Left Termination proof of /tmp/tmpXpTrag/flat-fb.pl

Left Termination of the query pattern flat_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

flat(a,g).

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

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

FLAT_IN_AG(.([], T), R) → U1_AG(T, R, flat_in_ag(T, R))
FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → U2_AG(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_aa(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.([], T), R) → U1_AA(T, R, flat_in_aa(T, R))
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → U2_AA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1)
U1_ag(x1, x2, x3) = U1_ag(x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
FLAT_IN_AG(x1, x2) = FLAT_IN_AG(x2)
U1_AA(x1, x2, x3) = U1_AA(x3)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x5)
U1_AG(x1, x2, x3) = U1_AG(x3)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U1_GA(x1, x2, x3) = U1_GA(x3)
FLAT_IN_AA(x1, x2) = FLAT_IN_AA

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

FLAT_IN_AG(.([], T), R) → U1_AG(T, R, flat_in_ag(T, R))
FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → U2_AG(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_aa(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.([], T), R) → U1_AA(T, R, flat_in_aa(T, R))
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → U2_AA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1)
U1_ag(x1, x2, x3) = U1_ag(x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
FLAT_IN_AA(x1, x2) = FLAT_IN_AA

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

  ↳ PrologToPiTRSProof

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

FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)

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

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

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

FLAT_IN_AA → FLAT_IN_GA(.)
FLAT_IN_GA(.) → FLAT_IN_GA(.)
FLAT_IN_AA → FLAT_IN_AA
FLAT_IN_GA(.) → FLAT_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

FLAT_IN_AA → FLAT_IN_GA(.)
FLAT_IN_GA(.) → FLAT_IN_GA(.)
FLAT_IN_AA → FLAT_IN_AA
FLAT_IN_GA(.) → FLAT_IN_AA

The TRS R consists of the following rules:none

s = FLAT_IN_GA(.) evaluates to t =FLAT_IN_GA(.)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLAT_IN_GA(.) to FLAT_IN_GA(.).

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1)
U1_ag(x1, x2, x3) = U1_ag(x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
FLAT_IN_AG(x1, x2) = FLAT_IN_AG(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

  ↳ PrologToPiTRSProof

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

FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)

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

                        ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

FLAT_IN_AG(R) → FLAT_IN_AG(R)

The TRS R consists of the following rules:none

s = FLAT_IN_AG(R) evaluates to t =FLAT_IN_AG(R)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLAT_IN_AG(R) to FLAT_IN_AG(R).

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

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

FLAT_IN_AG(.([], T), R) → U1_AG(T, R, flat_in_ag(T, R))
FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → U2_AG(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_aa(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.([], T), R) → U1_AA(T, R, flat_in_aa(T, R))
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → U2_AA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1, x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x1, x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x5)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
FLAT_IN_AG(x1, x2) = FLAT_IN_AG(x2)
U1_AA(x1, x2, x3) = U1_AA(x3)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x5)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5)
U1_GA(x1, x2, x3) = U1_GA(x3)
FLAT_IN_AA(x1, x2) = FLAT_IN_AA

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

FLAT_IN_AG(.([], T), R) → U1_AG(T, R, flat_in_ag(T, R))
FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → U2_AG(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AG(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.([], T), R) → U1_GA(T, R, flat_in_aa(T, R))
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.([], T), R) → U1_AA(T, R, flat_in_aa(T, R))
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → U2_AA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → U2_GA(H, T, TT, R, flat_in_ga(.(T, TT), R))
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1, x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x1, x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
FLAT_IN_GA(x1, x2) = FLAT_IN_GA(x1)
FLAT_IN_AA(x1, x2) = FLAT_IN_AA

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

FLAT_IN_AA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_GA(.(.(H, T), TT), .(H, R)) → FLAT_IN_GA(.(T, TT), R)
FLAT_IN_AA(.([], T), R) → FLAT_IN_AA(T, R)
FLAT_IN_GA(.([], T), R) → FLAT_IN_AA(T, R)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

FLAT_IN_AA → FLAT_IN_GA(.)
FLAT_IN_GA(.) → FLAT_IN_GA(.)
FLAT_IN_AA → FLAT_IN_AA
FLAT_IN_GA(.) → FLAT_IN_AA

The TRS R consists of the following rules:none

s = FLAT_IN_GA(.) evaluates to t =FLAT_IN_GA(.)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLAT_IN_GA(.) to FLAT_IN_GA(.).

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)

The TRS R consists of the following rules:

flat_in_ag([], []) → flat_out_ag([], [])
flat_in_ag(.([], T), R) → U1_ag(T, R, flat_in_ag(T, R))
flat_in_ag(.(.(H, T), TT), .(H, R)) → U2_ag(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga([], []) → flat_out_ga([], [])
flat_in_ga(.([], T), R) → U1_ga(T, R, flat_in_aa(T, R))
flat_in_aa([], []) → flat_out_aa([], [])
flat_in_aa(.([], T), R) → U1_aa(T, R, flat_in_aa(T, R))
flat_in_aa(.(.(H, T), TT), .(H, R)) → U2_aa(H, T, TT, R, flat_in_ga(.(T, TT), R))
flat_in_ga(.(.(H, T), TT), .(H, R)) → U2_ga(H, T, TT, R, flat_in_ga(.(T, TT), R))
U2_ga(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ga(.(.(H, T), TT), .(H, R))
U2_aa(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_aa(.(.(H, T), TT), .(H, R))
U1_aa(T, R, flat_out_aa(T, R)) → flat_out_aa(.([], T), R)
U1_ga(T, R, flat_out_aa(T, R)) → flat_out_ga(.([], T), R)
U2_ag(H, T, TT, R, flat_out_ga(.(T, TT), R)) → flat_out_ag(.(.(H, T), TT), .(H, R))
U1_ag(T, R, flat_out_ag(T, R)) → flat_out_ag(.([], T), R)

The argument filtering Pi contains the following mapping:
flat_in_ag(x1, x2) = flat_in_ag(x2)
[] = []
flat_out_ag(x1, x2) = flat_out_ag(x1, x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
.(x1, x2) = .
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x5)
flat_in_ga(x1, x2) = flat_in_ga(x1)
flat_out_ga(x1, x2) = flat_out_ga(x1, x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
flat_in_aa(x1, x2) = flat_in_aa
flat_out_aa(x1, x2) = flat_out_aa(x1, x2)
U1_aa(x1, x2, x3) = U1_aa(x3)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x5)
U2_ga(x1, x2, x3, x4, x5) = U2_ga(x5)
FLAT_IN_AG(x1, x2) = FLAT_IN_AG(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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

FLAT_IN_AG(.([], T), R) → FLAT_IN_AG(T, R)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

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

FLAT_IN_AG(R) → FLAT_IN_AG(R)

The TRS R consists of the following rules:none

s = FLAT_IN_AG(R) evaluates to t =FLAT_IN_AG(R)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FLAT_IN_AG(R) to FLAT_IN_AG(R).