Left Termination proof of /tmp/tmpX

Left Termination of the query pattern insert_in_3(a, a, g) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

insert(X, void, tree(X, void, void)).
insert(X, tree(X, Left, Right), tree(X, Left, Right)).
insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)).
insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

insert(a,a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
insert_in: (f,f,b) (b,f,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:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

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:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → 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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_AAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → 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_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → 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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → INSERT_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
insert_in_aag(x1, x2, x3) = insert_in_aag(x3)
tree(x1, x2, x3) = tree(x1, x2, x3)
void = void
insert_out_aag(x1, x2, x3) = insert_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5, x6) = U1_aag(x2, x4, x5, x6)
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_aag(x1, x2, x3, x4, x5, x6) = U2_aag(x1, x2, x4, x6)
insert_in_gag(x1, x2, x3) = insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3) = insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6) = U1_gag(x1, x2, x4, x5, x6)
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_gag(x1, x2, x3, x4, x5, x6) = U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6) = U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6) = U4_gag(x2, x3, x6)
U3_aag(x1, x2, x3, x4, x5, x6) = U3_aag(x2, x3, x5, x6)
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_aag(x1, x2, x3, x4, x5, x6) = U4_aag(x2, x3, x6)
U5_GG(x1, x2, x3) = U5_GG(x3)
U5_GA(x1, x2, x3) = U5_GA(x3)
U5_AG(x1, x2, x3) = U5_AG(x3)
U3_GAG(x1, x2, x3, x4, x5, x6) = U3_GAG(x1, x2, x3, x5, x6)
U4_GAG(x1, x2, x3, x4, x5, x6) = U4_GAG(x2, x3, x6)
INSERT_IN_AAG(x1, x2, x3) = INSERT_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5, x6) = U2_AAG(x1, x2, x4, x6)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
LESS_IN_AG(x1, x2) = LESS_IN_AG(x2)
INSERT_IN_GAG(x1, x2, x3) = INSERT_IN_GAG(x1, x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U1_AAG(x1, x2, x3, x4, x5, x6) = U1_AAG(x2, x4, x5, x6)
U4_AAG(x1, x2, x3, x4, x5, x6) = U4_AAG(x2, x3, x6)
U2_GAG(x1, x2, x3, x4, x5, x6) = U2_GAG(x2, x4, x6)
U1_GAG(x1, x2, x3, x4, x5, x6) = U1_GAG(x1, x2, x4, x5, x6)
U3_AAG(x1, x2, x3, x4, x5, x6) = U3_AAG(x2, x3, x5, x6)

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:

INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_AAG(X, Y, Left, Right, Left1, less_in_ag(X, Y))
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → 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_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_AAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_AAG(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → 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_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_GAG(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_GAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → 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_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → INSERT_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ 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:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ 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:

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

↳ 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:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ 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)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
insert_in_aag(x1, x2, x3) = insert_in_aag(x3)
tree(x1, x2, x3) = tree(x1, x2, x3)
void = void
insert_out_aag(x1, x2, x3) = insert_out_aag(x1, x2)
U1_aag(x1, x2, x3, x4, x5, x6) = U1_aag(x2, x4, x5, x6)
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_aag(x1, x2, x3, x4, x5, x6) = U2_aag(x1, x2, x4, x6)
insert_in_gag(x1, x2, x3) = insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3) = insert_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5, x6) = U1_gag(x1, x2, x4, x5, x6)
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_gag(x1, x2, x3, x4, x5, x6) = U2_gag(x2, x4, x6)
U3_gag(x1, x2, x3, x4, x5, x6) = U3_gag(x1, x2, x3, x5, x6)
U4_gag(x1, x2, x3, x4, x5, x6) = U4_gag(x2, x3, x6)
U3_aag(x1, x2, x3, x4, x5, x6) = U3_aag(x2, x3, x5, x6)
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_aag(x1, x2, x3, x4, x5, x6) = U4_aag(x2, x3, x6)
U3_GAG(x1, x2, x3, x4, x5, x6) = U3_GAG(x1, x2, x3, x5, x6)
INSERT_IN_GAG(x1, x2, x3) = INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6) = U1_GAG(x1, x2, x4, x5, x6)

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:

INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_GAG(X, Y, Left, Right, Left1, less_in_gg(X, Y))
INSERT_IN_GAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U1_GAG(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → INSERT_IN_GAG(X, Left, Left1)
U3_GAG(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → INSERT_IN_GAG(X, Right, Right1)

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_GAG(x1, x2, x3, x4, x5, x6) = U3_GAG(x1, x2, x3, x5, x6)
INSERT_IN_GAG(x1, x2, x3) = INSERT_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5, x6) = U1_GAG(x1, x2, x4, x5, x6)

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:

U3_GAG(X, Y, Left, Right1, less_out_gg) → INSERT_IN_GAG(X, Right1)
INSERT_IN_GAG(X, tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
U1_GAG(X, Y, Right, Left1, less_out_gg) → INSERT_IN_GAG(X, Left1)
INSERT_IN_GAG(X, tree(Y, Left1, Right)) → U1_GAG(X, Y, Right, Left1, less_in_gg(X, Y))

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:

U1_GAG(X, Y, Right, Left1, less_out_gg) → INSERT_IN_GAG(X, Left1)
The graph contains the following edges 1 >= 1, 4 >= 2
U3_GAG(X, Y, Left, Right1, less_out_gg) → INSERT_IN_GAG(X, Right1)
The graph contains the following edges 1 >= 1, 4 >= 2
INSERT_IN_GAG(X, tree(Y, Left1, Right)) → U1_GAG(X, Y, Right, Left1, less_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
INSERT_IN_GAG(X, tree(Y, Left, Right1)) → U3_GAG(X, Y, Left, Right1, less_in_gg(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

↳ 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:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ 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)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → INSERT_IN_AAG(X, Right, Right1)

The TRS R consists of the following rules:

insert_in_aag(X, void, tree(X, void, void)) → insert_out_aag(X, void, tree(X, void, void))
insert_in_aag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_aag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_aag(X, Y, Left, Right, Left1, 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_aag(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U2_aag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, void, tree(X, void, void)) → insert_out_gag(X, void, tree(X, void, void))
insert_in_gag(X, tree(X, Left, Right), tree(X, Left, Right)) → insert_out_gag(X, tree(X, Left, Right), tree(X, Left, Right))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U1_gag(X, Y, Left, Right, Left1, 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_gag(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U2_gag(X, Y, Left, Right, Left1, insert_in_gag(X, Left, Left1))
insert_in_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_gag(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U3_gag(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U4_gag(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U4_gag(X, Y, Left, Right, Right1, insert_out_gag(X, Right, Right1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U2_gag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_gag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U2_aag(X, Y, Left, Right, Left1, insert_out_gag(X, Left, Left1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left1, Right))
insert_in_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_aag(X, Y, Left, Right, Right1, 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_aag(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_aag(X, Y, Left, Right, Right1, insert_in_aag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_aag(X, Right, Right1)) → insert_out_aag(X, tree(Y, Left, Right), tree(Y, Left, Right1))

↳ 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:

INSERT_IN_AAG(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U3_AAG(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → INSERT_IN_AAG(X, Right, Right1)

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)
INSERT_IN_AAG(x1, x2, x3) = INSERT_IN_AAG(x3)
U3_AAG(x1, x2, x3, x4, x5, x6) = U3_AAG(x2, x3, x5, x6)

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_AAG(Y, Left, Right1, less_out_ga) → INSERT_IN_AAG(Right1)
INSERT_IN_AAG(tree(Y, Left, Right1)) → U3_AAG(Y, Left, Right1, 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:

U3_AAG(Y, Left, Right1, less_out_ga) → INSERT_IN_AAG(Right1)
The graph contains the following edges 3 >= 1
INSERT_IN_AAG(tree(Y, Left, Right1)) → U3_AAG(Y, Left, Right1, less_in_ga(Y))
The graph contains the following edges 1 > 1, 1 > 2, 1 > 3