Left Termination of the query pattern append_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
append([], Y, Y).

Queries:

append(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

append1(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) :- append1(T30, T31, T33).
append1(.(T8, []), T42, .([], T42)).
append1([], T44, T44).

Queries:

append1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
append1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

append1_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
append1_in_gga(.(T8, []), T42, .([], T42)) → append1_out_gga(.(T8, []), T42, .([], T42))
append1_in_gga([], T44, T44) → append1_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, append1_out_gga(T30, T31, T33)) → append1_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))

The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x2, x3, x6)
[]  =  []
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

append1_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
append1_in_gga(.(T8, []), T42, .([], T42)) → append1_out_gga(.(T8, []), T42, .([], T42))
append1_in_gga([], T44, T44) → append1_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, append1_out_gga(T30, T31, T33)) → append1_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))

The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x2, x3, x6)
[]  =  []
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

The TRS R consists of the following rules:

append1_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
append1_in_gga(.(T8, []), T42, .([], T42)) → append1_out_gga(.(T8, []), T42, .([], T42))
append1_in_gga([], T44, T44) → append1_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, append1_out_gga(T30, T31, T33)) → append1_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))

The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x2, x3, x6)
[]  =  []
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x2, x3, x6)

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

(6) Obligation:

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

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

The TRS R consists of the following rules:

append1_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
append1_in_gga(.(T8, []), T42, .([], T42)) → append1_out_gga(.(T8, []), T42, .([], T42))
append1_in_gga([], T44, T44) → append1_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, append1_out_gga(T30, T31, T33)) → append1_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))

The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x2, x3, x6)
[]  =  []
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x2, x3, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

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

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

The TRS R consists of the following rules:

append1_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, append1_in_gga(T30, T31, T33))
append1_in_gga(.(T8, []), T42, .([], T42)) → append1_out_gga(.(T8, []), T42, .([], T42))
append1_in_gga([], T44, T44) → append1_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, append1_out_gga(T30, T31, T33)) → append1_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))

The argument filtering Pi contains the following mapping:
append1_in_gga(x1, x2, x3)  =  append1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x2, x3, x6)
[]  =  []
append1_out_gga(x1, x2, x3)  =  append1_out_gga(x3)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPEND1_IN_GGA(T30, T31, T33)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3)  =  APPEND1_IN_GGA(x1, x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

APPEND1_IN_GGA(.(T8, .(T29, T30)), T31) → APPEND1_IN_GGA(T30, T31)

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

(13) 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:

  • APPEND1_IN_GGA(.(T8, .(T29, T30)), T31) → APPEND1_IN_GGA(T30, T31)
    The graph contains the following edges 1 > 1, 2 >= 2

(14) YES