Left Termination of the query pattern insert(b,f,f) w.r.t. the given Prolog program could not be shown:



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

insert3(X, void0, tree3(X, void0, void0)).
insert3(X, tree3(X, Left, Right), tree3(X, Left, Right)).
insert3(X, tree3(Y, Left, Right), tree3(Y, Left1, Right)) :- less2(X, Y), insert3(X, Left, Left1).
insert3(X, tree3(Y, Left, Right), tree3(Y, Left, Right1)) :- less2(Y, X), insert3(X, Right, Right1).
less2(00, s1(underscore)).
less2(s1(X), s1(Y)) :- less2(X, Y).


With regard to the inferred argument filtering the predicates were used in the following modes:
insert3: (b,f,f)
less2: (b,f) (f,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, 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_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, x6)


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

INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GA2(X, Y)
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_INSERT_3_IN_2_GAA6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AG2(Y, X)
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_INSERT_3_IN_4_GAA6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, x6)
IF_INSERT_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GAA2(x2, x6)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
LESS_2_IN_GA2(x1, x2)  =  LESS_2_IN_GA1(x1)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
IF_INSERT_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GAA2(x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)

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_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GA2(X, Y)
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_INSERT_3_IN_2_GAA6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AG2(Y, X)
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_INSERT_3_IN_4_GAA6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, x6)
IF_INSERT_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GAA2(x2, x6)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
LESS_2_IN_GA2(x1, x2)  =  LESS_2_IN_GA1(x1)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
IF_INSERT_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GAA2(x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 9 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_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_1
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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

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

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

LESS_2_IN_AA -> LESS_2_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LESS_2_IN_AA}.

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

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

IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa2(x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa2(x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa2(x2, x6)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))

The TRS R consists of the following rules:

less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))

The argument filtering Pi contains the following mapping:
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga1(x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag1(x1)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)

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

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

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

IF_INSERT_3_IN_1_GAA2(X, less_2_out_ga1(Y)) -> INSERT_3_IN_GAA1(X)
IF_INSERT_3_IN_3_GAA2(X, less_2_out_ag1(Y)) -> INSERT_3_IN_GAA1(X)
INSERT_3_IN_GAA1(X) -> IF_INSERT_3_IN_1_GAA2(X, less_2_in_ga1(X))
INSERT_3_IN_GAA1(X) -> IF_INSERT_3_IN_3_GAA2(X, less_2_in_ag1(X))

The TRS R consists of the following rules:

less_2_in_ga1(0_0) -> less_2_out_ga1(s_1)
less_2_in_ga1(s_1) -> if_less_2_in_1_ga1(less_2_in_aa)
less_2_in_ag1(s_1) -> less_2_out_ag1(0_0)
less_2_in_ag1(s_1) -> if_less_2_in_1_ag1(less_2_in_aa)
if_less_2_in_1_ga1(less_2_out_aa2(X, Y)) -> less_2_out_ga1(s_1)
if_less_2_in_1_ag1(less_2_out_aa2(X, Y)) -> less_2_out_ag1(s_1)
less_2_in_aa -> less_2_out_aa2(0_0, s_1)
less_2_in_aa -> if_less_2_in_1_aa1(less_2_in_aa)
if_less_2_in_1_aa1(less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_1, s_1)

The set Q consists of the following terms:

less_2_in_ga1(x0)
less_2_in_ag1(x0)
if_less_2_in_1_ga1(x0)
if_less_2_in_1_ag1(x0)
less_2_in_aa
if_less_2_in_1_aa1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {INSERT_3_IN_GAA1, IF_INSERT_3_IN_1_GAA2, IF_INSERT_3_IN_3_GAA2}.
With regard to the inferred argument filtering the predicates were used in the following modes:
insert3: (b,f,f)
less2: (b,f) (f,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, 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_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, x6)


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

INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GA2(X, Y)
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_INSERT_3_IN_2_GAA6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AG2(Y, X)
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_INSERT_3_IN_4_GAA6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, x6)
IF_INSERT_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GAA3(x1, x2, x6)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
LESS_2_IN_GA2(x1, x2)  =  LESS_2_IN_GA1(x1)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
IF_INSERT_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GAA3(x1, x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)

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_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GA2(X, Y)
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_GA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AA3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_INSERT_3_IN_2_GAA6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AG2(Y, X)
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_AG3(X, Y, less_2_in_aa2(X, Y))
LESS_2_IN_AG2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_INSERT_3_IN_4_GAA6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, x6)
IF_INSERT_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GAA3(x1, x2, x6)
LESS_2_IN_AG2(x1, x2)  =  LESS_2_IN_AG1(x2)
LESS_2_IN_GA2(x1, x2)  =  LESS_2_IN_GA1(x1)
IF_LESS_2_IN_1_AG3(x1, x2, x3)  =  IF_LESS_2_IN_1_AG1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
IF_INSERT_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GAA3(x1, x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 9 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_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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_2_IN_AA2(s_11(X), s_11(Y)) -> LESS_2_IN_AA2(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s_11(x1)  =  s_1
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA

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

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

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

LESS_2_IN_AA -> LESS_2_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {LESS_2_IN_AA}.

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

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

IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))

The TRS R consists of the following rules:

insert_3_in_gaa3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gaa3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_insert_3_in_1_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_in_gaa3(X, Left, Left1))
insert_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
if_insert_3_in_3_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_in_gaa3(X, Right, Right1))
if_insert_3_in_4_gaa6(X, Y, Left, Right, Right1, insert_3_out_gaa3(X, Right, Right1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gaa6(X, Y, Left, Right, Left1, insert_3_out_gaa3(X, Left, Left1)) -> insert_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gaa3(x1, x2, x3)  =  insert_3_in_gaa1(x1)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
insert_3_out_gaa3(x1, x2, x3)  =  insert_3_out_gaa3(x1, x2, x3)
if_insert_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gaa2(x1, x6)
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
if_insert_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gaa3(x1, x2, x6)
if_insert_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gaa2(x1, x6)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
if_insert_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gaa3(x1, x2, x6)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting 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:

IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA3(X, Left, Left1)
IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA3(X, Right, Right1)
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
INSERT_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))

The TRS R consists of the following rules:

less_2_in_ga2(0_0, s_11(underscore)) -> less_2_out_ga2(0_0, s_11(underscore))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(X, Y, less_2_in_aa2(X, Y))
less_2_in_ag2(0_0, s_11(underscore)) -> less_2_out_ag2(0_0, s_11(underscore))
less_2_in_ag2(s_11(X), s_11(Y)) -> if_less_2_in_1_ag3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_ga3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_11(X), s_11(Y))
if_less_2_in_1_ag3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_11(X), s_11(Y))
less_2_in_aa2(0_0, s_11(underscore)) -> less_2_out_aa2(0_0, s_11(underscore))
less_2_in_aa2(s_11(X), s_11(Y)) -> if_less_2_in_1_aa3(X, Y, less_2_in_aa2(X, Y))
if_less_2_in_1_aa3(X, Y, less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_11(X), s_11(Y))

The argument filtering Pi contains the following mapping:
tree_33(x1, x2, x3)  =  tree_31(x1)
0_0  =  0_0
s_11(x1)  =  s_1
less_2_in_ga2(x1, x2)  =  less_2_in_ga1(x1)
less_2_out_ga2(x1, x2)  =  less_2_out_ga2(x1, x2)
if_less_2_in_1_ga3(x1, x2, x3)  =  if_less_2_in_1_ga1(x3)
less_2_in_aa2(x1, x2)  =  less_2_in_aa
less_2_out_aa2(x1, x2)  =  less_2_out_aa2(x1, x2)
if_less_2_in_1_aa3(x1, x2, x3)  =  if_less_2_in_1_aa1(x3)
less_2_in_ag2(x1, x2)  =  less_2_in_ag1(x2)
less_2_out_ag2(x1, x2)  =  less_2_out_ag2(x1, x2)
if_less_2_in_1_ag3(x1, x2, x3)  =  if_less_2_in_1_ag1(x3)
IF_INSERT_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GAA2(x1, x6)
IF_INSERT_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GAA2(x1, x6)
INSERT_3_IN_GAA3(x1, x2, x3)  =  INSERT_3_IN_GAA1(x1)

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

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

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

IF_INSERT_3_IN_1_GAA2(X, less_2_out_ga2(X, Y)) -> INSERT_3_IN_GAA1(X)
IF_INSERT_3_IN_3_GAA2(X, less_2_out_ag2(Y, X)) -> INSERT_3_IN_GAA1(X)
INSERT_3_IN_GAA1(X) -> IF_INSERT_3_IN_1_GAA2(X, less_2_in_ga1(X))
INSERT_3_IN_GAA1(X) -> IF_INSERT_3_IN_3_GAA2(X, less_2_in_ag1(X))

The TRS R consists of the following rules:

less_2_in_ga1(0_0) -> less_2_out_ga2(0_0, s_1)
less_2_in_ga1(s_1) -> if_less_2_in_1_ga1(less_2_in_aa)
less_2_in_ag1(s_1) -> less_2_out_ag2(0_0, s_1)
less_2_in_ag1(s_1) -> if_less_2_in_1_ag1(less_2_in_aa)
if_less_2_in_1_ga1(less_2_out_aa2(X, Y)) -> less_2_out_ga2(s_1, s_1)
if_less_2_in_1_ag1(less_2_out_aa2(X, Y)) -> less_2_out_ag2(s_1, s_1)
less_2_in_aa -> less_2_out_aa2(0_0, s_1)
less_2_in_aa -> if_less_2_in_1_aa1(less_2_in_aa)
if_less_2_in_1_aa1(less_2_out_aa2(X, Y)) -> less_2_out_aa2(s_1, s_1)

The set Q consists of the following terms:

less_2_in_ga1(x0)
less_2_in_ag1(x0)
if_less_2_in_1_ga1(x0)
if_less_2_in_1_ag1(x0)
less_2_in_aa
if_less_2_in_1_aa1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {INSERT_3_IN_GAA1, IF_INSERT_3_IN_1_GAA2, IF_INSERT_3_IN_3_GAA2}.