(0) Obligation:
Clauses:
add(X, 0, X).
add(X, Y, s(Z)) :- ','(no(zero(Y)), ','(p(Y, P), add(X, P, Z))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X).
failure(b).
Query: add(a,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
addA(X1, s(X2), s(X3)) :- addA(X1, X2, X3).
Clauses:
addcA(X1, 0, X1).
addcA(X1, s(X2), s(X3)) :- addcA(X1, X2, X3).
Afs:
addA(x1, x2, x3) = addA(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:
addA_in: (f,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
ADDA_IN_AGA(X1, s(X2), s(X3)) → U1_AGA(X1, X2, X3, addA_in_aga(X1, X2, X3))
ADDA_IN_AGA(X1, s(X2), s(X3)) → ADDA_IN_AGA(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
addA_in_aga(
x1,
x2,
x3) =
addA_in_aga(
x2)
s(
x1) =
s(
x1)
ADDA_IN_AGA(
x1,
x2,
x3) =
ADDA_IN_AGA(
x2)
U1_AGA(
x1,
x2,
x3,
x4) =
U1_AGA(
x2,
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:
ADDA_IN_AGA(X1, s(X2), s(X3)) → U1_AGA(X1, X2, X3, addA_in_aga(X1, X2, X3))
ADDA_IN_AGA(X1, s(X2), s(X3)) → ADDA_IN_AGA(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
addA_in_aga(
x1,
x2,
x3) =
addA_in_aga(
x2)
s(
x1) =
s(
x1)
ADDA_IN_AGA(
x1,
x2,
x3) =
ADDA_IN_AGA(
x2)
U1_AGA(
x1,
x2,
x3,
x4) =
U1_AGA(
x2,
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:
ADDA_IN_AGA(X1, s(X2), s(X3)) → ADDA_IN_AGA(X1, X2, X3)
R is empty.
The argument filtering Pi contains the following mapping:
s(
x1) =
s(
x1)
ADDA_IN_AGA(
x1,
x2,
x3) =
ADDA_IN_AGA(
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:
ADDA_IN_AGA(s(X2)) → ADDA_IN_AGA(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:
- ADDA_IN_AGA(s(X2)) → ADDA_IN_AGA(X2)
The graph contains the following edges 1 > 1
(10) YES