Left Termination proof of /tmp/tmpQVE5wq/NJ4.pl

Left Termination of the query pattern p_in_4(g, g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

p(M, N, s(R), RES) :- p(M, R, N, RES).
p(M, s(N), R, RES) :- p(R, N, M, RES).
p(M, X, X1, M).

Queries:

p(g,g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (b,b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_ggga(M, N, s(R), RES) → U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
p_in_ggga(M, s(N), R, RES) → U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
p_in_ggga(M, X, X1, M) → p_out_ggga(M, X, X1, M)
U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) → p_out_ggga(M, s(N), R, RES)
U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) → p_out_ggga(M, N, s(R), RES)

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:

p_in_ggga(M, N, s(R), RES) → U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
p_in_ggga(M, s(N), R, RES) → U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
p_in_ggga(M, X, X1, M) → p_out_ggga(M, X, X1, M)
U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) → p_out_ggga(M, s(N), R, RES)
U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) → p_out_ggga(M, N, s(R), RES)

The argument filtering Pi contains the following mapping:
p_in_ggga(x1, x2, x3, x4) = p_in_ggga(x1, x2, x3)
s(x1) = s(x1)
U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x5)
U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5)
p_out_ggga(x1, x2, x3, x4) = p_out_ggga(x4)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN_GGGA(M, N, s(R), RES) → U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
P_IN_GGGA(M, N, s(R), RES) → P_IN_GGGA(M, R, N, RES)
P_IN_GGGA(M, s(N), R, RES) → U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
P_IN_GGGA(M, s(N), R, RES) → P_IN_GGGA(R, N, M, RES)

The TRS R consists of the following rules:

p_in_ggga(M, N, s(R), RES) → U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
p_in_ggga(M, s(N), R, RES) → U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
p_in_ggga(M, X, X1, M) → p_out_ggga(M, X, X1, M)
U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) → p_out_ggga(M, s(N), R, RES)
U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) → p_out_ggga(M, N, s(R), RES)

The argument filtering Pi contains the following mapping:
p_in_ggga(x1, x2, x3, x4) = p_in_ggga(x1, x2, x3)
s(x1) = s(x1)
U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x5)
U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5)
p_out_ggga(x1, x2, x3, x4) = p_out_ggga(x4)
P_IN_GGGA(x1, x2, x3, x4) = P_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5) = U1_GGGA(x5)
U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(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:

P_IN_GGGA(M, N, s(R), RES) → U1_GGGA(M, N, R, RES, p_in_ggga(M, R, N, RES))
P_IN_GGGA(M, N, s(R), RES) → P_IN_GGGA(M, R, N, RES)
P_IN_GGGA(M, s(N), R, RES) → U2_GGGA(M, N, R, RES, p_in_ggga(R, N, M, RES))
P_IN_GGGA(M, s(N), R, RES) → P_IN_GGGA(R, N, M, RES)

The TRS R consists of the following rules:

p_in_ggga(M, N, s(R), RES) → U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
p_in_ggga(M, s(N), R, RES) → U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
p_in_ggga(M, X, X1, M) → p_out_ggga(M, X, X1, M)
U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) → p_out_ggga(M, s(N), R, RES)
U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) → p_out_ggga(M, N, s(R), RES)

The argument filtering Pi contains the following mapping:
p_in_ggga(x1, x2, x3, x4) = p_in_ggga(x1, x2, x3)
s(x1) = s(x1)
U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x5)
U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5)
p_out_ggga(x1, x2, x3, x4) = p_out_ggga(x4)
P_IN_GGGA(x1, x2, x3, x4) = P_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5) = U1_GGGA(x5)
U2_GGGA(x1, x2, x3, x4, x5) = U2_GGGA(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

P_IN_GGGA(M, N, s(R), RES) → P_IN_GGGA(M, R, N, RES)
P_IN_GGGA(M, s(N), R, RES) → P_IN_GGGA(R, N, M, RES)

The TRS R consists of the following rules:

p_in_ggga(M, N, s(R), RES) → U1_ggga(M, N, R, RES, p_in_ggga(M, R, N, RES))
p_in_ggga(M, s(N), R, RES) → U2_ggga(M, N, R, RES, p_in_ggga(R, N, M, RES))
p_in_ggga(M, X, X1, M) → p_out_ggga(M, X, X1, M)
U2_ggga(M, N, R, RES, p_out_ggga(R, N, M, RES)) → p_out_ggga(M, s(N), R, RES)
U1_ggga(M, N, R, RES, p_out_ggga(M, R, N, RES)) → p_out_ggga(M, N, s(R), RES)

The argument filtering Pi contains the following mapping:
p_in_ggga(x1, x2, x3, x4) = p_in_ggga(x1, x2, x3)
s(x1) = s(x1)
U1_ggga(x1, x2, x3, x4, x5) = U1_ggga(x5)
U2_ggga(x1, x2, x3, x4, x5) = U2_ggga(x5)
p_out_ggga(x1, x2, x3, x4) = p_out_ggga(x4)
P_IN_GGGA(x1, x2, x3, x4) = P_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

P_IN_GGGA(M, N, s(R), RES) → P_IN_GGGA(M, R, N, RES)
P_IN_GGGA(M, s(N), R, RES) → P_IN_GGGA(R, N, M, RES)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
P_IN_GGGA(x1, x2, x3, x4) = P_IN_GGGA(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

P_IN_GGGA(M, s(N), R) → P_IN_GGGA(R, N, M)
P_IN_GGGA(M, N, s(R)) → P_IN_GGGA(M, R, N)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

P_IN_GGGA(M, N, s(R)) → P_IN_GGGA(M, R, N)
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3
P_IN_GGGA(M, s(N), R) → P_IN_GGGA(R, N, M)
The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3