Left Termination of the query pattern even_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇔)
    ↳10 QDP
    ↳11 MRRProof (⇔)
    ↳12 QDP
    ↳13 PisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 PrologToPiTRSProof (⇐)
    ↳16 PiTRS
    ↳17 DependencyPairsProof (⇔)
    ↳18 PiDP
    ↳19 DependencyGraphProof (⇔)
    ↳20 PiDP
    ↳21 UsableRulesProof (⇔)
    ↳22 PiDP
    ↳23 PiDPToQDPProof (⇔)
    ↳24 QDP
    ↳25 MRRProof (⇔)
    ↳26 QDP

(0) Obligation:

Clauses:

even(0).
even(s(s(0))).
even(s(s(s(X)))) :- odd(X).
odd(s(0)).
odd(s(X)) :- even(s(s(X))).

Queries:

even(g).

(1) PrologToPiTRSProof (SOUND transformation)

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

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

The argument filtering Pi contains the following mapping:
even_in_g(x1)  =  even_in_g(x1)
0  =  0
even_out_g(x1)  =  even_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
odd_in_g(x1)  =  odd_in_g(x1)
odd_out_g(x1)  =  odd_out_g
U2_g(x1, x2)  =  U2_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

The argument filtering Pi contains the following mapping:
even_in_g(x1)  =  even_in_g(x1)
0  =  0
even_out_g(x1)  =  even_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
odd_in_g(x1)  =  odd_in_g(x1)
odd_out_g(x1)  =  odd_out_g
U2_g(x1, x2)  =  U2_g(x2)

(3) 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:

EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

The argument filtering Pi contains the following mapping:
even_in_g(x1)  =  even_in_g(x1)
0  =  0
even_out_g(x1)  =  even_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
odd_in_g(x1)  =  odd_in_g(x1)
odd_out_g(x1)  =  odd_out_g
U2_g(x1, x2)  =  U2_g(x2)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
ODD_IN_G(x1)  =  ODD_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)

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

(4) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

The argument filtering Pi contains the following mapping:
even_in_g(x1)  =  even_in_g(x1)
0  =  0
even_out_g(x1)  =  even_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
odd_in_g(x1)  =  odd_in_g(x1)
odd_out_g(x1)  =  odd_out_g
U2_g(x1, x2)  =  U2_g(x2)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
ODD_IN_G(x1)  =  ODD_IN_G(x1)
U2_G(x1, x2)  =  U2_G(x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

The argument filtering Pi contains the following mapping:
even_in_g(x1)  =  even_in_g(x1)
0  =  0
even_out_g(x1)  =  even_out_g
s(x1)  =  s(x1)
U1_g(x1, x2)  =  U1_g(x2)
odd_in_g(x1)  =  odd_in_g(x1)
odd_out_g(x1)  =  odd_out_g
U2_g(x1, x2)  =  U2_g(x2)
EVEN_IN_G(x1)  =  EVEN_IN_G(x1)
ODD_IN_G(x1)  =  ODD_IN_G(x1)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

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

(11) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))


Used ordering: Polynomial interpretation [POLO]:

POL(EVEN_IN_G(x1)) = x1   
POL(ODD_IN_G(x1)) = 2 + x1   
POL(s(x1)) = 1 + x1   

(12) Obligation:

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

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) PrologToPiTRSProof (SOUND transformation)

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

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

Pi is empty.

(17) 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:

EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

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

(18) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

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

(23) PiDPToQDPProof (EQUIVALENT transformation)

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

(24) Obligation:

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

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))

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

(25) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))


Used ordering: Polynomial interpretation [POLO]:

POL(EVEN_IN_G(x1)) = x1   
POL(ODD_IN_G(x1)) = 2 + x1   
POL(s(x1)) = 1 + x1   

(26) Obligation:

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