Left Termination of the query pattern sublist_in_2(g, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 NonTerminationProof (⇔)
        ↳20 NO
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 YES

(0) Obligation:

Clauses:

append1([], Ys, Ys).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append2([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
sublist(X, Y) :- ','(append1(U, X, V), append2(V, W, Y)).

Queries:

sublist(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append27(.(T29, T30), X50, .(T29, T31)) :- append27(T30, X50, T31).
append117(.(X125, X126), T52, .(X125, X127)) :- append117(X126, T52, X127).
append227(.(T76, T79), X189, .(T76, T78)) :- append227(T79, X189, T78).
sublist1(T15, T6) :- append27(T15, X7, T6).
sublist1(T38, T6) :- append117(X88, T38, X89).
sublist1(T38, .(X154, T61)) :- ','(append1c17(T41, T38, T62), append227(T62, X155, T61)).

Clauses:

append2c7([], T22, T22).
append2c7(.(T29, T30), X50, .(T29, T31)) :- append2c7(T30, X50, T31).
append1c17([], T48, T48).
append1c17(.(X125, X126), T52, .(X125, X127)) :- append1c17(X126, T52, X127).
append2c27([], T69, T69).
append2c27(.(T76, T79), X189, .(T76, T78)) :- append2c27(T79, X189, T78).

Afs:

sublist1(x1, x2)  =  sublist1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
sublist1_in: (b,b)
append27_in: (b,f,b)
append117_in: (f,b,f)
append1c17_in: (f,b,f)
append227_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUBLIST1_IN_GG(T15, T6) → U4_GG(T15, T6, append27_in_gag(T15, X7, T6))
SUBLIST1_IN_GG(T15, T6) → APPEND27_IN_GAG(T15, X7, T6)
APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → U1_GAG(T29, T30, X50, T31, append27_in_gag(T30, X50, T31))
APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → APPEND27_IN_GAG(T30, X50, T31)
SUBLIST1_IN_GG(T38, T6) → U5_GG(T38, T6, append117_in_aga(X88, T38, X89))
SUBLIST1_IN_GG(T38, T6) → APPEND117_IN_AGA(X88, T38, X89)
APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → U2_AGA(X125, X126, T52, X127, append117_in_aga(X126, T52, X127))
APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → APPEND117_IN_AGA(X126, T52, X127)
SUBLIST1_IN_GG(T38, .(X154, T61)) → U6_GG(T38, X154, T61, append1c17_in_aga(T41, T38, T62))
U6_GG(T38, X154, T61, append1c17_out_aga(T41, T38, T62)) → U7_GG(T38, X154, T61, append227_in_gag(T62, X155, T61))
U6_GG(T38, X154, T61, append1c17_out_aga(T41, T38, T62)) → APPEND227_IN_GAG(T62, X155, T61)
APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → U3_GAG(T76, T79, X189, T78, append227_in_gag(T79, X189, T78))
APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → APPEND227_IN_GAG(T79, X189, T78)

The TRS R consists of the following rules:

append1c17_in_aga([], T48, T48) → append1c17_out_aga([], T48, T48)
append1c17_in_aga(.(X125, X126), T52, .(X125, X127)) → U10_aga(X125, X126, T52, X127, append1c17_in_aga(X126, T52, X127))
U10_aga(X125, X126, T52, X127, append1c17_out_aga(X126, T52, X127)) → append1c17_out_aga(.(X125, X126), T52, .(X125, X127))

The argument filtering Pi contains the following mapping:
append27_in_gag(x1, x2, x3)  =  append27_in_gag(x1, x3)
.(x1, x2)  =  .(x2)
append117_in_aga(x1, x2, x3)  =  append117_in_aga(x2)
append1c17_in_aga(x1, x2, x3)  =  append1c17_in_aga(x2)
append1c17_out_aga(x1, x2, x3)  =  append1c17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
append227_in_gag(x1, x2, x3)  =  append227_in_gag(x1, x3)
SUBLIST1_IN_GG(x1, x2)  =  SUBLIST1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
APPEND27_IN_GAG(x1, x2, x3)  =  APPEND27_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
APPEND117_IN_AGA(x1, x2, x3)  =  APPEND117_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)
U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)
APPEND227_IN_GAG(x1, x2, x3)  =  APPEND227_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SUBLIST1_IN_GG(T15, T6) → U4_GG(T15, T6, append27_in_gag(T15, X7, T6))
SUBLIST1_IN_GG(T15, T6) → APPEND27_IN_GAG(T15, X7, T6)
APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → U1_GAG(T29, T30, X50, T31, append27_in_gag(T30, X50, T31))
APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → APPEND27_IN_GAG(T30, X50, T31)
SUBLIST1_IN_GG(T38, T6) → U5_GG(T38, T6, append117_in_aga(X88, T38, X89))
SUBLIST1_IN_GG(T38, T6) → APPEND117_IN_AGA(X88, T38, X89)
APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → U2_AGA(X125, X126, T52, X127, append117_in_aga(X126, T52, X127))
APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → APPEND117_IN_AGA(X126, T52, X127)
SUBLIST1_IN_GG(T38, .(X154, T61)) → U6_GG(T38, X154, T61, append1c17_in_aga(T41, T38, T62))
U6_GG(T38, X154, T61, append1c17_out_aga(T41, T38, T62)) → U7_GG(T38, X154, T61, append227_in_gag(T62, X155, T61))
U6_GG(T38, X154, T61, append1c17_out_aga(T41, T38, T62)) → APPEND227_IN_GAG(T62, X155, T61)
APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → U3_GAG(T76, T79, X189, T78, append227_in_gag(T79, X189, T78))
APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → APPEND227_IN_GAG(T79, X189, T78)

The TRS R consists of the following rules:

append1c17_in_aga([], T48, T48) → append1c17_out_aga([], T48, T48)
append1c17_in_aga(.(X125, X126), T52, .(X125, X127)) → U10_aga(X125, X126, T52, X127, append1c17_in_aga(X126, T52, X127))
U10_aga(X125, X126, T52, X127, append1c17_out_aga(X126, T52, X127)) → append1c17_out_aga(.(X125, X126), T52, .(X125, X127))

The argument filtering Pi contains the following mapping:
append27_in_gag(x1, x2, x3)  =  append27_in_gag(x1, x3)
.(x1, x2)  =  .(x2)
append117_in_aga(x1, x2, x3)  =  append117_in_aga(x2)
append1c17_in_aga(x1, x2, x3)  =  append1c17_in_aga(x2)
append1c17_out_aga(x1, x2, x3)  =  append1c17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
append227_in_gag(x1, x2, x3)  =  append227_in_gag(x1, x3)
SUBLIST1_IN_GG(x1, x2)  =  SUBLIST1_IN_GG(x1, x2)
U4_GG(x1, x2, x3)  =  U4_GG(x1, x2, x3)
APPEND27_IN_GAG(x1, x2, x3)  =  APPEND27_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x2, x4, x5)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
APPEND117_IN_AGA(x1, x2, x3)  =  APPEND117_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x1, x3, x4)
U7_GG(x1, x2, x3, x4)  =  U7_GG(x1, x3, x4)
APPEND227_IN_GAG(x1, x2, x3)  =  APPEND227_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5)  =  U3_GAG(x2, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → APPEND227_IN_GAG(T79, X189, T78)

The TRS R consists of the following rules:

append1c17_in_aga([], T48, T48) → append1c17_out_aga([], T48, T48)
append1c17_in_aga(.(X125, X126), T52, .(X125, X127)) → U10_aga(X125, X126, T52, X127, append1c17_in_aga(X126, T52, X127))
U10_aga(X125, X126, T52, X127, append1c17_out_aga(X126, T52, X127)) → append1c17_out_aga(.(X125, X126), T52, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
append1c17_in_aga(x1, x2, x3)  =  append1c17_in_aga(x2)
append1c17_out_aga(x1, x2, x3)  =  append1c17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND227_IN_GAG(x1, x2, x3)  =  APPEND227_IN_GAG(x1, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

APPEND227_IN_GAG(.(T76, T79), X189, .(T76, T78)) → APPEND227_IN_GAG(T79, X189, T78)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND227_IN_GAG(x1, x2, x3)  =  APPEND227_IN_GAG(x1, x3)

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

APPEND227_IN_GAG(.(T79), .(T78)) → APPEND227_IN_GAG(T79, T78)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • APPEND227_IN_GAG(.(T79), .(T78)) → APPEND227_IN_GAG(T79, T78)
    The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → APPEND117_IN_AGA(X126, T52, X127)

The TRS R consists of the following rules:

append1c17_in_aga([], T48, T48) → append1c17_out_aga([], T48, T48)
append1c17_in_aga(.(X125, X126), T52, .(X125, X127)) → U10_aga(X125, X126, T52, X127, append1c17_in_aga(X126, T52, X127))
U10_aga(X125, X126, T52, X127, append1c17_out_aga(X126, T52, X127)) → append1c17_out_aga(.(X125, X126), T52, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
append1c17_in_aga(x1, x2, x3)  =  append1c17_in_aga(x2)
append1c17_out_aga(x1, x2, x3)  =  append1c17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND117_IN_AGA(x1, x2, x3)  =  APPEND117_IN_AGA(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

APPEND117_IN_AGA(.(X125, X126), T52, .(X125, X127)) → APPEND117_IN_AGA(X126, T52, X127)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND117_IN_AGA(x1, x2, x3)  =  APPEND117_IN_AGA(x2)

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

APPEND117_IN_AGA(T52) → APPEND117_IN_AGA(T52)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APPEND117_IN_AGA(T52) evaluates to t =APPEND117_IN_AGA(T52)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APPEND117_IN_AGA(T52) to APPEND117_IN_AGA(T52).



(20) NO

(21) Obligation:

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

APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → APPEND27_IN_GAG(T30, X50, T31)

The TRS R consists of the following rules:

append1c17_in_aga([], T48, T48) → append1c17_out_aga([], T48, T48)
append1c17_in_aga(.(X125, X126), T52, .(X125, X127)) → U10_aga(X125, X126, T52, X127, append1c17_in_aga(X126, T52, X127))
U10_aga(X125, X126, T52, X127, append1c17_out_aga(X126, T52, X127)) → append1c17_out_aga(.(X125, X126), T52, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
append1c17_in_aga(x1, x2, x3)  =  append1c17_in_aga(x2)
append1c17_out_aga(x1, x2, x3)  =  append1c17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND27_IN_GAG(x1, x2, x3)  =  APPEND27_IN_GAG(x1, x3)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

APPEND27_IN_GAG(.(T29, T30), X50, .(T29, T31)) → APPEND27_IN_GAG(T30, X50, T31)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND27_IN_GAG(x1, x2, x3)  =  APPEND27_IN_GAG(x1, x3)

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

APPEND27_IN_GAG(.(T30), .(T31)) → APPEND27_IN_GAG(T30, T31)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • APPEND27_IN_GAG(.(T30), .(T31)) → APPEND27_IN_GAG(T30, T31)
    The graph contains the following edges 1 > 1, 2 > 2

(27) YES