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].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))

The argument filtering Pi contains the following mapping:
flat_in(x1, x2)  =  flat_in(x2)
.(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(x1)

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(.(.(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(x2)
.(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(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:

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(x2)
.(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(x1)
U21(x1, x2, x3, x4, x5)  =  U21(x1, x5)
U11(x1, x2, x3)  =  U11(x3)
FLAT_IN(x1, x2)  =  FLAT_IN(x2)

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(.(.(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(x2)
.(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(x1)
U21(x1, x2, x3, x4, x5)  =  U21(x1, x5)
U11(x1, x2, x3)  =  U11(x3)
FLAT_IN(x1, x2)  =  FLAT_IN(x2)

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
  ↳ PrologToPiTRSProof

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(x2)
.(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(x1)
FLAT_IN(x1, x2)  =  FLAT_IN(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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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(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
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof
  ↳ PrologToPiTRSProof

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

FLAT_IN(R) → FLAT_IN(R)
FLAT_IN(.(H, R)) → FLAT_IN(R)

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(.(H, R)) → FLAT_IN(R)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof
                        ↳ QDP
                          ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

FLAT_IN(R) → FLAT_IN(R)

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(R) → FLAT_IN(R)

The TRS R consists of the following rules:none


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

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



Rewriting sequence

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




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))

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

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(.(.(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(x2)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x1, x4, x5)
[]  =  []
U1(x1, x2, x3)  =  U1(x2, x3)
flat_out(x1, x2)  =  flat_out(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:

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(x2)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x1, x4, x5)
[]  =  []
U1(x1, x2, x3)  =  U1(x2, x3)
flat_out(x1, x2)  =  flat_out(x1, x2)
U21(x1, x2, x3, x4, x5)  =  U21(x1, x4, x5)
U11(x1, x2, x3)  =  U11(x2, x3)
FLAT_IN(x1, x2)  =  FLAT_IN(x2)

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(.(.(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(x2)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x1, x4, x5)
[]  =  []
U1(x1, x2, x3)  =  U1(x2, x3)
flat_out(x1, x2)  =  flat_out(x1, x2)
U21(x1, x2, x3, x4, x5)  =  U21(x1, x4, x5)
U11(x1, x2, x3)  =  U11(x2, x3)
FLAT_IN(x1, x2)  =  FLAT_IN(x2)

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
  ↳ 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(x2)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x1, x4, x5)
[]  =  []
U1(x1, x2, x3)  =  U1(x2, x3)
flat_out(x1, x2)  =  flat_out(x1, x2)
FLAT_IN(x1, x2)  =  FLAT_IN(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
            ↳ 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(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof

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

FLAT_IN(R) → FLAT_IN(R)
FLAT_IN(.(H, R)) → FLAT_IN(R)

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(.(H, R)) → FLAT_IN(R)
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

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



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ UsableRulesReductionPairsProof
                        ↳ QDP
                          ↳ NonTerminationProof

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

FLAT_IN(R) → FLAT_IN(R)

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(R) → FLAT_IN(R)

The TRS R consists of the following rules:none


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

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



Rewriting sequence

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