(0) Obligation:

Clauses:

max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

max(a,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

less13(s(T36), s(T38)) :- less13(T36, T38).
less32(s(T75), s(T74)) :- less32(T75, T74).
max1(s(T24), s(T22), s(T24)) :- less13(T22, T24).
max1(s(T61), T60, T60) :- less32(T61, T60).
max1(s(T94), T93, T93) :- less32(T94, T93).

Clauses:

lessc13(0, s(T31)).
lessc13(s(T36), s(T38)) :- lessc13(T36, T38).
lessc32(0, s(T68)).
lessc32(s(T75), s(T74)) :- lessc32(T75, T74).

Afs:

max1(x1, x2, x3)  =  max1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
max1_in: (f,b,f)
less13_in: (b,f)
less32_in: (f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MAX1_IN_AGA(s(T24), s(T22), s(T24)) → U3_AGA(T24, T22, less13_in_ga(T22, T24))
MAX1_IN_AGA(s(T24), s(T22), s(T24)) → LESS13_IN_GA(T22, T24)
LESS13_IN_GA(s(T36), s(T38)) → U1_GA(T36, T38, less13_in_ga(T36, T38))
LESS13_IN_GA(s(T36), s(T38)) → LESS13_IN_GA(T36, T38)
MAX1_IN_AGA(s(T61), T60, T60) → U4_AGA(T61, T60, less32_in_ag(T61, T60))
MAX1_IN_AGA(s(T61), T60, T60) → LESS32_IN_AG(T61, T60)
LESS32_IN_AG(s(T75), s(T74)) → U2_AG(T75, T74, less32_in_ag(T75, T74))
LESS32_IN_AG(s(T75), s(T74)) → LESS32_IN_AG(T75, T74)
MAX1_IN_AGA(s(T94), T93, T93) → U5_AGA(T94, T93, less32_in_ag(T94, T93))

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less13_in_ga(x1, x2)  =  less13_in_ga(x1)
less32_in_ag(x1, x2)  =  less32_in_ag(x2)
MAX1_IN_AGA(x1, x2, x3)  =  MAX1_IN_AGA(x2)
U3_AGA(x1, x2, x3)  =  U3_AGA(x2, x3)
LESS13_IN_GA(x1, x2)  =  LESS13_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U4_AGA(x1, x2, x3)  =  U4_AGA(x2, x3)
LESS32_IN_AG(x1, x2)  =  LESS32_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U5_AGA(x1, x2, x3)  =  U5_AGA(x2, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MAX1_IN_AGA(s(T24), s(T22), s(T24)) → U3_AGA(T24, T22, less13_in_ga(T22, T24))
MAX1_IN_AGA(s(T24), s(T22), s(T24)) → LESS13_IN_GA(T22, T24)
LESS13_IN_GA(s(T36), s(T38)) → U1_GA(T36, T38, less13_in_ga(T36, T38))
LESS13_IN_GA(s(T36), s(T38)) → LESS13_IN_GA(T36, T38)
MAX1_IN_AGA(s(T61), T60, T60) → U4_AGA(T61, T60, less32_in_ag(T61, T60))
MAX1_IN_AGA(s(T61), T60, T60) → LESS32_IN_AG(T61, T60)
LESS32_IN_AG(s(T75), s(T74)) → U2_AG(T75, T74, less32_in_ag(T75, T74))
LESS32_IN_AG(s(T75), s(T74)) → LESS32_IN_AG(T75, T74)
MAX1_IN_AGA(s(T94), T93, T93) → U5_AGA(T94, T93, less32_in_ag(T94, T93))

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less13_in_ga(x1, x2)  =  less13_in_ga(x1)
less32_in_ag(x1, x2)  =  less32_in_ag(x2)
MAX1_IN_AGA(x1, x2, x3)  =  MAX1_IN_AGA(x2)
U3_AGA(x1, x2, x3)  =  U3_AGA(x2, x3)
LESS13_IN_GA(x1, x2)  =  LESS13_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U4_AGA(x1, x2, x3)  =  U4_AGA(x2, x3)
LESS32_IN_AG(x1, x2)  =  LESS32_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U5_AGA(x1, x2, x3)  =  U5_AGA(x2, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS32_IN_AG(s(T75), s(T74)) → LESS32_IN_AG(T75, T74)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS32_IN_AG(x1, x2)  =  LESS32_IN_AG(x2)

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

(8) PiDPToQDPProof (SOUND transformation)

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

(9) Obligation:

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

LESS32_IN_AG(s(T74)) → LESS32_IN_AG(T74)

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

(10) 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:

  • LESS32_IN_AG(s(T74)) → LESS32_IN_AG(T74)
    The graph contains the following edges 1 > 1

(11) YES

(12) Obligation:

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

LESS13_IN_GA(s(T36), s(T38)) → LESS13_IN_GA(T36, T38)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS13_IN_GA(x1, x2)  =  LESS13_IN_GA(x1)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

LESS13_IN_GA(s(T36)) → LESS13_IN_GA(T36)

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

(15) 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:

  • LESS13_IN_GA(s(T36)) → LESS13_IN_GA(T36)
    The graph contains the following edges 1 > 1

(16) YES