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

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
      ↳32 PiDP
        ↳33 UsableRulesProof (⇔)
        ↳34 PiDP

(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) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
max_in: (b,f,f)
less_in: (f,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)

(3) 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:

MAX_IN_GAA(X, Y, X) → U1_GAA(X, Y, less_in_ag(Y, X))
MAX_IN_GAA(X, Y, X) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U3_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
MAX_IN_GAA(X, Y, Y) → U2_GAA(X, Y, less_in_ga(X, s(Y)))
MAX_IN_GAA(X, Y, Y) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U3_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)
MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
U1_GAA(x1, x2, x3)  =  U1_GAA(x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x3)
U2_GAA(x1, x2, x3)  =  U2_GAA(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(4) Obligation:

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

MAX_IN_GAA(X, Y, X) → U1_GAA(X, Y, less_in_ag(Y, X))
MAX_IN_GAA(X, Y, X) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U3_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
MAX_IN_GAA(X, Y, Y) → U2_GAA(X, Y, less_in_ga(X, s(Y)))
MAX_IN_GAA(X, Y, Y) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U3_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)
MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
U1_GAA(x1, x2, x3)  =  U1_GAA(x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x3)
U2_GAA(x1, x2, x3)  =  U2_GAA(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(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 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

(12) 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:

  • LESS_IN_GA(s(X)) → LESS_IN_GA(X)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U3_ag(x1, x2, x3)  =  U3_ag(x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa
U2_gaa(x1, x2, x3)  =  U2_gaa(x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

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

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)

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

(19) 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:

  • LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
max_in: (b,f,f)
less_in: (f,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)

(23) 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:

MAX_IN_GAA(X, Y, X) → U1_GAA(X, Y, less_in_ag(Y, X))
MAX_IN_GAA(X, Y, X) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U3_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
MAX_IN_GAA(X, Y, Y) → U2_GAA(X, Y, less_in_ga(X, s(Y)))
MAX_IN_GAA(X, Y, Y) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U3_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
U1_GAA(x1, x2, x3)  =  U1_GAA(x1, x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x2, x3)
U2_GAA(x1, x2, x3)  =  U2_GAA(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(24) Obligation:

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

MAX_IN_GAA(X, Y, X) → U1_GAA(X, Y, less_in_ag(Y, X))
MAX_IN_GAA(X, Y, X) → LESS_IN_AG(Y, X)
LESS_IN_AG(s(X), s(Y)) → U3_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
MAX_IN_GAA(X, Y, Y) → U2_GAA(X, Y, less_in_ga(X, s(Y)))
MAX_IN_GAA(X, Y, Y) → LESS_IN_GA(X, s(Y))
LESS_IN_GA(s(X), s(Y)) → U3_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
MAX_IN_GAA(x1, x2, x3)  =  MAX_IN_GAA(x1)
U1_GAA(x1, x2, x3)  =  U1_GAA(x1, x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U3_AG(x1, x2, x3)  =  U3_AG(x2, x3)
U2_GAA(x1, x2, x3)  =  U2_GAA(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x1, x3)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

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

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

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

(32) Obligation:

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

max_in_gaa(X, Y, X) → U1_gaa(X, Y, less_in_ag(Y, X))
less_in_ag(0, s(X1)) → less_out_ag(0, s(X1))
less_in_ag(s(X), s(Y)) → U3_ag(X, Y, less_in_ag(X, Y))
U3_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_gaa(X, Y, less_out_ag(Y, X)) → max_out_gaa(X, Y, X)
max_in_gaa(X, Y, Y) → U2_gaa(X, Y, less_in_ga(X, s(Y)))
less_in_ga(0, s(X1)) → less_out_ga(0, s(X1))
less_in_ga(s(X), s(Y)) → U3_ga(X, Y, less_in_ga(X, Y))
U3_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U2_gaa(X, Y, less_out_ga(X, s(Y))) → max_out_gaa(X, Y, Y)

The argument filtering Pi contains the following mapping:
max_in_gaa(x1, x2, x3)  =  max_in_gaa(x1)
U1_gaa(x1, x2, x3)  =  U1_gaa(x1, x3)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U3_ag(x1, x2, x3)  =  U3_ag(x2, x3)
max_out_gaa(x1, x2, x3)  =  max_out_gaa(x1)
U2_gaa(x1, x2, x3)  =  U2_gaa(x1, x3)
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
less_out_ga(x1, x2)  =  less_out_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x1, x3)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

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

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