(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).

Queries:

add(a,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

add1(T4, 0, T4).
add1(T8, s(T12), s(T9)) :- add1(T8, T12, T9).

Queries:

add1(a,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
add1_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

add1_in_aga(T4, 0, T4) → add1_out_aga(T4, 0, T4)
add1_in_aga(T8, s(T12), s(T9)) → U1_aga(T8, T12, T9, add1_in_aga(T8, T12, T9))
U1_aga(T8, T12, T9, add1_out_aga(T8, T12, T9)) → add1_out_aga(T8, s(T12), s(T9))

The argument filtering Pi contains the following mapping:
add1_in_aga(x1, x2, x3)  =  add1_in_aga(x2)
0  =  0
add1_out_aga(x1, x2, x3)  =  add1_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

add1_in_aga(T4, 0, T4) → add1_out_aga(T4, 0, T4)
add1_in_aga(T8, s(T12), s(T9)) → U1_aga(T8, T12, T9, add1_in_aga(T8, T12, T9))
U1_aga(T8, T12, T9, add1_out_aga(T8, T12, T9)) → add1_out_aga(T8, s(T12), s(T9))

The argument filtering Pi contains the following mapping:
add1_in_aga(x1, x2, x3)  =  add1_in_aga(x2)
0  =  0
add1_out_aga(x1, x2, x3)  =  add1_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

ADD1_IN_AGA(T8, s(T12), s(T9)) → U1_AGA(T8, T12, T9, add1_in_aga(T8, T12, T9))
ADD1_IN_AGA(T8, s(T12), s(T9)) → ADD1_IN_AGA(T8, T12, T9)

The TRS R consists of the following rules:

add1_in_aga(T4, 0, T4) → add1_out_aga(T4, 0, T4)
add1_in_aga(T8, s(T12), s(T9)) → U1_aga(T8, T12, T9, add1_in_aga(T8, T12, T9))
U1_aga(T8, T12, T9, add1_out_aga(T8, T12, T9)) → add1_out_aga(T8, s(T12), s(T9))

The argument filtering Pi contains the following mapping:
add1_in_aga(x1, x2, x3)  =  add1_in_aga(x2)
0  =  0
add1_out_aga(x1, x2, x3)  =  add1_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADD1_IN_AGA(x1, x2, x3)  =  ADD1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD1_IN_AGA(T8, s(T12), s(T9)) → U1_AGA(T8, T12, T9, add1_in_aga(T8, T12, T9))
ADD1_IN_AGA(T8, s(T12), s(T9)) → ADD1_IN_AGA(T8, T12, T9)

The TRS R consists of the following rules:

add1_in_aga(T4, 0, T4) → add1_out_aga(T4, 0, T4)
add1_in_aga(T8, s(T12), s(T9)) → U1_aga(T8, T12, T9, add1_in_aga(T8, T12, T9))
U1_aga(T8, T12, T9, add1_out_aga(T8, T12, T9)) → add1_out_aga(T8, s(T12), s(T9))

The argument filtering Pi contains the following mapping:
add1_in_aga(x1, x2, x3)  =  add1_in_aga(x2)
0  =  0
add1_out_aga(x1, x2, x3)  =  add1_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADD1_IN_AGA(x1, x2, x3)  =  ADD1_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD1_IN_AGA(T8, s(T12), s(T9)) → ADD1_IN_AGA(T8, T12, T9)

The TRS R consists of the following rules:

add1_in_aga(T4, 0, T4) → add1_out_aga(T4, 0, T4)
add1_in_aga(T8, s(T12), s(T9)) → U1_aga(T8, T12, T9, add1_in_aga(T8, T12, T9))
U1_aga(T8, T12, T9, add1_out_aga(T8, T12, T9)) → add1_out_aga(T8, s(T12), s(T9))

The argument filtering Pi contains the following mapping:
add1_in_aga(x1, x2, x3)  =  add1_in_aga(x2)
0  =  0
add1_out_aga(x1, x2, x3)  =  add1_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADD1_IN_AGA(x1, x2, x3)  =  ADD1_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD1_IN_AGA(T8, s(T12), s(T9)) → ADD1_IN_AGA(T8, T12, T9)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD1_IN_AGA(x1, x2, x3)  =  ADD1_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD1_IN_AGA(s(T12)) → ADD1_IN_AGA(T12)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(13) 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:

  • ADD1_IN_AGA(s(T12)) → ADD1_IN_AGA(T12)
    The graph contains the following edges 1 > 1

(14) TRUE