Left Termination proof of /tmp/tmpX5SxVF/mult.pl

Left Termination of the query pattern mult_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

mult(X, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).

Queries:

mult(g,g,a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mult_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(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_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
mult_in(x1, x2, x3) = mult_in(x1, x2)
s(x1) = s(x1)
U1(x1, x2, x3, x4) = U1(x1, x4)
0 = 0
mult_out(x1, x2, x3) = mult_out(x3)
U2(x1, x2, x3, x4) = U2(x4)
sum_in(x1, x2, x3) = sum_in(x1, x2)
U3(x1, x2, x3, x4) = U3(x4)
sum_out(x1, x2, x3) = sum_out(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:

MULT_IN(X, s(Y), Z) → U1¹(X, Y, Z, mult_in(X, Y, W))
MULT_IN(X, s(Y), Z) → MULT_IN(X, Y, W)
U1¹(X, Y, Z, mult_out(X, Y, W)) → U2¹(X, Y, Z, sum_in(W, X, Z))
U1¹(X, Y, Z, mult_out(X, Y, W)) → SUM_IN(W, X, Z)
SUM_IN(X, s(Y), s(Z)) → U3¹(X, Y, Z, sum_in(X, Y, Z))
SUM_IN(X, s(Y), s(Z)) → SUM_IN(X, Y, Z)

The TRS R consists of the following rules:

mult_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
mult_in(x1, x2, x3) = mult_in(x1, x2)
s(x1) = s(x1)
U1(x1, x2, x3, x4) = U1(x1, x4)
0 = 0
mult_out(x1, x2, x3) = mult_out(x3)
U2(x1, x2, x3, x4) = U2(x4)
sum_in(x1, x2, x3) = sum_in(x1, x2)
U3(x1, x2, x3, x4) = U3(x4)
sum_out(x1, x2, x3) = sum_out(x3)
U3¹(x1, x2, x3, x4) = U3¹(x4)
SUM_IN(x1, x2, x3) = SUM_IN(x1, x2)
U2¹(x1, x2, x3, x4) = U2¹(x4)
MULT_IN(x1, x2, x3) = MULT_IN(x1, x2)
U1¹(x1, x2, x3, x4) = U1¹(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_IN(X, s(Y), Z) → U1¹(X, Y, Z, mult_in(X, Y, W))
MULT_IN(X, s(Y), Z) → MULT_IN(X, Y, W)
U1¹(X, Y, Z, mult_out(X, Y, W)) → U2¹(X, Y, Z, sum_in(W, X, Z))
U1¹(X, Y, Z, mult_out(X, Y, W)) → SUM_IN(W, X, Z)
SUM_IN(X, s(Y), s(Z)) → U3¹(X, Y, Z, sum_in(X, Y, Z))
SUM_IN(X, s(Y), s(Z)) → SUM_IN(X, Y, Z)

The TRS R consists of the following rules:

mult_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
mult_in(x1, x2, x3) = mult_in(x1, x2)
s(x1) = s(x1)
U1(x1, x2, x3, x4) = U1(x1, x4)
0 = 0
mult_out(x1, x2, x3) = mult_out(x3)
U2(x1, x2, x3, x4) = U2(x4)
sum_in(x1, x2, x3) = sum_in(x1, x2)
U3(x1, x2, x3, x4) = U3(x4)
sum_out(x1, x2, x3) = sum_out(x3)
U3¹(x1, x2, x3, x4) = U3¹(x4)
SUM_IN(x1, x2, x3) = SUM_IN(x1, x2)
U2¹(x1, x2, x3, x4) = U2¹(x4)
MULT_IN(x1, x2, x3) = MULT_IN(x1, x2)
U1¹(x1, x2, x3, x4) = U1¹(x1, x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] 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_IN(X, s(Y), s(Z)) → SUM_IN(X, Y, Z)

The TRS R consists of the following rules:

mult_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
mult_in(x1, x2, x3) = mult_in(x1, x2)
s(x1) = s(x1)
U1(x1, x2, x3, x4) = U1(x1, x4)
0 = 0
mult_out(x1, x2, x3) = mult_out(x3)
U2(x1, x2, x3, x4) = U2(x4)
sum_in(x1, x2, x3) = sum_in(x1, x2)
U3(x1, x2, x3, x4) = U3(x4)
sum_out(x1, x2, x3) = sum_out(x3)
SUM_IN(x1, x2, x3) = SUM_IN(x1, x2)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

SUM_IN(X, s(Y), s(Z)) → SUM_IN(X, Y, Z)

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

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

SUM_IN(X, s(Y)) → SUM_IN(X, Y)

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:

SUM_IN(X, s(Y)) → SUM_IN(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_IN(X, s(Y), Z) → MULT_IN(X, Y, W)

The TRS R consists of the following rules:

mult_in(X, s(Y), Z) → U1(X, Y, Z, mult_in(X, Y, W))
mult_in(X, 0, 0) → mult_out(X, 0, 0)
U1(X, Y, Z, mult_out(X, Y, W)) → U2(X, Y, Z, sum_in(W, X, Z))
sum_in(X, s(Y), s(Z)) → U3(X, Y, Z, sum_in(X, Y, Z))
sum_in(X, 0, X) → sum_out(X, 0, X)
U3(X, Y, Z, sum_out(X, Y, Z)) → sum_out(X, s(Y), s(Z))
U2(X, Y, Z, sum_out(W, X, Z)) → mult_out(X, s(Y), Z)

The argument filtering Pi contains the following mapping:
mult_in(x1, x2, x3) = mult_in(x1, x2)
s(x1) = s(x1)
U1(x1, x2, x3, x4) = U1(x1, x4)
0 = 0
mult_out(x1, x2, x3) = mult_out(x3)
U2(x1, x2, x3, x4) = U2(x4)
sum_in(x1, x2, x3) = sum_in(x1, x2)
U3(x1, x2, x3, x4) = U3(x4)
sum_out(x1, x2, x3) = sum_out(x3)
MULT_IN(x1, x2, x3) = MULT_IN(x1, x2)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

MULT_IN(X, s(Y), Z) → MULT_IN(X, Y, W)

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

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

MULT_IN(X, s(Y)) → MULT_IN(X, Y)

From the DPs we obtained the following set of size-change graphs:

MULT_IN(X, s(Y)) → MULT_IN(X, Y)
The graph contains the following edges 1 >= 1, 2 > 2