(0) Obligation:

Clauses:

add(X, 0, Y) :- ','(!, eq(X, Y)).
add(X, Y, s(Z)) :- ','(p(Y, P), add(X, P, Z)).
p(0, 0).
p(s(X), X).
eq(X, X).

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