Left Termination proof of /tmp/tmpCURZ0a/p

Left Termination of the query pattern p_in_1(g) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

p(cons(X, nil)).
p(cons(s(s(X)), cons(Y, Xs))) :- ','(p(cons(X, cons(Y, Xs))), ','(mult(X, Y, Z), p(cons(Z, Xs)))).
p(cons(0, Xs)) :- p(Xs).
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
mult(X, 0, 0).
mult(X, s(Y), Z) :- ','(mult(X, Y, W), sum(W, X, Z)).

Queries:

p(g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (b)
mult_in: (b,b,f)
sum_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

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:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))
P_IN_G(cons(0, Xs)) → U4_G(Xs, p_in_g(Xs))
P_IN_G(cons(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → MULT_IN_GGA(X, Y, Z)
MULT_IN_GGA(X, s(Y), Z) → U6_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U7_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U5_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_G(X, Y, Xs, p_in_g(cons(Z, Xs)))
U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) → P_IN_G(cons(Z, Xs))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1) = p_in_g(x1)
cons(x1, x2) = cons(x1, x2)
nil = nil
p_out_g(x1) = p_out_g
s(x1) = s(x1)
U1_g(x1, x2, x3, x4) = U1_g(x1, x2, x3, x4)
0 = 0
U4_g(x1, x2) = U4_g(x2)
U2_g(x1, x2, x3, x4) = U2_g(x3, x4)
mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2)
mult_out_gga(x1, x2, x3) = mult_out_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2)
sum_out_gga(x1, x2, x3) = sum_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
U3_g(x1, x2, x3, x4) = U3_g(x4)
P_IN_G(x1) = P_IN_G(x1)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x4)
U4_G(x1, x2) = U4_G(x2)
U1_G(x1, x2, x3, x4) = U1_G(x1, x2, x3, x4)
SUM_IN_GGA(x1, x2, x3) = SUM_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x4)
MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(x1, x2)
U3_G(x1, x2, x3, x4) = U3_G(x4)
U2_G(x1, x2, x3, x4) = U2_G(x3, 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:

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))
P_IN_G(cons(0, Xs)) → U4_G(Xs, p_in_g(Xs))
P_IN_G(cons(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → MULT_IN_GGA(X, Y, Z)
MULT_IN_GGA(X, s(Y), Z) → U6_GGA(X, Y, Z, mult_in_gga(X, Y, W))
MULT_IN_GGA(X, s(Y), Z) → MULT_IN_GGA(X, Y, W)
U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → U7_GGA(X, Y, Z, sum_in_gga(W, X, Z))
U6_GGA(X, Y, Z, mult_out_gga(X, Y, W)) → SUM_IN_GGA(W, X, Z)
SUM_IN_GGA(X, s(Y), s(Z)) → U5_GGA(X, Y, Z, sum_in_gga(X, Y, Z))
SUM_IN_GGA(X, s(Y), s(Z)) → SUM_IN_GGA(X, Y, Z)
U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_G(X, Y, Xs, p_in_g(cons(Z, Xs)))
U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) → P_IN_G(cons(Z, Xs))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

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

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1) = p_in_g(x1)
cons(x1, x2) = cons(x1, x2)
nil = nil
p_out_g(x1) = p_out_g
s(x1) = s(x1)
U1_g(x1, x2, x3, x4) = U1_g(x1, x2, x3, x4)
0 = 0
U4_g(x1, x2) = U4_g(x2)
U2_g(x1, x2, x3, x4) = U2_g(x3, x4)
mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2)
mult_out_gga(x1, x2, x3) = mult_out_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2)
sum_out_gga(x1, x2, x3) = sum_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
U3_g(x1, x2, x3, x4) = U3_g(x4)
SUM_IN_GGA(x1, x2, x3) = SUM_IN_GGA(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

              ↳ PiDP

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

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

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

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1) = p_in_g(x1)
cons(x1, x2) = cons(x1, x2)
nil = nil
p_out_g(x1) = p_out_g
s(x1) = s(x1)
U1_g(x1, x2, x3, x4) = U1_g(x1, x2, x3, x4)
0 = 0
U4_g(x1, x2) = U4_g(x2)
U2_g(x1, x2, x3, x4) = U2_g(x3, x4)
mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2)
mult_out_gga(x1, x2, x3) = mult_out_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2)
sum_out_gga(x1, x2, x3) = sum_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
U3_g(x1, x2, x3, x4) = U3_g(x4)
MULT_IN_GGA(x1, x2, x3) = MULT_IN_GGA(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

              ↳ PiDP

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

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

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

              ↳ PiDP

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

MULT_IN_GGA(X, s(Y)) → MULT_IN_GGA(X, Y)

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

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

U2_G(X, Y, Xs, mult_out_gga(X, Y, Z)) → P_IN_G(cons(Z, Xs))
P_IN_G(cons(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_G(X, Y, Xs, mult_in_gga(X, Y, Z))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g(cons(X, nil))
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(Xs, p_in_g(Xs))
U4_g(Xs, p_out_g(Xs)) → p_out_g(cons(0, Xs))
U1_g(X, Y, Xs, p_out_g(cons(X, cons(Y, Xs)))) → U2_g(X, Y, Xs, mult_in_gga(X, Y, Z))
mult_in_gga(X, 0, 0) → mult_out_gga(X, 0, 0)
mult_in_gga(X, s(Y), Z) → U6_gga(X, Y, Z, mult_in_gga(X, Y, W))
U6_gga(X, Y, Z, mult_out_gga(X, Y, W)) → U7_gga(X, Y, Z, sum_in_gga(W, X, Z))
sum_in_gga(X, 0, X) → sum_out_gga(X, 0, X)
sum_in_gga(X, s(Y), s(Z)) → U5_gga(X, Y, Z, sum_in_gga(X, Y, Z))
U5_gga(X, Y, Z, sum_out_gga(X, Y, Z)) → sum_out_gga(X, s(Y), s(Z))
U7_gga(X, Y, Z, sum_out_gga(W, X, Z)) → mult_out_gga(X, s(Y), Z)
U2_g(X, Y, Xs, mult_out_gga(X, Y, Z)) → U3_g(X, Y, Xs, p_in_g(cons(Z, Xs)))
U3_g(X, Y, Xs, p_out_g(cons(Z, Xs))) → p_out_g(cons(s(s(X)), cons(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1) = p_in_g(x1)
cons(x1, x2) = cons(x1, x2)
nil = nil
p_out_g(x1) = p_out_g
s(x1) = s(x1)
U1_g(x1, x2, x3, x4) = U1_g(x1, x2, x3, x4)
0 = 0
U4_g(x1, x2) = U4_g(x2)
U2_g(x1, x2, x3, x4) = U2_g(x3, x4)
mult_in_gga(x1, x2, x3) = mult_in_gga(x1, x2)
mult_out_gga(x1, x2, x3) = mult_out_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x4)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
sum_in_gga(x1, x2, x3) = sum_in_gga(x1, x2)
sum_out_gga(x1, x2, x3) = sum_out_gga(x3)
U5_gga(x1, x2, x3, x4) = U5_gga(x4)
U3_g(x1, x2, x3, x4) = U3_g(x4)
P_IN_G(x1) = P_IN_G(x1)
U1_G(x1, x2, x3, x4) = U1_G(x1, x2, x3, x4)
U2_G(x1, x2, x3, x4) = U2_G(x3, x4)

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

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

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

U1_G(X, Y, Xs, p_out_g) → U2_G(Xs, mult_in_gga(X, Y))
P_IN_G(cons(0, Xs)) → P_IN_G(Xs)
U2_G(Xs, mult_out_gga(Z)) → P_IN_G(cons(Z, Xs))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(p_in_g(Xs))
U4_g(p_out_g) → p_out_g
U1_g(X, Y, Xs, p_out_g) → U2_g(Xs, mult_in_gga(X, Y))
mult_in_gga(X, 0) → mult_out_gga(0)
mult_in_gga(X, s(Y)) → U6_gga(X, mult_in_gga(X, Y))
U6_gga(X, mult_out_gga(W)) → U7_gga(sum_in_gga(W, X))
sum_in_gga(X, 0) → sum_out_gga(X)
sum_in_gga(X, s(Y)) → U5_gga(sum_in_gga(X, Y))
U5_gga(sum_out_gga(Z)) → sum_out_gga(s(Z))
U7_gga(sum_out_gga(Z)) → mult_out_gga(Z)
U2_g(Xs, mult_out_gga(Z)) → U3_g(p_in_g(cons(Z, Xs)))
U3_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U4_g(x0)
U1_g(x0, x1, x2, x3)
mult_in_gga(x0, x1)
U6_gga(x0, x1)
sum_in_gga(x0, x1)
U5_gga(x0)
U7_gga(x0)
U2_g(x0, x1)
U3_g(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

P_IN_G(cons(0, Xs)) → P_IN_G(Xs)
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))

The remaining pairs can at least be oriented weakly.

U1_G(X, Y, Xs, p_out_g) → U2_G(Xs, mult_in_gga(X, Y))
U2_G(Xs, mult_out_gga(Z)) → P_IN_G(cons(Z, Xs))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(P_IN_G(x₁)) = x₁
POL(U1_G(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U1_g(x₁, x₂, x₃, x₄)) = 0
POL(U2_G(x₁, x₂)) = 1 + x₁
POL(U2_g(x₁, x₂)) = 0
POL(U3_g(x₁)) = 0
POL(U4_g(x₁)) = 0
POL(U5_gga(x₁)) = 0
POL(U6_gga(x₁, x₂)) = 0
POL(U7_gga(x₁)) = 0
POL(cons(x₁, x₂)) = 1 + x₂
POL(mult_in_gga(x₁, x₂)) = 0
POL(mult_out_gga(x₁)) = 0
POL(nil) = 0
POL(p_in_g(x₁)) = 0
POL(p_out_g) = 0
POL(s(x₁)) = 0
POL(sum_in_gga(x₁, x₂)) = 0
POL(sum_out_gga(x₁)) = 0

The following usable rules [17] were oriented: none

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

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

U1_G(X, Y, Xs, p_out_g) → U2_G(Xs, mult_in_gga(X, Y))
U2_G(Xs, mult_out_gga(Z)) → P_IN_G(cons(Z, Xs))
P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(p_in_g(Xs))
U4_g(p_out_g) → p_out_g
U1_g(X, Y, Xs, p_out_g) → U2_g(Xs, mult_in_gga(X, Y))
mult_in_gga(X, 0) → mult_out_gga(0)
mult_in_gga(X, s(Y)) → U6_gga(X, mult_in_gga(X, Y))
U6_gga(X, mult_out_gga(W)) → U7_gga(sum_in_gga(W, X))
sum_in_gga(X, 0) → sum_out_gga(X)
sum_in_gga(X, s(Y)) → U5_gga(sum_in_gga(X, Y))
U5_gga(sum_out_gga(Z)) → sum_out_gga(s(Z))
U7_gga(sum_out_gga(Z)) → mult_out_gga(Z)
U2_g(Xs, mult_out_gga(Z)) → U3_g(p_in_g(cons(Z, Xs)))
U3_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U4_g(x0)
U1_g(x0, x1, x2, x3)
mult_in_gga(x0, x1)
U6_gga(x0, x1)
sum_in_gga(x0, x1)
U5_gga(x0)
U7_gga(x0)
U2_g(x0, x1)
U3_g(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ UsableRulesProof

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

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(cons(X, nil)) → p_out_g
p_in_g(cons(s(s(X)), cons(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(cons(X, cons(Y, Xs))))
p_in_g(cons(0, Xs)) → U4_g(p_in_g(Xs))
U4_g(p_out_g) → p_out_g
U1_g(X, Y, Xs, p_out_g) → U2_g(Xs, mult_in_gga(X, Y))
mult_in_gga(X, 0) → mult_out_gga(0)
mult_in_gga(X, s(Y)) → U6_gga(X, mult_in_gga(X, Y))
U6_gga(X, mult_out_gga(W)) → U7_gga(sum_in_gga(W, X))
sum_in_gga(X, 0) → sum_out_gga(X)
sum_in_gga(X, s(Y)) → U5_gga(sum_in_gga(X, Y))
U5_gga(sum_out_gga(Z)) → sum_out_gga(s(Z))
U7_gga(sum_out_gga(Z)) → mult_out_gga(Z)
U2_g(Xs, mult_out_gga(Z)) → U3_g(p_in_g(cons(Z, Xs)))
U3_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U4_g(x0)
U1_g(x0, x1, x2, x3)
mult_in_gga(x0, x1)
U6_gga(x0, x1)
sum_in_gga(x0, x1)
U5_gga(x0)
U7_gga(x0)
U2_g(x0, x1)
U3_g(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

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

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

R is empty.
The set Q consists of the following terms:

p_in_g(x0)
U4_g(x0)
U1_g(x0, x1, x2, x3)
mult_in_gga(x0, x1)
U6_gga(x0, x1)
sum_in_gga(x0, x1)
U5_gga(x0)
U7_gga(x0)
U2_g(x0, x1)
U3_g(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

p_in_g(x0)
U4_g(x0)
U1_g(x0, x1, x2, x3)
mult_in_gga(x0, x1)
U6_gga(x0, x1)
sum_in_gga(x0, x1)
U5_gga(x0)
U7_gga(x0)
U2_g(x0, x1)
U3_g(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ UsableRulesReductionPairsProof

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

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P_IN_G(cons(s(s(X)), cons(Y, Xs))) → P_IN_G(cons(X, cons(Y, Xs)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(P_IN_G(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(s(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ PiDPToQDPProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ UsableRulesReductionPairsProof

                                      ↳ QDP

                                        ↳ PisEmptyProof

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.