(0) Obligation:

Clauses:

even(s(s(X))) :- even(X).
even(0).
lte(s(X), s(Y)) :- lte(X, Y).
lte(0, Y).
goal :- ','(lte(X, s(s(s(s(0))))), even(X)).

Queries:

goal().

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

even35(s(s(T13))) :- even35(T13).
goal1 :- ','(ltec8(s(T7)), even35(T7)).

Clauses:

evenc35(s(s(T13))) :- evenc35(T13).
evenc35(0).
ltec8(s(s(s(0)))).
ltec8(s(s(0))).
ltec8(s(0)).
ltec8(0).

Afs:

goal1  =  goal1

(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:
even35_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOAL1_IN_U2_1(ltec8_in_a(s(T7)))
U2_1(ltec8_out_a(s(T7))) → U3_1(even35_in_g(T7))
U2_1(ltec8_out_a(s(T7))) → EVEN35_IN_G(T7)
EVEN35_IN_G(s(s(T13))) → U1_G(T13, even35_in_g(T13))
EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)

The TRS R consists of the following rules:

ltec8_in_a(s(s(s(0)))) → ltec8_out_a(s(s(s(0))))
ltec8_in_a(s(s(0))) → ltec8_out_a(s(s(0)))
ltec8_in_a(s(0)) → ltec8_out_a(s(0))
ltec8_in_a(0) → ltec8_out_a(0)

The argument filtering Pi contains the following mapping:
ltec8_in_a(x1)  =  ltec8_in_a
ltec8_out_a(x1)  =  ltec8_out_a(x1)
s(x1)  =  s(x1)
even35_in_g(x1)  =  even35_in_g(x1)
GOAL1_IN_  =  GOAL1_IN_
U2_1(x1)  =  U2_1(x1)
U3_1(x1)  =  U3_1(x1)
EVEN35_IN_G(x1)  =  EVEN35_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

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:

GOAL1_IN_U2_1(ltec8_in_a(s(T7)))
U2_1(ltec8_out_a(s(T7))) → U3_1(even35_in_g(T7))
U2_1(ltec8_out_a(s(T7))) → EVEN35_IN_G(T7)
EVEN35_IN_G(s(s(T13))) → U1_G(T13, even35_in_g(T13))
EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)

The TRS R consists of the following rules:

ltec8_in_a(s(s(s(0)))) → ltec8_out_a(s(s(s(0))))
ltec8_in_a(s(s(0))) → ltec8_out_a(s(s(0)))
ltec8_in_a(s(0)) → ltec8_out_a(s(0))
ltec8_in_a(0) → ltec8_out_a(0)

The argument filtering Pi contains the following mapping:
ltec8_in_a(x1)  =  ltec8_in_a
ltec8_out_a(x1)  =  ltec8_out_a(x1)
s(x1)  =  s(x1)
even35_in_g(x1)  =  even35_in_g(x1)
GOAL1_IN_  =  GOAL1_IN_
U2_1(x1)  =  U2_1(x1)
U3_1(x1)  =  U3_1(x1)
EVEN35_IN_G(x1)  =  EVEN35_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, 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 4 less nodes.

(6) Obligation:

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

EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)

The TRS R consists of the following rules:

ltec8_in_a(s(s(s(0)))) → ltec8_out_a(s(s(s(0))))
ltec8_in_a(s(s(0))) → ltec8_out_a(s(s(0)))
ltec8_in_a(s(0)) → ltec8_out_a(s(0))
ltec8_in_a(0) → ltec8_out_a(0)

The argument filtering Pi contains the following mapping:
ltec8_in_a(x1)  =  ltec8_in_a
ltec8_out_a(x1)  =  ltec8_out_a(x1)
s(x1)  =  s(x1)
EVEN35_IN_G(x1)  =  EVEN35_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:

EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)

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:

EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)

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

(11) 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:

  • EVEN35_IN_G(s(s(T13))) → EVEN35_IN_G(T13)
    The graph contains the following edges 1 > 1

(12) YES