Left Termination of the query pattern len1_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

len1([], 0).
len1(.(X1, Ts), N) :- ','(len1(Ts, M), eq(N, s(M))).
eq(X, X).

Queries:

len1(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

len120(.(T24, T25), X49) :- len120(T25, X48).
len11(.(T6, .(T17, T18)), T9) :- len120(T18, X31).

Clauses:

len1c20([], 0).
len1c20(.(T24, T25), s(T29)) :- len1c20(T25, T29).

Afs:

len11(x1, x2)  =  len11(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:
len11_in: (b,f)
len120_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

LEN11_IN_GA(.(T6, .(T17, T18)), T9) → U2_GA(T6, T17, T18, T9, len120_in_ga(T18, X31))
LEN11_IN_GA(.(T6, .(T17, T18)), T9) → LEN120_IN_GA(T18, X31)
LEN120_IN_GA(.(T24, T25), X49) → U1_GA(T24, T25, X49, len120_in_ga(T25, X48))
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
LEN11_IN_GA(x1, x2)  =  LEN11_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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:

LEN11_IN_GA(.(T6, .(T17, T18)), T9) → U2_GA(T6, T17, T18, T9, len120_in_ga(T18, X31))
LEN11_IN_GA(.(T6, .(T17, T18)), T9) → LEN120_IN_GA(T18, X31)
LEN120_IN_GA(.(T24, T25), X49) → U1_GA(T24, T25, X49, len120_in_ga(T25, X48))
LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
len120_in_ga(x1, x2)  =  len120_in_ga(x1)
LEN11_IN_GA(x1, x2)  =  LEN11_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

LEN120_IN_GA(.(T24, T25), X49) → LEN120_IN_GA(T25, X48)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
LEN120_IN_GA(x1, x2)  =  LEN120_IN_GA(x1)

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:

LEN120_IN_GA(.(T24, T25)) → LEN120_IN_GA(T25)

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:

• LEN120_IN_GA(.(T24, T25)) → LEN120_IN_GA(T25)
  The graph contains the following edges 1 > 1

(10) YES