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)

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)

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)

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)


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)

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)

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
The approximation of the Dependency Graph 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_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: