Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
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").
| 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 l => let case_RJoin := tt in _
| RVoid _ _ l => let case_RVoid := tt in _
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.
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.
apply nd_rule.
apply SFlat.
apply RVoid.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RuCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuCanL ].
apply nd_rule.
apply SFlat.
apply RVoid.