(0) Obligation:
Clauses:
sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).
Query: sum(a,a,g)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
sumA(X1, s(s(X2)), s(s(X3))) :- sumA(X1, X2, X3).
sumA(X1, s(s(X2)), s(s(X3))) :- sumA(X1, X2, X3).
Clauses:
sumcA(X1, 0, X1).
sumcA(X1, s(0), s(X1)).
sumcA(X1, s(s(X2)), s(s(X3))) :- sumcA(X1, X2, X3).
sumcA(X1, s(0), s(X1)).
sumcA(X1, s(s(X2)), s(s(X3))) :- sumcA(X1, X2, X3).
Afs:
sumA(x1, x2, x3) = sumA(x3)
(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:
sumA_in: (f,f,b)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
SUMA_IN_AAG(X1, s(s(X2)), s(s(X3))) → U1_AAG(X1, X2, X3, sumA_in_aag(X1, X2, X3))
SUMA_IN_AAG(X1, s(s(X2)), s(s(X3))) → SUMA_IN_AAG(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
sumA_in_aag(
x1,
x2,
x3) =
sumA_in_aag(
x3)
s(
x1) =
s(
x1)
SUMA_IN_AAG(
x1,
x2,
x3) =
SUMA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4) =
U1_AAG(
x3,
x4)
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:
SUMA_IN_AAG(X1, s(s(X2)), s(s(X3))) → U1_AAG(X1, X2, X3, sumA_in_aag(X1, X2, X3))
SUMA_IN_AAG(X1, s(s(X2)), s(s(X3))) → SUMA_IN_AAG(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
sumA_in_aag(
x1,
x2,
x3) =
sumA_in_aag(
x3)
s(
x1) =
s(
x1)
SUMA_IN_AAG(
x1,
x2,
x3) =
SUMA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4) =
U1_AAG(
x3,
x4)
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:
SUMA_IN_AAG(X1, s(s(X2)), s(s(X3))) → SUMA_IN_AAG(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
SUMA_IN_AAG(
x1,
x2,
x3) =
SUMA_IN_AAG(
x3)
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:
SUMA_IN_AAG(s(s(X3))) → SUMA_IN_AAG(X3)
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:
- SUMA_IN_AAG(s(s(X3))) → SUMA_IN_AAG(X3)
The graph contains the following edges 1 > 1
(10) YES