Left Termination of the query pattern times_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

times(X, Y, Z) :- mult(X, Y, 0, Z).
mult(0, Y, 0, 0).
mult(s(U), Y, 0, Z) :- mult(U, Y, Y, Z).
mult(X, Y, s(W), s(Z)) :- mult(X, Y, W, Z).

Queries:

times(g,g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
times_in: (b,b,f)
mult_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:

times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
0  =  0
mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
s(x1)  =  s(x1)
U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

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:

times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
0  =  0
mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
s(x1)  =  s(x1)
U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)


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

TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)

The TRS R consists of the following rules:

times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
0  =  0
mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
s(x1)  =  s(x1)
U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(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:

TIMES_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
TIMES_IN_GGA(X, Y, Z) → MULT_IN_GGGA(X, Y, 0, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → U2_GGGA(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)
MULT_IN_GGGA(X, Y, s(W), s(Z)) → U3_GGGA(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)

The TRS R consists of the following rules:

times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
0  =  0
mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
s(x1)  =  s(x1)
U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)
U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_IN_GGGA(x1, x2, x3)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGGA(x1, x2, x3, x4)  =  U2_GGGA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)

The TRS R consists of the following rules:

times_in_gga(X, Y, Z) → U1_gga(X, Y, Z, mult_in_ggga(X, Y, 0, Z))
mult_in_ggga(0, Y, 0, 0) → mult_out_ggga(0, Y, 0, 0)
mult_in_ggga(s(U), Y, 0, Z) → U2_ggga(U, Y, Z, mult_in_ggga(U, Y, Y, Z))
mult_in_ggga(X, Y, s(W), s(Z)) → U3_ggga(X, Y, W, Z, mult_in_ggga(X, Y, W, Z))
U3_ggga(X, Y, W, Z, mult_out_ggga(X, Y, W, Z)) → mult_out_ggga(X, Y, s(W), s(Z))
U2_ggga(U, Y, Z, mult_out_ggga(U, Y, Y, Z)) → mult_out_ggga(s(U), Y, 0, Z)
U1_gga(X, Y, Z, mult_out_ggga(X, Y, 0, Z)) → times_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
mult_in_ggga(x1, x2, x3, x4)  =  mult_in_ggga(x1, x2, x3)
0  =  0
mult_out_ggga(x1, x2, x3, x4)  =  mult_out_ggga(x4)
s(x1)  =  s(x1)
U2_ggga(x1, x2, x3, x4)  =  U2_ggga(x4)
U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
times_out_gga(x1, x2, x3)  =  times_out_gga(x3)
MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_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:

MULT_IN_GGGA(X, Y, s(W), s(Z)) → MULT_IN_GGGA(X, Y, W, Z)
MULT_IN_GGGA(s(U), Y, 0, Z) → MULT_IN_GGGA(U, Y, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
MULT_IN_GGGA(x1, x2, x3, x4)  =  MULT_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:

MULT_IN_GGGA(s(U), Y, 0) → MULT_IN_GGGA(U, Y, Y)
MULT_IN_GGGA(X, Y, s(W)) → MULT_IN_GGGA(X, Y, W)

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: