Defined.
+ Lemma arrangePullback' {T Q}{f:T->Q}
+ : forall (Σ₁:Tree ??Q)(Σ₂:Tree ??Q), Arrange Σ₁ Σ₂ ->
+ forall Σ₂', Σ₂ = (mapOptionTree f Σ₂') ->
+ { Σ₁' : Tree ??T & prod (Σ₁ = (mapOptionTree f Σ₁')) (Arrange Σ₁' Σ₂') }
+ .
+
+ refine ((fix arrangePullback Σ₁ Σ₂ (arr:Arrange Σ₁ Σ₂) {struct arr} :
+ forall Σ₂', Σ₂ = (mapOptionTree f Σ₂') ->
+ { Σ₁' : Tree ??T & prod (Σ₁ = (mapOptionTree f Σ₁')) (Arrange Σ₁' Σ₂') }
+ :=
+ match arr as R in Arrange A B return
+ forall Σ₂', B = (mapOptionTree f Σ₂') ->
+ { Σ₁' : Tree ??T & prod (A = (mapOptionTree f Σ₁')) (Arrange Σ₁' Σ₂') }
+ with
+ | AId a => let case_AId := tt in _
+ | ACanL a => let case_ACanL := tt in _
+ | ACanR a => let case_ACanR := tt in _
+ | AuCanL a => let case_AuCanL := tt in _
+ | AuCanR a => let case_AuCanR := tt in _
+ | AAssoc a b c => let case_AAssoc := tt in _
+ | AuAssoc a b c => let case_AuAssoc := tt in _
+ | AExch a b => let case_AExch := tt in _
+ | AWeak a => let case_AWeak := tt in _
+ | ACont a => let case_ACont := tt in _
+ | ALeft a b c r' => let case_ALeft := tt in (fun rec => _) (arrangePullback _ _ r')
+ | ARight a b c r' => let case_ARight := tt in (fun rec => _) (arrangePullback _ _ r')
+ | AComp a b c r1 r2 => let case_AComp := tt in (fun rec1 rec2 => _) (arrangePullback _ _ r1) (arrangePullback _ _ r2)
+ end)); clear arrangePullback; intros.
+
+ destruct case_AId.
+ exists Σ₂'; split.
+ subst.
+ reflexivity.
+ apply AId.
+
+ destruct case_ACanL.
+ exists ([],,Σ₂'); split.
+ subst.
+ simpl.
+ reflexivity.
+ apply ACanL.
+
+ destruct case_ACanR.
+ exists (Σ₂',,[]); split.
+ subst.
+ simpl.
+ reflexivity.
+ apply ACanR.
+
+ destruct case_AuCanL.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ eexists; split.
+ reflexivity.
+ simpl in H.
+ inversion H.
+ destruct Σ₂'1; try destruct o; inversion H2.
+ apply AuCanL.
+
+ destruct case_AuCanR.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ eexists; split.
+ reflexivity.
+ simpl in H.
+ inversion H.
+ destruct Σ₂'2; try destruct o; inversion H2.
+ apply AuCanR.
+
+ destruct case_AAssoc.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ destruct Σ₂'1; try destruct o; inversion H; subst.
+ rewrite <- mapOptionTree_distributes.
+ rewrite <- mapOptionTree_distributes.
+ eexists; split.
+ reflexivity.
+ apply AAssoc.
+
+ destruct case_AuAssoc.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ destruct Σ₂'2; try destruct o; inversion H; subst.
+ rewrite <- mapOptionTree_distributes.
+ rewrite <- mapOptionTree_distributes.
+ eexists; split.
+ reflexivity.
+ apply AuAssoc.
+
+ destruct case_AExch.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ rewrite <- mapOptionTree_distributes.
+ eexists; split.
+ reflexivity.
+ apply AExch.
+
+ destruct case_AWeak.
+ exists []; split.
+ reflexivity.
+ apply AWeak.
+
+ destruct case_ACont.
+ exists (Σ₂',,Σ₂').
+ subst; split.
+ reflexivity.
+ apply ACont.
+
+ destruct case_ALeft.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ destruct (rec _ (refl_equal _)).
+ destruct p.
+ rewrite e.
+ rewrite <- mapOptionTree_distributes.
+ eexists; split.
+ reflexivity.
+ apply ALeft.
+ apply a0.
+
+ destruct case_ARight.
+ destruct Σ₂'; try destruct o; inversion H; subst.
+ destruct (rec _ (refl_equal _)).
+ destruct p.
+ rewrite e.
+ rewrite <- mapOptionTree_distributes.
+ eexists; split.
+ reflexivity.
+ apply ARight.
+ apply a0.
+
+ destruct case_AComp.
+ destruct (rec2 _ H).
+ destruct p.
+ destruct (rec1 _ e).
+ destruct p.
+ rewrite e0.
+ eexists; split.
+ reflexivity.
+ eapply AComp.
+ apply a1.
+ apply a0.
+ Defined.
+
+ Lemma arrangePullback {T Q}{f:T->Q}
+ : forall (Σ₁:Tree ??Q)(Σ₂:Tree ??T), Arrange Σ₁ (mapOptionTree f Σ₂) ->
+ { Σ₁' : Tree ??T & prod (Σ₁ = (mapOptionTree f Σ₁')) (Arrange Σ₁' Σ₂) }.
+ intros.
+ eapply arrangePullback'.
+ apply X.
+ reflexivity.
+ Defined.
+
(* given an Arrange from Σ₁ to Σ₂ and any predicate on tree nodes, we can construct an Arrange from (takeT Σ₁) to (takeT Σ₂) *)
(*
- Lemma arrangeTake {T} pred
+ Lemma arrangePullback {T} pred
: forall (Σ₁ Σ₂: Tree ??T), Arrange Σ₁ Σ₂ -> Arrange (takeT' (mkFlags pred Σ₁)) (takeT' (mkFlags pred Σ₂)).
unfold takeT'.
*)
+ (*
+ (* like Arrange, but without weakening or contraction *)
+ Inductive Permutation {T} : Tree ??T -> Tree ??T -> Type :=
+ | PId : forall a , Permutation a a
+ | PCanL : forall a , Permutation ( [],,a ) ( a )
+ | PCanR : forall a , Permutation ( a,,[] ) ( a )
+ | PuCanL : forall a , Permutation ( a ) ( [],,a )
+ | PuCanR : forall a , Permutation ( a ) ( a,,[] )
+ | PAssoc : forall a b c , Permutation (a,,(b,,c) ) ((a,,b),,c )
+ | PuAssoc : forall a b c , Permutation ((a,,b),,c ) ( a,,(b,,c) )
+ | PExch : forall a b , Permutation ( (b,,a) ) ( (a,,b) )
+ | PLeft : forall {h}{c} x , Permutation h c -> Permutation ( x,,h ) ( x,,c)
+ | PRight : forall {h}{c} x , Permutation h c -> Permutation ( h,,x ) ( c,,x)
+ | PComp : forall {a}{b}{c}, Permutation a b -> Permutation b c -> Permutation a c
+ .
+ Notation "a ≈ b" := (@Permutation _ a b) (at level 30).
+
+ Definition permutationToArrangement {T}{a b:Tree ??T} : a ≈ b -> a ⊆ b.
+ admit.
+ Defined.
+
+ Definition invertPermutation {T}{a b:Tree ??T} : a ≈ b -> b ≈ a.
+ admit.
+ Defined.
+
+ Definition partition {Γ}{ec:HaskEC Γ} Σ :
+ Σ ≈ ((dropEC ec Σ),, (takeEC ec Σ +@@@+ ec)).
+ admit.
+ Defined.
+ *)
+
End NaturalDeductionContext.
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