(0) Obligation:

Clauses:

num(0) :- !.
num(X) :- ','(p(X, Y), num(Y)).
p(0, 0).
p(s(X), X).

Query: num(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

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

numA_in_g(0) → numA_out_g(0)
numA_in_g(s(T6)) → U1_g(T6, numA_in_g(T6))
U1_g(T6, numA_out_g(T6)) → numA_out_g(s(T6))

Pi is empty.

(3) 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:

NUMA_IN_G(s(T6)) → U1_G(T6, numA_in_g(T6))
NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)

The TRS R consists of the following rules:

numA_in_g(0) → numA_out_g(0)
numA_in_g(s(T6)) → U1_g(T6, numA_in_g(T6))
U1_g(T6, numA_out_g(T6)) → numA_out_g(s(T6))

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

(4) Obligation:

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

NUMA_IN_G(s(T6)) → U1_G(T6, numA_in_g(T6))
NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)

The TRS R consists of the following rules:

numA_in_g(0) → numA_out_g(0)
numA_in_g(s(T6)) → U1_g(T6, numA_in_g(T6))
U1_g(T6, numA_out_g(T6)) → numA_out_g(s(T6))

Pi is empty.
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:

NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)

The TRS R consists of the following rules:

numA_in_g(0) → numA_out_g(0)
numA_in_g(s(T6)) → U1_g(T6, numA_in_g(T6))
U1_g(T6, numA_out_g(T6)) → numA_out_g(s(T6))

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)

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

(11) 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:

  • NUMA_IN_G(s(T6)) → NUMA_IN_G(T6)
    The graph contains the following edges 1 > 1

(12) YES