Left Termination proof of /tmp/tmp2GkN_Q/flatten

Left Termination of the query pattern count_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

flatten(atom(X), .(X, [])).
flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).
count(atom(X), s(0)).
count(cons(atom(X), Y), s(Z)) :- count(Y, Z).
count(cons(cons(U, V), W), Z) :- ','(flatten(cons(cons(U, V), W), X), count(X, Z)).

Queries:

count(g,a).

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

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(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:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2) = count_in_ga(x1)
atom(x1) = atom(x1)
count_out_ga(x1, x2) = count_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2) = count_in_ga(x1)
atom(x1) = atom(x1)
count_out_ga(x1, x2) = count_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1)

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:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → U3_GA(X, Y, Z, count_in_ga(Y, Z))
COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
COUNT_IN_GA(cons(cons(U, V), W), Z) → FLATTEN_IN_GA(cons(cons(U, V), W), X)
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → U1_GA(X, U, Y, flatten_in_ga(U, Y))
FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → U2_GA(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_GA(U, V, W, Z, count_in_ga(X, Z))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2) = count_in_ga(x1)
atom(x1) = atom(x1)
count_out_ga(x1, x2) = count_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2) = count_in_ga(x1)
atom(x1) = atom(x1)
count_out_ga(x1, x2) = count_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1)

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:

FLATTEN_IN_GA(cons(atom(X), U), .(X, Y)) → FLATTEN_IN_GA(U, Y)
FLATTEN_IN_GA(cons(cons(U, V), W), X) → FLATTEN_IN_GA(cons(U, cons(V, W)), X)

R is empty.
The argument filtering Pi contains the following mapping:
atom(x1) = atom(x1)
cons(x1, x2) = cons(x1, x2)
.(x1, x2) = .(x1, x2)
FLATTEN_IN_GA(x1, x2) = FLATTEN_IN_GA(x1)

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

                        ↳ UsableRulesReductionPairsProof

              ↳ PiDP

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

FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)
FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))

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:

FLATTEN_IN_GA(cons(atom(X), U)) → FLATTEN_IN_GA(U)
FLATTEN_IN_GA(cons(cons(U, V), W)) → FLATTEN_IN_GA(cons(U, cons(V, W)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(FLATTEN_IN_GA(x₁)) = 2·x₁
POL(atom(x₁)) = x₁
POL(cons(x₁, x₂)) = 1 + 2·x₁ + x₂

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ PiDP

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.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

count_in_ga(atom(X), s(0)) → count_out_ga(atom(X), s(0))
count_in_ga(cons(atom(X), Y), s(Z)) → U3_ga(X, Y, Z, count_in_ga(Y, Z))
count_in_ga(cons(cons(U, V), W), Z) → U4_ga(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
U4_ga(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → U5_ga(U, V, W, Z, count_in_ga(X, Z))
U5_ga(U, V, W, Z, count_out_ga(X, Z)) → count_out_ga(cons(cons(U, V), W), Z)
U3_ga(X, Y, Z, count_out_ga(Y, Z)) → count_out_ga(cons(atom(X), Y), s(Z))

The argument filtering Pi contains the following mapping:
count_in_ga(x1, x2) = count_in_ga(x1)
atom(x1) = atom(x1)
count_out_ga(x1, x2) = count_out_ga(x2)
cons(x1, x2) = cons(x1, x2)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1)

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:

COUNT_IN_GA(cons(atom(X), Y), s(Z)) → COUNT_IN_GA(Y, Z)
COUNT_IN_GA(cons(cons(U, V), W), Z) → U4_GA(U, V, W, Z, flatten_in_ga(cons(cons(U, V), W), X))
U4_GA(U, V, W, Z, flatten_out_ga(cons(cons(U, V), W), X)) → COUNT_IN_GA(X, Z)

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W), X) → U2_ga(U, V, W, X, flatten_in_ga(cons(U, cons(V, W)), X))
U2_ga(U, V, W, X, flatten_out_ga(cons(U, cons(V, W)), X)) → flatten_out_ga(cons(cons(U, V), W), X)
flatten_in_ga(cons(atom(X), U), .(X, Y)) → U1_ga(X, U, Y, flatten_in_ga(U, Y))
U1_ga(X, U, Y, flatten_out_ga(U, Y)) → flatten_out_ga(cons(atom(X), U), .(X, Y))
flatten_in_ga(atom(X), .(X, [])) → flatten_out_ga(atom(X), .(X, []))

The argument filtering Pi contains the following mapping:
atom(x1) = atom(x1)
cons(x1, x2) = cons(x1, x2)
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
COUNT_IN_GA(x1, x2) = COUNT_IN_GA(x1)

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

                        ↳ UsableRulesReductionPairsProof

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

COUNT_IN_GA(cons(atom(X), Y)) → COUNT_IN_GA(Y)
U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)
flatten_in_ga(cons(atom(X), U)) → U1_ga(X, flatten_in_ga(U))
U1_ga(X, flatten_out_ga(Y)) → flatten_out_ga(.(X, Y))
flatten_in_ga(atom(X)) → flatten_out_ga(.(X, []))

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)
U1_ga(x0, x1)

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:

COUNT_IN_GA(cons(atom(X), Y)) → COUNT_IN_GA(Y)

The following rules are removed from R:

flatten_in_ga(cons(atom(X), U)) → U1_ga(X, flatten_in_ga(U))
flatten_in_ga(atom(X)) → flatten_out_ga(.(X, []))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(COUNT_IN_GA(x₁)) = 2 + 2·x₁
POL(U1_ga(x₁, x₂)) = 2·x₁ + x₂
POL(U2_ga(x₁)) = x₁
POL(U4_GA(x₁)) = 2 + x₁
POL([]) = 0
POL(atom(x₁)) = 1 + 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(flatten_in_ga(x₁)) = x₁
POL(flatten_out_ga(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ UsableRulesProof

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

U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)
U1_ga(X, flatten_out_ga(Y)) → flatten_out_ga(.(X, Y))

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)
U1_ga(x0, x1)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

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

U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)
U1_ga(x0, x1)

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.

U1_ga(x0, x1)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ RuleRemovalProof

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

U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))

The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

U4_GA(flatten_out_ga(X)) → COUNT_IN_GA(X)
COUNT_IN_GA(cons(cons(U, V), W)) → U4_GA(flatten_in_ga(cons(cons(U, V), W)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(COUNT_IN_GA(x₁)) = 1 + 2·x₁
POL(U2_ga(x₁)) = x₁
POL(U4_GA(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(flatten_in_ga(x₁)) = x₁
POL(flatten_out_ga(x₁)) = 2 + 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ UsableRulesProof

                              ↳ QDP

                                ↳ QReductionProof

                                  ↳ QDP

                                    ↳ RuleRemovalProof

                                      ↳ QDP

                                        ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

flatten_in_ga(cons(cons(U, V), W)) → U2_ga(flatten_in_ga(cons(U, cons(V, W))))
U2_ga(flatten_out_ga(X)) → flatten_out_ga(X)

The set Q consists of the following terms:

flatten_in_ga(x0)
U2_ga(x0)

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