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



PROLOG
  ↳ PrologToPiTRSProof

times3(X, Y, Z) :- mult4(X, Y, 00, Z).
mult4(00, Y, 00, 00).
mult4(s1(U), Y, 00, Z) :- mult4(U, Y, Y, Z).
mult4(X, Y, s1(W), s1(Z)) :- mult4(X, Y, W, Z).


With regard to the inferred argument filtering the predicates were used in the following modes:
times3: (b,b,f)
mult4: (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_3_in_gga3(X, Y, Z) -> if_times_3_in_1_gga4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
mult_4_in_ggga4(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga4(0_0, Y, 0_0, 0_0)
mult_4_in_ggga4(s_11(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
mult_4_in_ggga4(X, Y, s_11(W), s_11(Z)) -> if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_out_ggga4(X, Y, W, Z)) -> mult_4_out_ggga4(X, Y, s_11(W), s_11(Z))
if_mult_4_in_1_ggga4(U, Y, Z, mult_4_out_ggga4(U, Y, Y, Z)) -> mult_4_out_ggga4(s_11(U), Y, 0_0, Z)
if_times_3_in_1_gga4(X, Y, Z, mult_4_out_ggga4(X, Y, 0_0, Z)) -> times_3_out_gga3(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga3(x1, x2, x3)  =  times_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_times_3_in_1_gga4(x1, x2, x3, x4)  =  if_times_3_in_1_gga1(x4)
mult_4_in_ggga4(x1, x2, x3, x4)  =  mult_4_in_ggga3(x1, x2, x3)
mult_4_out_ggga4(x1, x2, x3, x4)  =  mult_4_out_ggga1(x4)
if_mult_4_in_1_ggga4(x1, x2, x3, x4)  =  if_mult_4_in_1_ggga1(x4)
if_mult_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_mult_4_in_2_ggga1(x5)
times_3_out_gga3(x1, x2, x3)  =  times_3_out_gga1(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_3_in_gga3(X, Y, Z) -> if_times_3_in_1_gga4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
mult_4_in_ggga4(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga4(0_0, Y, 0_0, 0_0)
mult_4_in_ggga4(s_11(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
mult_4_in_ggga4(X, Y, s_11(W), s_11(Z)) -> if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_out_ggga4(X, Y, W, Z)) -> mult_4_out_ggga4(X, Y, s_11(W), s_11(Z))
if_mult_4_in_1_ggga4(U, Y, Z, mult_4_out_ggga4(U, Y, Y, Z)) -> mult_4_out_ggga4(s_11(U), Y, 0_0, Z)
if_times_3_in_1_gga4(X, Y, Z, mult_4_out_ggga4(X, Y, 0_0, Z)) -> times_3_out_gga3(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga3(x1, x2, x3)  =  times_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_times_3_in_1_gga4(x1, x2, x3, x4)  =  if_times_3_in_1_gga1(x4)
mult_4_in_ggga4(x1, x2, x3, x4)  =  mult_4_in_ggga3(x1, x2, x3)
mult_4_out_ggga4(x1, x2, x3, x4)  =  mult_4_out_ggga1(x4)
if_mult_4_in_1_ggga4(x1, x2, x3, x4)  =  if_mult_4_in_1_ggga1(x4)
if_mult_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_mult_4_in_2_ggga1(x5)
times_3_out_gga3(x1, x2, x3)  =  times_3_out_gga1(x3)


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

TIMES_3_IN_GGA3(X, Y, Z) -> IF_TIMES_3_IN_1_GGA4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
TIMES_3_IN_GGA3(X, Y, Z) -> MULT_4_IN_GGGA4(X, Y, 0_0, Z)
MULT_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> IF_MULT_4_IN_1_GGGA4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
MULT_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> MULT_4_IN_GGGA4(U, Y, Y, Z)
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> IF_MULT_4_IN_2_GGGA5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> MULT_4_IN_GGGA4(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga3(X, Y, Z) -> if_times_3_in_1_gga4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
mult_4_in_ggga4(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga4(0_0, Y, 0_0, 0_0)
mult_4_in_ggga4(s_11(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
mult_4_in_ggga4(X, Y, s_11(W), s_11(Z)) -> if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_out_ggga4(X, Y, W, Z)) -> mult_4_out_ggga4(X, Y, s_11(W), s_11(Z))
if_mult_4_in_1_ggga4(U, Y, Z, mult_4_out_ggga4(U, Y, Y, Z)) -> mult_4_out_ggga4(s_11(U), Y, 0_0, Z)
if_times_3_in_1_gga4(X, Y, Z, mult_4_out_ggga4(X, Y, 0_0, Z)) -> times_3_out_gga3(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga3(x1, x2, x3)  =  times_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_times_3_in_1_gga4(x1, x2, x3, x4)  =  if_times_3_in_1_gga1(x4)
mult_4_in_ggga4(x1, x2, x3, x4)  =  mult_4_in_ggga3(x1, x2, x3)
mult_4_out_ggga4(x1, x2, x3, x4)  =  mult_4_out_ggga1(x4)
if_mult_4_in_1_ggga4(x1, x2, x3, x4)  =  if_mult_4_in_1_ggga1(x4)
if_mult_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_mult_4_in_2_ggga1(x5)
times_3_out_gga3(x1, x2, x3)  =  times_3_out_gga1(x3)
IF_TIMES_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_TIMES_3_IN_1_GGA1(x4)
MULT_4_IN_GGGA4(x1, x2, x3, x4)  =  MULT_4_IN_GGGA3(x1, x2, x3)
TIMES_3_IN_GGA3(x1, x2, x3)  =  TIMES_3_IN_GGA2(x1, x2)
IF_MULT_4_IN_1_GGGA4(x1, x2, x3, x4)  =  IF_MULT_4_IN_1_GGGA1(x4)
IF_MULT_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_MULT_4_IN_2_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:

TIMES_3_IN_GGA3(X, Y, Z) -> IF_TIMES_3_IN_1_GGA4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
TIMES_3_IN_GGA3(X, Y, Z) -> MULT_4_IN_GGGA4(X, Y, 0_0, Z)
MULT_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> IF_MULT_4_IN_1_GGGA4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
MULT_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> MULT_4_IN_GGGA4(U, Y, Y, Z)
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> IF_MULT_4_IN_2_GGGA5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> MULT_4_IN_GGGA4(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga3(X, Y, Z) -> if_times_3_in_1_gga4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
mult_4_in_ggga4(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga4(0_0, Y, 0_0, 0_0)
mult_4_in_ggga4(s_11(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
mult_4_in_ggga4(X, Y, s_11(W), s_11(Z)) -> if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_out_ggga4(X, Y, W, Z)) -> mult_4_out_ggga4(X, Y, s_11(W), s_11(Z))
if_mult_4_in_1_ggga4(U, Y, Z, mult_4_out_ggga4(U, Y, Y, Z)) -> mult_4_out_ggga4(s_11(U), Y, 0_0, Z)
if_times_3_in_1_gga4(X, Y, Z, mult_4_out_ggga4(X, Y, 0_0, Z)) -> times_3_out_gga3(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga3(x1, x2, x3)  =  times_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_times_3_in_1_gga4(x1, x2, x3, x4)  =  if_times_3_in_1_gga1(x4)
mult_4_in_ggga4(x1, x2, x3, x4)  =  mult_4_in_ggga3(x1, x2, x3)
mult_4_out_ggga4(x1, x2, x3, x4)  =  mult_4_out_ggga1(x4)
if_mult_4_in_1_ggga4(x1, x2, x3, x4)  =  if_mult_4_in_1_ggga1(x4)
if_mult_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_mult_4_in_2_ggga1(x5)
times_3_out_gga3(x1, x2, x3)  =  times_3_out_gga1(x3)
IF_TIMES_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_TIMES_3_IN_1_GGA1(x4)
MULT_4_IN_GGGA4(x1, x2, x3, x4)  =  MULT_4_IN_GGGA3(x1, x2, x3)
TIMES_3_IN_GGA3(x1, x2, x3)  =  TIMES_3_IN_GGA2(x1, x2)
IF_MULT_4_IN_1_GGGA4(x1, x2, x3, x4)  =  IF_MULT_4_IN_1_GGGA1(x4)
IF_MULT_4_IN_2_GGGA5(x1, x2, x3, x4, x5)  =  IF_MULT_4_IN_2_GGGA1(x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph 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_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> MULT_4_IN_GGGA4(U, Y, Y, Z)
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> MULT_4_IN_GGGA4(X, Y, W, Z)

The TRS R consists of the following rules:

times_3_in_gga3(X, Y, Z) -> if_times_3_in_1_gga4(X, Y, Z, mult_4_in_ggga4(X, Y, 0_0, Z))
mult_4_in_ggga4(0_0, Y, 0_0, 0_0) -> mult_4_out_ggga4(0_0, Y, 0_0, 0_0)
mult_4_in_ggga4(s_11(U), Y, 0_0, Z) -> if_mult_4_in_1_ggga4(U, Y, Z, mult_4_in_ggga4(U, Y, Y, Z))
mult_4_in_ggga4(X, Y, s_11(W), s_11(Z)) -> if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_in_ggga4(X, Y, W, Z))
if_mult_4_in_2_ggga5(X, Y, W, Z, mult_4_out_ggga4(X, Y, W, Z)) -> mult_4_out_ggga4(X, Y, s_11(W), s_11(Z))
if_mult_4_in_1_ggga4(U, Y, Z, mult_4_out_ggga4(U, Y, Y, Z)) -> mult_4_out_ggga4(s_11(U), Y, 0_0, Z)
if_times_3_in_1_gga4(X, Y, Z, mult_4_out_ggga4(X, Y, 0_0, Z)) -> times_3_out_gga3(X, Y, Z)

The argument filtering Pi contains the following mapping:
times_3_in_gga3(x1, x2, x3)  =  times_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
if_times_3_in_1_gga4(x1, x2, x3, x4)  =  if_times_3_in_1_gga1(x4)
mult_4_in_ggga4(x1, x2, x3, x4)  =  mult_4_in_ggga3(x1, x2, x3)
mult_4_out_ggga4(x1, x2, x3, x4)  =  mult_4_out_ggga1(x4)
if_mult_4_in_1_ggga4(x1, x2, x3, x4)  =  if_mult_4_in_1_ggga1(x4)
if_mult_4_in_2_ggga5(x1, x2, x3, x4, x5)  =  if_mult_4_in_2_ggga1(x5)
times_3_out_gga3(x1, x2, x3)  =  times_3_out_gga1(x3)
MULT_4_IN_GGGA4(x1, x2, x3, x4)  =  MULT_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:

MULT_4_IN_GGGA4(s_11(U), Y, 0_0, Z) -> MULT_4_IN_GGGA4(U, Y, Y, Z)
MULT_4_IN_GGGA4(X, Y, s_11(W), s_11(Z)) -> MULT_4_IN_GGGA4(X, Y, W, Z)

R is empty.
The argument filtering Pi contains the following mapping:
0_0  =  0_0
s_11(x1)  =  s_11(x1)
MULT_4_IN_GGGA4(x1, x2, x3, x4)  =  MULT_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:

MULT_4_IN_GGGA3(s_11(U), Y, 0_0) -> MULT_4_IN_GGGA3(U, Y, Y)
MULT_4_IN_GGGA3(X, Y, s_11(W)) -> MULT_4_IN_GGGA3(X, Y, W)

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