Left Termination proof of /tmp/tmp0px5Di/flatten.pl

Left Termination of the query pattern flatten_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).

Queries:

flatten(g,a).

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

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))

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:

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))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
atom(x1) = atom(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
cons(x1, x2) = cons(x1, 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)

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

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)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
atom(x1) = atom(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
cons(x1, x2) = cons(x1, 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)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, 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:

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)

The TRS R consists of the following rules:

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))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
atom(x1) = atom(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
cons(x1, x2) = cons(x1, 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)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, 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 1 SCC with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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:

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))

The argument filtering Pi contains the following mapping:
flatten_in_ga(x1, x2) = flatten_in_ga(x1)
atom(x1) = atom(x1)
flatten_out_ga(x1, x2) = flatten_out_ga(x2)
cons(x1, x2) = cons(x1, 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)
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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ UsableRulesReductionPairsProof

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ 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.