Left Termination of the query pattern sublist_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

append([], Ys, Ys).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
sublist(X, Y) :- ','(append(P, X1, Y), append(X2, X, P)).

Queries:

sublist(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:

sublist_in(X, Y) → U2(X, Y, append_in(P, X1, Y))
append_in(.(X, Xs), Ys, .(X, Zs)) → U1(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
U1(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Y, append_out(P, X1, Y)) → U3(X, Y, append_in(X2, X, P))
U3(X, Y, append_out(X2, X, P)) → sublist_out(X, Y)

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

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:

sublist_in(X, Y) → U2(X, Y, append_in(P, X1, Y))
append_in(.(X, Xs), Ys, .(X, Zs)) → U1(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
U1(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Y, append_out(P, X1, Y)) → U3(X, Y, append_in(X2, X, P))
U3(X, Y, append_out(X2, X, P)) → sublist_out(X, Y)

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

SUBLIST_IN(X, Y) → U21(X, Y, append_in(P, X1, Y))
SUBLIST_IN(X, Y) → APPEND_IN(P, X1, Y)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U11(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U21(X, Y, append_out(P, X1, Y)) → U31(X, Y, append_in(X2, X, P))
U21(X, Y, append_out(P, X1, Y)) → APPEND_IN(X2, X, P)

The TRS R consists of the following rules:

sublist_in(X, Y) → U2(X, Y, append_in(P, X1, Y))
append_in(.(X, Xs), Ys, .(X, Zs)) → U1(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
U1(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Y, append_out(P, X1, Y)) → U3(X, Y, append_in(X2, X, P))
U3(X, Y, append_out(X2, X, P)) → sublist_out(X, Y)

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

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:

SUBLIST_IN(X, Y) → U21(X, Y, append_in(P, X1, Y))
SUBLIST_IN(X, Y) → APPEND_IN(P, X1, Y)
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → U11(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
APPEND_IN(.(X, Xs), Ys, .(X, Zs)) → APPEND_IN(Xs, Ys, Zs)
U21(X, Y, append_out(P, X1, Y)) → U31(X, Y, append_in(X2, X, P))
U21(X, Y, append_out(P, X1, Y)) → APPEND_IN(X2, X, P)

The TRS R consists of the following rules:

sublist_in(X, Y) → U2(X, Y, append_in(P, X1, Y))
append_in(.(X, Xs), Ys, .(X, Zs)) → U1(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
U1(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Y, append_out(P, X1, Y)) → U3(X, Y, append_in(X2, X, P))
U3(X, Y, append_out(X2, X, P)) → sublist_out(X, Y)

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

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

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:

sublist_in(X, Y) → U2(X, Y, append_in(P, X1, Y))
append_in(.(X, Xs), Ys, .(X, Zs)) → U1(X, Xs, Ys, Zs, append_in(Xs, Ys, Zs))
append_in([], Ys, Ys) → append_out([], Ys, Ys)
U1(X, Xs, Ys, Zs, append_out(Xs, Ys, Zs)) → append_out(.(X, Xs), Ys, .(X, Zs))
U2(X, Y, append_out(P, X1, Y)) → U3(X, Y, append_in(X2, X, P))
U3(X, Y, append_out(X2, X, P)) → sublist_out(X, Y)

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

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:

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)  =  .(x1, x2)
APPEND_IN(x1, x2, x3)  =  APPEND_IN(x3)

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
                      ↳ QDPSizeChangeProof

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

APPEND_IN(.(X, Zs)) → APPEND_IN(Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: