Left Termination of the query pattern in_in_2(a, g) 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 PiDP
        ↳22 UsableRulesProof (⇔)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇐)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔)
        ↳27 TRUE
      ↳28 PiDP
        ↳29 UsableRulesProof (⇔)
        ↳30 PiDP
        ↳31 PiDPToQDPProof (⇐)
        ↳32 QDP
        ↳33 QDPSizeChangeProof (⇔)
        ↳34 TRUE
      ↳35 PiDP
        ↳36 UsableRulesProof (⇔)
        ↳37 PiDP
        ↳38 PiDPToQDPProof (⇐)
        ↳39 QDP
        ↳40 QDPSizeChangeProof (⇔)
        ↳41 TRUE
  ↳42 PrologToPiTRSProof (⇐)
    ↳43 PiTRS
    ↳44 DependencyPairsProof (⇔)
    ↳45 PiDP
    ↳46 DependencyGraphProof (⇔)
    ↳47 AND
      ↳48 PiDP
        ↳49 UsableRulesProof (⇔)
        ↳50 PiDP
        ↳51 PiDPToQDPProof (⇐)
        ↳52 QDP
        ↳53 QDPSizeChangeProof (⇔)
        ↳54 TRUE
      ↳55 PiDP
        ↳56 UsableRulesProof (⇔)
        ↳57 PiDP
        ↳58 PiDPToQDPProof (⇔)
        ↳59 QDP
        ↳60 QDPSizeChangeProof (⇔)
        ↳61 TRUE
      ↳62 PiDP
        ↳63 UsableRulesProof (⇔)
        ↳64 PiDP
        ↳65 PiDPToQDPProof (⇔)
        ↳66 QDP
        ↳67 QDPSizeChangeProof (⇔)
        ↳68 TRUE
      ↳69 PiDP
        ↳70 UsableRulesProof (⇔)
        ↳71 PiDP
        ↳72 PiDPToQDPProof (⇐)
        ↳73 QDP
        ↳74 QDPSizeChangeProof (⇔)
        ↳75 TRUE
      ↳76 PiDP
        ↳77 UsableRulesProof (⇔)
        ↳78 PiDP
        ↳79 PiDPToQDPProof (⇐)
        ↳80 QDP

(0) Obligation:

Clauses:

in(X, tree(X, X1, X2)).
in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)).
in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)).
less(0, s(X5)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

in(a,g).

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

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)

(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:

IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x3, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)

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

(4) Obligation:

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

IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x3, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 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:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
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_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, 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_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(17) PiDPToQDPProof (EQUIVALENT 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_GG(s(X), s(Y)) → LESS_IN_GG(X, 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_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(20) TRUE

(21) Obligation:

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

U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

U1_GG(X, Left, less_out_gg) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Left, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Right, less_in_gg(Y, X))
U3_GG(X, Right, less_out_gg) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X5)) → less_out_gg
less_in_gg(s(X), s(Y)) → U5_gg(less_in_gg(X, Y))
U5_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0)

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

(26) 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:

  • IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Left, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2

  • IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Right, less_in_gg(Y, X))
    The graph contains the following edges 1 >= 1, 2 > 2

  • U1_GG(X, Left, less_out_gg) → IN_IN_GG(X, Left)
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U3_GG(X, Right, less_out_gg) → IN_IN_GG(X, Right)
    The graph contains the following edges 1 >= 1, 2 >= 2

(27) TRUE

(28) 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:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) 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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) 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.

(33) 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

(34) TRUE

(35) Obligation:

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

IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x3, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x3, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

IN_IN_AG(tree(Y, X4, Right)) → U3_AG(Right, less_in_ga(Y))
U3_AG(Right, less_out_ga) → IN_IN_AG(Right)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U5_ga(less_in_ga(X))
U5_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U5_ga(x0)

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

(40) 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:

  • U3_AG(Right, less_out_ga) → IN_IN_AG(Right)
    The graph contains the following edges 1 >= 1

  • IN_IN_AG(tree(Y, X4, Right)) → U3_AG(Right, less_in_ga(Y))
    The graph contains the following edges 1 > 1

(41) TRUE

(42) PrologToPiTRSProof (SOUND transformation)

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

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(43) Obligation:

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

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)

(44) 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:

IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x2, x3, x4, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x2, x3, x4, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x2, x3, x4, x5)

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

(45) Obligation:

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

IN_IN_AG(X, tree(Y, Left, X3)) → U1_AG(X, Y, Left, X3, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X3)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X3, in_in_gg(X, Left))
U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X3)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X3, in_in_gg(X, Left))
U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X4, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X4, Right, in_in_gg(X, Right))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X4, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X4, Right, in_in_ag(X, Right))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x2, x3, x4, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AG(x1, x2, x3)  =  U5_AG(x2, x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x2, x3, x4, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U5_GG(x1, x2, x3)  =  U5_GG(x1, x2, x3)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x1, x2, x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x1, x2, x3, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x2, x3, x4, x5)

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

(46) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes.

(47) Complex Obligation (AND)

(48) 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:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) 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

(51) PiDPToQDPProof (SOUND transformation)

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

(52) 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.

(53) 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

(54) TRUE

(55) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

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

(56) UsableRulesProof (EQUIVALENT transformation)

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

(57) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(58) PiDPToQDPProof (EQUIVALENT transformation)

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

(59) Obligation:

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

(60) 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_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(61) TRUE

(62) Obligation:

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

U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x2, x3, x4, x5)

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

(63) UsableRulesProof (EQUIVALENT transformation)

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

(64) Obligation:

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

U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

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

(65) PiDPToQDPProof (EQUIVALENT transformation)

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

(66) Obligation:

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

U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0, x1, x2)

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

(67) 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:

  • IN_IN_GG(X, tree(Y, Left, X3)) → U1_GG(X, Y, Left, X3, less_in_gg(X, Y))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

  • IN_IN_GG(X, tree(Y, X4, Right)) → U3_GG(X, Y, X4, Right, less_in_gg(Y, X))
    The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

  • U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
    The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2

  • U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
    The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2

(68) TRUE

(69) 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:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

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

(70) UsableRulesProof (EQUIVALENT transformation)

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

(71) 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

(72) PiDPToQDPProof (SOUND transformation)

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

(73) 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.

(74) 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

(75) TRUE

(76) Obligation:

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

IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X3)) → U1_ag(X, Y, Left, X3, less_in_ag(X, Y))
less_in_ag(0, s(X5)) → less_out_ag(0, s(X5))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_ag(X, Y))
U5_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X3)) → U1_gg(X, Y, Left, X3, less_in_gg(X, Y))
less_in_gg(0, s(X5)) → less_out_gg(0, s(X5))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X3, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X4, Right)) → U3_gg(X, Y, X4, Right, less_in_gg(Y, X))
U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X4, Right, in_in_gg(X, Right))
U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X4, Right))
U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X3))
U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X3))
in_in_ag(X, tree(Y, X4, Right)) → U3_ag(X, Y, X4, Right, less_in_ga(Y, X))
less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X4, Right, in_in_ag(X, Right))
U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X4, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2)  =  in_in_ag(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
in_out_ag(x1, x2)  =  in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2)  =  in_in_gg(x1, x2)
in_out_gg(x1, x2)  =  in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x1, x2, x3)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x2, x3, x4, x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)

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

(77) UsableRulesProof (EQUIVALENT transformation)

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

(78) Obligation:

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

IN_IN_AG(X, tree(Y, X4, Right)) → U3_AG(X, Y, X4, Right, less_in_ga(Y, X))
U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

less_in_ga(0, s(X5)) → less_out_ga(0, s(X5))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_ga(X, Y))
U5_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x2, x3, x4, x5)

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

(79) PiDPToQDPProof (SOUND transformation)

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

(80) Obligation:

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

IN_IN_AG(tree(Y, X4, Right)) → U3_AG(Y, X4, Right, less_in_ga(Y))
U3_AG(Y, X4, Right, less_out_ga(Y)) → IN_IN_AG(Right)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga(0)
less_in_ga(s(X)) → U5_ga(X, less_in_ga(X))
U5_ga(X, less_out_ga(X)) → less_out_ga(s(X))

The set Q consists of the following terms:

less_in_ga(x0)
U5_ga(x0, x1)

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