(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