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

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 QDPSizeChangeProof (⇔)
        ↳20 YES

(0) Obligation:

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).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append17(.(X97, X98), X99, .(X97, T45)) :- append17(X98, X99, T45).
append32(.(X158, X159), T91, .(X158, T90)) :- append32(X159, T91, T90).
sublist1(T7, .(X56, T27)) :- append17(X57, X58, T27).
sublist1(T77, .(X132, T27)) :- ','(appendc17(T76, T36, T27), append32(X133, T77, T76)).

Clauses:

appendc17([], T42, T42).
appendc17(.(X97, X98), X99, .(X97, T45)) :- appendc17(X98, X99, T45).
appendc32([], T84, T84).
appendc32(.(X158, X159), T91, .(X158, T90)) :- appendc32(X159, T91, T90).

Afs:

sublist1(x1, x2)  =  sublist1(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: (f,b)
append17_in: (f,f,b)
appendc17_in: (f,f,b)
append32_in: (f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SUBLIST1_IN_AG(T7, .(X56, T27)) → U3_AG(T7, X56, T27, append17_in_aag(X57, X58, T27))
SUBLIST1_IN_AG(T7, .(X56, T27)) → APPEND17_IN_AAG(X57, X58, T27)
APPEND17_IN_AAG(.(X97, X98), X99, .(X97, T45)) → U1_AAG(X97, X98, X99, T45, append17_in_aag(X98, X99, T45))
APPEND17_IN_AAG(.(X97, X98), X99, .(X97, T45)) → APPEND17_IN_AAG(X98, X99, T45)
SUBLIST1_IN_AG(T77, .(X132, T27)) → U4_AG(T77, X132, T27, appendc17_in_aag(T76, T36, T27))
U4_AG(T77, X132, T27, appendc17_out_aag(T76, T36, T27)) → U5_AG(T77, X132, T27, append32_in_aag(X133, T77, T76))
U4_AG(T77, X132, T27, appendc17_out_aag(T76, T36, T27)) → APPEND32_IN_AAG(X133, T77, T76)
APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → U2_AAG(X158, X159, T91, T90, append32_in_aag(X159, T91, T90))
APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → APPEND32_IN_AAG(X159, T91, T90)

The TRS R consists of the following rules:

appendc17_in_aag([], T42, T42) → appendc17_out_aag([], T42, T42)
appendc17_in_aag(.(X97, X98), X99, .(X97, T45)) → U7_aag(X97, X98, X99, T45, appendc17_in_aag(X98, X99, T45))
U7_aag(X97, X98, X99, T45, appendc17_out_aag(X98, X99, T45)) → appendc17_out_aag(.(X97, X98), X99, .(X97, T45))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append17_in_aag(x1, x2, x3)  =  append17_in_aag(x3)
appendc17_in_aag(x1, x2, x3)  =  appendc17_in_aag(x3)
appendc17_out_aag(x1, x2, x3)  =  appendc17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5)  =  U7_aag(x1, x4, x5)
append32_in_aag(x1, x2, x3)  =  append32_in_aag(x3)
SUBLIST1_IN_AG(x1, x2)  =  SUBLIST1_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
APPEND17_IN_AAG(x1, x2, x3)  =  APPEND17_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x4, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
APPEND32_IN_AAG(x1, x2, x3)  =  APPEND32_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, 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_AG(T7, .(X56, T27)) → U3_AG(T7, X56, T27, append17_in_aag(X57, X58, T27))
SUBLIST1_IN_AG(T7, .(X56, T27)) → APPEND17_IN_AAG(X57, X58, T27)
APPEND17_IN_AAG(.(X97, X98), X99, .(X97, T45)) → U1_AAG(X97, X98, X99, T45, append17_in_aag(X98, X99, T45))
APPEND17_IN_AAG(.(X97, X98), X99, .(X97, T45)) → APPEND17_IN_AAG(X98, X99, T45)
SUBLIST1_IN_AG(T77, .(X132, T27)) → U4_AG(T77, X132, T27, appendc17_in_aag(T76, T36, T27))
U4_AG(T77, X132, T27, appendc17_out_aag(T76, T36, T27)) → U5_AG(T77, X132, T27, append32_in_aag(X133, T77, T76))
U4_AG(T77, X132, T27, appendc17_out_aag(T76, T36, T27)) → APPEND32_IN_AAG(X133, T77, T76)
APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → U2_AAG(X158, X159, T91, T90, append32_in_aag(X159, T91, T90))
APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → APPEND32_IN_AAG(X159, T91, T90)

The TRS R consists of the following rules:

appendc17_in_aag([], T42, T42) → appendc17_out_aag([], T42, T42)
appendc17_in_aag(.(X97, X98), X99, .(X97, T45)) → U7_aag(X97, X98, X99, T45, appendc17_in_aag(X98, X99, T45))
U7_aag(X97, X98, X99, T45, appendc17_out_aag(X98, X99, T45)) → appendc17_out_aag(.(X97, X98), X99, .(X97, T45))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append17_in_aag(x1, x2, x3)  =  append17_in_aag(x3)
appendc17_in_aag(x1, x2, x3)  =  appendc17_in_aag(x3)
appendc17_out_aag(x1, x2, x3)  =  appendc17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5)  =  U7_aag(x1, x4, x5)
append32_in_aag(x1, x2, x3)  =  append32_in_aag(x3)
SUBLIST1_IN_AG(x1, x2)  =  SUBLIST1_IN_AG(x2)
U3_AG(x1, x2, x3, x4)  =  U3_AG(x2, x3, x4)
APPEND17_IN_AAG(x1, x2, x3)  =  APPEND17_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5)  =  U1_AAG(x1, x4, x5)
U4_AG(x1, x2, x3, x4)  =  U4_AG(x2, x3, x4)
U5_AG(x1, x2, x3, x4)  =  U5_AG(x2, x3, x4)
APPEND32_IN_AAG(x1, x2, x3)  =  APPEND32_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → APPEND32_IN_AAG(X159, T91, T90)

The TRS R consists of the following rules:

appendc17_in_aag([], T42, T42) → appendc17_out_aag([], T42, T42)
appendc17_in_aag(.(X97, X98), X99, .(X97, T45)) → U7_aag(X97, X98, X99, T45, appendc17_in_aag(X98, X99, T45))
U7_aag(X97, X98, X99, T45, appendc17_out_aag(X98, X99, T45)) → appendc17_out_aag(.(X97, X98), X99, .(X97, T45))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appendc17_in_aag(x1, x2, x3)  =  appendc17_in_aag(x3)
appendc17_out_aag(x1, x2, x3)  =  appendc17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5)  =  U7_aag(x1, x4, x5)
APPEND32_IN_AAG(x1, x2, x3)  =  APPEND32_IN_AAG(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:

APPEND32_IN_AAG(.(X158, X159), T91, .(X158, T90)) → APPEND32_IN_AAG(X159, T91, T90)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND32_IN_AAG(x1, x2, x3)  =  APPEND32_IN_AAG(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:

APPEND32_IN_AAG(.(X158, T90)) → APPEND32_IN_AAG(T90)

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:

  • APPEND32_IN_AAG(.(X158, T90)) → APPEND32_IN_AAG(T90)
    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_AAG(.(X97, X98), X99, .(X97, T45)) → APPEND17_IN_AAG(X98, X99, T45)

The TRS R consists of the following rules:

appendc17_in_aag([], T42, T42) → appendc17_out_aag([], T42, T42)
appendc17_in_aag(.(X97, X98), X99, .(X97, T45)) → U7_aag(X97, X98, X99, T45, appendc17_in_aag(X98, X99, T45))
U7_aag(X97, X98, X99, T45, appendc17_out_aag(X98, X99, T45)) → appendc17_out_aag(.(X97, X98), X99, .(X97, T45))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
appendc17_in_aag(x1, x2, x3)  =  appendc17_in_aag(x3)
appendc17_out_aag(x1, x2, x3)  =  appendc17_out_aag(x1, x2, x3)
U7_aag(x1, x2, x3, x4, x5)  =  U7_aag(x1, x4, x5)
APPEND17_IN_AAG(x1, x2, x3)  =  APPEND17_IN_AAG(x3)

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_AAG(.(X97, X98), X99, .(X97, T45)) → APPEND17_IN_AAG(X98, X99, T45)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND17_IN_AAG(x1, x2, x3)  =  APPEND17_IN_AAG(x3)

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_AAG(.(X97, T45)) → APPEND17_IN_AAG(T45)

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

(19) 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:

  • APPEND17_IN_AAG(.(X97, T45)) → APPEND17_IN_AAG(T45)
    The graph contains the following edges 1 > 1

(20) YES