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



PROLOG
  ↳ PrologToPiTRSProof

mult3(underscore, 00, 00).
mult3(X, s1(Y), Z) :- mult3(X, Y, W), sum3(W, X, Z).
sum3(X, 00, X).
sum3(X, s1(Y), s1(Z)) :- sum3(X, Y, Z).


With regard to the inferred argument filtering the predicates were used in the following modes:
mult3: (b,b,f)
sum3: (b,b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(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:

mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(x4)


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

MULT_3_IN_GGA3(X, s_11(Y), Z) -> IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_in_gga3(X, Y, W))
MULT_3_IN_GGA3(X, s_11(Y), Z) -> MULT_3_IN_GGA3(X, Y, W)
IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> IF_MULT_3_IN_2_GGA5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> SUM_3_IN_GGA3(W, X, Z)
SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> IF_SUM_3_IN_1_GGA4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> SUM_3_IN_GGA3(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(x4)
IF_SUM_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_SUM_3_IN_1_GGA1(x4)
IF_MULT_3_IN_2_GGA5(x1, x2, x3, x4, x5)  =  IF_MULT_3_IN_2_GGA1(x5)
MULT_3_IN_GGA3(x1, x2, x3)  =  MULT_3_IN_GGA2(x1, x2)
SUM_3_IN_GGA3(x1, x2, x3)  =  SUM_3_IN_GGA2(x1, x2)
IF_MULT_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_MULT_3_IN_1_GGA2(x1, 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:

MULT_3_IN_GGA3(X, s_11(Y), Z) -> IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_in_gga3(X, Y, W))
MULT_3_IN_GGA3(X, s_11(Y), Z) -> MULT_3_IN_GGA3(X, Y, W)
IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> IF_MULT_3_IN_2_GGA5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
IF_MULT_3_IN_1_GGA4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> SUM_3_IN_GGA3(W, X, Z)
SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> IF_SUM_3_IN_1_GGA4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> SUM_3_IN_GGA3(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(x4)
IF_SUM_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_SUM_3_IN_1_GGA1(x4)
IF_MULT_3_IN_2_GGA5(x1, x2, x3, x4, x5)  =  IF_MULT_3_IN_2_GGA1(x5)
MULT_3_IN_GGA3(x1, x2, x3)  =  MULT_3_IN_GGA2(x1, x2)
SUM_3_IN_GGA3(x1, x2, x3)  =  SUM_3_IN_GGA2(x1, x2)
IF_MULT_3_IN_1_GGA4(x1, x2, x3, x4)  =  IF_MULT_3_IN_1_GGA2(x1, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 4 less nodes.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> SUM_3_IN_GGA3(X, Y, Z)

The TRS R consists of the following rules:

mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(x4)
SUM_3_IN_GGA3(x1, x2, x3)  =  SUM_3_IN_GGA2(x1, x2)

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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

SUM_3_IN_GGA3(X, s_11(Y), s_11(Z)) -> SUM_3_IN_GGA3(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_11(x1)
SUM_3_IN_GGA3(x1, x2, x3)  =  SUM_3_IN_GGA2(x1, x2)

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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

SUM_3_IN_GGA2(X, s_11(Y)) -> SUM_3_IN_GGA2(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SUM_3_IN_GGA2}.
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:



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
PiDP
                ↳ UsableRulesProof

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

MULT_3_IN_GGA3(X, s_11(Y), Z) -> MULT_3_IN_GGA3(X, Y, W)

The TRS R consists of the following rules:

mult_3_in_gga3(underscore, 0_0, 0_0) -> mult_3_out_gga3(underscore, 0_0, 0_0)
mult_3_in_gga3(X, s_11(Y), Z) -> if_mult_3_in_1_gga4(X, Y, Z, mult_3_in_gga3(X, Y, W))
if_mult_3_in_1_gga4(X, Y, Z, mult_3_out_gga3(X, Y, W)) -> if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_in_gga3(W, X, Z))
sum_3_in_gga3(X, 0_0, X) -> sum_3_out_gga3(X, 0_0, X)
sum_3_in_gga3(X, s_11(Y), s_11(Z)) -> if_sum_3_in_1_gga4(X, Y, Z, sum_3_in_gga3(X, Y, Z))
if_sum_3_in_1_gga4(X, Y, Z, sum_3_out_gga3(X, Y, Z)) -> sum_3_out_gga3(X, s_11(Y), s_11(Z))
if_mult_3_in_2_gga5(X, Y, Z, W, sum_3_out_gga3(W, X, Z)) -> mult_3_out_gga3(X, s_11(Y), Z)

The argument filtering Pi contains the following mapping:
mult_3_in_gga3(x1, x2, x3)  =  mult_3_in_gga2(x1, x2)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
mult_3_out_gga3(x1, x2, x3)  =  mult_3_out_gga1(x3)
if_mult_3_in_1_gga4(x1, x2, x3, x4)  =  if_mult_3_in_1_gga2(x1, x4)
if_mult_3_in_2_gga5(x1, x2, x3, x4, x5)  =  if_mult_3_in_2_gga1(x5)
sum_3_in_gga3(x1, x2, x3)  =  sum_3_in_gga2(x1, x2)
sum_3_out_gga3(x1, x2, x3)  =  sum_3_out_gga1(x3)
if_sum_3_in_1_gga4(x1, x2, x3, x4)  =  if_sum_3_in_1_gga1(x4)
MULT_3_IN_GGA3(x1, x2, x3)  =  MULT_3_IN_GGA2(x1, x2)

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
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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

MULT_3_IN_GGA3(X, s_11(Y), Z) -> MULT_3_IN_GGA3(X, Y, W)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_11(x1)
MULT_3_IN_GGA3(x1, x2, x3)  =  MULT_3_IN_GGA2(x1, x2)

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
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof

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

MULT_3_IN_GGA2(X, s_11(Y)) -> MULT_3_IN_GGA2(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {MULT_3_IN_GGA2}.
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: