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 HaskProof.
Require Import HaskStrongToProof.
Require Import HaskProofToStrong.
+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.
- 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).
-
- 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.
-
- (* take and drop are not exact inverses *)
-
- (* 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 getlev {Γ}{κ}(lht:LeveledHaskType Γ κ) : HaskLevel Γ :=
+ match lht with t @@ l => l end.
Definition arrange :
forall {T} (Σ:Tree ??T) (f:T -> bool),
- Arrange Σ (drop (mkFlags (setNone false f) Σ),,( (drop (mkFlags (setNone false (bnot ○ f)) Σ)))).
+ Arrange Σ (dropT (mkFlags (liftBoolFunc false f) Σ),,( (dropT (mkFlags (liftBoolFunc false (bnot ○ f)) Σ)))).
intros.
induction Σ.
simpl.
simpl.
apply RuCanL.
simpl in *.
- eapply RComp; [ idtac | apply swapMiddle ].
+ eapply RComp; [ idtac | apply arrangeSwapMiddle ].
eapply RComp.
eapply RLeft.
apply IHΣ2.
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.
+ Arrange (dropT (mkFlags (liftBoolFunc false f) Σ),,( (dropT (mkFlags (liftBoolFunc false (bnot ○ f)) Σ)))) Σ.
intros.
induction Σ.
simpl.
simpl.
apply RCanL.
simpl in *.
- eapply RComp; [ apply swapMiddle | idtac ].
+ eapply RComp; [ apply arrangeSwapMiddle | idtac ].
eapply RComp.
eapply RLeft.
apply IHΣ2.
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] ] =>
[ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ]
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 ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }.
- Context {gaTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }.
-
- Definition ga_mk_tree {Γ} (tr:Tree ??(HaskType Γ ★)) : HaskType Γ ★ :=
- fun TV ite => reduceTree (unitTy TV) (prodTy TV) (mapOptionTree (fun x => x TV ite) tr).
-
- Definition ga_mk {Γ}(ec:HaskType Γ ECKind )(ant suc:Tree ??(HaskType Γ ★)) : HaskType Γ ★ :=
- fun TV ite => gaTy TV (ec TV ite) (ga_mk_tree ant TV ite) (ga_mk_tree suc TV ite).
-
- Class garrow :=
- { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a @@ l] ]
- ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a @@ l] ]
- ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a @@ l] ]
- ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) @@ l] ]
- ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) @@ l] ]
- ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ]
- ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ]
- ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) @@ l] ]
- ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] @@ l] ]
- ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) @@ l] ]
- ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (a,,x) (b,,x) @@ l] ]
- ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (x,,a) (x,,b) @@ l] ]
- ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] @@ l] ]
- ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] @@ l] ]
- ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c @@ l] ]
- ; ga_apply : ∀ Γ Δ ec l a a' b c, ND Rule []
- [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] @@ l] ]
- ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule
- [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ]
- [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
- }.
- Context `(gar:garrow).
-
- Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20).
-
- (*
- * The story:
- * - code types <[t]>@c become garrows c () t
- * - free variables of type t at a level lev deeper than the succedent become garrows c () t
- * - free variables at the level of the succedent become
- *)
- Fixpoint garrowfy_raw_codetypes {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ :=
- match exp with
- | TVar _ x => TVar x
- | TAll _ y => TAll _ (fun v => garrowfy_raw_codetypes (y v))
- | TApp _ _ x y => TApp (garrowfy_raw_codetypes x) (garrowfy_raw_codetypes y)
- | TCon tc => TCon tc
- | TCoerc _ t1 t2 t => TCoerc (garrowfy_raw_codetypes t1) (garrowfy_raw_codetypes t2)
- (garrowfy_raw_codetypes t)
- | TArrow => TArrow
- | TCode v e => gaTy TV v (unitTy TV) (garrowfy_raw_codetypes e)
- | TyFunApp tfc kl k lt => TyFunApp tfc kl k (garrowfy_raw_codetypes_list _ lt)
- end
- with garrowfy_raw_codetypes_list {TV}(lk:list Kind)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk :=
- match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with
- | TyFunApp_nil => TyFunApp_nil
- | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (garrowfy_raw_codetypes t) (garrowfy_raw_codetypes_list _ rest)
- end.
- Definition garrowfy_code_types {Γ}{κ}(ht:HaskType Γ κ) : HaskType Γ κ :=
- fun TV ite =>
- garrowfy_raw_codetypes (ht TV ite).
-
- Definition v2t {Γ}(ec:HaskTyVar Γ ECKind) := fun TV ite => TVar (ec TV ite).
-
- Fixpoint garrowfy_leveled_code_types' {Γ}(ht:HaskType Γ ★)(lev:HaskLevel Γ) : HaskType Γ ★ :=
- match lev with
- | nil => garrowfy_code_types ht
- | ec::lev' => @ga_mk _ (v2t ec) [] [garrowfy_leveled_code_types' ht lev']
- end.
-
- Definition garrowfy_leveled_code_types {Γ}(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
- match ht with
- htt @@ lev =>
- garrowfy_leveled_code_types' htt lev @@ nil
- end.
+ Definition v2t {Γ}(ec:HaskTyVar Γ ECKind) : HaskType Γ ECKind := fun TV ite => TVar (ec TV ite).
Definition levelMatch {Γ}(lev:HaskLevel Γ) : LeveledHaskType Γ ★ -> bool :=
fun t => match t with ttype@@tlev => if eqd_dec tlev lev then true else false end.
(* 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.
+ mkFlags (liftBoolFunc false (levelMatch lev)) tt.
Definition drop_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
- drop (mkDropFlags lev tt).
+ 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 (setNone true (bnot ○ levelMatch lev)) tt.
+ mkFlags (liftBoolFunc true (bnot ○ levelMatch lev)) tt.
- Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) :=
- mapOptionTree (fun x => garrowfy_code_types (unlev x)) (drop (mkTakeFlags lev tt)).
+ Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
+ dropT (mkTakeFlags lev tt).
(*
- mapOptionTree (fun x => garrowfy_code_types (unlev x))
- (maybeTree (take tt (mkFlags (
+ 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_lev_lemma : forall Γ (lev:HaskLevel Γ) x, drop_lev lev [x @@ lev] = [].
auto.
Qed.
- Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [garrowfy_code_types x].
+ Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [x @@ lev].
intros; simpl.
Opaque eqd_dec.
unfold take_lev.
auto.
Qed.
+ Lemma take_lemma' : forall Γ (lev:HaskLevel Γ) x, take_lev lev (x @@@ lev) = x @@@ lev.
+ intros.
+ induction x.
+ destruct a; simpl; try reflexivity.
+ apply take_lemma.
+ simpl.
+ rewrite <- IHx1 at 2.
+ rewrite <- IHx2 at 2.
+ reflexivity.
+ Qed.
+(*
+ Lemma drop_lev_lemma' : forall Γ (lev:HaskLevel Γ) x, drop_lev lev (x @@@ lev) = [].
+ intros.
+ induction x.
+ destruct a; simpl; try reflexivity.
+ apply drop_lev_lemma.
+ simpl.
+ change (@drop_lev _ lev (x1 @@@ lev ,, x2 @@@ lev))
+ with ((@drop_lev _ lev (x1 @@@ lev)) ,, (@drop_lev _ lev (x2 @@@ lev))).
+ simpl.
+ rewrite IHx1.
+ rewrite IHx2.
+ reflexivity.
+ Qed.
+*)
Ltac drop_simplify :=
match goal with
| [ |- context[@drop_lev ?G ?L [ ?X @@ ?L ] ] ] =>
rewrite (drop_lev_lemma G L X)
+(*
+ | [ |- context[@drop_lev ?G ?L [ ?X @@@ ?L ] ] ] =>
+ rewrite (drop_lev_lemma' G L X)
+*)
| [ |- context[@drop_lev ?G (?E :: ?L) [ ?X @@ (?E :: ?L) ] ] ] =>
rewrite (drop_lev_lemma_s G L E X)
| [ |- context[@drop_lev ?G ?N (?A,,?B)] ] =>
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.
- Axiom literal_types_unchanged : forall Γ l, garrowfy_code_types (literalType l) = literalType(Γ:=Γ) l.
+
+ (*******************************************************************************)
+
+
+ Context (hetmet_flatten : WeakExprVar).
+ Context (hetmet_unflatten : WeakExprVar).
+ Context (hetmet_id : WeakExprVar).
+ Context {unitTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ }.
+ Context {prodTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }.
+ Context {gaTy : forall TV, RawHaskType TV ECKind -> RawHaskType TV ★ -> RawHaskType TV ★ -> RawHaskType TV ★ }.
+
+ Definition ga_mk_tree' {TV}(ec:RawHaskType TV ECKind)(tr:Tree ??(RawHaskType TV ★)) : RawHaskType TV ★ :=
+ reduceTree (unitTy TV ec) (prodTy TV ec) tr.
+
+ Definition ga_mk_tree {Γ}(ec:HaskType Γ ECKind)(tr:Tree ??(HaskType Γ ★)) : HaskType Γ ★ :=
+ fun TV ite => ga_mk_tree' (ec TV ite) (mapOptionTree (fun x => x TV ite) tr).
+
+ Definition ga_mk_raw {TV}(ec:RawHaskType TV ECKind)(ant suc:Tree ??(RawHaskType TV ★)) : RawHaskType TV ★ :=
+ gaTy TV ec
+ (ga_mk_tree' ec ant)
+ (ga_mk_tree' ec suc).
+
+ Definition ga_mk {Γ}(ec:HaskType Γ ECKind)(ant suc:Tree ??(HaskType Γ ★)) : HaskType Γ ★ :=
+ fun TV ite => gaTy TV (ec TV ite) (ga_mk_tree ec ant TV ite) (ga_mk_tree ec suc TV ite).
+
+ (*
+ * The story:
+ * - code types <[t]>@c become garrows c () t
+ * - free variables of type t at a level lev deeper than the succedent become garrows c () t
+ * - free variables at the level of the succedent become
+ *)
+ Fixpoint flatten_rawtype {TV}{κ}(exp: RawHaskType TV κ) : RawHaskType TV κ :=
+ match exp with
+ | TVar _ x => TVar x
+ | TAll _ y => TAll _ (fun v => flatten_rawtype (y v))
+ | TApp _ _ x y => TApp (flatten_rawtype x) (flatten_rawtype y)
+ | TCon tc => TCon tc
+ | TCoerc _ t1 t2 t => TCoerc (flatten_rawtype t1) (flatten_rawtype t2) (flatten_rawtype t)
+ | TArrow => TArrow
+ | TCode ec e => let e' := flatten_rawtype e
+ in ga_mk_raw ec (unleaves_ (take_arg_types e')) [drop_arg_types e']
+ | TyFunApp tfc kl k lt => TyFunApp tfc kl k (flatten_rawtype_list _ lt)
+ end
+ with flatten_rawtype_list {TV}(lk:list Kind)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk :=
+ match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with
+ | TyFunApp_nil => TyFunApp_nil
+ | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flatten_rawtype t) (flatten_rawtype_list _ rest)
+ end.
+
+ Definition flatten_type {Γ}{κ}(ht:HaskType Γ κ) : HaskType Γ κ :=
+ fun TV ite => flatten_rawtype (ht TV ite).
+
+ Fixpoint levels_to_tcode {Γ}(ht:HaskType Γ ★)(lev:HaskLevel Γ) : HaskType Γ ★ :=
+ match lev with
+ | nil => flatten_type ht
+ | ec::lev' => @ga_mk _ (v2t ec) [] [levels_to_tcode ht lev']
+ end.
+
+ Definition flatten_leveled_type {Γ}(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
+ levels_to_tcode (unlev ht) (getlev ht) @@ nil.
+
+ (* AXIOMS *)
+
+ Axiom literal_types_unchanged : forall Γ l, flatten_type (literalType l) = literalType(Γ:=Γ) l.
Axiom flatten_coercion : forall Γ Δ κ (σ τ:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)),
- HaskCoercion Γ Δ (garrowfy_code_types σ ∼∼∼ garrowfy_code_types τ).
+ HaskCoercion Γ Δ (flatten_type σ ∼∼∼ flatten_type τ).
+
+ Axiom 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.
(* 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
end
end.
- Definition unlev' {Γ} (x:LeveledHaskType Γ ★) :=
- garrowfy_code_types (unlev x).
-
(* "n" is the maximum depth remaining AFTER flattening *)
Definition flatten_judgment (j:Judg) :=
match j as J return Judg with
Γ > Δ > ant |- suc =>
match getjlev suc with
- | nil => Γ > Δ > mapOptionTree garrowfy_leveled_code_types ant
- |- mapOptionTree garrowfy_leveled_code_types suc
+ | nil => Γ > Δ > mapOptionTree flatten_leveled_type ant
+ |- mapOptionTree flatten_leveled_type suc
- | (ec::lev') => Γ > Δ > mapOptionTree garrowfy_leveled_code_types (drop_lev (ec::lev') ant)
+ | (ec::lev') => Γ > Δ > mapOptionTree flatten_leveled_type (drop_lev (ec::lev') ant)
|- [ga_mk (v2t ec)
- (take_lev (ec::lev') ant)
- (mapOptionTree unlev' suc) (* we know the level of all of suc *)
- @@ nil]
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec::lev') ant))
+ (mapOptionTree (flatten_type ○ unlev) suc )
+ @@ nil] (* we know the level of all of suc *)
end
end.
+ Class garrow :=
+ { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a @@ l] ]
+ ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a @@ l] ]
+ ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a @@ l] ]
+ ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) @@ l] ]
+ ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) @@ l] ]
+ ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ]
+ ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ]
+ ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) @@ l] ]
+ ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] @@ l] ]
+ ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) @@ l] ]
+ ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (a,,x) (b,,x) @@ l] ]
+ ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (x,,a) (x,,b) @@ l] ]
+ ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] @@ l] ]
+ ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] @@ l] ]
+ ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c @@ l] ]
+ ; ga_apply : ∀ Γ Δ ec l a a' b c,
+ ND Rule [] [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] @@ l] ]
+ ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule
+ [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ]
+ [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
+ }.
+ Context `(gar:garrow).
+
+ Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20).
+
Definition boost : forall Γ Δ ant x y {lev},
ND Rule [] [ Γ > Δ > [x@@lev] |- [y@@lev] ] ->
ND Rule [ Γ > Δ > ant |- [x@@lev] ] [ Γ > Δ > ant |- [y@@lev] ].
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; [ 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.
apply ga_comp.
Defined.
+ Definition precompose Γ Δ ec : forall a x y z lev,
+ ND Rule
+ [ 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.
+ eapply nd_comp.
+ apply nd_rlecnac.
+ eapply nd_comp.
+ eapply nd_prod.
+ apply nd_id.
+ eapply ga_comp.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
+
+ apply nd_rule.
+ apply RLet.
+ Defined.
+
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 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.
+ eapply nd_comp.
apply X.
- apply ga_comp.
+ apply precompose.
Defined.
Definition postcompose : ∀ Γ Δ ec lev a b c,
apply ga_second.
Defined.
- Lemma ga_unkappa : ∀ Γ Δ ec l a b Σ,
+ Lemma ga_unkappa : ∀ Γ Δ ec l z a b Σ,
ND Rule
- [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
- [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ].
+ [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
+ [Γ > Δ > Σ,,[@ga_mk Γ ec z a @@ l] |- [@ga_mk Γ ec z b @@ l] ].
intros.
- set (ga_comp Γ Δ ec l [] a b) as q.
+ set (ga_comp Γ Δ ec l z a b) as q.
set (@RLet Γ Δ) as q'.
- set (@RLet Γ Δ [@ga_mk _ ec [] a @@ l] Σ (@ga_mk _ ec [] b) (@ga_mk _ ec a b) l) as q''.
+ set (@RLet Γ Δ [@ga_mk _ ec z a @@ l] Σ (@ga_mk _ ec z b) (@ga_mk _ ec a b) l) as q''.
eapply nd_comp.
Focus 2.
eapply nd_rule.
apply q.
Defined.
-(*
+ Lemma ga_unkappa' : ∀ Γ Δ ec l a b Σ x,
+ ND Rule
+ [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b @@ l] ]
+ [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b @@ l] ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ apply ga_first.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ apply postcompose.
+ apply ga_uncancell.
+ apply precompose.
+ Defined.
+
+ Lemma ga_kappa' : ∀ Γ Δ ec l a b Σ x,
+ ND Rule
+ [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b @@ l] ]
+ [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b @@ l] ].
+ apply (Prelude_error "ga_kappa not supported yet (BIG FIXME)").
+ Defined.
+
+ (* useful for cutting down on the pretty-printed noise
+
Notation "` x" := (take_lev _ x) (at level 20).
Notation "`` x" := (mapOptionTree unlev x) (at level 20).
Notation "``` x" := (drop_lev _ x) (at level 20).
-*)
- Definition garrowfy_arrangement' :
+ *)
+ Definition flatten_arrangement' :
forall Γ (Δ:CoercionEnv Γ)
(ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2),
- ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ nil] ].
+ ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil] ].
intros Γ Δ ec lev.
- refine (fix garrowfy ant1 ant2 (r:Arrange ant1 ant2):
- ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ nil]] :=
+ refine (fix flatten ant1 ant2 (r:Arrange ant1 ant2):
+ ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil]] :=
match r as R in Arrange A B return
- ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ nil]]
+ ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) B))
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) A)) @@ nil]]
with
+ | RId a => let case_RId := tt in ga_id _ _ _ _ _
| RCanL a => let case_RCanL := tt in ga_uncancell _ _ _ _ _
| RCanR a => let case_RCanR := tt in ga_uncancelr _ _ _ _ _
| RuCanL a => let case_RuCanL := tt in ga_cancell _ _ _ _ _
| 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 _ _ _ _ _
+ | RWeak a => let case_RWeak := tt in ga_drop _ _ _ _ _
| RCont a => let case_RCont := tt in ga_copy _ _ _ _ _
- | RLeft a b c r' => let case_RLeft := tt in garrowfy _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _)
- | RRight a b c r' => let case_RRight := tt in garrowfy _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _)
- | RComp c b a r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (garrowfy _ _ r1) (garrowfy _ _ r2)
- end); clear garrowfy; repeat take_simplify; repeat drop_simplify; intros.
+ | RLeft a b c r' => let case_RLeft := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _)
+ | RRight a b c r' => let case_RRight := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _)
+ | RComp c b a r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (flatten _ _ r1) (flatten _ _ r2)
+ end); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
destruct case_RComp.
- set (take_lev (ec :: lev) a) as a' in *.
- set (take_lev (ec :: lev) b) as b' in *.
- set (take_lev (ec :: lev) c) as c' in *.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) a)) as a' in *.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) b)) as b' in *.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) c)) as c' in *.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ].
eapply nd_comp; [ idtac | eapply nd_rule; apply
(@RLet Γ Δ [] [] (@ga_mk _ (v2t ec) a' c') (@ga_mk _ (v2t ec) a' b')) ].
apply ga_comp.
Defined.
- Definition garrowfy_arrangement :
+ Definition flatten_arrangement :
forall Γ (Δ:CoercionEnv Γ) n
(ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ,
ND Rule
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev n ant1)
- |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant1) (mapOptionTree (unlev' ) succ) @@ nil]]
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev n ant2)
- |- [@ga_mk _ (v2t ec) (take_lev (ec :: lev) ant2) (mapOptionTree (unlev' ) succ) @@ nil]].
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant1)
+ |- [@ga_mk _ (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1))
+ (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant2)
+ |- [@ga_mk _ (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
+ (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]].
intros.
- refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (garrowfy_arrangement' Γ Δ ec lev ant1 ant2 r)))).
+ refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (flatten_arrangement' Γ Δ ec lev ant1 ant2 r)))).
apply nd_rule.
apply RArrange.
- refine ((fix garrowfy ant1 ant2 (r:Arrange ant1 ant2) :=
+ refine ((fix flatten ant1 ant2 (r:Arrange ant1 ant2) :=
match r as R in Arrange A B return
- Arrange (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev _ A))
- (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev _ B)) with
+ Arrange (mapOptionTree (flatten_leveled_type ) (drop_lev _ A))
+ (mapOptionTree (flatten_leveled_type ) (drop_lev _ B)) with
+ | RId a => let case_RId := tt in RId _
| RCanL a => let case_RCanL := tt in RCanL _
| RCanR a => let case_RCanR := tt in RCanR _
| RuCanL a => let case_RuCanL := tt in RuCanL _
| RExch a b => let case_RExch := tt in RExch _ _
| RWeak a => let case_RWeak := tt in RWeak _
| RCont a => let case_RCont := tt in RCont _
- | RLeft a b c r' => let case_RLeft := tt in RLeft _ (garrowfy _ _ r')
- | RRight a b c r' => let case_RRight := tt in RRight _ (garrowfy _ _ r')
- | RComp a b c r1 r2 => let case_RComp := tt in RComp (garrowfy _ _ r1) (garrowfy _ _ r2)
- end) ant1 ant2 r); clear garrowfy; repeat take_simplify; repeat drop_simplify; intros.
+ | RLeft a b c r' => let case_RLeft := tt in RLeft _ (flatten _ _ r')
+ | RRight a b c r' => let case_RRight := tt in RRight _ (flatten _ _ r')
+ | RComp a b c r1 r2 => let case_RComp := tt in RComp (flatten _ _ r1) (flatten _ _ r2)
+ end) ant1 ant2 r); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
Defined.
- Definition flatten_arrangement :
+ Definition flatten_arrangement'' :
forall Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2),
ND Rule (mapOptionTree (flatten_judgment ) [Γ > Δ > ant1 |- succ])
(mapOptionTree (flatten_judgment ) [Γ > Δ > ant2 |- succ]).
apply nd_rule.
apply RArrange.
induction r; simpl.
+ apply RId.
apply RCanL.
apply RCanR.
apply RuCanL.
apply RRight; auto.
eapply RComp; [ apply IHr1 | apply IHr2 ].
- apply garrowfy_arrangement.
+ apply flatten_arrangement.
apply r.
Defined.
Definition arrange_brak : forall Γ Δ ec succ t,
ND Rule
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: nil) succ),,
- [(@ga_mk _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]]
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) succ |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]].
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ),,
+ [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [t @@ nil]]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]].
intros.
unfold drop_lev.
set (@arrange' _ succ (levelMatch (ec::nil))) as q.
- set (arrangeMap _ _ garrowfy_leveled_code_types q) as y.
+ set (arrangeMap _ _ flatten_leveled_type q) as y.
eapply nd_comp.
Focus 2.
eapply nd_rule.
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 IHsucc2.
Defined.
+ Definition arrange_empty_tree : forall {T}{A}(q:Tree A)(t:Tree ??T),
+ t = mapTree (fun _:A => None) q ->
+ Arrange t [].
+ intros T A q.
+ induction q; intros.
+ simpl in H.
+ rewrite H.
+ apply RId.
+ simpl in *.
+ destruct t; try destruct o; inversion H.
+ set (IHq1 _ H1) as x1.
+ set (IHq2 _ H2) as x2.
+ eapply RComp.
+ eapply RRight.
+ rewrite <- H1.
+ apply x1.
+ eapply RComp.
+ apply RCanL.
+ rewrite <- H2.
+ apply x2.
+ Defined.
+
+(* Definition unarrange_empty_tree : forall {T}{A}(t:Tree ??T)(q:Tree A),
+ t = mapTree (fun _:A => None) q ->
+ Arrange [] t.
+ Defined.*)
+
+ Definition decide_tree_empty : forall {T:Type}(t:Tree ??T),
+ sum { q:Tree unit & t = mapTree (fun _ => None) q } unit.
+ intro T.
+ refine (fix foo t :=
+ match t with
+ | T_Leaf x => _
+ | T_Branch b1 b2 => let b1' := foo b1 in let b2' := foo b2 in _
+ end).
+ intros.
+ destruct x.
+ right; apply tt.
+ left.
+ exists (T_Leaf tt).
+ auto.
+ destruct b1'.
+ destruct b2'.
+ destruct s.
+ destruct s0.
+ subst.
+ left.
+ exists (x,,x0).
+ reflexivity.
+ right; auto.
+ right; auto.
+ Defined.
+
Definition arrange_esc : forall Γ Δ ec succ t,
ND Rule
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) succ |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]]
- [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: nil) succ),,
- [(@ga_mk _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga_mk _ (v2t ec) [] [garrowfy_code_types t]) @@ nil]].
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ),,
+ [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [t @@ nil]].
intros.
- unfold drop_lev.
set (@arrange _ succ (levelMatch (ec::nil))) as q.
- set (arrangeMap _ _ garrowfy_leveled_code_types q) as y.
+ set (@drop_lev Γ (ec::nil) succ) as q'.
+ assert (@drop_lev Γ (ec::nil) succ=q') as H.
+ reflexivity.
+ unfold drop_lev in H.
+ unfold mkDropFlags in H.
+ rewrite H in q.
+ clear H.
+ set (arrangeMap _ _ flatten_leveled_type q) as y.
eapply nd_comp.
eapply nd_rule.
eapply RArrange.
apply y.
- idtac.
clear y q.
+ set (mapOptionTree flatten_leveled_type (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ))) as q.
+ destruct (decide_tree_empty q); [ idtac | apply (Prelude_error "escapifying open code not yet supported") ].
+ destruct s.
+
+ simpl.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
+ set (fun z z' => @RLet Γ Δ z (mapOptionTree flatten_leveled_type q') t z' nil) as q''.
+ eapply nd_comp; [ idtac | eapply nd_rule; apply q'' ].
+ clear q''.
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_rule.
+ apply RArrange.
+ eapply RComp; [ idtac | apply RCanR ].
+ apply RLeft.
+ apply (@arrange_empty_tree _ _ _ _ e).
+
+ eapply nd_comp.
+ eapply nd_rule.
+ eapply (@RVar Γ Δ t nil).
+ apply nd_rule.
+ apply RArrange.
+ eapply RComp.
+ apply RuCanL.
+ apply RRight.
+ apply RWeak.
+(*
+ eapply decide_tree_empty.
+
+ simpl.
+ set (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ)) as escapified.
+ destruct (decide_tree_empty escapified).
+
induction succ.
destruct a.
+ unfold drop_lev.
destruct l.
simpl.
unfold mkDropFlags; simpl.
simpl.
destruct (General.list_eq_dec h0 (ec :: nil)).
simpl.
- unfold garrowfy_leveled_code_types'.
rewrite e.
apply nd_id.
simpl.
apply RLeft.
apply RWeak.
apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported").
+*)
Defined.
Lemma mapOptionTree_distributes
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).
+ 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.
- 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).
+ Lemma tree_of_nothing : forall Γ ec t a,
+ Arrange (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))) a.
+ intros.
+ induction t; try destruct o; try destruct a0.
+ simpl.
+ drop_simplify.
+ simpl.
+ apply RCanR.
+ simpl.
+ apply RCanR.
+ Opaque drop_lev.
+ simpl.
+ Transparent drop_lev.
+ drop_simplify.
+ simpl.
+ eapply RComp.
+ eapply RComp.
+ eapply RAssoc.
+ eapply RRight.
+ apply IHt1.
+ apply IHt2.
+ Defined.
- Axiom globals_do_not_have_code_types : forall (Γ:TypeEnv) (g:Global Γ) v,
- garrowfy_code_types (g v) = g v.
+ Lemma tree_of_nothing' : forall Γ ec t a,
+ Arrange a (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))).
+ intros.
+ induction t; try destruct o; try destruct a0.
+ simpl.
+ drop_simplify.
+ simpl.
+ apply RuCanR.
+ simpl.
+ apply RuCanR.
+ Opaque drop_lev.
+ simpl.
+ Transparent drop_lev.
+ drop_simplify.
+ simpl.
+ eapply RComp.
+ Focus 2.
+ eapply RComp.
+ Focus 2.
+ eapply RCossa.
+ Focus 2.
+ eapply RRight.
+ apply IHt1.
+ apply IHt2.
+ Defined.
- Definition decide_tree_empty : forall {T:Type}(t:Tree ??T),
- sum { q:Tree unit & t = mapTree (fun _ => None) q } unit.
- intro T.
- refine (fix foo t :=
- match t with
- | T_Leaf x => _
- | T_Branch b1 b2 => let b1' := foo b1 in let b2' := foo b2 in _
- end).
+ Lemma 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.
- destruct x.
- right; apply tt.
- left.
- exists (T_Leaf tt).
- auto.
- destruct b1'.
- destruct b2'.
- destruct s.
- destruct s0.
- subst.
- left.
- exists (x,,x0).
+ 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.
- right; auto.
- right; auto.
- Defined.
+ unfold flatten_type.
+ clear hetmet_flatten.
+ clear hetmet_unflatten.
+ clear hetmet_id.
+ clear gar.
+ set (t tv ite) as x.
+ admit.
+ admit.
+ Qed.
-
Definition flatten_proof :
forall {h}{c},
- ND Rule h c ->
- ND Rule (mapOptionTree (flatten_judgment ) h) (mapOptionTree (flatten_judgment ) c).
+ ND SRule h c ->
+ ND Rule (mapOptionTree (flatten_judgment ) h) (mapOptionTree (flatten_judgment ) c).
intros.
eapply nd_map'; [ idtac | apply X ].
clear h c X.
intros.
simpl in *.
- refine (match X as R in Rule H C with
+ refine
+ (match X as R in SRule H C with
+ | SBrak Γ Δ t ec succ lev => let case_SBrak := tt in _
+ | SEsc Γ Δ t ec succ lev => let case_SEsc := tt in _
+ | SFlat h c r => let case_SFlat := tt in _
+ end).
+
+ destruct case_SFlat.
+ refine (match r as R in Rule H C with
| RArrange Γ Δ a b x d => let case_RArrange := tt in _
| RNote Γ Δ Σ τ l n => let case_RNote := tt in _
| RLit Γ Δ l _ => let case_RLit := tt in _
end); clear X h c.
destruct case_RArrange.
- apply (flatten_arrangement Γ Δ a b x d).
+ apply (flatten_arrangement'' Γ Δ a b x d).
destruct case_RBrak.
- simpl.
- destruct lev.
- change ([garrowfy_code_types (<[ ec |- t ]>) @@ nil])
- with ([ga_mk (v2t ec) [] [garrowfy_code_types t] @@ nil]).
- refine (ga_unkappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) _
- (mapOptionTree (garrowfy_leveled_code_types) (drop_lev (ec::nil) succ)) ;; _ ).
- apply arrange_brak.
- apply (Prelude_error "found Brak at depth >0 (a)").
+ apply (Prelude_error "found unskolemized Brak rule; this shouldn't happen").
destruct case_REsc.
- simpl.
- destruct lev.
- simpl.
- change ([garrowfy_code_types (<[ ec |- t ]>) @@ nil])
- with ([ga_mk (v2t ec) [] [garrowfy_code_types t] @@ nil]).
- eapply nd_comp; [ apply arrange_esc | idtac ].
- set (decide_tree_empty (take_lev (ec :: nil) succ)) as q'.
- destruct q'.
- destruct s.
- rewrite e.
- clear e.
-
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod; [ idtac | eapply boost ].
- induction x.
- apply ga_id.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
- apply ga_join.
- apply IHx1.
- apply IHx2.
- unfold unlev'.
- simpl.
- apply postcompose.
- apply ga_drop.
-
- (* environment has non-empty leaves *)
- apply (ga_kappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) _ _).
- apply (Prelude_error "found Esc at depth >0 (a)").
+ apply (Prelude_error "found unskolemized Esc rule; this shouldn't happen").
destruct case_RNote.
simpl.
destruct case_RLit.
simpl.
destruct l0; simpl.
+ unfold flatten_leveled_type.
+ simpl.
rewrite literal_types_unchanged.
apply nd_rule; apply RLit.
unfold take_lev; simpl.
unfold drop_lev; simpl.
- unfold unlev'.
simpl.
rewrite literal_types_unchanged.
apply ga_lit.
destruct lev.
apply nd_rule. apply RVar.
repeat drop_simplify.
- unfold unlev'.
repeat take_simplify.
simpl.
apply ga_id.
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 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_lev.
simpl.
apply RCanR.
apply boost.
+ simpl.
apply ga_curry.
destruct case_RCast.
simpl.
destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RCast; auto | idtac ].
+ simpl.
apply flatten_coercion; auto.
- apply (Prelude_error "RCast at level >0").
+ apply (Prelude_error "RCast at level >0; casting inside of code brackets is currently not supported").
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").
+ destruct (getjlev x); destruct (getjlev q);
+ [ apply nd_rule; apply RJoin | idtac | idtac | idtac ];
+ apply (Prelude_error "RJoin at depth >0").
destruct case_RApp.
simpl.
destruct lev as [|ec lev]. simpl. apply nd_rule.
- replace (garrowfy_code_types (tx ---> te)) with ((garrowfy_code_types tx) ---> (garrowfy_code_types te)).
+ unfold flatten_leveled_type at 4.
+ unfold flatten_leveled_type at 2.
+ simpl.
+ replace (flatten_type (tx ---> te))
+ with (flatten_type tx ---> flatten_type te).
apply RApp.
reflexivity.
repeat drop_simplify.
repeat take_simplify.
rewrite mapOptionTree_distributes.
- set (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: lev) Σ₁)) as Σ₁'.
- set (mapOptionTree (garrowfy_leveled_code_types ) (drop_lev (ec :: lev) Σ₂)) as Σ₂'.
+ set (mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: lev) Σ₁)) as Σ₁'.
+ set (mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: lev) Σ₂)) as Σ₂'.
set (take_lev (ec :: lev) Σ₁) as Σ₁''.
set (take_lev (ec :: lev) Σ₂) as Σ₂''.
- replace (garrowfy_code_types (tx ---> te)) with ((garrowfy_code_types tx) ---> (garrowfy_code_types te)).
+ replace (flatten_type (tx ---> te)) with ((flatten_type tx) ---> (flatten_type te)).
apply (Prelude_error "FIXME: ga_apply").
reflexivity.
+
(*
- Notation "` x" := (take_lev _ x) (at level 20).
+ Notation "` x" := (take_lev _ x).
Notation "`` x" := (mapOptionTree unlev x) (at level 20).
Notation "``` x" := ((drop_lev _ x)) (at level 20).
- Notation "!<[]> x" := (garrowfy_code_types _ x) (at level 1).
- Notation "!<[@]>" := (garrowfy_leveled_code_types _) (at level 1).
+ Notation "!<[]> x" := (flatten_type _ x) (at level 1).
+ Notation "!<[@]> x" := (mapOptionTree flatten_leveled_type x) (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.
- 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 nd_prod; [ idtac | apply nd_id ].
eapply boost.
- apply ga_comp.
+ apply ga_second.
eapply nd_comp.
- eapply nd_rule.
- eapply RArrange.
- eapply RCanR.
-
+ eapply nd_prod.
+ apply nd_id.
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.
+ apply RCanR.
+ eapply precompose.
- eapply nd_rule.
- eapply RVar.
+ apply nd_rule.
+ apply RLet.
- apply nd_id.
-*)
destruct case_RVoid.
simpl.
apply nd_rule.
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 "ga_apply").
+ 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.
+ unfold flatten_leveled_type at 4.
+ unfold flatten_leveled_type at 2.
+ simpl.
+ rewrite flatten_commutes_with_HaskTAll.
+ rewrite flatten_commutes_with_HaskTApp.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RAbsT ].
simpl.
- set (mapOptionTree (garrowfy_leveled_code_types ) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a.
- set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (garrowfy_leveled_code_types ) Σ)) as q'.
+ set (mapOptionTree (flatten_leveled_type ) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a.
+ set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (flatten_leveled_type ) Σ)) as q'.
assert (a=q').
unfold a.
unfold q'.
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_leveled_type at 4.
+ unfold flatten_leveled_type at 2.
+ 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.
+ simpl.
apply (Prelude_error "LetRec not supported (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.
+ unfold flatten_leveled_type at 4.
+ simpl.
+ rewrite krunk.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host.
+ set (mapOptionTree (flatten_type ○ unlev)(take_lev (ec :: nil) succ)) as succ_guest.
+ set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args.
+ unfold empty_tree.
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; apply tree_of_nothing | idtac ].
+ refine (ga_unkappa' Γ Δ (v2t ec) nil _ _ _ _ ;; _).
+ unfold succ_host.
+ unfold succ_guest.
+ apply arrange_brak.
+ apply (Prelude_error "found Brak at depth >0 indicating 3-level code; only two-level code is currently supported").
+
+ destruct case_SEsc.
+ simpl.
+ destruct lev.
+ simpl.
+ unfold flatten_leveled_type at 2.
+ simpl.
+ rewrite krunk.
+ rewrite mapOptionTree_compose.
+ take_simplify.
+ drop_simplify.
+ simpl.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply tree_of_nothing' ].
+ simpl.
+ rewrite take_lemma'.
+ rewrite unlev_relev.
+ rewrite <- mapOptionTree_compose.
+ eapply nd_comp; [ apply (arrange_esc _ _ ec) | idtac ].
+
+ set (decide_tree_empty (take_lev (ec :: nil) succ)) as q'.
+ destruct q'.
+ destruct s.
+ rewrite e.
+ clear e.
+
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host.
+ set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod; [ idtac | eapply boost ].
+ induction x.
+ apply ga_id.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
+ simpl.
+ apply ga_join.
+ apply IHx1.
+ apply IHx2.
+ simpl.
+ apply postcompose.
+
+ refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))).
+ apply ga_cancell.
+ apply firstify.
+
+ induction x.
+ destruct a; simpl.
+ apply ga_id.
+ simpl.
+ refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))).
+ apply ga_cancell.
+ refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ _))).
+ eapply firstify.
+ apply IHx1.
+ apply secondify.
+ apply IHx2.
+
+ (* environment has non-empty leaves *)
+ apply ga_kappa'.
+
+ (* nesting too deep *)
+ apply (Prelude_error "found Esc at depth >0 indicating 3-level code; only two-level code is currently supported").
Defined.
+ Definition skolemize_and_flatten_proof :
+ forall {h}{c},
+ ND Rule h c ->
+ ND Rule
+ (mapOptionTree (flatten_judgment ○ skolemize_judgment) h)
+ (mapOptionTree (flatten_judgment ○ skolemize_judgment) c).
+ intros.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ apply flatten_proof.
+ apply skolemize_proof.
+ apply X.
+ Defined.
+
(* to do: establish some metric on judgments (max length of level of any succedent type, probably), show how to
* calculate it, and show that the flattening procedure above drives it down by one *)
projects / coq-hetmet.git / blobdiff