(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(g,a,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

less13(s(T38), s(T37)) :- less13(T38, T37).
less32(s(T73), s(T75)) :- less32(T73, T75).
max1(s(T23), s(T24), s(T23)) :- less13(T24, T23).
max1(s(T59), T61, T61) :- less32(T59, T61).
max1(s(T92), T94, T94) :- less32(T92, T94).

Clauses:

lessc13(0, s(T31)).
lessc13(s(T38), s(T37)) :- lessc13(T38, T37).
lessc32(0, s(T68)).
lessc32(s(T73), s(T75)) :- lessc32(T73, T75).

Afs:

max1(x1, x2, x3)  =  max1(x1)

(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: (b,f,f)
less13_in: (f,b)
less32_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MAX1_IN_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_GAA(s(T23), s(T24), s(T23)) → LESS13_IN_AG(T24, T23)
LESS13_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, less13_in_ag(T38, T37))
LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)
MAX1_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, less32_in_ga(T59, T61))
MAX1_IN_GAA(s(T59), T61, T61) → LESS32_IN_GA(T59, T61)
LESS32_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, less32_in_ga(T73, T75))
LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)
MAX1_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, less32_in_ga(T92, T94))

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
MAX1_IN_GAA(x1, x2, x3)  =  MAX1_IN_GAA(x1)
U3_GAA(x1, x2, x3)  =  U3_GAA(x1, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_GAA(x1, x2, x3)  =  U4_GAA(x1, x3)
LESS32_IN_GA(x1, x2)  =  LESS32_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U5_GAA(x1, x2, x3)  =  U5_GAA(x1, 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_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_GAA(s(T23), s(T24), s(T23)) → LESS13_IN_AG(T24, T23)
LESS13_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, less13_in_ag(T38, T37))
LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)
MAX1_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, less32_in_ga(T59, T61))
MAX1_IN_GAA(s(T59), T61, T61) → LESS32_IN_GA(T59, T61)
LESS32_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, less32_in_ga(T73, T75))
LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)
MAX1_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, less32_in_ga(T92, T94))

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
MAX1_IN_GAA(x1, x2, x3)  =  MAX1_IN_GAA(x1)
U3_GAA(x1, x2, x3)  =  U3_GAA(x1, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U4_GAA(x1, x2, x3)  =  U4_GAA(x1, x3)
LESS32_IN_GA(x1, x2)  =  LESS32_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U5_GAA(x1, x2, x3)  =  U5_GAA(x1, 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_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS32_IN_GA(x1, x2)  =  LESS32_IN_GA(x1)

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_GA(s(T73)) → LESS32_IN_GA(T73)

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_GA(s(T73)) → LESS32_IN_GA(T73)
    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_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)

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_AG(s(T37)) → LESS13_IN_AG(T37)

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_AG(s(T37)) → LESS13_IN_AG(T37)
    The graph contains the following edges 1 > 1

(16) YES