Left Termination proof of /tmp/tmp2hgoDn/suffix-bf.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

suffix₂(Xs, Ys) :- app₃(underscore, Xs, Ys).
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:
suffix₂: (b,f)
app₃: (f,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_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:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

SUFFIX_2_IN_GA₂(Xs, Ys) -> IF_SUFFIX_2_IN_1_GA₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
SUFFIX_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AGA₃(underscore, Xs, Ys)
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AGA₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_2_in_ga₂(x1, x2) = suffix_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_suffix_2_in_1_ga₃(x1, x2, x3) = if_suffix_2_in_1_ga₁(x3)
app_3_in_aga₃(x1, x2, x3) = app_3_in_aga₁(x2)
app_3_out_aga₃(x1, x2, x3) = app_3_out_aga₂(x1, x3)
if_app_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aga₁(x5)
suffix_2_out_ga₂(x1, x2) = suffix_2_out_ga₁(x2)
SUFFIX_2_IN_GA₂(x1, x2) = SUFFIX_2_IN_GA₁(x1)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(x2)
IF_APP_3_IN_1_AGA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AGA₁(x5)
IF_SUFFIX_2_IN_1_GA₃(x1, x2, x3) = IF_SUFFIX_2_IN_1_GA₁(x3)

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:

SUFFIX_2_IN_GA₂(Xs, Ys) -> IF_SUFFIX_2_IN_1_GA₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
SUFFIX_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AGA₃(underscore, Xs, Ys)
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AGA₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_2_in_ga₂(x1, x2) = suffix_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_suffix_2_in_1_ga₃(x1, x2, x3) = if_suffix_2_in_1_ga₁(x3)
app_3_in_aga₃(x1, x2, x3) = app_3_in_aga₁(x2)
app_3_out_aga₃(x1, x2, x3) = app_3_out_aga₂(x1, x3)
if_app_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aga₁(x5)
suffix_2_out_ga₂(x1, x2) = suffix_2_out_ga₁(x2)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₁(x2)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

  ↳ PrologToPiTRSProof

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

APP_3_IN_AGA₁(Ys) -> APP_3_IN_AGA₁(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_AGA₁}.
With regard to the inferred argument filtering the predicates were used in the following modes:
suffix₂: (b,f)
app₃: (f,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_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:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

SUFFIX_2_IN_GA₂(Xs, Ys) -> IF_SUFFIX_2_IN_1_GA₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
SUFFIX_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AGA₃(underscore, Xs, Ys)
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AGA₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_2_in_ga₂(x1, x2) = suffix_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_suffix_2_in_1_ga₃(x1, x2, x3) = if_suffix_2_in_1_ga₂(x1, x3)
app_3_in_aga₃(x1, x2, x3) = app_3_in_aga₁(x2)
app_3_out_aga₃(x1, x2, x3) = app_3_out_aga₃(x1, x2, x3)
if_app_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aga₂(x3, x5)
suffix_2_out_ga₂(x1, x2) = suffix_2_out_ga₂(x1, x2)
SUFFIX_2_IN_GA₂(x1, x2) = SUFFIX_2_IN_GA₁(x1)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(x2)
IF_APP_3_IN_1_AGA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AGA₂(x3, x5)
IF_SUFFIX_2_IN_1_GA₃(x1, x2, x3) = IF_SUFFIX_2_IN_1_GA₂(x1, x3)

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:

SUFFIX_2_IN_GA₂(Xs, Ys) -> IF_SUFFIX_2_IN_1_GA₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
SUFFIX_2_IN_GA₂(Xs, Ys) -> APP_3_IN_AGA₃(underscore, Xs, Ys)
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> IF_APP_3_IN_1_AGA₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_2_in_ga₂(x1, x2) = suffix_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_suffix_2_in_1_ga₃(x1, x2, x3) = if_suffix_2_in_1_ga₂(x1, x3)
app_3_in_aga₃(x1, x2, x3) = app_3_in_aga₁(x2)
app_3_out_aga₃(x1, x2, x3) = app_3_out_aga₃(x1, x2, x3)
if_app_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aga₂(x3, x5)
suffix_2_out_ga₂(x1, x2) = suffix_2_out_ga₂(x1, x2)
SUFFIX_2_IN_GA₂(x1, x2) = SUFFIX_2_IN_GA₁(x1)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(x2)
IF_APP_3_IN_1_AGA₅(x1, x2, x3, x4, x5) = IF_APP_3_IN_1_AGA₂(x3, x5)
IF_SUFFIX_2_IN_1_GA₃(x1, x2, x3) = IF_SUFFIX_2_IN_1_GA₂(x1, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 1 SCC with 3 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

The TRS R consists of the following rules:

suffix_2_in_ga₂(Xs, Ys) -> if_suffix_2_in_1_ga₃(Xs, Ys, app_3_in_aga₃(underscore, Xs, Ys))
app_3_in_aga₃([]_0, X, X) -> app_3_out_aga₃([]_0, X, X)
app_3_in_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_in_aga₃(Xs, Ys, Zs))
if_app_3_in_1_aga₅(X, Xs, Ys, Zs, app_3_out_aga₃(Xs, Ys, Zs)) -> app_3_out_aga₃(._2₂(X, Xs), Ys, ._2₂(X, Zs))
if_suffix_2_in_1_ga₃(Xs, Ys, app_3_out_aga₃(underscore, Xs, Ys)) -> suffix_2_out_ga₂(Xs, Ys)

The argument filtering Pi contains the following mapping:
suffix_2_in_ga₂(x1, x2) = suffix_2_in_ga₁(x1)
[]_0 = []_0
._2₂(x1, x2) = ._2₁(x2)
if_suffix_2_in_1_ga₃(x1, x2, x3) = if_suffix_2_in_1_ga₂(x1, x3)
app_3_in_aga₃(x1, x2, x3) = app_3_in_aga₁(x2)
app_3_out_aga₃(x1, x2, x3) = app_3_out_aga₃(x1, x2, x3)
if_app_3_in_1_aga₅(x1, x2, x3, x4, x5) = if_app_3_in_1_aga₂(x3, x5)
suffix_2_out_ga₂(x1, x2) = suffix_2_out_ga₂(x1, x2)
APP_3_IN_AGA₃(x1, x2, x3) = APP_3_IN_AGA₁(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

APP_3_IN_AGA₃(._2₂(X, Xs), Ys, ._2₂(X, Zs)) -> APP_3_IN_AGA₃(Xs, Ys, Zs)

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

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

APP_3_IN_AGA₁(Ys) -> APP_3_IN_AGA₁(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_AGA₁}.