(0) Obligation:

Clauses:

less(0, s(X2)).
less(X, Y) :- ','(no(zero(X)), ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1)))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X3).
failure(b).

Query: less(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

lessA(s(T33)) :- lessA(T33).
lessB(0, s(T4)).
lessB(s(T21), 0) :- lessA(T21).
lessB(s(T21), s(T37)) :- lessB(T21, T37).

Query: lessB(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:
lessB_in: (b,f)
lessA_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)

(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:

LESSB_IN_GA(s(T21), 0) → U2_GA(T21, lessA_in_g(T21))
LESSB_IN_GA(s(T21), 0) → LESSA_IN_G(T21)
LESSA_IN_G(s(T33)) → U1_G(T33, lessA_in_g(T33))
LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
LESSB_IN_GA(s(T21), s(T37)) → U3_GA(T21, T37, lessB_in_ga(T21, T37))
LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(6) Obligation:

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

LESSB_IN_GA(s(T21), 0) → U2_GA(T21, lessA_in_g(T21))
LESSB_IN_GA(s(T21), 0) → LESSA_IN_G(T21)
LESSA_IN_G(s(T33)) → U1_G(T33, lessA_in_g(T33))
LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
LESSB_IN_GA(s(T21), s(T37)) → U3_GA(T21, T37, lessB_in_ga(T21, T37))
LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

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

(14) 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:

  • LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

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

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LESSB_IN_GA(s(T21)) → LESSB_IN_GA(T21)

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

(21) 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:

  • LESSB_IN_GA(s(T21)) → LESSB_IN_GA(T21)
    The graph contains the following edges 1 > 1

(22) YES