Left Termination proof of /tmp/tmp8rEEeW/mult.pl

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)

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)

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)

↳ 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:

SUM_3_IN_GGA₂(X, s_1₁(Y)) -> SUM_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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:

MULT_3_IN_GGA₂(X, s_1₁(Y)) -> MULT_3_IN_GGA₂(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2