Left Termination proof of /tmp/tmp8hQLUW/sublist-bf.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

sublist₂(Xs, Ys) :- app₃(underscore, Zs, Ys), app₃(Xs, underscore1, Zs).
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:
sublist₂: (b,f)
app₃: (f,f,f) (b,f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

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:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

SUBLIST_2_IN_GA₂(Xs, Ys) -> IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
SUBLIST_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AAA₃(underscore, Zs, Ys)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> IF_SUBLIST_2_IN_2_GA₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> APP_3_IN_GAA₃(Xs, underscore1, Zs)
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_GAA₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga₂(x1, x2) = sublist_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_sublist_2_in_1_ga₃(x1, x2, x3) = if_sublist_2_in_1_ga₂(x1, x3)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_sublist_2_in_2_ga₄(x1, x2, x3, x4) = if_sublist_2_in_2_ga₁(x4)
app_3_in_gaa₃(x1, x2, x3) = app_3_in_gaa₁(x1)
app_3_out_gaa₃(x1, x2, x3) = app_3_out_gaa
if_app_3_in_1_gaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_gaa₁(x5)
sublist_2_out_ga₂(x1, x2) = sublist_2_out_ga
SUBLIST_2_IN_GA₂(x1, x2) = SUBLIST_2_IN_GA₁(x1)
IF_APP_3_IN_1_GAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_GAA₁(x5)
IF_SUBLIST_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBLIST_2_IN_2_GA₁(x4)
APP_3_IN_GAA₃(x1, x2, x3) = APP_3_IN_GAA₁(x1)
IF_SUBLIST_2_IN_1_GA₃(x1, x2, x3) = IF_SUBLIST_2_IN_1_GA₂(x1, x3)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)

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:

SUBLIST_2_IN_GA₂(Xs, Ys) -> IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
SUBLIST_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AAA₃(underscore, Zs, Ys)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> IF_SUBLIST_2_IN_2_GA₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> APP_3_IN_GAA₃(Xs, underscore1, Zs)
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_GAA₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APP_3_IN_GAA₃(x1, x2, x3) = APP_3_IN_GAA₁(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

APP_3_IN_GAA₁(._2₁(Xs)) -> APP_3_IN_GAA₁(Xs)

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_GAA₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

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

APP_3_IN_GAA₁(._2₁(Xs)) -> APP_3_IN_GAA₁(Xs)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

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

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

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

  ↳ PrologToPiTRSProof

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

APP_3_IN_AAA -> APP_3_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {APP_3_IN_AAA}.
With regard to the inferred argument filtering the predicates were used in the following modes:
sublist₂: (b,f)
app₃: (f,f,f) (b,f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

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:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

SUBLIST_2_IN_GA₂(Xs, Ys) -> IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
SUBLIST_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AAA₃(underscore, Zs, Ys)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> IF_SUBLIST_2_IN_2_GA₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> APP_3_IN_GAA₃(Xs, underscore1, Zs)
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_GAA₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga₂(x1, x2) = sublist_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_sublist_2_in_1_ga₃(x1, x2, x3) = if_sublist_2_in_1_ga₂(x1, x3)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_sublist_2_in_2_ga₄(x1, x2, x3, x4) = if_sublist_2_in_2_ga₂(x1, x4)
app_3_in_gaa₃(x1, x2, x3) = app_3_in_gaa₁(x1)
app_3_out_gaa₃(x1, x2, x3) = app_3_out_gaa₁(x1)
if_app_3_in_1_gaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_gaa₂(x2, x5)
sublist_2_out_ga₂(x1, x2) = sublist_2_out_ga₁(x1)
SUBLIST_2_IN_GA₂(x1, x2) = SUBLIST_2_IN_GA₁(x1)
IF_APP_3_IN_1_GAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_GAA₂(x2, x5)
IF_SUBLIST_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBLIST_2_IN_2_GA₂(x1, x4)
APP_3_IN_GAA₃(x1, x2, x3) = APP_3_IN_GAA₁(x1)
IF_SUBLIST_2_IN_1_GA₃(x1, x2, x3) = IF_SUBLIST_2_IN_1_GA₂(x1, x3)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)

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:

SUBLIST_2_IN_GA₂(Xs, Ys) -> IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
SUBLIST_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AAA₃(underscore, Zs, Ys)
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AAA₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
APP_3_IN_AAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AAA₃(Xs, Ys, Zs)
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> IF_SUBLIST_2_IN_2_GA₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
IF_SUBLIST_2_IN_1_GA₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> APP_3_IN_GAA₃(Xs, underscore1, Zs)
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_GAA₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
sublist_2_in_ga₂(x1, x2) = sublist_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_sublist_2_in_1_ga₃(x1, x2, x3) = if_sublist_2_in_1_ga₂(x1, x3)
app_3_in_aaa₃(x1, x2, x3) = app_3_in_aaa
app_3_out_aaa₃(x1, x2, x3) = app_3_out_aaa₁(x1)
if_app_3_in_1_aaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aaa₁(x5)
if_sublist_2_in_2_ga₄(x1, x2, x3, x4) = if_sublist_2_in_2_ga₂(x1, x4)
app_3_in_gaa₃(x1, x2, x3) = app_3_in_gaa₁(x1)
app_3_out_gaa₃(x1, x2, x3) = app_3_out_gaa₁(x1)
if_app_3_in_1_gaa₅(x1, x2, x3, x4, x5) = if_app_3_in_1_gaa₂(x2, x5)
sublist_2_out_ga₂(x1, x2) = sublist_2_out_ga₁(x1)
SUBLIST_2_IN_GA₂(x1, x2) = SUBLIST_2_IN_GA₁(x1)
IF_APP_3_IN_1_GAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_GAA₂(x2, x5)
IF_SUBLIST_2_IN_2_GA₄(x1, x2, x3, x4) = IF_SUBLIST_2_IN_2_GA₂(x1, x4)
APP_3_IN_GAA₃(x1, x2, x3) = APP_3_IN_GAA₁(x1)
IF_SUBLIST_2_IN_1_GA₃(x1, x2, x3) = IF_SUBLIST_2_IN_1_GA₂(x1, x3)
APP_3_IN_AAA₃(x1, x2, x3) = APP_3_IN_AAA
IF_APP_3_IN_1_AAA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AAA₁(x5)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APP_3_IN_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ 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_GAA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_GAA₃(Xs, Ys, Zs)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ 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_GAA₁(._2₁(Xs)) -> APP_3_IN_GAA₁(Xs)

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

APP_3_IN_GAA₁(._2₁(Xs)) -> APP_3_IN_GAA₁(Xs)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

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

The TRS R consists of the following rules:

sublist_2_in_ga₂(Xs, Ys) -> if_sublist_2_in_1_ga₃(Xs, Ys, app_3_in_aaa₃(underscore, Zs, Ys))
app_3_in_aaa₃([]_0, X, X) -> app_3_out_aaa₃([]_0, X, X)
app_3_in_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_in_aaa₃(Xs, Ys, Zs))
if_app_3_in_1_aaa₅(X, Xs, Ys, Zs, app_3_out_aaa₃(Xs, Ys, Zs)) -> app_3_out_aaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_1_ga₃(Xs, Ys, app_3_out_aaa₃(underscore, Zs, Ys)) -> if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_in_gaa₃(Xs, underscore1, Zs))
app_3_in_gaa₃([]_0, X, X) -> app_3_out_gaa₃([]_0, X, X)
app_3_in_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_in_gaa₃(Xs, Ys, Zs))
if_app_3_in_1_gaa₅(X, Xs, Ys, Zs, app_3_out_gaa₃(Xs, Ys, Zs)) -> app_3_out_gaa₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_sublist_2_in_2_ga₄(Xs, Ys, Zs, app_3_out_gaa₃(Xs, underscore1, Zs)) -> sublist_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

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

APP_3_IN_AAA -> APP_3_IN_AAA

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