(0) Obligation:
Clauses:
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).
Query: less(a,g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
lessA(s(s(X1)), s(s(X2))) :- lessA(X1, X2).
lessA(s(s(X1)), s(s(X2))) :- lessA(X1, X2).
Clauses:
lesscA(0, s(X1)).
lesscA(s(0), s(s(X1))).
lesscA(s(s(X1)), s(s(X2))) :- lesscA(X1, X2).
lesscA(s(0), s(s(X1))).
lesscA(s(s(X1)), s(s(X2))) :- lesscA(X1, X2).
Afs:
lessA(x1, x2) = lessA(x2)
(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:
lessA_in: (f,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
LESSA_IN_AG(s(s(X1)), s(s(X2))) → U1_AG(X1, X2, lessA_in_ag(X1, X2))
LESSA_IN_AG(s(s(X1)), s(s(X2))) → LESSA_IN_AG(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
lessA_in_ag(
x1,
x2) =
lessA_in_ag(
x2)
s(
x1) =
s(
x1)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3) =
U1_AG(
x2,
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:
LESSA_IN_AG(s(s(X1)), s(s(X2))) → U1_AG(X1, X2, lessA_in_ag(X1, X2))
LESSA_IN_AG(s(s(X1)), s(s(X2))) → LESSA_IN_AG(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
lessA_in_ag(
x1,
x2) =
lessA_in_ag(
x2)
s(
x1) =
s(
x1)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3) =
U1_AG(
x2,
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 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LESSA_IN_AG(s(s(X1)), s(s(X2))) → LESSA_IN_AG(X1, X2)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
LESSA_IN_AG(
x1,
x2) =
LESSA_IN_AG(
x2)
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:
LESSA_IN_AG(s(s(X2))) → LESSA_IN_AG(X2)
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:
- LESSA_IN_AG(s(s(X2))) → LESSA_IN_AG(X2)
The graph contains the following edges 1 > 1
(10) YES