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



↳ PROLOG
  ↳ PrologToPiTRSProof

rotate2(X, Y) :- append23(A, B, X), append13(B, A, Y).
append13(.2(X, Xs), Ys, .2(X, Zs)) :- append13(Xs, Ys, Zs).
append13({}0, Ys, Ys).
append23(.2(X, Xs), Ys, .2(X, Zs)) :- append23(Xs, Ys, Zs).
append23({}0, Ys, Ys).


With regard to the inferred argument filtering the predicates were used in the following modes:
rotate2: (b,f)
append23: (f,f,b)
append13: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)

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:

rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)


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

ROTATE_2_IN_GA2(X, Y) -> IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_in_aag3(A, B, X))
ROTATE_2_IN_GA2(X, Y) -> APPEND2_3_IN_AAG3(A, B, X)
APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND2_3_IN_1_AAG5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND2_3_IN_AAG3(Xs, Ys, Zs)
IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_out_aag3(A, B, X)) -> IF_ROTATE_2_IN_2_GA5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_out_aag3(A, B, X)) -> APPEND1_3_IN_GGA3(B, A, Y)
APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND1_3_IN_1_GGA5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND1_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)
IF_ROTATE_2_IN_1_GA3(x1, x2, x3)  =  IF_ROTATE_2_IN_1_GA1(x3)
ROTATE_2_IN_GA2(x1, x2)  =  ROTATE_2_IN_GA1(x1)
IF_APPEND1_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APPEND1_3_IN_1_GGA2(x1, x5)
APPEND1_3_IN_GGA3(x1, x2, x3)  =  APPEND1_3_IN_GGA2(x1, x2)
IF_ROTATE_2_IN_2_GA5(x1, x2, x3, x4, x5)  =  IF_ROTATE_2_IN_2_GA1(x5)
APPEND2_3_IN_AAG3(x1, x2, x3)  =  APPEND2_3_IN_AAG1(x3)
IF_APPEND2_3_IN_1_AAG5(x1, x2, x3, x4, x5)  =  IF_APPEND2_3_IN_1_AAG2(x1, x5)

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:

ROTATE_2_IN_GA2(X, Y) -> IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_in_aag3(A, B, X))
ROTATE_2_IN_GA2(X, Y) -> APPEND2_3_IN_AAG3(A, B, X)
APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND2_3_IN_1_AAG5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND2_3_IN_AAG3(Xs, Ys, Zs)
IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_out_aag3(A, B, X)) -> IF_ROTATE_2_IN_2_GA5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
IF_ROTATE_2_IN_1_GA3(X, Y, append2_3_out_aag3(A, B, X)) -> APPEND1_3_IN_GGA3(B, A, Y)
APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> IF_APPEND1_3_IN_1_GGA5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND1_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)
IF_ROTATE_2_IN_1_GA3(x1, x2, x3)  =  IF_ROTATE_2_IN_1_GA1(x3)
ROTATE_2_IN_GA2(x1, x2)  =  ROTATE_2_IN_GA1(x1)
IF_APPEND1_3_IN_1_GGA5(x1, x2, x3, x4, x5)  =  IF_APPEND1_3_IN_1_GGA2(x1, x5)
APPEND1_3_IN_GGA3(x1, x2, x3)  =  APPEND1_3_IN_GGA2(x1, x2)
IF_ROTATE_2_IN_2_GA5(x1, x2, x3, x4, x5)  =  IF_ROTATE_2_IN_2_GA1(x5)
APPEND2_3_IN_AAG3(x1, x2, x3)  =  APPEND2_3_IN_AAG1(x3)
IF_APPEND2_3_IN_1_AAG5(x1, x2, x3, x4, x5)  =  IF_APPEND2_3_IN_1_AAG2(x1, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 6 less nodes.

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

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

APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND1_3_IN_GGA3(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)
APPEND1_3_IN_GGA3(x1, x2, x3)  =  APPEND1_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:

APPEND1_3_IN_GGA3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND1_3_IN_GGA3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
APPEND1_3_IN_GGA3(x1, x2, x3)  =  APPEND1_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:

APPEND1_3_IN_GGA2(._22(X, Xs), Ys) -> APPEND1_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 {APPEND1_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
                ↳ UsableRulesProof

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

APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND2_3_IN_AAG3(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_2_in_ga2(X, Y) -> if_rotate_2_in_1_ga3(X, Y, append2_3_in_aag3(A, B, X))
append2_3_in_aag3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_in_aag3(Xs, Ys, Zs))
append2_3_in_aag3([]_0, Ys, Ys) -> append2_3_out_aag3([]_0, Ys, Ys)
if_append2_3_in_1_aag5(X, Xs, Ys, Zs, append2_3_out_aag3(Xs, Ys, Zs)) -> append2_3_out_aag3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_1_ga3(X, Y, append2_3_out_aag3(A, B, X)) -> if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_in_gga3(B, A, Y))
append1_3_in_gga3(._22(X, Xs), Ys, ._22(X, Zs)) -> if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_in_gga3(Xs, Ys, Zs))
append1_3_in_gga3([]_0, Ys, Ys) -> append1_3_out_gga3([]_0, Ys, Ys)
if_append1_3_in_1_gga5(X, Xs, Ys, Zs, append1_3_out_gga3(Xs, Ys, Zs)) -> append1_3_out_gga3(._22(X, Xs), Ys, ._22(X, Zs))
if_rotate_2_in_2_ga5(X, Y, A, B, append1_3_out_gga3(B, A, Y)) -> rotate_2_out_ga2(X, Y)

The argument filtering Pi contains the following mapping:
rotate_2_in_ga2(x1, x2)  =  rotate_2_in_ga1(x1)
._22(x1, x2)  =  ._22(x1, x2)
[]_0  =  []_0
if_rotate_2_in_1_ga3(x1, x2, x3)  =  if_rotate_2_in_1_ga1(x3)
append2_3_in_aag3(x1, x2, x3)  =  append2_3_in_aag1(x3)
if_append2_3_in_1_aag5(x1, x2, x3, x4, x5)  =  if_append2_3_in_1_aag2(x1, x5)
append2_3_out_aag3(x1, x2, x3)  =  append2_3_out_aag2(x1, x2)
if_rotate_2_in_2_ga5(x1, x2, x3, x4, x5)  =  if_rotate_2_in_2_ga1(x5)
append1_3_in_gga3(x1, x2, x3)  =  append1_3_in_gga2(x1, x2)
if_append1_3_in_1_gga5(x1, x2, x3, x4, x5)  =  if_append1_3_in_1_gga2(x1, x5)
append1_3_out_gga3(x1, x2, x3)  =  append1_3_out_gga1(x3)
rotate_2_out_ga2(x1, x2)  =  rotate_2_out_ga1(x2)
APPEND2_3_IN_AAG3(x1, x2, x3)  =  APPEND2_3_IN_AAG1(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

APPEND2_3_IN_AAG3(._22(X, Xs), Ys, ._22(X, Zs)) -> APPEND2_3_IN_AAG3(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._22(x1, x2)  =  ._22(x1, x2)
APPEND2_3_IN_AAG3(x1, x2, x3)  =  APPEND2_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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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

APPEND2_3_IN_AAG1(._22(X, Zs)) -> APPEND2_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 {APPEND2_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: