(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

less13(0, s(T31)).
less13(s(T38), s(T37)) :- less13(T38, T37).
less32(0, s(T68)).
less32(s(T73), s(T75)) :- less32(T73, T75).
max1(s(T13), 0, s(T13)).
max1(s(T23), s(T24), s(T23)) :- less13(T24, T23).
max1(0, T54, T54).
max1(s(T59), T61, T61) :- less32(T59, T61).
max1(0, T87, T87).
max1(s(T92), T94, T94) :- less32(T92, T94).

Queries:

max1(g,a,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:
max1_in: (b,f,f)
less13_in: (f,b)
less32_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, 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:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, 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:

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))

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
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

(6) 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))

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
LESS32_IN_GA(x1, x2)  =  LESS32_IN_GA(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:

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

(12) PiDPToQDPProof (SOUND 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:

LESS32_IN_GA(s(T73)) → LESS32_IN_GA(T73)

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:

  • LESS32_IN_GA(s(T73)) → LESS32_IN_GA(T73)
    The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3)  =  max1_in_gaa(x1)
s(x1)  =  s(x1)
max1_out_gaa(x1, x2, x3)  =  max1_out_gaa(x1)
U3_gaa(x1, x2, x3)  =  U3_gaa(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1, x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
0  =  0
U4_gaa(x1, x2, x3)  =  U4_gaa(x1, x3)
less32_in_ga(x1, x2)  =  less32_in_ga(x1)
less32_out_ga(x1, x2)  =  less32_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
U5_gaa(x1, x2, x3)  =  U5_gaa(x1, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)

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:

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

(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:

LESS13_IN_AG(s(T37)) → LESS13_IN_AG(T37)

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:

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

(22) YES