Left Termination of the query pattern rotate_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

rotate(X, Y) :- ','(append(A, B, X), append(B, A, Y)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
append([], Ys, Ys).

Queries:

rotate(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:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x3)
rotate_out(x1, x2)  =  rotate_out

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:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x3)
rotate_out(x1, x2)  =  rotate_out


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

ROTATE_IN(X, Y) → U11(X, Y, append_in(A, B, X))
ROTATE_IN(X, Y) → APPEND_IN(A, B, X)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U11(X, Y, append_out(A, B, X)) → U21(X, Y, append_in(B, A, Y))
U11(X, Y, append_out(A, B, X)) → APPEND_IN(B, A, Y)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x3)
rotate_out(x1, x2)  =  rotate_out
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN
U21(x1, x2, x3)  =  U21(x3)
U11(x1, x2, x3)  =  U11(x3)
ROTATE_IN(x1, x2)  =  ROTATE_IN(x1)

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:

ROTATE_IN(X, Y) → U11(X, Y, append_in(A, B, X))
ROTATE_IN(X, Y) → APPEND_IN(A, B, X)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U11(X, Y, append_out(A, B, X)) → U21(X, Y, append_in(B, A, Y))
U11(X, Y, append_out(A, B, X)) → APPEND_IN(B, A, Y)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x3)
rotate_out(x1, x2)  =  rotate_out
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN
U21(x1, x2, x3)  =  U21(x3)
U11(x1, x2, x3)  =  U11(x3)
ROTATE_IN(x1, x2)  =  ROTATE_IN(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x3)
rotate_out(x1, x2)  =  rotate_out
APPEND_IN(x1, x2, x3)  =  APPEND_IN

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:

APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND_IN(x1, x2, x3)  =  APPEND_IN

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
                      ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

APPEND_IN → APPEND_IN

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:

APPEND_IN → APPEND_IN

The TRS R consists of the following rules:none


s = APPEND_IN evaluates to t =APPEND_IN

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 APPEND_IN to APPEND_IN.




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:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x1, x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x1, x3)
rotate_out(x1, x2)  =  rotate_out(x1)

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:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x1, x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x1, x3)
rotate_out(x1, x2)  =  rotate_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:

ROTATE_IN(X, Y) → U11(X, Y, append_in(A, B, X))
ROTATE_IN(X, Y) → APPEND_IN(A, B, X)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U11(X, Y, append_out(A, B, X)) → U21(X, Y, append_in(B, A, Y))
U11(X, Y, append_out(A, B, X)) → APPEND_IN(B, A, Y)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x1, x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x1, x3)
rotate_out(x1, x2)  =  rotate_out(x1)
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN
U21(x1, x2, x3)  =  U21(x1, x3)
U11(x1, x2, x3)  =  U11(x1, x3)
ROTATE_IN(x1, x2)  =  ROTATE_IN(x1)

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:

ROTATE_IN(X, Y) → U11(X, Y, append_in(A, B, X))
ROTATE_IN(X, Y) → APPEND_IN(A, B, X)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U31(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U11(X, Y, append_out(A, B, X)) → U21(X, Y, append_in(B, A, Y))
U11(X, Y, append_out(A, B, X)) → APPEND_IN(B, A, Y)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x1, x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x1, x3)
rotate_out(x1, x2)  =  rotate_out(x1)
U31(x1, x2, x3, x4, x5)  =  U31(x5)
APPEND_IN(x1, x2, x3)  =  APPEND_IN
U21(x1, x2, x3)  =  U21(x1, x3)
U11(x1, x2, x3)  =  U11(x1, x3)
ROTATE_IN(x1, x2)  =  ROTATE_IN(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in(X, Y) → U1(X, Y, append_in(A, B, X))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
append_in(.(X, Xs), Ys, .(X, Zs)) → U3(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
U3(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U1(X, Y, append_out(A, B, X)) → U2(X, Y, append_in(B, A, Y))
U2(X, Y, append_out(B, A, Y)) → rotate_out(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in(x1, x2)  =  rotate_in(x1)
U1(x1, x2, x3)  =  U1(x1, x3)
append_in(x1, x2, x3)  =  append_in
[]  =  []
append_out(x1, x2, x3)  =  append_out(x1)
.(x1, x2)  =  .(x2)
U3(x1, x2, x3, x4, x5)  =  U3(x5)
U2(x1, x2, x3)  =  U2(x1, x3)
rotate_out(x1, x2)  =  rotate_out(x1)
APPEND_IN(x1, x2, x3)  =  APPEND_IN

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:

APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND_IN(x1, x2, x3)  =  APPEND_IN

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
                      ↳ NonTerminationProof

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

APPEND_IN → APPEND_IN

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:

APPEND_IN → APPEND_IN

The TRS R consists of the following rules:none


s = APPEND_IN evaluates to t =APPEND_IN

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 APPEND_IN to APPEND_IN.