(0) Obligation:

Clauses:

num(0).
num(X) :- ','(no(zero(0)), ','(p(X, Y), num(Y))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: num(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

numA(0).

Query: numA(g)

(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:
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

numA_in_g(0) → numA_out_g(0)

The argument filtering Pi contains the following mapping:
numA_in_g(x1)  =  numA_in_g(x1)
0  =  0
numA_out_g(x1)  =  numA_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

numA_in_g(0) → numA_out_g(0)

The argument filtering Pi contains the following mapping:
numA_in_g(x1)  =  numA_in_g(x1)
0  =  0
numA_out_g(x1)  =  numA_out_g

(5) DependencyPairsProof (EQUIVALENT transformation)

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

numA_in_g(0) → numA_out_g(0)

The argument filtering Pi contains the following mapping:
numA_in_g(x1)  =  numA_in_g(x1)
0  =  0
numA_out_g(x1)  =  numA_out_g

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

(6) Obligation:

Pi DP problem:
P is empty.
The TRS R consists of the following rules:

numA_in_g(0) → numA_out_g(0)

The argument filtering Pi contains the following mapping:
numA_in_g(x1)  =  numA_in_g(x1)
0  =  0
numA_out_g(x1)  =  numA_out_g

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,R,Pi) chain.

(8) YES