(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).

Queries:

less(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

less25(s(T13)) :- less25(T13).
less1(0, s(T3)).
less1(s(s(T13)), 0) :- less25(T13).
less1(s(T9), s(T15)) :- less1(T9, T15).
less1(s(T9), s(T17)) :- less1(T9, T17).

Queries:

less1(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:
less1_in: (b,f)
less25_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_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:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_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:

LESS1_IN_GA(s(s(T13)), 0) → U2_GA(T13, less25_in_g(T13))
LESS1_IN_GA(s(s(T13)), 0) → LESS25_IN_G(T13)
LESS25_IN_G(s(T13)) → U1_G(T13, less25_in_g(T13))
LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
LESS1_IN_GA(s(T9), s(T15)) → U3_GA(T9, T15, less1_in_ga(T9, T15))
LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)
LESS1_IN_GA(s(T9), s(T17)) → U4_GA(T9, T17, less1_in_ga(T9, T17))

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESS25_IN_G(x1)  =  LESS25_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)

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

(6) Obligation:

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

LESS1_IN_GA(s(s(T13)), 0) → U2_GA(T13, less25_in_g(T13))
LESS1_IN_GA(s(s(T13)), 0) → LESS25_IN_G(T13)
LESS25_IN_G(s(T13)) → U1_G(T13, less25_in_g(T13))
LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
LESS1_IN_GA(s(T9), s(T15)) → U3_GA(T9, T15, less1_in_ga(T9, T15))
LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)
LESS1_IN_GA(s(T9), s(T17)) → U4_GA(T9, T17, less1_in_ga(T9, T17))

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESS25_IN_G(x1)  =  LESS25_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U4_GA(x1, x2, x3)  =  U4_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 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS25_IN_G(x1)  =  LESS25_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:

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

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:

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

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:

  • LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
    The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_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:

LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS1_IN_GA(x1, x2)  =  LESS1_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:

LESS1_IN_GA(s(T9)) → LESS1_IN_GA(T9)

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:

  • LESS1_IN_GA(s(T9)) → LESS1_IN_GA(T9)
    The graph contains the following edges 1 > 1

(22) TRUE