Left Termination proof of /tmp/tmpyzpCgB/weight.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

sum₃(.₂(s₁(N), XS), .₂(M, YS), ZS) :- sum₃(.₂(N, XS), .₂(s₁(M), YS), ZS).
sum₃(.₂(0₀, XS), YS, ZS) :- sum₃(XS, YS, ZS).
sum₃({}₀, YS, YS).
weight₂(.₂(N, .₂(M, XS)), X) :- sum₃(.₂(N, .₂(M, XS)), .₂(0₀, XS), YS), weight₂(YS, X).
weight₂(.₂(X, {}₀), X).

With regard to the inferred argument filtering the predicates were used in the following modes:
weight₂: (b,f)
sum₃: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

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:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> SUM_3_IN_GGA₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> IF_SUM_3_IN_1_GGA₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> SUM_3_IN_GGA₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)
SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> IF_SUM_3_IN_2_GGA₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> SUM_3_IN_GGA₃(XS, YS, ZS)
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> IF_WEIGHT_2_IN_2_GA₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> WEIGHT_2_IN_GA₂(YS, X)

The TRS R consists of the following rules:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

The argument filtering Pi contains the following mapping:
weight_2_in_ga₂(x1, x2) = weight_2_in_ga₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
if_weight_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_weight_2_in_1_ga₁(x5)
sum_3_in_gga₃(x1, x2, x3) = sum_3_in_gga₂(x1, x2)
if_sum_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_sum_3_in_1_gga₁(x6)
if_sum_3_in_2_gga₄(x1, x2, x3, x4) = if_sum_3_in_2_gga₁(x4)
sum_3_out_gga₃(x1, x2, x3) = sum_3_out_gga₁(x3)
if_weight_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_weight_2_in_2_ga₁(x6)
weight_2_out_ga₂(x1, x2) = weight_2_out_ga₁(x2)
WEIGHT_2_IN_GA₂(x1, x2) = WEIGHT_2_IN_GA₁(x1)
IF_SUM_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_SUM_3_IN_1_GGA₁(x6)
SUM_3_IN_GGA₃(x1, x2, x3) = SUM_3_IN_GGA₂(x1, x2)
IF_WEIGHT_2_IN_2_GA₆(x1, x2, x3, x4, x5, x6) = IF_WEIGHT_2_IN_2_GA₁(x6)
IF_WEIGHT_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_WEIGHT_2_IN_1_GA₁(x5)
IF_SUM_3_IN_2_GGA₄(x1, x2, x3, x4) = IF_SUM_3_IN_2_GGA₁(x4)

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:

WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> SUM_3_IN_GGA₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> IF_SUM_3_IN_1_GGA₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> SUM_3_IN_GGA₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)
SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> IF_SUM_3_IN_2_GGA₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> SUM_3_IN_GGA₃(XS, YS, ZS)
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> IF_WEIGHT_2_IN_2_GA₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> WEIGHT_2_IN_GA₂(YS, X)

The TRS R consists of the following rules:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> SUM_3_IN_GGA₃(XS, YS, ZS)
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> SUM_3_IN_GGA₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)

The TRS R consists of the following rules:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

The argument filtering Pi contains the following mapping:
weight_2_in_ga₂(x1, x2) = weight_2_in_ga₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
if_weight_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_weight_2_in_1_ga₁(x5)
sum_3_in_gga₃(x1, x2, x3) = sum_3_in_gga₂(x1, x2)
if_sum_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_sum_3_in_1_gga₁(x6)
if_sum_3_in_2_gga₄(x1, x2, x3, x4) = if_sum_3_in_2_gga₁(x4)
sum_3_out_gga₃(x1, x2, x3) = sum_3_out_gga₁(x3)
if_weight_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_weight_2_in_2_ga₁(x6)
weight_2_out_ga₂(x1, x2) = weight_2_out_ga₁(x2)
SUM_3_IN_GGA₃(x1, x2, x3) = SUM_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:

SUM_3_IN_GGA₃(._2₂(0_0, XS), YS, ZS) -> SUM_3_IN_GGA₃(XS, YS, ZS)
SUM_3_IN_GGA₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> SUM_3_IN_GGA₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)

R is empty.
The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
SUM_3_IN_GGA₃(x1, x2, x3) = SUM_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:

SUM_3_IN_GGA₂(._2₂(0_0, XS), YS) -> SUM_3_IN_GGA₂(XS, YS)
SUM_3_IN_GGA₂(._2₂(s_1₁(N), XS), ._2₂(M, YS)) -> SUM_3_IN_GGA₂(._2₂(N, XS), ._2₂(s_1₁(M), YS))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SUM_3_IN_GGA₂}.
We used the following order and afs together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)

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

SUM_3_IN_GGA₂(._2₂(0_0, XS), YS) -> SUM_3_IN_GGA₂(XS, YS) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
SUM_3_IN_GGA₂(._2₂(s_1₁(N), XS), ._2₂(M, YS)) -> SUM_3_IN_GGA₂(._2₂(N, XS), ._2₂(s_1₁(M), YS)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1

We oriented the following set of usable rules. none

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> WEIGHT_2_IN_GA₂(YS, X)

The TRS R consists of the following rules:

weight_2_in_ga₂(._2₂(N, ._2₂(M, XS)), X) -> if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_weight_2_in_1_ga₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_in_ga₂(YS, X))
weight_2_in_ga₂(._2₂(X, []_0), X) -> weight_2_out_ga₂(._2₂(X, []_0), X)
if_weight_2_in_2_ga₆(N, M, XS, X, YS, weight_2_out_ga₂(YS, X)) -> weight_2_out_ga₂(._2₂(N, ._2₂(M, XS)), X)

The argument filtering Pi contains the following mapping:
weight_2_in_ga₂(x1, x2) = weight_2_in_ga₁(x1)
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
if_weight_2_in_1_ga₅(x1, x2, x3, x4, x5) = if_weight_2_in_1_ga₁(x5)
sum_3_in_gga₃(x1, x2, x3) = sum_3_in_gga₂(x1, x2)
if_sum_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_sum_3_in_1_gga₁(x6)
if_sum_3_in_2_gga₄(x1, x2, x3, x4) = if_sum_3_in_2_gga₁(x4)
sum_3_out_gga₃(x1, x2, x3) = sum_3_out_gga₁(x3)
if_weight_2_in_2_ga₆(x1, x2, x3, x4, x5, x6) = if_weight_2_in_2_ga₁(x6)
weight_2_out_ga₂(x1, x2) = weight_2_out_ga₁(x2)
WEIGHT_2_IN_GA₂(x1, x2) = WEIGHT_2_IN_GA₁(x1)
IF_WEIGHT_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_WEIGHT_2_IN_1_GA₁(x5)

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:

WEIGHT_2_IN_GA₂(._2₂(N, ._2₂(M, XS)), X) -> IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_in_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS))
IF_WEIGHT_2_IN_1_GA₅(N, M, XS, X, sum_3_out_gga₃(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS), YS)) -> WEIGHT_2_IN_GA₂(YS, X)

The TRS R consists of the following rules:

sum_3_in_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS) -> if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_in_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS))
sum_3_in_gga₃(._2₂(0_0, XS), YS, ZS) -> if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_in_gga₃(XS, YS, ZS))
if_sum_3_in_1_gga₆(N, XS, M, YS, ZS, sum_3_out_gga₃(._2₂(N, XS), ._2₂(s_1₁(M), YS), ZS)) -> sum_3_out_gga₃(._2₂(s_1₁(N), XS), ._2₂(M, YS), ZS)
if_sum_3_in_2_gga₄(XS, YS, ZS, sum_3_out_gga₃(XS, YS, ZS)) -> sum_3_out_gga₃(._2₂(0_0, XS), YS, ZS)
sum_3_in_gga₃([]_0, YS, YS) -> sum_3_out_gga₃([]_0, YS, YS)

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
s_1₁(x1) = s_1₁(x1)
0_0 = 0_0
[]_0 = []_0
sum_3_in_gga₃(x1, x2, x3) = sum_3_in_gga₂(x1, x2)
if_sum_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_sum_3_in_1_gga₁(x6)
if_sum_3_in_2_gga₄(x1, x2, x3, x4) = if_sum_3_in_2_gga₁(x4)
sum_3_out_gga₃(x1, x2, x3) = sum_3_out_gga₁(x3)
WEIGHT_2_IN_GA₂(x1, x2) = WEIGHT_2_IN_GA₁(x1)
IF_WEIGHT_2_IN_1_GA₅(x1, x2, x3, x4, x5) = IF_WEIGHT_2_IN_1_GA₁(x5)

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:

WEIGHT_2_IN_GA₁(._2₂(N, ._2₂(M, XS))) -> IF_WEIGHT_2_IN_1_GA₁(sum_3_in_gga₂(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS)))
IF_WEIGHT_2_IN_1_GA₁(sum_3_out_gga₁(YS)) -> WEIGHT_2_IN_GA₁(YS)

The TRS R consists of the following rules:

sum_3_in_gga₂(._2₂(s_1₁(N), XS), ._2₂(M, YS)) -> if_sum_3_in_1_gga₁(sum_3_in_gga₂(._2₂(N, XS), ._2₂(s_1₁(M), YS)))
sum_3_in_gga₂(._2₂(0_0, XS), YS) -> if_sum_3_in_2_gga₁(sum_3_in_gga₂(XS, YS))
if_sum_3_in_1_gga₁(sum_3_out_gga₁(ZS)) -> sum_3_out_gga₁(ZS)
if_sum_3_in_2_gga₁(sum_3_out_gga₁(ZS)) -> sum_3_out_gga₁(ZS)
sum_3_in_gga₂([]_0, YS) -> sum_3_out_gga₁(YS)

The set Q consists of the following terms:

sum_3_in_gga₂(x0, x1)
if_sum_3_in_1_gga₁(x0)
if_sum_3_in_2_gga₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_WEIGHT_2_IN_1_GA₁, WEIGHT_2_IN_GA₁}.
We used the following order together with the size-change analysis to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation:

POL(0_0) = 1
POL(._2₂(x₁, x₂)) = 1 + x₂
POL(sum_3_out_gga₁(x₁)) = x₁
POL(if_sum_3_in_1_gga₁(x₁)) = x₁
POL(if_sum_3_in_2_gga₁(x₁)) = x₁
POL(sum_3_in_gga₂(x₁, x₂)) = x₂
POL(s_1₁(x₁)) = 0
POL([]_0) = 0

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

IF_WEIGHT_2_IN_1_GA₁(sum_3_out_gga₁(YS)) -> WEIGHT_2_IN_GA₁(YS) (allowed arguments on rhs = {1})
The graph contains the following edges 1 >= 1
WEIGHT_2_IN_GA₁(._2₂(N, ._2₂(M, XS))) -> IF_WEIGHT_2_IN_1_GA₁(sum_3_in_gga₂(._2₂(N, ._2₂(M, XS)), ._2₂(0_0, XS))) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1

We oriented the following set of usable rules.

sum_3_in_gga₂([]_0, YS) -> sum_3_out_gga₁(YS)
sum_3_in_gga₂(._2₂(s_1₁(N), XS), ._2₂(M, YS)) -> if_sum_3_in_1_gga₁(sum_3_in_gga₂(._2₂(N, XS), ._2₂(s_1₁(M), YS)))
sum_3_in_gga₂(._2₂(0_0, XS), YS) -> if_sum_3_in_2_gga₁(sum_3_in_gga₂(XS, YS))
if_sum_3_in_2_gga₁(sum_3_out_gga₁(ZS)) -> sum_3_out_gga₁(ZS)
if_sum_3_in_1_gga₁(sum_3_out_gga₁(ZS)) -> sum_3_out_gga₁(ZS)