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



↳ PROLOG
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

delete3(X, tree3(X, void0, Right), Right).
delete3(X, tree3(X, Left, void0), Left).
delete3(X, tree3(X, Left, Right), tree3(Y, Left, Right1)) :- delmin3(Right, Y, Right1).
delete3(X, tree3(Y, Left, Right), tree3(Y, Left1, Right)) :- less2(X, Y), delete3(X, Left, Left1).
delete3(X, tree3(Y, Left, Right), tree3(Y, Left, Right1)) :- less2(Y, X), delete3(X, Right, Right1).
delmin3(tree3(Y, void0, Right), Y, Right).
delmin3(tree3(X, Left, underscore), Y, tree3(X, Left1, underscore1)) :- delmin3(Left, Y, Left1).
less2(00, s1(underscore2)).
less2(s1(X), s1(Y)) :- less2(X, Y).


With regard to the inferred argument filtering the predicates were used in the following modes:
delete3: (f,b,f) (b,f,f)
delmin3: (f,f,f)
less2: (f,f) (b,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(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:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)


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

DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_AGA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> IF_DELMIN_3_IN_1_AAA7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> 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_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> IF_DELETE_3_IN_3_AGA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_GAA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_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)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_DELETE_3_IN_3_GAA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
DELETE_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_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_DELETE_3_IN_5_GAA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AA2(Y, X)
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> IF_DELETE_3_IN_5_AGA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
IF_DELETE_3_IN_3_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_AGA1(x6)
IF_DELETE_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_GAA1(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)
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_DELETE_3_IN_5_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_GAA1(x6)
IF_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)
IF_DELETE_3_IN_1_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_AGA1(x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, x6)
IF_DELETE_3_IN_2_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_AGA1(x6)
DELETE_3_IN_AGA3(x1, x2, x3)  =  DELETE_3_IN_AGA1(x2)
IF_DELETE_3_IN_5_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_AGA1(x6)
IF_DELETE_3_IN_4_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_AGA1(x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_DELMIN_3_IN_1_AAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_DELMIN_3_IN_1_AAA1(x7)
IF_DELETE_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_GAA1(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:

DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_AGA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> IF_DELMIN_3_IN_1_AAA7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> 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_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> IF_DELETE_3_IN_3_AGA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_GAA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_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)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_DELETE_3_IN_3_GAA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
DELETE_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_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_DELETE_3_IN_5_GAA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AA2(Y, X)
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> IF_DELETE_3_IN_5_AGA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
IF_DELETE_3_IN_3_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_AGA1(x6)
IF_DELETE_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_GAA1(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)
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_DELETE_3_IN_5_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_GAA1(x6)
IF_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)
IF_DELETE_3_IN_1_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_AGA1(x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, x6)
IF_DELETE_3_IN_2_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_AGA1(x6)
DELETE_3_IN_AGA3(x1, x2, x3)  =  DELETE_3_IN_AGA1(x2)
IF_DELETE_3_IN_5_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_AGA1(x6)
IF_DELETE_3_IN_4_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_AGA1(x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_DELMIN_3_IN_1_AAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_DELMIN_3_IN_1_AAA1(x7)
IF_DELETE_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_GAA1(x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 3 SCCs with 22 less nodes.

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

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(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
              ↳ 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
              ↳ 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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)

R is empty.
The argument filtering Pi contains the following mapping:
tree_33(x1, x2, x3)  =  tree_3
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA

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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

DELMIN_3_IN_AAA -> DELMIN_3_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {DELMIN_3_IN_AAA}.

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

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

DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa1(x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa1(x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa1(x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa1(x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
IF_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, 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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)

The TRS R consists of the following rules:

less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
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(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_3
0_0  =  0_0
s_11(x1)  =  s_1
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_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_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_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, 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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
  ↳ PrologToPiTRSProof

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

DELETE_3_IN_GAA1(X) -> IF_DELETE_3_IN_2_GAA2(X, less_2_in_ga1(X))
DELETE_3_IN_GAA1(X) -> IF_DELETE_3_IN_4_GAA2(X, less_2_in_ag1(X))
IF_DELETE_3_IN_4_GAA2(X, less_2_out_ag1(Y)) -> DELETE_3_IN_GAA1(X)
IF_DELETE_3_IN_2_GAA2(X, less_2_out_ga1(Y)) -> DELETE_3_IN_GAA1(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 {IF_DELETE_3_IN_2_GAA2, DELETE_3_IN_GAA1, IF_DELETE_3_IN_4_GAA2}.
With regard to the inferred argument filtering the predicates were used in the following modes:
delete3: (f,b,f) (b,f,f)
delmin3: (f,f,f)
less2: (f,f) (b,f) (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga1(x2)
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa2(x1, x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa2(x1, x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa2(x1, x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa2(x1, x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(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:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga1(x2)
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa2(x1, x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa2(x1, x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa2(x1, x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa2(x1, x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)


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

DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_AGA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> IF_DELMIN_3_IN_1_AAA7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> 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_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> IF_DELETE_3_IN_3_AGA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_GAA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_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)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_DELETE_3_IN_3_GAA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
DELETE_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_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_DELETE_3_IN_5_GAA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AA2(Y, X)
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> IF_DELETE_3_IN_5_AGA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga1(x2)
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa2(x1, x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa2(x1, x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa2(x1, x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa2(x1, x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
IF_DELETE_3_IN_3_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_AGA1(x6)
IF_DELETE_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_GAA2(x1, 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)
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_DELETE_3_IN_5_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_GAA2(x1, x6)
IF_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)
IF_DELETE_3_IN_1_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_AGA1(x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, x6)
IF_DELETE_3_IN_2_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_AGA1(x6)
DELETE_3_IN_AGA3(x1, x2, x3)  =  DELETE_3_IN_AGA1(x2)
IF_DELETE_3_IN_5_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_AGA1(x6)
IF_DELETE_3_IN_4_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_AGA1(x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_DELMIN_3_IN_1_AAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_DELMIN_3_IN_1_AAA1(x7)
IF_DELETE_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_GAA2(x1, x6)

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

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

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

DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_AGA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_AGA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> IF_DELMIN_3_IN_1_AAA7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
DELMIN_3_IN_AAA3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> DELMIN_3_IN_AAA3(Left, Y, Left1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> 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_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> IF_DELETE_3_IN_3_AGA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_AGA6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_1_GAA6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
DELETE_3_IN_GAA3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> DELMIN_3_IN_AAA3(Right, Y, Right1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
DELETE_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)
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> IF_DELETE_3_IN_3_GAA6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
IF_DELETE_3_IN_2_GAA6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> DELETE_3_IN_GAA3(X, Left, Left1)
DELETE_3_IN_GAA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
DELETE_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_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> IF_DELETE_3_IN_5_GAA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_GAA6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
DELETE_3_IN_AGA3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> LESS_2_IN_AA2(Y, X)
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> IF_DELETE_3_IN_5_AGA6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
IF_DELETE_3_IN_4_AGA6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> DELETE_3_IN_GAA3(X, Right, Right1)

The TRS R consists of the following rules:

delete_3_in_aga3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_aga3(X, tree_33(X, void_0, Right), Right)
delete_3_in_aga3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_aga3(X, tree_33(X, Left, void_0), Left)
delete_3_in_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
delmin_3_in_aaa3(tree_33(Y, void_0, Right), Y, Right) -> delmin_3_out_aaa3(tree_33(Y, void_0, Right), Y, Right)
delmin_3_in_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1)) -> if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_in_aaa3(Left, Y, Left1))
if_delmin_3_in_1_aaa7(X, Left, underscore, Y, Left1, underscore1, delmin_3_out_aaa3(Left, Y, Left1)) -> delmin_3_out_aaa3(tree_33(X, Left, underscore), Y, tree_33(X, Left1, underscore1))
if_delete_3_in_1_aga6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_aga3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_in_aa2(X, Y))
less_2_in_aa2(0_0, s_11(underscore2)) -> less_2_out_aa2(0_0, s_11(underscore2))
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_delete_3_in_2_aga6(X, Y, Left, Right, Left1, less_2_out_aa2(X, Y)) -> if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(X, void_0, Right), Right) -> delete_3_out_gaa3(X, tree_33(X, void_0, Right), Right)
delete_3_in_gaa3(X, tree_33(X, Left, void_0), Left) -> delete_3_out_gaa3(X, tree_33(X, Left, void_0), Left)
delete_3_in_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_in_aaa3(Right, Y, Right1))
if_delete_3_in_1_gaa6(X, Left, Right, Y, Right1, delmin_3_out_aaa3(Right, Y, Right1)) -> delete_3_out_gaa3(X, tree_33(X, Left, Right), tree_33(Y, Left, Right1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right)) -> if_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_in_ga2(X, Y))
less_2_in_ga2(0_0, s_11(underscore2)) -> less_2_out_ga2(0_0, s_11(underscore2))
less_2_in_ga2(s_11(X), s_11(Y)) -> if_less_2_in_1_ga3(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_delete_3_in_2_gaa6(X, Y, Left, Right, Left1, less_2_out_ga2(X, Y)) -> if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_in_gaa3(X, Left, Left1))
delete_3_in_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_in_ag2(Y, X))
less_2_in_ag2(0_0, s_11(underscore2)) -> less_2_out_ag2(0_0, s_11(underscore2))
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_delete_3_in_4_gaa6(X, Y, Left, Right, Right1, less_2_out_ag2(Y, X)) -> if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_gaa6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))
if_delete_3_in_3_gaa6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_gaa3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
if_delete_3_in_3_aga6(X, Y, Left, Right, Left1, delete_3_out_gaa3(X, Left, Left1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left1, Right))
delete_3_in_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1)) -> if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_in_aa2(Y, X))
if_delete_3_in_4_aga6(X, Y, Left, Right, Right1, less_2_out_aa2(Y, X)) -> if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_in_gaa3(X, Right, Right1))
if_delete_3_in_5_aga6(X, Y, Left, Right, Right1, delete_3_out_gaa3(X, Right, Right1)) -> delete_3_out_aga3(X, tree_33(Y, Left, Right), tree_33(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_3_in_aga3(x1, x2, x3)  =  delete_3_in_aga1(x2)
tree_33(x1, x2, x3)  =  tree_3
void_0  =  void_0
0_0  =  0_0
s_11(x1)  =  s_1
delete_3_out_aga3(x1, x2, x3)  =  delete_3_out_aga1(x2)
if_delete_3_in_1_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_aga1(x6)
delmin_3_in_aaa3(x1, x2, x3)  =  delmin_3_in_aaa
delmin_3_out_aaa3(x1, x2, x3)  =  delmin_3_out_aaa1(x1)
if_delmin_3_in_1_aaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_delmin_3_in_1_aaa1(x7)
if_delete_3_in_2_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_aga1(x6)
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_delete_3_in_3_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_aga1(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_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)
delete_3_in_gaa3(x1, x2, x3)  =  delete_3_in_gaa1(x1)
delete_3_out_gaa3(x1, x2, x3)  =  delete_3_out_gaa2(x1, x2)
if_delete_3_in_1_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_1_gaa2(x1, x6)
if_delete_3_in_2_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_2_gaa2(x1, x6)
if_delete_3_in_3_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_3_gaa2(x1, x6)
if_delete_3_in_4_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_gaa2(x1, x6)
if_delete_3_in_5_gaa6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_gaa2(x1, x6)
if_delete_3_in_4_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_4_aga1(x6)
if_delete_3_in_5_aga6(x1, x2, x3, x4, x5, x6)  =  if_delete_3_in_5_aga1(x6)
IF_DELETE_3_IN_3_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_AGA1(x6)
IF_DELETE_3_IN_1_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_GAA2(x1, 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)
DELMIN_3_IN_AAA3(x1, x2, x3)  =  DELMIN_3_IN_AAA
IF_LESS_2_IN_1_AA3(x1, x2, x3)  =  IF_LESS_2_IN_1_AA1(x3)
IF_DELETE_3_IN_5_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_GAA2(x1, x6)
IF_DELETE_3_IN_4_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_GAA2(x1, x6)
IF_LESS_2_IN_1_GA3(x1, x2, x3)  =  IF_LESS_2_IN_1_GA1(x3)
IF_DELETE_3_IN_1_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_1_AGA1(x6)
DELETE_3_IN_GAA3(x1, x2, x3)  =  DELETE_3_IN_GAA1(x1)
IF_DELETE_3_IN_2_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_GAA2(x1, x6)
IF_DELETE_3_IN_2_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_2_AGA1(x6)
DELETE_3_IN_AGA3(x1, x2, x3)  =  DELETE_3_IN_AGA1(x2)
IF_DELETE_3_IN_5_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_5_AGA1(x6)
IF_DELETE_3_IN_4_AGA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_4_AGA1(x6)
LESS_2_IN_AA2(x1, x2)  =  LESS_2_IN_AA
IF_DELMIN_3_IN_1_AAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_DELMIN_3_IN_1_AAA1(x7)
IF_DELETE_3_IN_3_GAA6(x1, x2, x3, x4, x5, x6)  =  IF_DELETE_3_IN_3_GAA2(x1, x6)

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