Left Termination of the query pattern in_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

in(a,g).

We use the technique of [30]. 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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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


Using Dependency Pairs [1,30] 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, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → 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, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → 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, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, 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, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_ag(X, Right))
U3_AG(X, Y, X1, 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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x3, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

IN_IN_AG(X, tree(Y, Left, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → 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, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → 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, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, 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, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_ag(X, Right))
U3_AG(X, Y, X1, 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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U5_AG(x1, x2, x3)  =  U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x1, x5)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x3, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 5 SCCs with 13 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
              ↳ PiDP

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, 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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ PiDP

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x1, x4, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x3, x5)
IN_IN_GG(x1, x2)  =  IN_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP

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

IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Left, less_in_gg(X, Y))
U3_GG(X, Right, less_out_gg) → IN_IN_GG(X, Right)
U1_GG(X, Left, less_out_gg) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Right, less_in_gg(Y, X))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → 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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: 



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
U3_AG(X, Y, X1, 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, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
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, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, 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, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
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, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_ag(X, Right))
U4_ag(X, Y, X1, Right, in_out_ag(X, Right)) → in_out_ag(X, tree(Y, X1, 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)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → IN_IN_AG(X, Right)

The TRS R consists of the following rules:

less_in_ga(0, s(X)) → less_out_ga(0, s(X))
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)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x4, x5)
IN_IN_AG(x1, x2)  =  IN_IN_AG(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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

U3_AG(Right, less_out_ga) → IN_IN_AG(Right)
IN_IN_AG(tree(Y, X1, Right)) → U3_AG(Right, less_in_ga(Y))

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: