Left Termination of the query pattern insert(b,b,f) w.r.t. the given Prolog program could successfully be proven:



↳ PROLOG
  ↳ 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,b,f)
less2: (b,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)


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

INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GG2(X, Y)
LESS_2_IN_GG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GG3(X, Y, less_2_in_gg2(X, Y))
LESS_2_IN_GG2(s_11(X), s_11(Y)) -> LESS_2_IN_GG2(X, Y)
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> IF_INSERT_3_IN_2_GGA6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> INSERT_3_IN_GGA3(X, Left, Left1)
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_GG2(Y, X)
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> IF_INSERT_3_IN_4_GGA6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> INSERT_3_IN_GGA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)
INSERT_3_IN_GGA3(x1, x2, x3)  =  INSERT_3_IN_GGA2(x1, x2)
IF_INSERT_3_IN_4_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GGA3(x2, x3, x6)
IF_INSERT_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_2_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GGA3(x2, x4, x6)
IF_LESS_2_IN_1_GG3(x1, x2, x3)  =  IF_LESS_2_IN_1_GG1(x3)
LESS_2_IN_GG2(x1, x2)  =  LESS_2_IN_GG2(x1, x2)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> LESS_2_IN_GG2(X, Y)
LESS_2_IN_GG2(s_11(X), s_11(Y)) -> IF_LESS_2_IN_1_GG3(X, Y, less_2_in_gg2(X, Y))
LESS_2_IN_GG2(s_11(X), s_11(Y)) -> LESS_2_IN_GG2(X, Y)
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> IF_INSERT_3_IN_2_GGA6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> INSERT_3_IN_GGA3(X, Left, Left1)
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_GG2(Y, X)
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> IF_INSERT_3_IN_4_GGA6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> INSERT_3_IN_GGA3(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)
INSERT_3_IN_GGA3(x1, x2, x3)  =  INSERT_3_IN_GGA2(x1, x2)
IF_INSERT_3_IN_4_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_4_GGA3(x2, x3, x6)
IF_INSERT_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_2_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_2_GGA3(x2, x4, x6)
IF_LESS_2_IN_1_GG3(x1, x2, x3)  =  IF_LESS_2_IN_1_GG1(x3)
LESS_2_IN_GG2(x1, x2)  =  LESS_2_IN_GG2(x1, x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 5 less nodes.

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

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

LESS_2_IN_GG2(s_11(X), s_11(Y)) -> LESS_2_IN_GG2(X, Y)

The TRS R consists of the following rules:

insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)
LESS_2_IN_GG2(x1, x2)  =  LESS_2_IN_GG2(x1, x2)

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

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

LESS_2_IN_GG2(s_11(X), s_11(Y)) -> LESS_2_IN_GG2(X, Y)

R is empty.
Pi is empty.
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
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

LESS_2_IN_GG2(s_11(X), s_11(Y)) -> LESS_2_IN_GG2(X, Y)

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_GG2}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> INSERT_3_IN_GGA3(X, Right, Right1)
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> INSERT_3_IN_GGA3(X, Left, Left1)

The TRS R consists of the following rules:

insert_3_in_gga3(X, void_0, tree_33(X, void_0, void_0)) -> insert_3_out_gga3(X, void_0, tree_33(X, void_0, void_0))
insert_3_in_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right)) -> insert_3_out_gga3(X, tree_33(X, Left, Right), tree_33(X, Left, Right))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))
if_insert_3_in_1_gga6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_in_gga3(X, Left, Left1))
insert_3_in_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
if_insert_3_in_3_gga6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_in_gga3(X, Right, Right1))
if_insert_3_in_4_gga6(X, Y, Left, Right, Right1, insert_3_out_gga3(X, Right, Right1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_insert_3_in_2_gga6(X, Y, Left, Right, Left1, insert_3_out_gga3(X, Left, Left1)) -> insert_3_out_gga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))

The argument filtering Pi contains the following mapping:
insert_3_in_gga3(x1, x2, x3)  =  insert_3_in_gga2(x1, x2)
void_0  =  void_0
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
insert_3_out_gga3(x1, x2, x3)  =  insert_3_out_gga1(x3)
if_insert_3_in_1_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_1_gga5(x1, x2, x3, x4, x6)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
if_insert_3_in_2_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_2_gga3(x2, x4, x6)
if_insert_3_in_3_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_3_gga5(x1, x2, x3, x4, x6)
if_insert_3_in_4_gga6(x1, x2, x3, x4, x5, x6)  =  if_insert_3_in_4_gga3(x2, x3, x6)
INSERT_3_IN_GGA3(x1, x2, x3)  =  INSERT_3_IN_GGA2(x1, x2)
IF_INSERT_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GGA5(x1, x2, x3, x4, x6)

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

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

INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_in_gg2(Y, X))
IF_INSERT_3_IN_3_GGA6(X, Y, Left, Right, Right1, less_2_out_gg2(Y, X)) -> INSERT_3_IN_GGA3(X, Right, Right1)
INSERT_3_IN_GGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_in_gg2(X, Y))
IF_INSERT_3_IN_1_GGA6(X, Y, Left, Right, Left1, less_2_out_gg2(X, Y)) -> INSERT_3_IN_GGA3(X, Left, Left1)

The TRS R consists of the following rules:

less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg2(0_0, s_11(underscore))
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg3(X, Y, less_2_in_gg2(X, Y))
if_less_2_in_1_gg3(X, Y, less_2_out_gg2(X, Y)) -> less_2_out_gg2(s_11(X), s_11(Y))

The argument filtering Pi contains the following mapping:
tree_33(x1, x2, x3)  =  tree_33(x1, x2, x3)
0_0  =  0_0
s_11(x1)  =  s_11(x1)
less_2_in_gg2(x1, x2)  =  less_2_in_gg2(x1, x2)
less_2_out_gg2(x1, x2)  =  less_2_out_gg
if_less_2_in_1_gg3(x1, x2, x3)  =  if_less_2_in_1_gg1(x3)
INSERT_3_IN_GGA3(x1, x2, x3)  =  INSERT_3_IN_GGA2(x1, x2)
IF_INSERT_3_IN_1_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_1_GGA5(x1, x2, x3, x4, x6)
IF_INSERT_3_IN_3_GGA6(x1, x2, x3, x4, x5, x6)  =  IF_INSERT_3_IN_3_GGA5(x1, x2, x3, x4, x6)

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
                        ↳ QDPSizeChangeProof

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

INSERT_3_IN_GGA2(X, tree_33(Y, Left, Right)) -> IF_INSERT_3_IN_3_GGA5(X, Y, Left, Right, less_2_in_gg2(Y, X))
IF_INSERT_3_IN_3_GGA5(X, Y, Left, Right, less_2_out_gg) -> INSERT_3_IN_GGA2(X, Right)
INSERT_3_IN_GGA2(X, tree_33(Y, Left, Right)) -> IF_INSERT_3_IN_1_GGA5(X, Y, Left, Right, less_2_in_gg2(X, Y))
IF_INSERT_3_IN_1_GGA5(X, Y, Left, Right, less_2_out_gg) -> INSERT_3_IN_GGA2(X, Left)

The TRS R consists of the following rules:

less_2_in_gg2(0_0, s_11(underscore)) -> less_2_out_gg
less_2_in_gg2(s_11(X), s_11(Y)) -> if_less_2_in_1_gg1(less_2_in_gg2(X, Y))
if_less_2_in_1_gg1(less_2_out_gg) -> less_2_out_gg

The set Q consists of the following terms:

less_2_in_gg2(x0, x1)
if_less_2_in_1_gg1(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_INSERT_3_IN_3_GGA5, INSERT_3_IN_GGA2, IF_INSERT_3_IN_1_GGA5}.
By using the subterm criterion together with the size-change analysis 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: