(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

addA(T5, 0, T5).
addA(T20, s(T36), s(T21)) :- addA(T20, T36, T21).

Query: addA(a,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
addA_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3)  =  addA_in_aga(x2)
0  =  0
addA_out_aga(x1, x2, x3)  =  addA_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:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3)  =  addA_in_aga(x2)
0  =  0
addA_out_aga(x1, x2, x3)  =  addA_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:

ADDA_IN_AGA(T20, s(T36), s(T21)) → U1_AGA(T20, T36, T21, addA_in_aga(T20, T36, T21))
ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3)  =  addA_in_aga(x2)
0  =  0
addA_out_aga(x1, x2, x3)  =  addA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADDA_IN_AGA(x1, x2, x3)  =  ADDA_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:

ADDA_IN_AGA(T20, s(T36), s(T21)) → U1_AGA(T20, T36, T21, addA_in_aga(T20, T36, T21))
ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3)  =  addA_in_aga(x2)
0  =  0
addA_out_aga(x1, x2, x3)  =  addA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADDA_IN_AGA(x1, x2, x3)  =  ADDA_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:

ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

The TRS R consists of the following rules:

addA_in_aga(T5, 0, T5) → addA_out_aga(T5, 0, T5)
addA_in_aga(T20, s(T36), s(T21)) → U1_aga(T20, T36, T21, addA_in_aga(T20, T36, T21))
U1_aga(T20, T36, T21, addA_out_aga(T20, T36, T21)) → addA_out_aga(T20, s(T36), s(T21))

The argument filtering Pi contains the following mapping:
addA_in_aga(x1, x2, x3)  =  addA_in_aga(x2)
0  =  0
addA_out_aga(x1, x2, x3)  =  addA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
ADDA_IN_AGA(x1, x2, x3)  =  ADDA_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:

ADDA_IN_AGA(T20, s(T36), s(T21)) → ADDA_IN_AGA(T20, T36, T21)

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

(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:

ADDA_IN_AGA(s(T36)) → ADDA_IN_AGA(T36)

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:

  • ADDA_IN_AGA(s(T36)) → ADDA_IN_AGA(T36)
    The graph contains the following edges 1 > 1

(14) YES