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



PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

front2(void0, {}0).
front2(tree3(X, void0, void0), .2(X, {}0)).
front2(tree3(underscore, L, R), Xs) :- front2(L, Ls), front2(R, Rs), app3(Ls, 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:
front2: (f,b) (f,f)
app3: (b,b,f) (b,b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)

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:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)


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

FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AA7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGA3(Ls, 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)
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AG7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGG3(Ls, Rs, Xs)
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGG5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)
IF_FRONT_2_IN_3_AG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AG3(x2, x3, x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA1(x5)
IF_FRONT_2_IN_1_AG5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AG2(x4, x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_FRONT_2_IN_2_AA6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AA3(x2, x5, x6)
IF_APP_3_IN_1_GGG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGG1(x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AG2(x1, x2)  =  FRONT_2_IN_AG1(x2)
IF_FRONT_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AG4(x2, x4, x5, x6)
IF_FRONT_2_IN_3_AA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AA3(x2, x3, x7)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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:

FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AA7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGA3(Ls, 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)
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AG7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGG3(Ls, Rs, Xs)
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGG5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)
IF_FRONT_2_IN_3_AG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AG3(x2, x3, x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA1(x5)
IF_FRONT_2_IN_1_AG5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AG2(x4, x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_FRONT_2_IN_2_AA6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AA3(x2, x5, x6)
IF_APP_3_IN_1_GGG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGG1(x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AG2(x1, x2)  =  FRONT_2_IN_AG1(x2)
IF_FRONT_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AG4(x2, x4, x5, x6)
IF_FRONT_2_IN_3_AA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AA3(x2, x3, x7)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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

↳ 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_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, 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_GGG3(._21(Xs), Ys, ._21(Zs)) -> APP_3_IN_GGG3(Xs, Ys, 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_GGG3}.
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
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(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
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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)  =  ._21(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APP_3_IN_GGA2(._21(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
PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag1(x1)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag3(x2, x3, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg1(x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))

The TRS R consists of the following rules:

front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, Rs, Xs))
if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), 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))

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)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, 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_gga1(x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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_FRONT_2_IN_1_AA1(front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA
FRONT_2_IN_AA -> FRONT_2_IN_AA
FRONT_2_IN_AA -> IF_FRONT_2_IN_1_AA1(front_2_in_aa)

The TRS R consists of the following rules:

front_2_in_aa -> front_2_out_aa2(void_0, []_0)
front_2_in_aa -> front_2_out_aa2(tree_32(void_0, void_0), ._21([]_0))
front_2_in_aa -> if_front_2_in_1_aa1(front_2_in_aa)
if_front_2_in_1_aa1(front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa3(L, Ls, front_2_in_aa)
if_front_2_in_2_aa3(L, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa3(L, R, app_3_in_gga2(Ls, Rs))
if_front_2_in_3_aa3(L, R, app_3_out_gga1(Xs)) -> front_2_out_aa2(tree_32(L, R), Xs)
app_3_in_gga2([]_0, X) -> app_3_out_gga1(X)
app_3_in_gga2(._21(Xs), Ys) -> if_app_3_in_1_gga1(app_3_in_gga2(Xs, Ys))
if_app_3_in_1_gga1(app_3_out_gga1(Zs)) -> app_3_out_gga1(._21(Zs))

The set Q consists of the following terms:

front_2_in_aa
if_front_2_in_1_aa1(x0)
if_front_2_in_2_aa3(x0, x1, x2)
if_front_2_in_3_aa3(x0, x1, x2)
app_3_in_gga2(x0, x1)
if_app_3_in_1_gga1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {FRONT_2_IN_AA, IF_FRONT_2_IN_1_AA1}.
With regard to the inferred argument filtering the predicates were used in the following modes:
front2: (f,b) (f,f)
app3: (b,b,f) (b,b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)

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:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)


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

FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AA7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGA3(Ls, 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)
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AG7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGG3(Ls, Rs, Xs)
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGG5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)
IF_FRONT_2_IN_3_AG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AG4(x2, x3, x4, x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA3(x2, x3, x5)
IF_FRONT_2_IN_1_AG5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AG2(x4, x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_FRONT_2_IN_2_AA6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AA3(x2, x5, x6)
IF_APP_3_IN_1_GGG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGG4(x2, x3, x4, x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AG2(x1, x2)  =  FRONT_2_IN_AG1(x2)
IF_FRONT_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AG4(x2, x4, x5, x6)
IF_FRONT_2_IN_3_AA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AA3(x2, x3, x7)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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:

FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AG2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AA7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AA6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGA3(Ls, 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)
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
IF_FRONT_2_IN_1_AG5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> IF_FRONT_2_IN_3_AG7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
IF_FRONT_2_IN_2_AG6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> APP_3_IN_GGG3(Ls, Rs, Xs)
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APP_3_IN_1_GGG5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
APP_3_IN_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)
IF_FRONT_2_IN_3_AG7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AG4(x2, x3, x4, x7)
IF_APP_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGA3(x2, x3, x5)
IF_FRONT_2_IN_1_AG5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AG2(x4, x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)
IF_FRONT_2_IN_2_AA6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AA3(x2, x5, x6)
IF_APP_3_IN_1_GGG5(x1, x2, x3, x4, x5)  =  IF_APP_3_IN_1_GGG4(x2, x3, x4, x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AG2(x1, x2)  =  FRONT_2_IN_AG1(x2)
IF_FRONT_2_IN_2_AG6(x1, x2, x3, x4, x5, x6)  =  IF_FRONT_2_IN_2_AG4(x2, x4, x5, x6)
IF_FRONT_2_IN_3_AA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_FRONT_2_IN_3_AA3(x2, x3, x7)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, x3)

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

↳ 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_GGG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGG3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._21(x2)
APP_3_IN_GGG3(x1, x2, x3)  =  APP_3_IN_GGG3(x1, x2, 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_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APP_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)
APP_3_IN_GGA3(x1, x2, x3)  =  APP_3_IN_GGA2(x1, x2)

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:

IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> FRONT_2_IN_AA2(R, Rs)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> FRONT_2_IN_AA2(L, Ls)
FRONT_2_IN_AA2(tree_33(underscore, L, R), Xs) -> IF_FRONT_2_IN_1_AA5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))

The TRS R consists of the following rules:

front_2_in_ag2(void_0, []_0) -> front_2_out_ag2(void_0, []_0)
front_2_in_ag2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_ag2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_ag2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
front_2_in_aa2(void_0, []_0) -> front_2_out_aa2(void_0, []_0)
front_2_in_aa2(tree_33(X, void_0, void_0), ._22(X, []_0)) -> front_2_out_aa2(tree_33(X, void_0, void_0), ._22(X, []_0))
front_2_in_aa2(tree_33(underscore, L, R), Xs) -> if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_in_aa2(L, Ls))
if_front_2_in_1_aa5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_aa6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_in_gga3(Ls, 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_front_2_in_3_aa7(underscore, L, R, Xs, Ls, Rs, app_3_out_gga3(Ls, Rs, Xs)) -> front_2_out_aa2(tree_33(underscore, L, R), Xs)
if_front_2_in_1_ag5(underscore, L, R, Xs, front_2_out_aa2(L, Ls)) -> if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_in_aa2(R, Rs))
if_front_2_in_2_ag6(underscore, L, R, Xs, Ls, front_2_out_aa2(R, Rs)) -> if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_in_ggg3(Ls, Rs, Xs))
app_3_in_ggg3([]_0, X, X) -> app_3_out_ggg3([]_0, X, X)
app_3_in_ggg3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_in_ggg3(Xs, Ys, Zs))
if_app_3_in_1_ggg5(X, Xs, Ys, Zs, app_3_out_ggg3(Xs, Ys, Zs)) -> app_3_out_ggg3(._22(X, Xs), Ys, ._22(X, Zs))
if_front_2_in_3_ag7(underscore, L, R, Xs, Ls, Rs, app_3_out_ggg3(Ls, Rs, Xs)) -> front_2_out_ag2(tree_33(underscore, L, R), Xs)

The argument filtering Pi contains the following mapping:
front_2_in_ag2(x1, x2)  =  front_2_in_ag1(x2)
void_0  =  void_0
[]_0  =  []_0
tree_33(x1, x2, x3)  =  tree_32(x2, x3)
._22(x1, x2)  =  ._21(x2)
front_2_out_ag2(x1, x2)  =  front_2_out_ag2(x1, x2)
if_front_2_in_1_ag5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_ag2(x4, x5)
front_2_in_aa2(x1, x2)  =  front_2_in_aa
front_2_out_aa2(x1, x2)  =  front_2_out_aa2(x1, x2)
if_front_2_in_1_aa5(x1, x2, x3, x4, x5)  =  if_front_2_in_1_aa1(x5)
if_front_2_in_2_aa6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_aa3(x2, x5, x6)
if_front_2_in_3_aa7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_aa3(x2, x3, x7)
app_3_in_gga3(x1, x2, x3)  =  app_3_in_gga2(x1, x2)
app_3_out_gga3(x1, x2, x3)  =  app_3_out_gga3(x1, x2, x3)
if_app_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_gga3(x2, x3, x5)
if_front_2_in_2_ag6(x1, x2, x3, x4, x5, x6)  =  if_front_2_in_2_ag4(x2, x4, x5, x6)
if_front_2_in_3_ag7(x1, x2, x3, x4, x5, x6, x7)  =  if_front_2_in_3_ag4(x2, x3, x4, x7)
app_3_in_ggg3(x1, x2, x3)  =  app_3_in_ggg3(x1, x2, x3)
app_3_out_ggg3(x1, x2, x3)  =  app_3_out_ggg3(x1, x2, x3)
if_app_3_in_1_ggg5(x1, x2, x3, x4, x5)  =  if_app_3_in_1_ggg4(x2, x3, x4, x5)
IF_FRONT_2_IN_1_AA5(x1, x2, x3, x4, x5)  =  IF_FRONT_2_IN_1_AA1(x5)
FRONT_2_IN_AA2(x1, x2)  =  FRONT_2_IN_AA

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