Left Termination of the query pattern in_order(b,f) w.r.t. the given Prolog program could successfully be proven:



PROLOG
  ↳ PrologToPiTRSProof

inorder2(void0, {}0).
inorder2(tree3(X, Left, Right), Xs) :- inorder2(Left, Ls), inorder2(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: (b,f)
app3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof

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

in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)


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

IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IN_ORDER_2_IN_GA2(Left, Ls)
IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> IN_ORDER_2_IN_GA2(Right, Rs)
IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> IF_IN_ORDER_2_IN_3_GA7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> APP_3_IN_GGA3(Ls, ._22(X, Rs), Xs)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
IF_IN_ORDER_2_IN_3_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_IN_ORDER_2_IN_3_GA1(x7)
IF_IN_ORDER_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_IN_ORDER_2_IN_1_GA3(x1, x3, x5)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
IN_ORDER_2_IN_GA2(x1, x2)  =  IN_ORDER_2_IN_GA1(x1)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_IN_ORDER_2_IN_2_GA6(x1, x2, x3, x4, x5, x6)  =  IF_IN_ORDER_2_IN_2_GA3(x1, x5, x6)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IN_ORDER_2_IN_GA2(Left, Ls)
IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> IN_ORDER_2_IN_GA2(Right, Rs)
IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> IF_IN_ORDER_2_IN_3_GA7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
IF_IN_ORDER_2_IN_2_GA6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> APP_3_IN_GGA3(Ls, ._22(X, Rs), Xs)
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGA5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
IF_IN_ORDER_2_IN_3_GA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_IN_ORDER_2_IN_3_GA1(x7)
IF_IN_ORDER_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_IN_ORDER_2_IN_1_GA3(x1, x3, x5)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA2(x1, x5)
IN_ORDER_2_IN_GA2(x1, x2)  =  IN_ORDER_2_IN_GA1(x1)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_IN_ORDER_2_IN_2_GA6(x1, x2, x3, x4, x5, x6)  =  IF_IN_ORDER_2_IN_2_GA3(x1, x5, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 4 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

APP_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

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

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

APP_3_IN_GGA2(._22(X, Xs), Ys) -> APP_3_IN_GGA2(Xs, Ys)

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_GGA2}.
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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ PiDPToQDPProof

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

IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IN_ORDER_2_IN_GA2(Left, Ls)
IN_ORDER_2_IN_GA2(tree_33(X, Left, Right), Xs) -> IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
IF_IN_ORDER_2_IN_1_GA5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> IN_ORDER_2_IN_GA2(Right, Rs)

The TRS R consists of the following rules:

in_order_2_in_ga2(void_0, []_0) -> in_order_2_out_ga2(void_0, []_0)
in_order_2_in_ga2(tree_33(X, Left, Right), Xs) -> if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_in_ga2(Left, Ls))
if_in_order_2_in_1_ga5(X, Left, Right, Xs, in_order_2_out_ga2(Left, Ls)) -> if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_in_ga2(Right, Rs))
if_in_order_2_in_2_ga6(X, Left, Right, Xs, Ls, in_order_2_out_ga2(Right, Rs)) -> if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_in_gga3(Ls, ._22(X, Rs), Xs))
app_3_in_gga3([]_0, X, X) -> app_3_out_gga3([]_0, X, X)
app_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_in_gga3(Xs, Ys, Zs))
if_app_3_in_1_gga5(X, Xs, Ys, Zs, app_3_out_gga3(Xs, Ys, Zs)) -> app_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_in_order_2_in_3_ga7(X, Left, Right, Xs, Ls, Rs, app_3_out_gga3(Ls, ._22(X, Rs), Xs)) -> in_order_2_out_ga2(tree_33(X, Left, Right), Xs)

The argument filtering Pi contains the following mapping:
in_order_2_in_ga2(x1, x2)  =  in_order_2_in_ga1(x1)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
._22(x1, x2)  =  ._22(x1, x2)
in_order_2_out_ga2(x1, x2)  =  in_order_2_out_ga1(x2)
if_in_order_2_in_1_ga5(x1, x2, x3, x4, x5)  =  if_in_order_2_in_1_ga3(x1, x3, x5)
if_in_order_2_in_2_ga6(x1, x2, x3, x4, x5, x6)  =  if_in_order_2_in_2_ga3(x1, x5, x6)
if_in_order_2_in_3_ga7(x1, x2, x3, x4, x5, x6, x7)  =  if_in_order_2_in_3_ga1(x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga1(x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga2(x1, x5)
IF_IN_ORDER_2_IN_1_GA5(x1, x2, x3, x4, x5)  =  IF_IN_ORDER_2_IN_1_GA3(x1, x3, x5)
IN_ORDER_2_IN_GA2(x1, x2)  =  IN_ORDER_2_IN_GA1(x1)

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
                ↳ PiDPToQDPProof
QDP
                    ↳ QDPSizeChangeProof

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

IN_ORDER_2_IN_GA1(tree_33(X, Left, Right)) -> IN_ORDER_2_IN_GA1(Left)
IN_ORDER_2_IN_GA1(tree_33(X, Left, Right)) -> IF_IN_ORDER_2_IN_1_GA3(X, Right, in_order_2_in_ga1(Left))
IF_IN_ORDER_2_IN_1_GA3(X, Right, in_order_2_out_ga1(Ls)) -> IN_ORDER_2_IN_GA1(Right)

The TRS R consists of the following rules:

in_order_2_in_ga1(void_0) -> in_order_2_out_ga1([]_0)
in_order_2_in_ga1(tree_33(X, Left, Right)) -> if_in_order_2_in_1_ga3(X, Right, in_order_2_in_ga1(Left))
if_in_order_2_in_1_ga3(X, Right, in_order_2_out_ga1(Ls)) -> if_in_order_2_in_2_ga3(X, Ls, in_order_2_in_ga1(Right))
if_in_order_2_in_2_ga3(X, Ls, in_order_2_out_ga1(Rs)) -> if_in_order_2_in_3_ga1(app_3_in_gga2(Ls, ._22(X, Rs)))
app_3_in_gga2([]_0, X) -> app_3_out_gga1(X)
app_3_in_gga2(._22(X, Xs), Ys) -> if_app_3_in_1_gga2(X, app_3_in_gga2(Xs, Ys))
if_app_3_in_1_gga2(X, app_3_out_gga1(Zs)) -> app_3_out_gga1(._22(X, Zs))
if_in_order_2_in_3_ga1(app_3_out_gga1(Xs)) -> in_order_2_out_ga1(Xs)

The set Q consists of the following terms:

in_order_2_in_ga1(x0)
if_in_order_2_in_1_ga3(x0, x1, x2)
if_in_order_2_in_2_ga3(x0, x1, x2)
app_3_in_gga2(x0, x1)
if_app_3_in_1_gga2(x0, x1)
if_in_order_2_in_3_ga1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IN_ORDER_2_IN_GA1, IF_IN_ORDER_2_IN_1_GA3}.
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: