Left Termination proof of /tmp/tmp3B8gvM/NJ4.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

p₄(M, N, s₁(R), RES) :- p₄(M, R, N, RES).
p₄(M, s₁(N), R, RES) :- p₄(R, N, M, RES).
p₄(M, underscore, underscore1, M).

With regard to the inferred argument filtering the predicates were used in the following modes:
p₄: (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_4_in_ggga₄(M, N, s_1₁(R), RES) -> if_p_4_in_1_ggga₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
p_4_in_ggga₄(M, s_1₁(N), R, RES) -> if_p_4_in_2_ggga₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
p_4_in_ggga₄(M, underscore, underscore1, M) -> p_4_out_ggga₄(M, underscore, underscore1, M)
if_p_4_in_2_ggga₅(M, N, R, RES, p_4_out_ggga₄(R, N, M, RES)) -> p_4_out_ggga₄(M, s_1₁(N), R, RES)
if_p_4_in_1_ggga₅(M, N, R, RES, p_4_out_ggga₄(M, R, N, RES)) -> p_4_out_ggga₄(M, N, s_1₁(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_4_in_ggga₄(M, N, s_1₁(R), RES) -> if_p_4_in_1_ggga₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
p_4_in_ggga₄(M, s_1₁(N), R, RES) -> if_p_4_in_2_ggga₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
p_4_in_ggga₄(M, underscore, underscore1, M) -> p_4_out_ggga₄(M, underscore, underscore1, M)
if_p_4_in_2_ggga₅(M, N, R, RES, p_4_out_ggga₄(R, N, M, RES)) -> p_4_out_ggga₄(M, s_1₁(N), R, RES)
if_p_4_in_1_ggga₅(M, N, R, RES, p_4_out_ggga₄(M, R, N, RES)) -> p_4_out_ggga₄(M, N, s_1₁(R), RES)

P_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> IF_P_4_IN_1_GGGA₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
P_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> P_4_IN_GGGA₄(M, R, N, RES)
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> IF_P_4_IN_2_GGGA₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> P_4_IN_GGGA₄(R, N, M, RES)

The TRS R consists of the following rules:

p_4_in_ggga₄(M, N, s_1₁(R), RES) -> if_p_4_in_1_ggga₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
p_4_in_ggga₄(M, s_1₁(N), R, RES) -> if_p_4_in_2_ggga₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
p_4_in_ggga₄(M, underscore, underscore1, M) -> p_4_out_ggga₄(M, underscore, underscore1, M)
if_p_4_in_2_ggga₅(M, N, R, RES, p_4_out_ggga₄(R, N, M, RES)) -> p_4_out_ggga₄(M, s_1₁(N), R, RES)
if_p_4_in_1_ggga₅(M, N, R, RES, p_4_out_ggga₄(M, R, N, RES)) -> p_4_out_ggga₄(M, N, s_1₁(R), RES)

The argument filtering Pi contains the following mapping:
p_4_in_ggga₄(x1, x2, x3, x4) = p_4_in_ggga₃(x1, x2, x3)
s_1₁(x1) = s_1₁(x1)
if_p_4_in_1_ggga₅(x1, x2, x3, x4, x5) = if_p_4_in_1_ggga₁(x5)
if_p_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_p_4_in_2_ggga₁(x5)
p_4_out_ggga₄(x1, x2, x3, x4) = p_4_out_ggga₁(x4)
P_4_IN_GGGA₄(x1, x2, x3, x4) = P_4_IN_GGGA₃(x1, x2, x3)
IF_P_4_IN_2_GGGA₅(x1, x2, x3, x4, x5) = IF_P_4_IN_2_GGGA₁(x5)
IF_P_4_IN_1_GGGA₅(x1, x2, x3, x4, x5) = IF_P_4_IN_1_GGGA₁(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_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> IF_P_4_IN_1_GGGA₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
P_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> P_4_IN_GGGA₄(M, R, N, RES)
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> IF_P_4_IN_2_GGGA₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> P_4_IN_GGGA₄(R, N, M, RES)

The TRS R consists of the following rules:

p_4_in_ggga₄(M, N, s_1₁(R), RES) -> if_p_4_in_1_ggga₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
p_4_in_ggga₄(M, s_1₁(N), R, RES) -> if_p_4_in_2_ggga₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
p_4_in_ggga₄(M, underscore, underscore1, M) -> p_4_out_ggga₄(M, underscore, underscore1, M)
if_p_4_in_2_ggga₅(M, N, R, RES, p_4_out_ggga₄(R, N, M, RES)) -> p_4_out_ggga₄(M, s_1₁(N), R, RES)
if_p_4_in_1_ggga₅(M, N, R, RES, p_4_out_ggga₄(M, R, N, RES)) -> p_4_out_ggga₄(M, N, s_1₁(R), RES)

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

P_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> P_4_IN_GGGA₄(M, R, N, RES)
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> P_4_IN_GGGA₄(R, N, M, RES)

The TRS R consists of the following rules:

p_4_in_ggga₄(M, N, s_1₁(R), RES) -> if_p_4_in_1_ggga₅(M, N, R, RES, p_4_in_ggga₄(M, R, N, RES))
p_4_in_ggga₄(M, s_1₁(N), R, RES) -> if_p_4_in_2_ggga₅(M, N, R, RES, p_4_in_ggga₄(R, N, M, RES))
p_4_in_ggga₄(M, underscore, underscore1, M) -> p_4_out_ggga₄(M, underscore, underscore1, M)
if_p_4_in_2_ggga₅(M, N, R, RES, p_4_out_ggga₄(R, N, M, RES)) -> p_4_out_ggga₄(M, s_1₁(N), R, RES)
if_p_4_in_1_ggga₅(M, N, R, RES, p_4_out_ggga₄(M, R, N, RES)) -> p_4_out_ggga₄(M, N, s_1₁(R), RES)

The argument filtering Pi contains the following mapping:
p_4_in_ggga₄(x1, x2, x3, x4) = p_4_in_ggga₃(x1, x2, x3)
s_1₁(x1) = s_1₁(x1)
if_p_4_in_1_ggga₅(x1, x2, x3, x4, x5) = if_p_4_in_1_ggga₁(x5)
if_p_4_in_2_ggga₅(x1, x2, x3, x4, x5) = if_p_4_in_2_ggga₁(x5)
p_4_out_ggga₄(x1, x2, x3, x4) = p_4_out_ggga₁(x4)
P_4_IN_GGGA₄(x1, x2, x3, x4) = P_4_IN_GGGA₃(x1, x2, 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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

P_4_IN_GGGA₄(M, N, s_1₁(R), RES) -> P_4_IN_GGGA₄(M, R, N, RES)
P_4_IN_GGGA₄(M, s_1₁(N), R, RES) -> P_4_IN_GGGA₄(R, N, M, RES)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
P_4_IN_GGGA₄(x1, x2, x3, x4) = P_4_IN_GGGA₃(x1, x2, 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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

P_4_IN_GGGA₃(M, N, s_1₁(R)) -> P_4_IN_GGGA₃(M, R, N)
P_4_IN_GGGA₃(M, s_1₁(N), R) -> P_4_IN_GGGA₃(R, N, M)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {P_4_IN_GGGA₃}.
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:

P_4_IN_GGGA₃(M, N, s_1₁(R)) -> P_4_IN_GGGA₃(M, R, N)
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3
P_4_IN_GGGA₃(M, s_1₁(N), R) -> P_4_IN_GGGA₃(R, N, M)
The graph contains the following edges 3 >= 1, 2 > 2, 1 >= 3