Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
end.
(* rules of skolemized proofs *)
- 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
- end.
+ Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ @ _ => Γ end.
Fixpoint take_trustme {Γ}
(n:nat)
destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
rewrite <- e.
simpl.
- apply RId.
+ apply AId.
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 ACanR.
apply phoas_extensionality.
reflexivity.
apply (Prelude_error "should not be possible").
destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
rewrite <- e.
simpl.
- apply RId.
+ apply AId.
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 AuCanR.
apply phoas_extensionality.
reflexivity.
apply (Prelude_error "should not be possible").
| SFlat : forall h c, Rule h c -> SRule h c
| SBrak : forall Γ Δ t ec Σ l,
SRule
- [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t @@ (ec::l) ]]
- [Γ > Δ > Σ |- [<[ec |- t]> @@ l ]]
+ [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
+ [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
| SEsc : forall Γ Δ t ec Σ l,
SRule
- [Γ > Δ > Σ |- [<[ec |- t]> @@ l ]]
- [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t @@ (ec::l) ]]
+ [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
+ [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
.
Definition take_arg_types_as_tree' {Γ}(lt:LeveledHaskType Γ ★) :=
Definition skolemize_judgment (j:Judg) : Judg :=
match j with
- Γ > Δ > Σ₁ |- Σ₂ =>
- match getjlev Σ₂ with
- | nil => j
- | lev => Γ > Δ > Σ₁,,(mapOptionTreeAndFlatten take_arg_types_as_tree' Σ₂) |- mapOptionTree drop_arg_types_as_tree' Σ₂
- end
+ | Γ > Δ > Σ₁ |- Σ₂ @ nil => j
+ | Γ > Δ > Σ₁ |- Σ₂ @ lev =>
+ Γ > Δ > Σ₁,,(mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂ @@@ lev) |- mapOptionTree drop_arg_types_as_tree Σ₂ @ lev
end.
Definition check_hof : forall {Γ}(t:HaskType Γ ★),
intros.
refine (match X as R in Rule H C with
- | RArrange Γ Δ a b x d => let case_RArrange := tt in _
+ | 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 _
+ | RCut Γ Δ 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 _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := 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 _
destruct case_RArrange.
simpl.
- destruct (getjlev x).
+ destruct l.
apply nd_rule.
apply SFlat.
apply RArrange.
apply nd_rule.
apply SFlat.
apply RArrange.
- apply RRight.
+ apply ARight.
apply d.
destruct case_RBrak.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply RuCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply AuCanL ].
apply nd_rule.
apply SFlat.
apply RLit.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
apply nd_rule.
apply SFlat.
apply RVar.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
apply nd_rule.
apply SFlat.
apply RGlobal.
apply nd_rule.
apply SFlat.
apply RArrange.
- eapply RComp.
- eapply RCossa.
- eapply RLeft.
+ eapply AComp.
+ eapply AuAssoc.
+ eapply ALeft.
apply take_arrange.
destruct case_RCast.
apply γ.
apply (Prelude_error "found RCast at level >0").
- destruct case_RJoin.
- simpl.
- destruct (getjlev x).
- destruct (getjlev q).
- apply nd_rule.
- apply SFlat.
- apply RJoin.
- apply (Prelude_error "found RJoin at level >0").
- apply (Prelude_error "found RJoin at level >0").
-
destruct case_RApp.
simpl.
destruct lev.
rewrite H0.
simpl.
eapply nd_comp.
- eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
+ eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
eapply nd_rule.
eapply SFlat.
eapply RArrange.
- eapply RLeft.
+ eapply ALeft.
eapply take_unarrange.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
eapply nd_rule; eapply SFlat; apply RWhere.
destruct case_RLet.
rewrite H0.
eapply nd_comp.
- eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR | eapply nd_id ].
+ eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR | eapply nd_id ].
set (@RLet Γ Δ Σ₁ (Σ₂,,(take_arg_types_as_tree σ₂ @@@ (h::lev))) σ₁ (drop_arg_types_as_tree σ₂) (h::lev)) as q.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply q ].
apply nd_prod.
apply nd_id.
apply nd_rule.
eapply SFlat.
eapply RArrange.
- eapply RCossa.
+ eapply AuAssoc.
destruct case_RWhere.
simpl.
rewrite H0.
eapply nd_comp.
- eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RAssoc ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RLeft; eapply RAssoc ].
+ eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AAssoc ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RWhere ].
apply nd_prod; [ idtac | eapply nd_id ].
eapply nd_rule; apply SFlat; eapply RArrange.
- eapply RComp.
- eapply RCossa.
- apply RLeft.
- eapply RCossa.
+ eapply AComp.
+ eapply AuAssoc.
+ apply ALeft.
+ eapply AuAssoc.
+
+ destruct case_RCut.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RCut | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₃) as Σ₃''.
+ set (mapOptionTree drop_arg_types_as_tree Σ₃) as Σ₃'''.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₁₂) as Σ₁₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ₁₂) as Σ₁₂'''.
+ destruct (decide_tree_empty Σ₁₂''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
+ destruct (eqd_dec Σ₁₂ Σ₁₂'''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
+ rewrite <- e.
+ eapply nd_comp.
+ eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RCut ].
+ apply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
+ apply nd_rule.
+ apply SFlat.
+ apply RArrange.
+ apply ALeft.
+ destruct s.
+ eapply arrangeCancelEmptyTree with (q:=x).
+ rewrite e0.
+ admit. (* FIXME, but not serious *)
+ apply nd_id.
+
+ destruct case_RLeft.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RLeft | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
+ set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
+ destruct (decide_tree_empty (Σ' @@@ (h::l)));
+ [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
+ destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
+ rewrite <- e.
+ clear Σ'' e.
+ destruct s.
+ set (arrangeUnCancelEmptyTree _ _ e) as q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ARight; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanL; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ apply nd_rule.
+ eapply SFlat.
+ eapply RLeft.
+
+ destruct case_RRight.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RRight | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
+ set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
+ destruct (decide_tree_empty (Σ' @@@ (h::l)));
+ [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
+ destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
+ rewrite <- e.
+ clear Σ'' e.
+ destruct s.
+ set (arrangeUnCancelEmptyTree _ _ e) as q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ALeft; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AExch ]. (* yuck *)
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
+ eapply nd_rule.
+ eapply SFlat.
+ apply RRight.
destruct case_RVoid.
simpl.
+ destruct l.
+ apply nd_rule.
+ apply SFlat.
+ apply RVoid.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuCanL ].
apply nd_rule.
apply SFlat.
apply RVoid.
destruct case_RLetRec.
simpl.
destruct t.
- destruct (getjlev (y @@@ nil)).
apply nd_rule.
apply SFlat.
apply (@RLetRec Γ Δ lri x y nil).
apply (Prelude_error "RLetRec at depth>0").
- apply (Prelude_error "RLetRec at depth>0").
destruct case_RCase.
simpl.
apply (Prelude_error "CASE: BIG FIXME").
Defined.
+
Transparent take_arg_types_as_tree.
End HaskSkolemizer.
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