Left Termination proof of /tmp/tmpICOrAq/inorder-fb.pl

Left Termination of the query pattern in_order(f,b) w.r.t. the given Prolog program could not be shown:

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

in_order₂(void₀, {}₀).
in_order₂(tree₃(X, Left, Right), Xs) :- in_order₂(Left, Ls), in_order₂(Right, Rs), app₃(Ls, .₂(X, Rs), Xs).
app₃({}₀, X, X).
app₃(.₂(X, Xs), Ys, .₂(X, Zs)) :- app₃(Xs, Ys, Zs).

With regard to the inferred argument filtering the predicates were used in the following modes:
in_order₂: (f,b) (f,f)
app₃: (f,f,f) (f,f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AA₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAA₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AG₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAG₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAG₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ag₂(x1, x2) = in_order_2_in_ag₁(x2)
void_0 = void_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
._2₂(x1, x2) = ._2₁(x2)
in_order_2_out_ag₂(x1, x2) = in_order_2_out_ag₁(x1)
if_in_order_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_ag₂(x4, x5)
in_order_2_in_aa₂(x1, x2) = in_order_2_in_aa
in_order_2_out_aa₂(x1, x2) = in_order_2_out_aa₁(x1)
if_in_order_2_in_1_aa₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_aa₁(x5)
if_in_order_2_in_2_aa₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_aa₂(x2, x6)
if_in_order_2_in_3_aa₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_aa₃(x2, x3, x7)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_in_order_2_in_2_ag₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_ag₃(x2, x4, x6)
if_in_order_2_in_3_ag₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_ag₃(x2, x3, x7)
app_3_in_aag₃(x1, x2, x3) = app_3_in_aag₁(x3)
app_3_out_aag₃(x1, x2, x3) = app_3_out_aag₂(x1, x2)
if_app_3_in_1_aag₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aag₁(x5)
IF_IN_ORDER_2_IN_3_AA₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AA₃(x2, x3, x7)
IN_ORDER_2_IN_AA₂(x1, x2) = IN_ORDER_2_IN_AA
IF_IN_ORDER_2_IN_2_AG₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AG₃(x2, x4, x6)
IF_IN_ORDER_2_IN_1_AA₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AA₁(x5)
IN_ORDER_2_IN_AG₂(x1, x2) = IN_ORDER_2_IN_AG₁(x2)
IF_IN_ORDER_2_IN_3_AG₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AG₃(x2, x3, x7)
IF_APP_3_IN_1_AAG₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAG₁(x5)
APP_3_IN_AAG₃(x1, x2, x3) = APP_3_IN_AAG₁(x3)
IF_IN_ORDER_2_IN_2_AA₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AA₂(x2, x6)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)
IF_IN_ORDER_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AG₂(x4, x5)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AA₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAA₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AG₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAG₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAG₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APP_3_IN_AAG₃(x1, x2, x3) = APP_3_IN_AAG₁(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAG₁(._2₁(Zs)) -> APP_3_IN_AAG₁(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APP_3_IN_AAG₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP_3_IN_AAG₁(._2₁(Zs)) -> APP_3_IN_AAG₁(Zs)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAA -> APP_3_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APP_3_IN_AAA}.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)

The TRS R consists of the following rules:

in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))

The argument filtering Pi contains the following mapping:
void_0 = void_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
._2₂(x1, x2) = ._2₁(x2)
in_order_2_in_aa₂(x1, x2) = in_order_2_in_aa
in_order_2_out_aa₂(x1, x2) = in_order_2_out_aa₁(x1)
if_in_order_2_in_1_aa₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_aa₁(x5)
if_in_order_2_in_2_aa₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_aa₂(x2, x6)
if_in_order_2_in_3_aa₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_aa₃(x2, x3, x7)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
IN_ORDER_2_IN_AA₂(x1, x2) = IN_ORDER_2_IN_AA
IF_IN_ORDER_2_IN_1_AA₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AA₁(x5)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

IF_IN_ORDER_2_IN_1_AA₁(in_order_2_out_aa₁(Left)) -> IN_ORDER_2_IN_AA
IN_ORDER_2_IN_AA -> IF_IN_ORDER_2_IN_1_AA₁(in_order_2_in_aa)
IN_ORDER_2_IN_AA -> IN_ORDER_2_IN_AA

The TRS R consists of the following rules:

in_order_2_in_aa -> in_order_2_out_aa₁(void_0)
in_order_2_in_aa -> if_in_order_2_in_1_aa₁(in_order_2_in_aa)
if_in_order_2_in_1_aa₁(in_order_2_out_aa₁(Left)) -> if_in_order_2_in_2_aa₂(Left, in_order_2_in_aa)
if_in_order_2_in_2_aa₂(Left, in_order_2_out_aa₁(Right)) -> if_in_order_2_in_3_aa₃(Left, Right, app_3_in_aaa)
if_in_order_2_in_3_aa₃(Left, Right, app_3_out_aaa₁(Ls)) -> in_order_2_out_aa₁(tree_3₂(Left, Right))
app_3_in_aaa -> app_3_out_aaa₁([]_0)
app_3_in_aaa -> if_app_3_in_1_aaa₁(app_3_in_aaa)
if_app_3_in_1_aaa₁(app_3_out_aaa₁(Xs)) -> app_3_out_aaa₁(._2₁(Xs))

The set Q consists of the following terms:

in_order_2_in_aa
if_in_order_2_in_1_aa₁(x0)
if_in_order_2_in_2_aa₂(x0, x1)
if_in_order_2_in_3_aa₃(x0, x1, x2)
app_3_in_aaa
if_app_3_in_1_aaa₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IN_ORDER_2_IN_AA, IF_IN_ORDER_2_IN_1_AA₁}.
With regard to the inferred argument filtering the predicates were used in the following modes:
in_order₂: (f,b) (f,f)
app₃: (f,f,f) (f,f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AA₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAA₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AG₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAG₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAG₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ag₂(x1, x2) = in_order_2_in_ag₁(x2)
void_0 = void_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
._2₂(x1, x2) = ._2₁(x2)
in_order_2_out_ag₂(x1, x2) = in_order_2_out_ag₂(x1, x2)
if_in_order_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_ag₂(x4, x5)
in_order_2_in_aa₂(x1, x2) = in_order_2_in_aa
in_order_2_out_aa₂(x1, x2) = in_order_2_out_aa₁(x1)
if_in_order_2_in_1_aa₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_aa₁(x5)
if_in_order_2_in_2_aa₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_aa₂(x2, x6)
if_in_order_2_in_3_aa₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_aa₃(x2, x3, x7)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_in_order_2_in_2_ag₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_ag₃(x2, x4, x6)
if_in_order_2_in_3_ag₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_ag₄(x2, x3, x4, x7)
app_3_in_aag₃(x1, x2, x3) = app_3_in_aag₁(x3)
app_3_out_aag₃(x1, x2, x3) = app_3_out_aag₃(x1, x2, x3)
if_app_3_in_1_aag₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aag₂(x4, x5)
IF_IN_ORDER_2_IN_3_AA₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AA₃(x2, x3, x7)
IN_ORDER_2_IN_AA₂(x1, x2) = IN_ORDER_2_IN_AA
IF_IN_ORDER_2_IN_2_AG₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AG₃(x2, x4, x6)
IF_IN_ORDER_2_IN_1_AA₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AA₁(x5)
IN_ORDER_2_IN_AG₂(x1, x2) = IN_ORDER_2_IN_AG₁(x2)
IF_IN_ORDER_2_IN_3_AG₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AG₄(x2, x3, x4, x7)
IF_APP_3_IN_1_AAG₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAG₂(x4, x5)
APP_3_IN_AAG₃(x1, x2, x3) = APP_3_IN_AAG₁(x3)
IF_IN_ORDER_2_IN_2_AA₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AA₂(x2, x6)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)
IF_IN_ORDER_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AG₂(x4, x5)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AG₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AA₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AA₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAA₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
IF_IN_ORDER_2_IN_1_AG₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> IF_IN_ORDER_2_IN_3_AG₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_AG₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> APP_3_IN_AAG₃(Ls, ._2₂(X, Rs), Xs)
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAG₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ag₂(x1, x2) = in_order_2_in_ag₁(x2)
void_0 = void_0
[]_0 = []_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
._2₂(x1, x2) = ._2₁(x2)
in_order_2_out_ag₂(x1, x2) = in_order_2_out_ag₂(x1, x2)
if_in_order_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_ag₂(x4, x5)
in_order_2_in_aa₂(x1, x2) = in_order_2_in_aa
in_order_2_out_aa₂(x1, x2) = in_order_2_out_aa₁(x1)
if_in_order_2_in_1_aa₅(x1, x2, x3, x4, x5) = if_in_order_2_in_1_aa₁(x5)
if_in_order_2_in_2_aa₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_aa₂(x2, x6)
if_in_order_2_in_3_aa₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_aa₃(x2, x3, x7)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_in_order_2_in_2_ag₆(x1, x2, x3, x4, x5, x6) = if_in_order_2_in_2_ag₃(x2, x4, x6)
if_in_order_2_in_3_ag₇(x1, x2, x3, x4, x5, x6, x7) = if_in_order_2_in_3_ag₄(x2, x3, x4, x7)
app_3_in_aag₃(x1, x2, x3) = app_3_in_aag₁(x3)
app_3_out_aag₃(x1, x2, x3) = app_3_out_aag₃(x1, x2, x3)
if_app_3_in_1_aag₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aag₂(x4, x5)
IF_IN_ORDER_2_IN_3_AA₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AA₃(x2, x3, x7)
IN_ORDER_2_IN_AA₂(x1, x2) = IN_ORDER_2_IN_AA
IF_IN_ORDER_2_IN_2_AG₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AG₃(x2, x4, x6)
IF_IN_ORDER_2_IN_1_AA₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AA₁(x5)
IN_ORDER_2_IN_AG₂(x1, x2) = IN_ORDER_2_IN_AG₁(x2)
IF_IN_ORDER_2_IN_3_AG₇(x1, x2, x3, x4, x5, x6, x7) = IF_IN_ORDER_2_IN_3_AG₄(x2, x3, x4, x7)
IF_APP_3_IN_1_AAG₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAG₂(x4, x5)
APP_3_IN_AAG₃(x1, x2, x3) = APP_3_IN_AAG₁(x3)
IF_IN_ORDER_2_IN_2_AA₆(x1, x2, x3, x4, x5, x6) = IF_IN_ORDER_2_IN_2_AA₂(x2, x6)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)
IF_IN_ORDER_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_ORDER_2_IN_1_AG₂(x4, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 3 SCCs with 11 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAG₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAG₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APP_3_IN_AAG₃(x1, x2, x3) = APP_3_IN_AAG₁(x3)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> IN_ORDER_2_IN_AA₂(Right, Rs)
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_AA₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
IN_ORDER_2_IN_AA₂(tree_3₃(X, Left, Right), Xs) -> IN_ORDER_2_IN_AA₂(Left, Ls)

The TRS R consists of the following rules:

in_order_2_in_ag₂(void_0, []_0) -> in_order_2_out_ag₂(void_0, []_0)
in_order_2_in_ag₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
in_order_2_in_aa₂(void_0, []_0) -> in_order_2_out_aa₂(void_0, []_0)
in_order_2_in_aa₂(tree_3₃(X, Left, Right), Xs) -> if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_in_aa₂(Left, Ls))
if_in_order_2_in_1_aa₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_aa₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aaa₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_aa₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aaa₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_aa₂(tree_3₃(X, Left, Right), Xs)
if_in_order_2_in_1_ag₅(X, Left, Right, Xs, in_order_2_out_aa₂(Left, Ls)) -> if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_in_aa₂(Right, Rs))
if_in_order_2_in_2_ag₆(X, Left, Right, Xs, Ls, in_order_2_out_aa₂(Right, Rs)) -> if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_in_aag₃(Ls, ._2₂(X, Rs), Xs))
app_3_in_aag₃([]_0, X, X) -> app_3_out_aag₃([]_0, X, X)
app_3_in_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_in_aag₃(Xs, Ys, Zs))
if_app_3_in_1_aag₅(X, Xs, Ys, Zs, app_3_out_aag₃(Xs, Ys, Zs)) -> app_3_out_aag₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_in_order_2_in_3_ag₇(X, Left, Right, Xs, Ls, Rs, app_3_out_aag₃(Ls, ._2₂(X, Rs), Xs)) -> in_order_2_out_ag₂(tree_3₃(X, Left, Right), Xs)