Left Termination proof of /tmp/tmp2SGnG4/evenodd2.pl

Left Termination of the query pattern even_in_1(g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 CutEliminatorProof (⇐)

↳2 Prolog

↳3 UndefinedPredicateHandlerProof (⇐)

↳4 Prolog

  ↳5 PrologToPiTRSProof (⇐)

    ↳6 PiTRS

    ↳7 DependencyPairsProof (⇔)

    ↳8 PiDP

    ↳9 DependencyGraphProof (⇔)

    ↳10 PiDP

    ↳11 UsableRulesProof (⇔)

    ↳12 PiDP

    ↳13 PiDPToQDPProof (⇐)

    ↳14 QDP

    ↳15 Narrowing (⇐)

    ↳16 QDP

    ↳17 Narrowing (⇐)

    ↳18 QDP

    ↳19 UsableRulesProof (⇔)

    ↳20 QDP

    ↳21 QReductionProof (⇔)

    ↳22 QDP

    ↳23 Instantiation (⇔)

    ↳24 QDP

    ↳25 Instantiation (⇔)

    ↳26 QDP

    ↳27 DependencyGraphProof (⇔)

    ↳28 AND

      ↳29 QDP

        ↳30 NonTerminationProof (⇔)

        ↳31 FALSE

      ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔)

        ↳34 TRUE

  ↳35 PrologToPiTRSProof (⇐)

    ↳36 PiTRS

    ↳37 DependencyPairsProof (⇔)

    ↳38 PiDP

    ↳39 DependencyGraphProof (⇔)

    ↳40 PiDP

    ↳41 UsableRulesProof (⇔)

    ↳42 PiDP

    ↳43 PiDPToQDPProof (⇐)

    ↳44 QDP

    ↳45 UsableRulesReductionPairsProof (⇔)

    ↳46 QDP

    ↳47 Narrowing (⇐)

    ↳48 QDP

    ↳49 Narrowing (⇐)

    ↳50 QDP

    ↳51 UsableRulesProof (⇔)

    ↳52 QDP

    ↳53 QReductionProof (⇔)

    ↳54 QDP

    ↳55 Instantiation (⇔)

    ↳56 QDP

    ↳57 Instantiation (⇔)

    ↳58 QDP

    ↳59 NonTerminationProof (⇔)

    ↳60 FALSE

(0) Obligation:

Clauses:

even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).

Queries:

even(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

even(0) :- true.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- true.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).

Queries:

even(g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

even(0) :- true.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- true.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).
true.

Queries:

even(g).

(5) 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) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

(7) 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(0) → U1_G(true_in_)
EVEN_IN_G(0) → TRUE_IN_
EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
EVEN_IN_G(N) → P_IN_GA(N, P)
U2_G(N, p_out_ga(N, P)) → U3_G(N, odd_in_g(P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(s(0)) → U4_G(true_in_)
ODD_IN_G(s(0)) → TRUE_IN_
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
ODD_IN_G(N) → P_IN_GA(N, P)
U5_G(N, p_out_ga(N, P)) → U6_G(N, even_in_g(P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
U1_g(x1) = U1_g(x1)
true_in_ = true_in_
true_out_ = true_out_
even_out_g(x1) = even_out_g(x1)
U2_g(x1, x2) = U2_g(x1, x2)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
U3_g(x1, x2) = U3_g(x1, x2)
odd_in_g(x1) = odd_in_g(x1)
U4_g(x1) = U4_g(x1)
odd_out_g(x1) = odd_out_g(x1)
U5_g(x1, x2) = U5_g(x1, x2)
U6_g(x1, x2) = U6_g(x1, x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U1_G(x1) = U1_G(x1)
TRUE_IN_ = TRUE_IN_
U2_G(x1, x2) = U2_G(x1, x2)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_G(x1, x2) = U3_G(x1, x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U4_G(x1) = U4_G(x1)
U5_G(x1, x2) = U5_G(x1, x2)
U6_G(x1, x2) = U6_G(x1, x2)

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

(8) Obligation:

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

EVEN_IN_G(0) → U1_G(true_in_)
EVEN_IN_G(0) → TRUE_IN_
EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
EVEN_IN_G(N) → P_IN_GA(N, P)
U2_G(N, p_out_ga(N, P)) → U3_G(N, odd_in_g(P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(s(0)) → U4_G(true_in_)
ODD_IN_G(s(0)) → TRUE_IN_
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
ODD_IN_G(N) → P_IN_GA(N, P)
U5_G(N, p_out_ga(N, P)) → U6_G(N, even_in_g(P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes.

(10) Obligation:

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

EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
U1_g(x1) = U1_g(x1)
true_in_ = true_in_
true_out_ = true_out_
even_out_g(x1) = even_out_g(x1)
U2_g(x1, x2) = U2_g(x1, x2)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
U3_g(x1, x2) = U3_g(x1, x2)
odd_in_g(x1) = odd_in_g(x1)
U4_g(x1) = U4_g(x1)
odd_out_g(x1) = odd_out_g(x1)
U5_g(x1, x2) = U5_g(x1, x2)
U6_g(x1, x2) = U6_g(x1, x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U2_G(x1, x2) = U2_G(x1, x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U5_G(x1, x2) = U5_G(x1, x2)

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

(11) UsableRulesProof (EQUIVALENT transformation)

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

(12) Obligation:

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

EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U2_G(x1, x2) = U2_G(x1, x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U5_G(x1, x2) = U5_G(x1, x2)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

EVEN_IN_G(N) → U2_G(N, p_in_ga(N))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(15) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule EVEN_IN_G(N) → U2_G(N, p_in_ga(N)) at position [1] we obtained the following new rules [LPAR04]:

EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))

(16) Obligation:

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

U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(17) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule ODD_IN_G(N) → U5_G(N, p_in_ga(N)) at position [1] we obtained the following new rules [LPAR04]:

ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))

(18) Obligation:

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

U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(19) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(20) Obligation:

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

U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(21) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(22) Obligation:

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

U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))

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

(23) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P) we obtained the following new rules [LPAR04]:

U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)
U2_G(s(z0), p_out_ga(s(z0), z0)) → ODD_IN_G(z0)

(24) Obligation:

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

U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))
U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)
U2_G(s(z0), p_out_ga(s(z0), z0)) → ODD_IN_G(z0)

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

(25) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P) we obtained the following new rules [LPAR04]:

U5_G(0, p_out_ga(0, 0)) → EVEN_IN_G(0)
U5_G(s(z0), p_out_ga(s(z0), z0)) → EVEN_IN_G(z0)

(26) Obligation:

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

EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))
U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)
U2_G(s(z0), p_out_ga(s(z0), z0)) → ODD_IN_G(z0)
U5_G(0, p_out_ga(0, 0)) → EVEN_IN_G(0)
U5_G(s(z0), p_out_ga(s(z0), z0)) → EVEN_IN_G(z0)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(28) Complex Obligation (AND)

(29) Obligation:

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

U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)
ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
U5_G(0, p_out_ga(0, 0)) → EVEN_IN_G(0)
EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))

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

(30) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = ODD_IN_G(0) evaluates to t =ODD_IN_G(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0))
with rule ODD_IN_G(0) → U5_G(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U5_G(0, p_out_ga(0, 0)) → EVEN_IN_G(0)
with rule U5_G(0, p_out_ga(0, 0)) → EVEN_IN_G(0) at position [] and matcher [ ]

EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0))
with rule EVEN_IN_G(0) → U2_G(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)
with rule U2_G(0, p_out_ga(0, 0)) → ODD_IN_G(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(31) FALSE

(32) Obligation:

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

ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))
U5_G(s(z0), p_out_ga(s(z0), z0)) → EVEN_IN_G(z0)
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
U2_G(s(z0), p_out_ga(s(z0), z0)) → ODD_IN_G(z0)

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

(33) 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:

U5_G(s(z0), p_out_ga(s(z0), z0)) → EVEN_IN_G(z0)
The graph contains the following edges 1 > 1, 2 > 1
U2_G(s(z0), p_out_ga(s(z0), z0)) → ODD_IN_G(z0)
The graph contains the following edges 1 > 1, 2 > 1
EVEN_IN_G(s(x0)) → U2_G(s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1
ODD_IN_G(s(x0)) → U5_G(s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1

(34) TRUE

(35) PrologToPiTRSProof (SOUND transformation)

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(36) Obligation:

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

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

(37) 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(0) → U1_G(true_in_)
EVEN_IN_G(0) → TRUE_IN_
EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
EVEN_IN_G(N) → P_IN_GA(N, P)
U2_G(N, p_out_ga(N, P)) → U3_G(N, odd_in_g(P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(s(0)) → U4_G(true_in_)
ODD_IN_G(s(0)) → TRUE_IN_
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
ODD_IN_G(N) → P_IN_GA(N, P)
U5_G(N, p_out_ga(N, P)) → U6_G(N, even_in_g(P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
U1_g(x1) = U1_g(x1)
true_in_ = true_in_
true_out_ = true_out_
even_out_g(x1) = even_out_g
U2_g(x1, x2) = U2_g(x2)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U3_g(x1, x2) = U3_g(x2)
odd_in_g(x1) = odd_in_g(x1)
U4_g(x1) = U4_g(x1)
odd_out_g(x1) = odd_out_g
U5_g(x1, x2) = U5_g(x2)
U6_g(x1, x2) = U6_g(x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U1_G(x1) = U1_G(x1)
TRUE_IN_ = TRUE_IN_
U2_G(x1, x2) = U2_G(x2)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_G(x1, x2) = U3_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U4_G(x1) = U4_G(x1)
U5_G(x1, x2) = U5_G(x2)
U6_G(x1, x2) = U6_G(x2)

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

(38) Obligation:

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

EVEN_IN_G(0) → U1_G(true_in_)
EVEN_IN_G(0) → TRUE_IN_
EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
EVEN_IN_G(N) → P_IN_GA(N, P)
U2_G(N, p_out_ga(N, P)) → U3_G(N, odd_in_g(P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(s(0)) → U4_G(true_in_)
ODD_IN_G(s(0)) → TRUE_IN_
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
ODD_IN_G(N) → P_IN_GA(N, P)
U5_G(N, p_out_ga(N, P)) → U6_G(N, even_in_g(P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
U1_g(x1) = U1_g(x1)
true_in_ = true_in_
true_out_ = true_out_
even_out_g(x1) = even_out_g
U2_g(x1, x2) = U2_g(x2)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U3_g(x1, x2) = U3_g(x2)
odd_in_g(x1) = odd_in_g(x1)
U4_g(x1) = U4_g(x1)
odd_out_g(x1) = odd_out_g
U5_g(x1, x2) = U5_g(x2)
U6_g(x1, x2) = U6_g(x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U1_G(x1) = U1_G(x1)
TRUE_IN_ = TRUE_IN_
U2_G(x1, x2) = U2_G(x2)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_G(x1, x2) = U3_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U4_G(x1) = U4_G(x1)
U5_G(x1, x2) = U5_G(x2)
U6_G(x1, x2) = U6_G(x2)

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes.

(40) Obligation:

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

EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

even_in_g(0) → U1_g(true_in_)
true_in_ → true_out_
U1_g(true_out_) → even_out_g(0)
even_in_g(N) → U2_g(N, p_in_ga(N, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_g(N, p_out_ga(N, P)) → U3_g(N, odd_in_g(P))
odd_in_g(s(0)) → U4_g(true_in_)
U4_g(true_out_) → odd_out_g(s(0))
odd_in_g(N) → U5_g(N, p_in_ga(N, P))
U5_g(N, p_out_ga(N, P)) → U6_g(N, even_in_g(P))
U6_g(N, even_out_g(P)) → odd_out_g(N)
U3_g(N, odd_out_g(P)) → even_out_g(N)

The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
U1_g(x1) = U1_g(x1)
true_in_ = true_in_
true_out_ = true_out_
even_out_g(x1) = even_out_g
U2_g(x1, x2) = U2_g(x2)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U3_g(x1, x2) = U3_g(x2)
odd_in_g(x1) = odd_in_g(x1)
U4_g(x1) = U4_g(x1)
odd_out_g(x1) = odd_out_g
U5_g(x1, x2) = U5_g(x2)
U6_g(x1, x2) = U6_g(x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U2_G(x1, x2) = U2_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U5_G(x1, x2) = U5_G(x2)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

EVEN_IN_G(N) → U2_G(N, p_in_ga(N, P))
U2_G(N, p_out_ga(N, P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(N, p_in_ga(N, P))
U5_G(N, p_out_ga(N, P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
U2_G(x1, x2) = U2_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U5_G(x1, x2) = U5_G(x2)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

EVEN_IN_G(N) → U2_G(p_in_ga(N))
U2_G(p_out_ga(P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(p_in_ga(N))
U5_G(p_out_ga(P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(45) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0
POL(EVEN_IN_G(x₁)) = 2·x₁
POL(ODD_IN_G(x₁)) = 2·x₁
POL(U2_G(x₁)) = x₁
POL(U5_G(x₁)) = 2·x₁
POL(p_in_ga(x₁)) = x₁
POL(p_out_ga(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

(46) Obligation:

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

EVEN_IN_G(N) → U2_G(p_in_ga(N))
U2_G(p_out_ga(P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(p_in_ga(N))
U5_G(p_out_ga(P)) → EVEN_IN_G(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(47) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule EVEN_IN_G(N) → U2_G(p_in_ga(N)) at position [0] we obtained the following new rules [LPAR04]:

EVEN_IN_G(0) → U2_G(p_out_ga(0))

(48) Obligation:

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

U2_G(p_out_ga(P)) → ODD_IN_G(P)
ODD_IN_G(N) → U5_G(p_in_ga(N))
U5_G(p_out_ga(P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule ODD_IN_G(N) → U5_G(p_in_ga(N)) at position [0] we obtained the following new rules [LPAR04]:

ODD_IN_G(0) → U5_G(p_out_ga(0))

(50) Obligation:

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

U2_G(p_out_ga(P)) → ODD_IN_G(P)
U5_G(p_out_ga(P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(p_out_ga(0))
ODD_IN_G(0) → U5_G(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(51) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(52) Obligation:

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

U2_G(p_out_ga(P)) → ODD_IN_G(P)
U5_G(p_out_ga(P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(p_out_ga(0))
ODD_IN_G(0) → U5_G(p_out_ga(0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(53) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(54) Obligation:

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

U2_G(p_out_ga(P)) → ODD_IN_G(P)
U5_G(p_out_ga(P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(p_out_ga(0))
ODD_IN_G(0) → U5_G(p_out_ga(0))

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

(55) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_G(p_out_ga(P)) → ODD_IN_G(P) we obtained the following new rules [LPAR04]:

U2_G(p_out_ga(0)) → ODD_IN_G(0)

(56) Obligation:

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

U5_G(p_out_ga(P)) → EVEN_IN_G(P)
EVEN_IN_G(0) → U2_G(p_out_ga(0))
ODD_IN_G(0) → U5_G(p_out_ga(0))
U2_G(p_out_ga(0)) → ODD_IN_G(0)

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

(57) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U5_G(p_out_ga(P)) → EVEN_IN_G(P) we obtained the following new rules [LPAR04]:

U5_G(p_out_ga(0)) → EVEN_IN_G(0)

(58) Obligation:

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

EVEN_IN_G(0) → U2_G(p_out_ga(0))
ODD_IN_G(0) → U5_G(p_out_ga(0))
U2_G(p_out_ga(0)) → ODD_IN_G(0)
U5_G(p_out_ga(0)) → EVEN_IN_G(0)

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

(59) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U2_G(p_out_ga(0)) evaluates to t =U2_G(p_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

U2_G(p_out_ga(0)) → ODD_IN_G(0)
with rule U2_G(p_out_ga(0)) → ODD_IN_G(0) at position [] and matcher [ ]

ODD_IN_G(0) → U5_G(p_out_ga(0))
with rule ODD_IN_G(0) → U5_G(p_out_ga(0)) at position [] and matcher [ ]

U5_G(p_out_ga(0)) → EVEN_IN_G(0)
with rule U5_G(p_out_ga(0)) → EVEN_IN_G(0) at position [] and matcher [ ]

EVEN_IN_G(0) → U2_G(p_out_ga(0))
with rule EVEN_IN_G(0) → U2_G(p_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(0) Obligation:

(1) CutEliminatorProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateHandlerProof (SOUND transformation)

(4) Obligation:

(5) PrologToPiTRSProof (SOUND transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) UsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

(13) PiDPToQDPProof (SOUND transformation)

(14) Obligation:

(15) Narrowing (SOUND transformation)

(16) Obligation:

(17) Narrowing (SOUND transformation)

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) QReductionProof (EQUIVALENT transformation)

(22) Obligation:

(23) Instantiation (EQUIVALENT transformation)

(24) Obligation:

(25) Instantiation (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Complex Obligation (AND)

(29) Obligation:

(30) NonTerminationProof (EQUIVALENT transformation)

(31) FALSE

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) TRUE

(35) PrologToPiTRSProof (SOUND transformation)

(36) Obligation:

(37) DependencyPairsProof (EQUIVALENT transformation)

(38) Obligation:

(39) DependencyGraphProof (EQUIVALENT transformation)

(40) Obligation:

(41) UsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

(43) PiDPToQDPProof (SOUND transformation)

(44) Obligation:

(45) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(46) Obligation:

(47) Narrowing (SOUND transformation)

(48) Obligation:

(49) Narrowing (SOUND transformation)

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) QReductionProof (EQUIVALENT transformation)

(54) Obligation:

(55) Instantiation (EQUIVALENT transformation)

(56) Obligation:

(57) Instantiation (EQUIVALENT transformation)

(58) Obligation:

(59) NonTerminationProof (EQUIVALENT transformation)

(60) FALSE