Left Termination of the query pattern select_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

select(X1, [], X2) :- ','(!, failure(a)).
select(X, Y, Zs) :- ','(head(Y, X), tail(Y, Zs)).
select(X, Y, .(H, Zs)) :- ','(head(Y, H), ','(tail(Y, T), select(X, T, Zs))).
head([], X3).
head(.(H, X4), H).
tail([], []).
tail(.(X5, T), T).
failure(b).

Queries:

select(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

select1(T40, .(T64, T65), .(T64, T55)) :- select1(T40, T65, T55).

Clauses:

selectc1(T30, .(T30, T31), T31).
selectc1(T40, .(T64, T65), .(T64, T55)) :- selectc1(T40, T65, T55).

Afs:

select1(x1, x2, x3)  =  select1(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:
select1_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SELECT1_IN_GGA(T40, .(T64, T65), .(T64, T55)) → U1_GGA(T40, T64, T65, T55, select1_in_gga(T40, T65, T55))
SELECT1_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECT1_IN_GGA(T40, T65, T55)

R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, 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:

SELECT1_IN_GGA(T40, .(T64, T65), .(T64, T55)) → U1_GGA(T40, T64, T65, T55, select1_in_gga(T40, T65, T55))
SELECT1_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECT1_IN_GGA(T40, T65, T55)

R is empty.
The argument filtering Pi contains the following mapping:
select1_in_gga(x1, x2, x3)  =  select1_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

SELECT1_IN_GGA(T40, .(T64, T65), .(T64, T55)) → SELECT1_IN_GGA(T40, T65, T55)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECT1_IN_GGA(x1, x2, x3)  =  SELECT1_IN_GGA(x1, x2)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

SELECT1_IN_GGA(T40, .(T64, T65)) → SELECT1_IN_GGA(T40, T65)

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

(9) 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:

• SELECT1_IN_GGA(T40, .(T64, T65)) → SELECT1_IN_GGA(T40, T65)
  The graph contains the following edges 1 >= 1, 2 > 2

(10) YES