(0) Obligation:
Clauses:
p(0).
p(s(X)) :- ','(geq(X, Y), p(Y)).
geq(X, X).
geq(s(X), Y) :- geq(X, Y).
Query: p(g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
pB(X1, X1) :- pA(X1).
pB(s(X1), X2) :- pB(X1, X2).
pA(s(X1)) :- pB(X1, X2).
Clauses:
pcA(0).
pcA(s(X1)) :- qcB(X1, X2).
qcB(X1, X1) :- pcA(X1).
qcB(s(X1), X2) :- qcB(X1, X2).
Afs:
pA(x1) = pA(x1)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
pA_in: (b)
pB_in: (b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(s(X1)) → U3_G(X1, pB_in_ga(X1, X2))
PA_IN_G(s(X1)) → PB_IN_GA(X1, X2)
PB_IN_GA(X1, X1) → U1_GA(X1, pA_in_g(X1))
PB_IN_GA(X1, X1) → PA_IN_G(X1)
PB_IN_GA(s(X1), X2) → U2_GA(X1, X2, pB_in_ga(X1, X2))
PB_IN_GA(s(X1), X2) → PB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
pA_in_g(
x1) =
pA_in_g(
x1)
s(
x1) =
s(
x1)
pB_in_ga(
x1,
x2) =
pB_in_ga(
x1)
PA_IN_G(
x1) =
PA_IN_G(
x1)
U3_G(
x1,
x2) =
U3_G(
x1,
x2)
PB_IN_GA(
x1,
x2) =
PB_IN_GA(
x1)
U1_GA(
x1,
x2) =
U1_GA(
x1,
x2)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
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:
PA_IN_G(s(X1)) → U3_G(X1, pB_in_ga(X1, X2))
PA_IN_G(s(X1)) → PB_IN_GA(X1, X2)
PB_IN_GA(X1, X1) → U1_GA(X1, pA_in_g(X1))
PB_IN_GA(X1, X1) → PA_IN_G(X1)
PB_IN_GA(s(X1), X2) → U2_GA(X1, X2, pB_in_ga(X1, X2))
PB_IN_GA(s(X1), X2) → PB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
pA_in_g(
x1) =
pA_in_g(
x1)
s(
x1) =
s(
x1)
pB_in_ga(
x1,
x2) =
pB_in_ga(
x1)
PA_IN_G(
x1) =
PA_IN_G(
x1)
U3_G(
x1,
x2) =
U3_G(
x1,
x2)
PB_IN_GA(
x1,
x2) =
PB_IN_GA(
x1)
U1_GA(
x1,
x2) =
U1_GA(
x1,
x2)
U2_GA(
x1,
x2,
x3) =
U2_GA(
x1,
x3)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(s(X1)) → PB_IN_GA(X1, X2)
PB_IN_GA(X1, X1) → PA_IN_G(X1)
PB_IN_GA(s(X1), X2) → PB_IN_GA(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
PA_IN_G(
x1) =
PA_IN_G(
x1)
PB_IN_GA(
x1,
x2) =
PB_IN_GA(
x1)
We have to consider all (P,R,Pi)-chains
(7) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PA_IN_G(s(X1)) → PB_IN_GA(X1)
PB_IN_GA(X1) → PA_IN_G(X1)
PB_IN_GA(s(X1)) → PB_IN_GA(X1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(9) 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:
- PB_IN_GA(X1) → PA_IN_G(X1)
The graph contains the following edges 1 >= 1
- PB_IN_GA(s(X1)) → PB_IN_GA(X1)
The graph contains the following edges 1 > 1
- PA_IN_G(s(X1)) → PB_IN_GA(X1)
The graph contains the following edges 1 > 1
(10) YES