as d.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ)) as e.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ)) as f.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod.
+ simpl.
+ eapply secondify.
+ apply ga_first.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ALeft; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuAssoc ].
+ simpl.
+ apply precompose.
- eapply nd_rule.
- eapply RVar.
-
- apply nd_id.
+ destruct case_RLeft.
+ simpl.
+ destruct l as [|ec lev].
+ simpl.
+ replace (mapOptionTree flatten_leveled_type (Σ @@@ nil)) with (mapOptionTree flatten_type Σ @@@ nil).
+ apply nd_rule.
+ apply RLeft.
+ induction Σ; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ repeat drop_simplify.
+ rewrite drop_to_nothing.
+ simpl.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply ARight.
+ apply arrangeUnCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ)).
+ admit (* FIXME *).
+ idtac.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanL ].
+ apply boost.
+ take_simplify.
+ simpl.
+ replace (take_lev (ec :: lev) (Σ @@@ (ec :: lev))) with (Σ @@@ (ec::lev)).
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ apply ga_second.
+ rewrite take_lemma'.
+ reflexivity.
+
+ destruct case_RRight.
+ simpl.
+ destruct l as [|ec lev].
+ simpl.
+ replace (mapOptionTree flatten_leveled_type (Σ @@@ nil)) with (mapOptionTree flatten_type Σ @@@ nil).
+ apply nd_rule.
+ apply RRight.
+ induction Σ; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ repeat drop_simplify.
+ rewrite drop_to_nothing.
+ simpl.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply ALeft.
+ apply arrangeUnCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ)).
+ admit (* FIXME *).
+ idtac.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply boost.
+ take_simplify.
+ simpl.
+ replace (take_lev (ec :: lev) (Σ @@@ (ec :: lev))) with (Σ @@@ (ec::lev)).
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ apply ga_first.
+ rewrite take_lemma'.
+ reflexivity.
destruct case_RVoid.
simpl.
apply nd_rule.
+ destruct l.
apply RVoid.
+ apply (Prelude_error "RVoid at level >0").
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.
apply Δ.
- apply (Prelude_error "AppT at depth>0").
+ apply (Prelude_error "found type application at level >0; this is not supported").
destruct case_RAbsT.
simpl. destruct lev; simpl.
- rewrite garrowfy_commutes_with_HaskTAll.
- rewrite garrowfy_commutes_with_HaskTApp.
+ 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 n) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a.
- set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (garrowfy_leveled_code_types n) 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'.
induction Σ.
destruct a.
simpl.
- rewrite garrowfy_commutes_with_weakLT.
+ rewrite flatten_commutes_with_weakLT.
reflexivity.
reflexivity.
simpl.
apply nd_id.
apply Δ.
apply Δ.
- apply (Prelude_error "AbsT at depth>0").
+ apply (Prelude_error "found type abstraction at level >0; this is not supported").
destruct case_RAppCo.
simpl. destruct lev; simpl.
- unfold garrowfy_code_types.
+ unfold flatten_type.
simpl.
apply nd_rule.
apply RAppCo.
apply flatten_coercion.
apply γ.
- apply (Prelude_error "AppCo at depth>0").
+ apply (Prelude_error "found coercion application at level >0; this is not supported").
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 "AbsCo at depth>0").
+ apply (Prelude_error "found coercion abstraction at level >0; this is not supported").
destruct case_RLetRec.
rename t into lev.
- apply (Prelude_error "LetRec not supported (FIXME)").
+ simpl. destruct lev; simpl.
+ apply nd_rule.
+ set (@RLetRec Γ Δ (mapOptionTree flatten_leveled_type lri) (flatten_type x) (mapOptionTree flatten_type y) nil) as q.
+ replace (mapOptionTree flatten_leveled_type (y @@@ nil)) with (mapOptionTree flatten_type y @@@ nil).
+ apply q.
+ induction y; try destruct a; auto.
+ simpl.
+ rewrite IHy1.
+ rewrite IHy2.
+ reflexivity.
+ apply (Prelude_error "LetRec not supported inside brackets yet (FIXME)").
destruct case_RCase.
simpl.
- apply (Prelude_error "Case not supported (FIXME)").
+ 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.
+ 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; eapply ALeft; apply tree_of_nothing | idtac ].
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ACanR | idtac ].
+ refine (ga_unkappa Γ Δ (v2t ec) nil _ _ _ _ ;; _).
+ eapply nd_comp; [ idtac | eapply arrange_brak ].
+ unfold succ_host.
+ unfold succ_guest.
+ eapply nd_rule.
+ eapply RArrange.
+ apply AExch.
+ 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; eapply ALeft; apply tree_of_nothing' ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ 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 AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanL ].
+ 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 ACanL ].
+ 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 (Prelude_error "ga_kappa not supported yet (BIG FIXME)").
+
+ (* nesting too deep *)
+ apply (Prelude_error "found Esc at depth >0 indicating 3-level code; only two-level code is currently supported").
Defined.
+ Definition flatten_proof :
+ forall {h}{c},
+ ND Rule h c ->
+ ND Rule h c.
+ apply (Prelude_error "sorry, non-skolemized flattening isn't implemented").
+ 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_skolemized_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 *)
projects / coq-hetmet.git / blobdiff