Left Termination of the query pattern max_in_3(a, a, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(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,a,g).

(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).
max1(s(T13), 0, s(T13)).
max1(s(T23), s(T24), s(T23)) :- less13(T24, T23).
max1(T47, T46, T46) :- less13(T47, s(T46)).
max1(0, T59, T59).
max1(s(T66), T65, T65) :- less13(T66, T65).

Queries:

max1(a,a,g).

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

max1_in_aag(s(T13), 0, s(T13)) → max1_out_aag(s(T13), 0, s(T13))
max1_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, less13_out_ag(T24, T23)) → max1_out_aag(s(T23), s(T24), s(T23))
max1_in_aag(T47, T46, T46) → U3_aag(T47, T46, less13_in_ag(T47, s(T46)))
U3_aag(T47, T46, less13_out_ag(T47, s(T46))) → max1_out_aag(T47, T46, T46)
max1_in_aag(0, T59, T59) → max1_out_aag(0, T59, T59)
max1_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, less13_in_ag(T66, T65))
U4_aag(T66, T65, less13_out_ag(T66, T65)) → max1_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
max1_in_aag(x1, x2, x3)  =  max1_in_aag(x3)
s(x1)  =  s(x1)
max1_out_aag(x1, x2, x3)  =  max1_out_aag(x1, x2)
U2_aag(x1, x2, x3)  =  U2_aag(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
U3_aag(x1, x2, x3)  =  U3_aag(x2, x3)
U4_aag(x1, x2, x3)  =  U4_aag(x2, 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_aag(s(T13), 0, s(T13)) → max1_out_aag(s(T13), 0, s(T13))
max1_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, less13_out_ag(T24, T23)) → max1_out_aag(s(T23), s(T24), s(T23))
max1_in_aag(T47, T46, T46) → U3_aag(T47, T46, less13_in_ag(T47, s(T46)))
U3_aag(T47, T46, less13_out_ag(T47, s(T46))) → max1_out_aag(T47, T46, T46)
max1_in_aag(0, T59, T59) → max1_out_aag(0, T59, T59)
max1_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, less13_in_ag(T66, T65))
U4_aag(T66, T65, less13_out_ag(T66, T65)) → max1_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
max1_in_aag(x1, x2, x3)  =  max1_in_aag(x3)
s(x1)  =  s(x1)
max1_out_aag(x1, x2, x3)  =  max1_out_aag(x1, x2)
U2_aag(x1, x2, x3)  =  U2_aag(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
U3_aag(x1, x2, x3)  =  U3_aag(x2, x3)
U4_aag(x1, x2, x3)  =  U4_aag(x2, 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_AAG(s(T23), s(T24), s(T23)) → U2_AAG(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_AAG(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_AAG(T47, T46, T46) → U3_AAG(T47, T46, less13_in_ag(T47, s(T46)))
MAX1_IN_AAG(T47, T46, T46) → LESS13_IN_AG(T47, s(T46))
MAX1_IN_AAG(s(T66), T65, T65) → U4_AAG(T66, T65, less13_in_ag(T66, T65))
MAX1_IN_AAG(s(T66), T65, T65) → LESS13_IN_AG(T66, T65)

The TRS R consists of the following rules:

max1_in_aag(s(T13), 0, s(T13)) → max1_out_aag(s(T13), 0, s(T13))
max1_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, less13_out_ag(T24, T23)) → max1_out_aag(s(T23), s(T24), s(T23))
max1_in_aag(T47, T46, T46) → U3_aag(T47, T46, less13_in_ag(T47, s(T46)))
U3_aag(T47, T46, less13_out_ag(T47, s(T46))) → max1_out_aag(T47, T46, T46)
max1_in_aag(0, T59, T59) → max1_out_aag(0, T59, T59)
max1_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, less13_in_ag(T66, T65))
U4_aag(T66, T65, less13_out_ag(T66, T65)) → max1_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
max1_in_aag(x1, x2, x3)  =  max1_in_aag(x3)
s(x1)  =  s(x1)
max1_out_aag(x1, x2, x3)  =  max1_out_aag(x1, x2)
U2_aag(x1, x2, x3)  =  U2_aag(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
U3_aag(x1, x2, x3)  =  U3_aag(x2, x3)
U4_aag(x1, x2, x3)  =  U4_aag(x2, x3)
MAX1_IN_AAG(x1, x2, x3)  =  MAX1_IN_AAG(x3)
U2_AAG(x1, x2, x3)  =  U2_AAG(x1, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
U3_AAG(x1, x2, x3)  =  U3_AAG(x2, x3)
U4_AAG(x1, x2, x3)  =  U4_AAG(x2, 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_AAG(s(T23), s(T24), s(T23)) → U2_AAG(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_AAG(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_AAG(T47, T46, T46) → U3_AAG(T47, T46, less13_in_ag(T47, s(T46)))
MAX1_IN_AAG(T47, T46, T46) → LESS13_IN_AG(T47, s(T46))
MAX1_IN_AAG(s(T66), T65, T65) → U4_AAG(T66, T65, less13_in_ag(T66, T65))
MAX1_IN_AAG(s(T66), T65, T65) → LESS13_IN_AG(T66, T65)

The TRS R consists of the following rules:

max1_in_aag(s(T13), 0, s(T13)) → max1_out_aag(s(T13), 0, s(T13))
max1_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, less13_out_ag(T24, T23)) → max1_out_aag(s(T23), s(T24), s(T23))
max1_in_aag(T47, T46, T46) → U3_aag(T47, T46, less13_in_ag(T47, s(T46)))
U3_aag(T47, T46, less13_out_ag(T47, s(T46))) → max1_out_aag(T47, T46, T46)
max1_in_aag(0, T59, T59) → max1_out_aag(0, T59, T59)
max1_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, less13_in_ag(T66, T65))
U4_aag(T66, T65, less13_out_ag(T66, T65)) → max1_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
max1_in_aag(x1, x2, x3)  =  max1_in_aag(x3)
s(x1)  =  s(x1)
max1_out_aag(x1, x2, x3)  =  max1_out_aag(x1, x2)
U2_aag(x1, x2, x3)  =  U2_aag(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
U3_aag(x1, x2, x3)  =  U3_aag(x2, x3)
U4_aag(x1, x2, x3)  =  U4_aag(x2, x3)
MAX1_IN_AAG(x1, x2, x3)  =  MAX1_IN_AAG(x3)
U2_AAG(x1, x2, x3)  =  U2_AAG(x1, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x3)
U3_AAG(x1, x2, x3)  =  U3_AAG(x2, x3)
U4_AAG(x1, x2, x3)  =  U4_AAG(x2, x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(8) 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_aag(s(T13), 0, s(T13)) → max1_out_aag(s(T13), 0, s(T13))
max1_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, less13_out_ag(T24, T23)) → max1_out_aag(s(T23), s(T24), s(T23))
max1_in_aag(T47, T46, T46) → U3_aag(T47, T46, less13_in_ag(T47, s(T46)))
U3_aag(T47, T46, less13_out_ag(T47, s(T46))) → max1_out_aag(T47, T46, T46)
max1_in_aag(0, T59, T59) → max1_out_aag(0, T59, T59)
max1_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, less13_in_ag(T66, T65))
U4_aag(T66, T65, less13_out_ag(T66, T65)) → max1_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
max1_in_aag(x1, x2, x3)  =  max1_in_aag(x3)
s(x1)  =  s(x1)
max1_out_aag(x1, x2, x3)  =  max1_out_aag(x1, x2)
U2_aag(x1, x2, x3)  =  U2_aag(x1, x3)
less13_in_ag(x1, x2)  =  less13_in_ag(x2)
less13_out_ag(x1, x2)  =  less13_out_ag(x1)
U1_ag(x1, x2, x3)  =  U1_ag(x3)
U3_aag(x1, x2, x3)  =  U3_aag(x2, x3)
U4_aag(x1, x2, x3)  =  U4_aag(x2, x3)
LESS13_IN_AG(x1, x2)  =  LESS13_IN_AG(x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) 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.

(13) 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

(14) YES