Left Termination of the query pattern append3_in_4(g, g, g, a) 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([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).

Queries:

append3(g,g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

append7(.(T34, T35), T36, .(T34, T38)) :- append7(T35, T36, T38).
append19(.(T66, T67), T68, .(T66, X85)) :- append19(T67, T68, X85).
append31([], T18, T11, T13) :- append7(T18, T11, T13).
append31(.(T47, T48), T49, T11, T13) :- append19(T48, T49, X56).
append31(.(T47, T48), T49, T11, T13) :- ','(appendc19(T48, T49, T52), append7(.(T47, T52), T11, T13)).

Clauses:

appendc7([], T25, T25).
appendc7(.(T34, T35), T36, .(T34, T38)) :- appendc7(T35, T36, T38).
appendc19([], T59, T59).
appendc19(.(T66, T67), T68, .(T66, X85)) :- appendc19(T67, T68, X85).

Afs:

append31(x1, x2, x3, x4)  =  append31(x1, x2, x3)

(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:
append31_in: (b,b,b,f)
append7_in: (b,b,f)
append19_in: (b,b,f)
appendc19_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

APPEND31_IN_GGGA([], T18, T11, T13) → U3_GGGA(T18, T11, T13, append7_in_gga(T18, T11, T13))
APPEND31_IN_GGGA([], T18, T11, T13) → APPEND7_IN_GGA(T18, T11, T13)
APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, append7_in_gga(T35, T36, T38))
APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPEND7_IN_GGA(T35, T36, T38)
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → U4_GGGA(T47, T48, T49, T11, T13, append19_in_gga(T48, T49, X56))
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → APPEND19_IN_GGA(T48, T49, X56)
APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, append19_in_gga(T67, T68, X85))
APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APPEND19_IN_GGA(T67, T68, X85)
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → U5_GGGA(T47, T48, T49, T11, T13, appendc19_in_gga(T48, T49, T52))
U5_GGGA(T47, T48, T49, T11, T13, appendc19_out_gga(T48, T49, T52)) → U6_GGGA(T47, T48, T49, T11, T13, append7_in_gga(.(T47, T52), T11, T13))
U5_GGGA(T47, T48, T49, T11, T13, appendc19_out_gga(T48, T49, T52)) → APPEND7_IN_GGA(.(T47, T52), T11, T13)

The TRS R consists of the following rules:

appendc19_in_gga([], T59, T59) → appendc19_out_gga([], T59, T59)
appendc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appendc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appendc19_out_gga(T67, T68, X85)) → appendc19_out_gga(.(T66, T67), T68, .(T66, X85))

The argument filtering Pi contains the following mapping:
[]  =  []
append7_in_gga(x1, x2, x3)  =  append7_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
append19_in_gga(x1, x2, x3)  =  append19_in_gga(x1, x2)
appendc19_in_gga(x1, x2, x3)  =  appendc19_in_gga(x1, x2)
appendc19_out_gga(x1, x2, x3)  =  appendc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
APPEND31_IN_GGGA(x1, x2, x3, x4)  =  APPEND31_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x1, x2, x4)
APPEND7_IN_GGA(x1, x2, x3)  =  APPEND7_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
APPEND19_IN_GGA(x1, x2, x3)  =  APPEND19_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x3, x4, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)

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:

APPEND31_IN_GGGA([], T18, T11, T13) → U3_GGGA(T18, T11, T13, append7_in_gga(T18, T11, T13))
APPEND31_IN_GGGA([], T18, T11, T13) → APPEND7_IN_GGA(T18, T11, T13)
APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, append7_in_gga(T35, T36, T38))
APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPEND7_IN_GGA(T35, T36, T38)
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → U4_GGGA(T47, T48, T49, T11, T13, append19_in_gga(T48, T49, X56))
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → APPEND19_IN_GGA(T48, T49, X56)
APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, append19_in_gga(T67, T68, X85))
APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APPEND19_IN_GGA(T67, T68, X85)
APPEND31_IN_GGGA(.(T47, T48), T49, T11, T13) → U5_GGGA(T47, T48, T49, T11, T13, appendc19_in_gga(T48, T49, T52))
U5_GGGA(T47, T48, T49, T11, T13, appendc19_out_gga(T48, T49, T52)) → U6_GGGA(T47, T48, T49, T11, T13, append7_in_gga(.(T47, T52), T11, T13))
U5_GGGA(T47, T48, T49, T11, T13, appendc19_out_gga(T48, T49, T52)) → APPEND7_IN_GGA(.(T47, T52), T11, T13)

The TRS R consists of the following rules:

appendc19_in_gga([], T59, T59) → appendc19_out_gga([], T59, T59)
appendc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appendc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appendc19_out_gga(T67, T68, X85)) → appendc19_out_gga(.(T66, T67), T68, .(T66, X85))

The argument filtering Pi contains the following mapping:
[]  =  []
append7_in_gga(x1, x2, x3)  =  append7_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
append19_in_gga(x1, x2, x3)  =  append19_in_gga(x1, x2)
appendc19_in_gga(x1, x2, x3)  =  appendc19_in_gga(x1, x2)
appendc19_out_gga(x1, x2, x3)  =  appendc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
APPEND31_IN_GGGA(x1, x2, x3, x4)  =  APPEND31_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4)  =  U3_GGGA(x1, x2, x4)
APPEND7_IN_GGA(x1, x2, x3)  =  APPEND7_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x1, x2, x3, x4, x6)
APPEND19_IN_GGA(x1, x2, x3)  =  APPEND19_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U5_GGGA(x1, x2, x3, x4, x5, x6)  =  U5_GGGA(x1, x2, x3, x4, x6)
U6_GGGA(x1, x2, x3, x4, x5, x6)  =  U6_GGGA(x1, x2, x3, x4, x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APPEND19_IN_GGA(T67, T68, X85)

The TRS R consists of the following rules:

appendc19_in_gga([], T59, T59) → appendc19_out_gga([], T59, T59)
appendc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appendc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appendc19_out_gga(T67, T68, X85)) → appendc19_out_gga(.(T66, T67), T68, .(T66, X85))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
appendc19_in_gga(x1, x2, x3)  =  appendc19_in_gga(x1, x2)
appendc19_out_gga(x1, x2, x3)  =  appendc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
APPEND19_IN_GGA(x1, x2, x3)  =  APPEND19_IN_GGA(x1, x2)

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:

APPEND19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APPEND19_IN_GGA(T67, T68, X85)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND19_IN_GGA(x1, x2, x3)  =  APPEND19_IN_GGA(x1, x2)

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:

APPEND19_IN_GGA(.(T66, T67), T68) → APPEND19_IN_GGA(T67, T68)

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:

  • APPEND19_IN_GGA(.(T66, T67), T68) → APPEND19_IN_GGA(T67, T68)
    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:

APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPEND7_IN_GGA(T35, T36, T38)

The TRS R consists of the following rules:

appendc19_in_gga([], T59, T59) → appendc19_out_gga([], T59, T59)
appendc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appendc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appendc19_out_gga(T67, T68, X85)) → appendc19_out_gga(.(T66, T67), T68, .(T66, X85))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
appendc19_in_gga(x1, x2, x3)  =  appendc19_in_gga(x1, x2)
appendc19_out_gga(x1, x2, x3)  =  appendc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)
APPEND7_IN_GGA(x1, x2, x3)  =  APPEND7_IN_GGA(x1, 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:

APPEND7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APPEND7_IN_GGA(T35, T36, T38)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND7_IN_GGA(x1, x2, x3)  =  APPEND7_IN_GGA(x1, 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:

APPEND7_IN_GGA(.(T34, T35), T36) → APPEND7_IN_GGA(T35, T36)

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:

  • APPEND7_IN_GGA(.(T34, T35), T36) → APPEND7_IN_GGA(T35, T36)
    The graph contains the following edges 1 > 1, 2 >= 2

(20) YES