| a::b => mkArrows b (a ---> t)
end.
+(*
Fixpoint unleaves_ {Γ}(t:Tree ??(LeveledHaskType Γ ★))(l:list (HaskType Γ ★)) lev : Tree ??(LeveledHaskType Γ ★) :=
match l with
| nil => t
| a::b => unleaves_ (t,,[a @@ lev]) b lev
end.
+*)
+ (* weak inverse of "leaves" *)
+ Fixpoint unleaves_ {A:Type}(l:list A) : Tree (option A) :=
+ match l with
+ | nil => []
+ | (a::nil) => [a]
+ | (a::b) => [a],,(unleaves_ b)
+ end.
(* rules of skolemized proofs *)
Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ => Γ end.
end
end.
- Definition ite_unit : ∀ Γ, InstantiatedTypeEnv (fun _ => unit) Γ.
- intros.
- induction Γ.
- apply INil.
- apply ICons; auto.
- apply tt.
- Defined.
-
- Fixpoint grab (n:nat) {T} (l:list T) : T :=
- match l with
- | nil => Prelude_error "grab failed"
- | h::t => match n with
- | 0 => h
- | S n' => grab n' t
- end
- end.
+ Fixpoint take_trustme {Γ}
+ (n:nat)
+ (l:forall TV, InstantiatedTypeEnv TV Γ -> list (RawHaskType TV ★))
+ : list (HaskType Γ ★) :=
- Fixpoint count' (n:nat)(ln:list nat) : list nat :=
match n with
- | 0 => ln
- | S n' => count' n' (n'::ln)
+ | 0 => nil
+ | S n' => (fun TV ite => match l TV ite with
+ | nil => Prelude_error "impossible"
+ | a::b => a
+ end)
+ ::
+ take_trustme n' (fun TV ite => match l TV ite with
+ | nil => Prelude_error "impossible"
+ | a::b => b
+ end)
end.
+
+ Axiom phoas_extensionality : forall Γ Q (f g:forall TV, InstantiatedTypeEnv TV Γ -> Q TV),
+ (forall tv ite, f tv ite = g tv ite) -> f=g.
- Definition count (n:nat) := count' n nil.
-
- Definition rebundle {Γ}(X:forall TV, InstantiatedTypeEnv TV Γ -> list (RawHaskType TV ★)) : list (HaskType Γ ★ ) :=
- map (fun n => fun TV ite => grab n (X TV ite)) (count (length (X _ (ite_unit _)))).
-
- Definition take_arg_types_as_tree Γ (ht:HaskType Γ ★) :=
- (unleaves' (rebundle (fun TV ite => (take_arg_types (ht TV ite))))).
+ Definition take_arg_types_as_tree {Γ}(ht:HaskType Γ ★) : Tree ??(HaskType Γ ★ ) :=
+ unleaves_
+ (take_trustme
+ (count_arg_types (ht _ (ite_unit _)))
+ (fun TV ite => take_arg_types (ht TV ite))).
Definition drop_arg_types_as_tree {Γ} (ht:HaskType Γ ★) : HaskType Γ ★ :=
fun TV ite => drop_arg_types (ht TV ite).
Implicit Arguments take_arg_types_as_tree [[Γ]].
Implicit Arguments drop_arg_types_as_tree [[Γ]].
- Lemma take_works : forall {Γ}(t1 t2:HaskType Γ ★),
- take_arg_types_as_tree (t1 ---> t2) = [t1],,(take_arg_types_as_tree t2).
+ Definition take_arrange : forall {Γ} (tx te:HaskType Γ ★) lev,
+ Arrange ([tx @@ lev],, take_arg_types_as_tree te @@@ lev)
+ (take_arg_types_as_tree (tx ---> te) @@@ lev).
intros.
- unfold take_arg_types_as_tree; simpl.
- unfold rebundle at 1.
- admit.
- Qed.
+ destruct (eqd_dec ([tx],, take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
+ rewrite <- e.
+ simpl.
+ apply RId.
+ unfold take_arg_types_as_tree.
+ Opaque take_arg_types_as_tree.
+ simpl.
+ destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
+ simpl.
+ replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
+ apply RCanR.
+ apply phoas_extensionality.
+ reflexivity.
+ apply (Prelude_error "should not be possible").
+ Defined.
+ Transparent take_arg_types_as_tree.
+
+ Definition take_unarrange : forall {Γ} (tx te:HaskType Γ ★) lev,
+ Arrange (take_arg_types_as_tree (tx ---> te) @@@ lev)
+ ([tx @@ lev],, take_arg_types_as_tree te @@@ lev).
+ intros.
+ destruct (eqd_dec ([tx],, take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
+ rewrite <- e.
+ simpl.
+ apply RId.
+ unfold take_arg_types_as_tree.
+ Opaque take_arg_types_as_tree.
+ simpl.
+ destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
+ simpl.
+ replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
+ apply RuCanR.
+ apply phoas_extensionality.
+ reflexivity.
+ apply (Prelude_error "should not be possible").
+ Defined.
+ Transparent take_arg_types_as_tree.
Lemma drop_works : forall {Γ}(t1 t2:HaskType Γ ★),
drop_arg_types_as_tree (t1 ---> t2) = (drop_arg_types_as_tree t2).
left; auto.
Defined.
+ Opaque take_arg_types_as_tree.
Definition skolemize_proof :
forall {h}{c},
ND Rule h c ->
apply RGlobal.
destruct case_RLam.
- simpl.
destruct lev.
- apply nd_rule.
- apply SFlat.
- apply RLam.
- rewrite take_works.
+ apply nd_rule.
+ apply SFlat.
+ simpl.
+ apply RLam.
+ simpl.
rewrite drop_works.
apply nd_rule.
- apply SFlat.
- apply RArrange.
- apply RCossa.
+ apply SFlat.
+ apply RArrange.
+ eapply RComp.
+ apply RCossa.
+ apply RLeft.
+ apply take_arrange.
destruct case_RCast.
simpl.
apply nd_rule.
apply SFlat.
apply RApp.
- rewrite take_works.
rewrite drop_works.
set (check_hof tx) as hof_tx.
destruct hof_tx; [ apply (Prelude_error "attempt tp apply a higher-order function at depth>0") | idtac ].
apply SFlat.
apply RArrange.
apply RCanR.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
- apply nd_rule; apply SFlat; apply RArrange; apply RLeft; eapply RExch.
+ apply nd_rule.
+ apply SFlat.
+ apply RArrange.
+ eapply RComp; [ idtac | eapply RAssoc ].
+ apply RLeft.
+ eapply RComp; [ idtac | apply RExch ].
+ apply take_unarrange.
destruct case_RLet.
simpl.
simpl.
apply (Prelude_error "CASE: BIG FIXME").
Defined.
+ Transparent take_arg_types_as_tree.
End HaskSkolemizer.
projects / coq-hetmet.git / blobdiff