Left Termination proof of /tmp/tmpKuWJci/flat-bf.pl

Left Termination of the query pattern flat_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

flat([], []).
flat(.([], T), R) :- flat(T, R).
flat(.(.(H, T), TT), .(H, R)) :- flat(.(T, TT), R).

Queries:

flat(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:

flat_in(.(.(H, T), TT), .(H, R)) → U2(H, T, TT, R, flat_in(.(T, TT), R))
flat_in(.([], T), R) → U1(T, R, flat_in(T, R))
flat_in([], []) → flat_out([], [])
U1(T, R, flat_out(T, R)) → flat_out(.([], T), R)
U2(H, T, TT, R, flat_out(.(T, TT), R)) → flat_out(.(.(H, T), TT), .(H, R))

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:

flat_in(.(.(H, T), TT), .(H, R)) → U2(H, T, TT, R, flat_in(.(T, TT), R))
flat_in(.([], T), R) → U1(T, R, flat_in(T, R))
flat_in([], []) → flat_out([], [])
U1(T, R, flat_out(T, R)) → flat_out(.([], T), R)
U2(H, T, TT, R, flat_out(.(T, TT), R)) → flat_out(.(.(H, T), TT), .(H, R))

The argument filtering Pi contains the following mapping:
flat_in(x1, x2) = flat_in(x1)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5) = U2(x1, x5)
[] = []
U1(x1, x2, x3) = U1(x3)
flat_out(x1, x2) = flat_out(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:

FLAT_IN(.(.(H, T), TT), .(H, R)) → U2¹(H, T, TT, R, flat_in(.(T, TT), R))
FLAT_IN(.(.(H, T), TT), .(H, R)) → FLAT_IN(.(T, TT), R)
FLAT_IN(.([], T), R) → U1¹(T, R, flat_in(T, R))
FLAT_IN(.([], T), R) → FLAT_IN(T, R)

The TRS R consists of the following rules:

flat_in(.(.(H, T), TT), .(H, R)) → U2(H, T, TT, R, flat_in(.(T, TT), R))
flat_in(.([], T), R) → U1(T, R, flat_in(T, R))
flat_in([], []) → flat_out([], [])
U1(T, R, flat_out(T, R)) → flat_out(.([], T), R)
U2(H, T, TT, R, flat_out(.(T, TT), R)) → flat_out(.(.(H, T), TT), .(H, R))

The argument filtering Pi contains the following mapping:
flat_in(x1, x2) = flat_in(x1)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5) = U2(x1, x5)
[] = []
U1(x1, x2, x3) = U1(x3)
flat_out(x1, x2) = flat_out(x2)
U2¹(x1, x2, x3, x4, x5) = U2¹(x1, x5)
U1¹(x1, x2, x3) = U1¹(x3)
FLAT_IN(x1, x2) = FLAT_IN(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:

FLAT_IN(.(.(H, T), TT), .(H, R)) → U2¹(H, T, TT, R, flat_in(.(T, TT), R))
FLAT_IN(.(.(H, T), TT), .(H, R)) → FLAT_IN(.(T, TT), R)
FLAT_IN(.([], T), R) → U1¹(T, R, flat_in(T, R))
FLAT_IN(.([], T), R) → FLAT_IN(T, R)

The TRS R consists of the following rules:

flat_in(.(.(H, T), TT), .(H, R)) → U2(H, T, TT, R, flat_in(.(T, TT), R))
flat_in(.([], T), R) → U1(T, R, flat_in(T, R))
flat_in([], []) → flat_out([], [])
U1(T, R, flat_out(T, R)) → flat_out(.([], T), R)
U2(H, T, TT, R, flat_out(.(T, TT), R)) → flat_out(.(.(H, T), TT), .(H, R))

The argument filtering Pi contains the following mapping:
flat_in(x1, x2) = flat_in(x1)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5) = U2(x1, x5)
[] = []
U1(x1, x2, x3) = U1(x3)
flat_out(x1, x2) = flat_out(x2)
U2¹(x1, x2, x3, x4, x5) = U2¹(x1, x5)
U1¹(x1, x2, x3) = U1¹(x3)
FLAT_IN(x1, x2) = FLAT_IN(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:

FLAT_IN(.(.(H, T), TT), .(H, R)) → FLAT_IN(.(T, TT), R)
FLAT_IN(.([], T), R) → FLAT_IN(T, R)

The TRS R consists of the following rules:

flat_in(.(.(H, T), TT), .(H, R)) → U2(H, T, TT, R, flat_in(.(T, TT), R))
flat_in(.([], T), R) → U1(T, R, flat_in(T, R))
flat_in([], []) → flat_out([], [])
U1(T, R, flat_out(T, R)) → flat_out(.([], T), R)
U2(H, T, TT, R, flat_out(.(T, TT), R)) → flat_out(.(.(H, T), TT), .(H, R))

The argument filtering Pi contains the following mapping:
flat_in(x1, x2) = flat_in(x1)
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5) = U2(x1, x5)
[] = []
U1(x1, x2, x3) = U1(x3)
flat_out(x1, x2) = flat_out(x2)
FLAT_IN(x1, x2) = FLAT_IN(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:

FLAT_IN(.(.(H, T), TT), .(H, R)) → FLAT_IN(.(T, TT), R)
FLAT_IN(.([], T), R) → FLAT_IN(T, R)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
FLAT_IN(x1, x2) = FLAT_IN(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:

FLAT_IN(.(.(H, T), TT)) → FLAT_IN(.(T, TT))
FLAT_IN(.([], T)) → FLAT_IN(T)

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:

FLAT_IN(.([], T)) → FLAT_IN(T)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(FLAT_IN(x₁)) = 2·x₁
POL([]) = 0

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ UsableRulesReductionPairsProof

                        ↳ QDP

                          ↳ RuleRemovalProof

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

FLAT_IN(.(.(H, T), TT)) → FLAT_IN(.(T, TT))

R is empty.
Q is empty.
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:

FLAT_IN(.(.(H, T), TT)) → FLAT_IN(.(T, TT))

Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(FLAT_IN(x₁)) = 2·x₁

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ UsableRulesReductionPairsProof

                        ↳ QDP

                          ↳ RuleRemovalProof

                            ↳ 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.