Left Termination of the query pattern sublist_in_2(g, a) 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:

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

Queries:

sublist(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append7(.(T30, T31), X50, .(T30, T33)) :- append7(T31, X50, T33).
append17(.(X125, X126), T54, .(X125, X127)) :- append17(X126, T54, X127).
append27(.(T79, T82), X189, .(T79, T83)) :- append27(T82, X189, T83).
sublist1(T16, T7) :- append7(T16, X7, T7).
sublist1(T40, T7) :- append17(X88, T40, X89).
sublist1(T40, .(X154, T65)) :- ','(appendc17(T43, T40, T64), append27(T64, X155, T65)).

Clauses:

appendc7([], T23, T23).
appendc7(.(T30, T31), X50, .(T30, T33)) :- appendc7(T31, X50, T33).
appendc17([], T50, T50).
appendc17(.(X125, X126), T54, .(X125, X127)) :- appendc17(X126, T54, X127).
appendc27([], T72, T72).
appendc27(.(T79, T82), X189, .(T79, T83)) :- appendc27(T82, X189, T83).

Afs:

sublist1(x1, x2)  =  sublist1(x1)

(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,f)
append7_in: (b,f,f)
append17_in: (f,b,f)
appendc17_in: (f,b,f)
append27_in: (b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUBLIST1_IN_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, appendc17_in_aga(T43, T40, T64))
U6_GA(T40, X154, T65, appendc17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, appendc17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

appendc17_in_aga([], T50, T50) → appendc17_out_aga([], T50, T50)
appendc17_in_aga(.(X125, X126), T54, .(X125, X127)) → U10_aga(X125, X126, T54, X127, appendc17_in_aga(X126, T54, X127))
U10_aga(X125, X126, T54, X127, appendc17_out_aga(X126, T54, X127)) → appendc17_out_aga(.(X125, X126), T54, .(X125, X127))

The argument filtering Pi contains the following mapping:
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
.(x1, x2)  =  .(x2)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
appendc17_in_aga(x1, x2, x3)  =  appendc17_in_aga(x2)
appendc17_out_aga(x1, x2, x3)  =  appendc17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, 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_GA(T16, T7) → U4_GA(T16, T7, append7_in_gaa(T16, X7, T7))
SUBLIST1_IN_GA(T16, T7) → APPEND7_IN_GAA(T16, X7, T7)
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → U1_GAA(T30, T31, X50, T33, append7_in_gaa(T31, X50, T33))
APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)
SUBLIST1_IN_GA(T40, T7) → U5_GA(T40, T7, append17_in_aga(X88, T40, X89))
SUBLIST1_IN_GA(T40, T7) → APPEND17_IN_AGA(X88, T40, X89)
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → U2_AGA(X125, X126, T54, X127, append17_in_aga(X126, T54, X127))
APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)
SUBLIST1_IN_GA(T40, .(X154, T65)) → U6_GA(T40, X154, T65, appendc17_in_aga(T43, T40, T64))
U6_GA(T40, X154, T65, appendc17_out_aga(T43, T40, T64)) → U7_GA(T40, X154, T65, append27_in_gaa(T64, X155, T65))
U6_GA(T40, X154, T65, appendc17_out_aga(T43, T40, T64)) → APPEND27_IN_GAA(T64, X155, T65)
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → U3_GAA(T79, T82, X189, T83, append27_in_gaa(T82, X189, T83))
APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

appendc17_in_aga([], T50, T50) → appendc17_out_aga([], T50, T50)
appendc17_in_aga(.(X125, X126), T54, .(X125, X127)) → U10_aga(X125, X126, T54, X127, appendc17_in_aga(X126, T54, X127))
U10_aga(X125, X126, T54, X127, appendc17_out_aga(X126, T54, X127)) → appendc17_out_aga(.(X125, X126), T54, .(X125, X127))

The argument filtering Pi contains the following mapping:
append7_in_gaa(x1, x2, x3)  =  append7_in_gaa(x1)
.(x1, x2)  =  .(x2)
append17_in_aga(x1, x2, x3)  =  append17_in_aga(x2)
appendc17_in_aga(x1, x2, x3)  =  appendc17_in_aga(x2)
appendc17_out_aga(x1, x2, x3)  =  appendc17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
append27_in_gaa(x1, x2, x3)  =  append27_in_gaa(x1)
SUBLIST1_IN_GA(x1, x2)  =  SUBLIST1_IN_GA(x1)
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_IN_AGA(x2)
U2_AGA(x1, x2, x3, x4, x5)  =  U2_AGA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U7_GA(x1, x2, x3, x4)  =  U7_GA(x1, x4)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5)  =  U3_GAA(x2, 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:

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

The TRS R consists of the following rules:

appendc17_in_aga([], T50, T50) → appendc17_out_aga([], T50, T50)
appendc17_in_aga(.(X125, X126), T54, .(X125, X127)) → U10_aga(X125, X126, T54, X127, appendc17_in_aga(X126, T54, X127))
U10_aga(X125, X126, T54, X127, appendc17_out_aga(X126, T54, X127)) → appendc17_out_aga(.(X125, X126), T54, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
appendc17_in_aga(x1, x2, x3)  =  appendc17_in_aga(x2)
appendc17_out_aga(x1, x2, x3)  =  appendc17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T79, T82), X189, .(T79, T83)) → APPEND27_IN_GAA(T82, X189, T83)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND27_IN_GAA(x1, x2, x3)  =  APPEND27_IN_GAA(x1)

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:

APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)

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:

  • APPEND27_IN_GAA(.(T82)) → APPEND27_IN_GAA(T82)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

The TRS R consists of the following rules:

appendc17_in_aga([], T50, T50) → appendc17_out_aga([], T50, T50)
appendc17_in_aga(.(X125, X126), T54, .(X125, X127)) → U10_aga(X125, X126, T54, X127, appendc17_in_aga(X126, T54, X127))
U10_aga(X125, X126, T54, X127, appendc17_out_aga(X126, T54, X127)) → appendc17_out_aga(.(X125, X126), T54, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
appendc17_in_aga(x1, x2, x3)  =  appendc17_in_aga(x2)
appendc17_out_aga(x1, x2, x3)  =  appendc17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_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:

APPEND17_IN_AGA(.(X125, X126), T54, .(X125, X127)) → APPEND17_IN_AGA(X126, T54, X127)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND17_IN_AGA(x1, x2, x3)  =  APPEND17_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:

APPEND17_IN_AGA(T54) → APPEND17_IN_AGA(T54)

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 = APPEND17_IN_AGA(T54) evaluates to t =APPEND17_IN_AGA(T54)

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 APPEND17_IN_AGA(T54) to APPEND17_IN_AGA(T54).



(20) NO

(21) Obligation:

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

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

The TRS R consists of the following rules:

appendc17_in_aga([], T50, T50) → appendc17_out_aga([], T50, T50)
appendc17_in_aga(.(X125, X126), T54, .(X125, X127)) → U10_aga(X125, X126, T54, X127, appendc17_in_aga(X126, T54, X127))
U10_aga(X125, X126, T54, X127, appendc17_out_aga(X126, T54, X127)) → appendc17_out_aga(.(X125, X126), T54, .(X125, X127))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
appendc17_in_aga(x1, x2, x3)  =  appendc17_in_aga(x2)
appendc17_out_aga(x1, x2, x3)  =  appendc17_out_aga(x1, x2, x3)
U10_aga(x1, x2, x3, x4, x5)  =  U10_aga(x3, x5)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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:

APPEND7_IN_GAA(.(T30, T31), X50, .(T30, T33)) → APPEND7_IN_GAA(T31, X50, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
APPEND7_IN_GAA(x1, x2, x3)  =  APPEND7_IN_GAA(x1)

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:

APPEND7_IN_GAA(.(T31)) → APPEND7_IN_GAA(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:

  • APPEND7_IN_GAA(.(T31)) → APPEND7_IN_GAA(T31)
    The graph contains the following edges 1 > 1

(27) YES