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 PrologToPiTRSProof (⇐)
    ↳12 PiTRS
    ↳13 DependencyPairsProof (⇔)
    ↳14 PiDP
    ↳15 DependencyGraphProof (⇔)
    ↳16 PiDP
    ↳17 UsableRulesProof (⇔)
    ↳18 PiDP
    ↳19 PiDPToQDPProof (⇔)
    ↳20 QDP
    ↳21 QDPSizeChangeProof (⇔)
    ↳22 TRUE

(0) Obligation:

Clauses:

even(0).
even(s(X)) :- odd(X).
odd(s(X)) :- even(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(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))

Pi is empty.

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

Pi is empty.

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

The TRS R consists of the following rules:

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

Pi is empty.
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(X)) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(X))
ODD_IN_G(s(X)) → EVEN_IN_G(X)

The TRS R consists of the following rules:

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

Pi is empty.
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(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(X)

The TRS R consists of the following rules:

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

Pi is empty.
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(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(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(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(X)

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

(11) 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(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(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)
U2_g(x1, x2)  =  U2_g(x2)
odd_out_g(x1)  =  odd_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) 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(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(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)
U2_g(x1, x2)  =  U2_g(x2)
odd_out_g(x1)  =  odd_out_g

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

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(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)
U2_g(x1, x2)  =  U2_g(x2)
odd_out_g(x1)  =  odd_out_g
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

(14) Obligation:

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

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

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(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)
U2_g(x1, x2)  =  U2_g(x2)
odd_out_g(x1)  =  odd_out_g
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

(15) DependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

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

The TRS R consists of the following rules:

even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(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)
U2_g(x1, x2)  =  U2_g(x2)
odd_out_g(x1)  =  odd_out_g
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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

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

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

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

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

(21) 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:

  • ODD_IN_G(s(X)) → EVEN_IN_G(X)
    The graph contains the following edges 1 > 1

  • EVEN_IN_G(s(X)) → ODD_IN_G(X)
    The graph contains the following edges 1 > 1

(22) TRUE