Left Termination of the query pattern insert_in_3(a, a, g) w.r.t. the given Prolog program could not be shown:



Prolog
  ↳ PrologToPiTRSProof
  ↳ 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) (f,f) (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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
PiTRS
      ↳ DependencyPairsProof
  ↳ PrologToPiTRSProof

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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x6)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
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_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x6)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
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(x1, x2, x3, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x2, x4, x6)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x3, x5, x6)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)

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

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

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_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x6)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
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(x1, x2, x3, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x2, x4, x6)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x3, x5, x6)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 19 less nodes.

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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
  ↳ PrologToPiTRSProof

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LESS_IN_AALESS_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN_AALESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA.





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

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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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
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(x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, 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
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s
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)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof
  ↳ PrologToPiTRSProof

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) → less_out_gg
less_in_gg(s, s) → U5_gg(less_in_aa)
U5_gg(less_out_aa(X, Y)) → less_out_gg
less_in_aaless_out_aa(0, s)
less_in_aaU5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0)
less_in_aa
U5_aa(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:


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) (f,f) (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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ 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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, x6)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
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_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, x6)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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(x1, x2, x3, x5, x6)
INSERT_IN_AAG(x1, x2, x3)  =  INSERT_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x1, x2, x4, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
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(x1, x2, x3, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x3, x5, x6)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ 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_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U4_AAG(X, Y, Left, Right, Right1, insert_in_gag(X, Right, Right1))
U3_AAG(X, Y, Left, Right, Right1, less_out_ga(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, x6)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA
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(x1, x2, x3, x5, x6)
INSERT_IN_AAG(x1, x2, x3)  =  INSERT_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5, x6)  =  U2_AAG(x1, x2, x4, x5, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U5_AA(x1, x2, x3)  =  U5_AA(x3)
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(x1, x2, x3, x5, x6)
U2_GAG(x1, x2, x3, x4, x5, x6)  =  U2_GAG(x1, x2, x4, x5, x6)
U3_AAG(x1, x2, x3, x4, x5, x6)  =  U3_AAG(x2, x3, x5, x6)
U1_GAG(x1, x2, x3, x4, x5, x6)  =  U1_GAG(x1, x2, x4, x5, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 19 less nodes.

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, x6)
LESS_IN_AA(x1, x2)  =  LESS_IN_AA

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

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

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

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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

LESS_IN_AALESS_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN_AALESS_IN_AA

The TRS R consists of the following rules:none


s = LESS_IN_AA evaluates to t =LESS_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:




Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA.





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

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_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(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_aa(X, Y))
U5_gg(X, Y, less_out_aa(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_aa(X, Y))
U5_ga(X, Y, less_out_aa(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_gag(X, Right, Right1))
U4_aag(X, Y, Left, Right, Right1, insert_out_gag(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, x3)
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
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U5_ag(x1, x2, x3)  =  U5_ag(x3)
U2_aag(x1, x2, x3, x4, x5, x6)  =  U2_aag(x1, x2, x4, x5, x6)
insert_in_gag(x1, x2, x3)  =  insert_in_gag(x1, x3)
insert_out_gag(x1, x2, x3)  =  insert_out_gag(x1, x2, x3)
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(x1, x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_gag(x1, x2, x3, x4, x5, x6)  =  U2_gag(x1, x2, x4, x5, 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(x1, x2, x3, x5, 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(x1, x2)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(x3)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U4_aag(x1, x2, x3, x4, x5, x6)  =  U4_aag(x1, x2, x3, x5, 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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
PiDP
                    ↳ PiDPToQDPProof

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_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s
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(x3)
less_in_aa(x1, x2)  =  less_in_aa
less_out_aa(x1, x2)  =  less_out_aa(x1, x2)
U5_aa(x1, x2, x3)  =  U5_aa(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
QDP
                        ↳ QDPSizeChangeProof

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

U3_GAG(X, Y, Left, Right1, less_out_gg(Y, X)) → 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(X, Y)) → 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) → less_out_gg(0, s)
less_in_gg(s, s) → U5_gg(less_in_aa)
U5_gg(less_out_aa(X, Y)) → less_out_gg(s, s)
less_in_aaless_out_aa(0, s)
less_in_aaU5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0)
less_in_aa
U5_aa(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: