Left Termination of the query pattern f_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

f(X) :- g(s(s(s(X)))).
f(s(X)) :- f(X).
g(s(s(s(s(X))))) :- f(X).

Queries:

f(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

f1(s(T8)) :- f1(T8).
f1(s(T12)) :- f1(T12).

Queries:

f1(g).

(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:
f1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

f1_in_g(s(T8)) → U1_g(T8, f1_in_g(T8))
f1_in_g(s(T12)) → U2_g(T12, f1_in_g(T12))
U2_g(T12, f1_out_g(T12)) → f1_out_g(s(T12))
U1_g(T8, f1_out_g(T8)) → f1_out_g(s(T8))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f1_out_g(x1)  =  f1_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

f1_in_g(s(T8)) → U1_g(T8, f1_in_g(T8))
f1_in_g(s(T12)) → U2_g(T12, f1_in_g(T12))
U2_g(T12, f1_out_g(T12)) → f1_out_g(s(T12))
U1_g(T8, f1_out_g(T8)) → f1_out_g(s(T8))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f1_out_g(x1)  =  f1_out_g

(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:

F1_IN_G(s(T8)) → U1_G(T8, f1_in_g(T8))
F1_IN_G(s(T8)) → F1_IN_G(T8)
F1_IN_G(s(T12)) → U2_G(T12, f1_in_g(T12))

The TRS R consists of the following rules:

f1_in_g(s(T8)) → U1_g(T8, f1_in_g(T8))
f1_in_g(s(T12)) → U2_g(T12, f1_in_g(T12))
U2_g(T12, f1_out_g(T12)) → f1_out_g(s(T12))
U1_g(T8, f1_out_g(T8)) → f1_out_g(s(T8))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f1_out_g(x1)  =  f1_out_g
F1_IN_G(x1)  =  F1_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U2_G(x1, x2)  =  U2_G(x2)

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

(6) Obligation:

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

F1_IN_G(s(T8)) → U1_G(T8, f1_in_g(T8))
F1_IN_G(s(T8)) → F1_IN_G(T8)
F1_IN_G(s(T12)) → U2_G(T12, f1_in_g(T12))

The TRS R consists of the following rules:

f1_in_g(s(T8)) → U1_g(T8, f1_in_g(T8))
f1_in_g(s(T12)) → U2_g(T12, f1_in_g(T12))
U2_g(T12, f1_out_g(T12)) → f1_out_g(s(T12))
U1_g(T8, f1_out_g(T8)) → f1_out_g(s(T8))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f1_out_g(x1)  =  f1_out_g
F1_IN_G(x1)  =  F1_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U2_G(x1, x2)  =  U2_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

F1_IN_G(s(T8)) → F1_IN_G(T8)

The TRS R consists of the following rules:

f1_in_g(s(T8)) → U1_g(T8, f1_in_g(T8))
f1_in_g(s(T12)) → U2_g(T12, f1_in_g(T12))
U2_g(T12, f1_out_g(T12)) → f1_out_g(s(T12))
U1_g(T8, f1_out_g(T8)) → f1_out_g(s(T8))

The argument filtering Pi contains the following mapping:
f1_in_g(x1)  =  f1_in_g(x1)
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
U2_g(x1, x2)  =  U2_g(x2)
f1_out_g(x1)  =  f1_out_g
F1_IN_G(x1)  =  F1_IN_G(x1)

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:

F1_IN_G(s(T8)) → F1_IN_G(T8)

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

(11) PiDPToQDPProof (EQUIVALENT 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:

F1_IN_G(s(T8)) → F1_IN_G(T8)

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:

• F1_IN_G(s(T8)) → F1_IN_G(T8)
  The graph contains the following edges 1 > 1

(14) YES