Left Termination proof of /tmp/tmpJM-Jn2/insert-fbf.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

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

With regard to the inferred argument filtering the predicates were used in the following modes:
insert₃: (f,b,f) (b,b,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:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
INSERT_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_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> IF_INSERT_3_IN_2_AGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
INSERT_3_IN_GGA₃(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_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> IF_INSERT_3_IN_2_GGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AG₂(Y, X)
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_AG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> IF_INSERT_3_IN_4_GGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AA₂(Y, X)
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> IF_INSERT_3_IN_4_AGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
insert_3_in_aga₃(x1, x2, x3) = insert_3_in_aga₁(x2)
void_0 = void_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
insert_3_out_aga₃(x1, x2, x3) = insert_3_out_aga₁(x3)
if_insert_3_in_1_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_aga₃(x3, x4, 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_insert_3_in_2_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_aga₂(x4, x6)
insert_3_in_gga₃(x1, x2, x3) = insert_3_in_gga₂(x1, x2)
insert_3_out_gga₃(x1, x2, x3) = insert_3_out_gga₁(x3)
if_insert_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_gga₄(x1, x3, x4, 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)
if_insert_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_gga₂(x4, x6)
if_insert_3_in_3_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_gga₄(x1, x3, x4, x6)
less_2_in_ag₂(x1, x2) = less_2_in_ag₁(x2)
less_2_out_ag₂(x1, x2) = less_2_out_ag₁(x1)
if_less_2_in_1_ag₃(x1, x2, x3) = if_less_2_in_1_ag₁(x3)
if_insert_3_in_4_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_gga₂(x3, x6)
if_insert_3_in_3_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_aga₃(x3, x4, x6)
if_insert_3_in_4_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_aga₂(x3, x6)
IF_INSERT_3_IN_2_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_AGA₂(x4, x6)
INSERT_3_IN_GGA₃(x1, x2, x3) = INSERT_3_IN_GGA₂(x1, x2)
LESS_2_IN_GA₂(x1, x2) = LESS_2_IN_GA₁(x1)
LESS_2_IN_AG₂(x1, x2) = LESS_2_IN_AG₁(x2)
IF_LESS_2_IN_1_AG₃(x1, x2, x3) = IF_LESS_2_IN_1_AG₁(x3)
IF_INSERT_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_GGA₂(x4, x6)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_INSERT_3_IN_3_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_AGA₃(x3, x4, x6)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_INSERT_3_IN_4_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_GGA₂(x3, x6)
IF_INSERT_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GGA₄(x1, x3, x4, x6)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
INSERT_3_IN_AGA₃(x1, x2, x3) = INSERT_3_IN_AGA₁(x2)
IF_INSERT_3_IN_4_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_AGA₂(x3, x6)
IF_INSERT_3_IN_1_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_AGA₃(x3, x4, x6)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
INSERT_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_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> IF_INSERT_3_IN_2_AGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
INSERT_3_IN_GGA₃(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_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> IF_INSERT_3_IN_2_GGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AG₂(Y, X)
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_AG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> IF_INSERT_3_IN_4_GGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AA₂(Y, X)
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> IF_INSERT_3_IN_4_AGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_2_IN_AA₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_2_IN_AA₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)

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

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_2_IN_AA -> LESS_2_IN_AA

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
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)
INSERT_3_IN_GGA₃(x1, x2, x3) = INSERT_3_IN_GGA₂(x1, x2)
IF_INSERT_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GGA₄(x1, x3, x4, x6)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

IF_INSERT_3_IN_3_GGA₄(X, Left, Right, less_2_out_ag₁(Y)) -> INSERT_3_IN_GGA₂(X, Right)
INSERT_3_IN_GGA₂(X, tree_3₂(Left, Right)) -> IF_INSERT_3_IN_1_GGA₄(X, Left, Right, less_2_in_ga₁(X))
IF_INSERT_3_IN_1_GGA₄(X, Left, Right, less_2_out_ga₁(Y)) -> INSERT_3_IN_GGA₂(X, Left)
INSERT_3_IN_GGA₂(X, tree_3₂(Left, Right)) -> IF_INSERT_3_IN_3_GGA₄(X, Left, Right, less_2_in_ag₁(X))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {INSERT_3_IN_GGA₂, IF_INSERT_3_IN_3_GGA₄, IF_INSERT_3_IN_1_GGA₄}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

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

INSERT_3_IN_GGA₂(X, tree_3₂(Left, Right)) -> IF_INSERT_3_IN_3_GGA₄(X, Left, Right, less_2_in_ag₁(X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
INSERT_3_IN_GGA₂(X, tree_3₂(Left, Right)) -> IF_INSERT_3_IN_1_GGA₄(X, Left, Right, less_2_in_ga₁(X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
IF_INSERT_3_IN_1_GGA₄(X, Left, Right, less_2_out_ga₁(Y)) -> INSERT_3_IN_GGA₂(X, Left)
The graph contains the following edges 1 >= 1, 2 >= 2
IF_INSERT_3_IN_3_GGA₄(X, Left, Right, less_2_out_ag₁(Y)) -> INSERT_3_IN_GGA₂(X, Right)
The graph contains the following edges 1 >= 1, 3 >= 2

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
INSERT_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_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> IF_INSERT_3_IN_2_AGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
INSERT_3_IN_GGA₃(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_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> IF_INSERT_3_IN_2_GGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AG₂(Y, X)
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_AG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> IF_INSERT_3_IN_4_GGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AA₂(Y, X)
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> IF_INSERT_3_IN_4_AGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
insert_3_in_aga₃(x1, x2, x3) = insert_3_in_aga₁(x2)
void_0 = void_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
insert_3_out_aga₃(x1, x2, x3) = insert_3_out_aga₂(x2, x3)
if_insert_3_in_1_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_aga₃(x3, x4, 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_insert_3_in_2_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_aga₃(x3, x4, x6)
insert_3_in_gga₃(x1, x2, x3) = insert_3_in_gga₂(x1, x2)
insert_3_out_gga₃(x1, x2, x3) = insert_3_out_gga₃(x1, x2, x3)
if_insert_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_gga₄(x1, x3, x4, 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)
if_insert_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_gga₄(x1, x3, x4, x6)
if_insert_3_in_3_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_gga₄(x1, x3, x4, x6)
less_2_in_ag₂(x1, x2) = less_2_in_ag₁(x2)
less_2_out_ag₂(x1, x2) = less_2_out_ag₂(x1, x2)
if_less_2_in_1_ag₃(x1, x2, x3) = if_less_2_in_1_ag₁(x3)
if_insert_3_in_4_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_gga₄(x1, x3, x4, x6)
if_insert_3_in_3_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_aga₃(x3, x4, x6)
if_insert_3_in_4_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_aga₃(x3, x4, x6)
IF_INSERT_3_IN_2_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_AGA₃(x3, x4, x6)
INSERT_3_IN_GGA₃(x1, x2, x3) = INSERT_3_IN_GGA₂(x1, x2)
LESS_2_IN_GA₂(x1, x2) = LESS_2_IN_GA₁(x1)
LESS_2_IN_AG₂(x1, x2) = LESS_2_IN_AG₁(x2)
IF_LESS_2_IN_1_AG₃(x1, x2, x3) = IF_LESS_2_IN_1_AG₁(x3)
IF_INSERT_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_GGA₄(x1, x3, x4, x6)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_INSERT_3_IN_3_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_AGA₃(x3, x4, x6)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_INSERT_3_IN_4_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GGA₄(x1, x3, x4, x6)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
INSERT_3_IN_AGA₃(x1, x2, x3) = INSERT_3_IN_AGA₁(x2)
IF_INSERT_3_IN_4_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_AGA₃(x3, x4, x6)
IF_INSERT_3_IN_1_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_AGA₃(x3, x4, x6)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
INSERT_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_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> IF_INSERT_3_IN_2_AGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_AGA₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
INSERT_3_IN_GGA₃(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_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> IF_INSERT_3_IN_2_GGA₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AG₂(Y, X)
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_AG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_AG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> IF_INSERT_3_IN_4_GGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
INSERT_3_IN_AGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> LESS_2_IN_AA₂(Y, X)
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> IF_INSERT_3_IN_4_AGA₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
IF_INSERT_3_IN_3_AGA₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
insert_3_in_aga₃(x1, x2, x3) = insert_3_in_aga₁(x2)
void_0 = void_0
tree_3₃(x1, x2, x3) = tree_3₂(x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
insert_3_out_aga₃(x1, x2, x3) = insert_3_out_aga₂(x2, x3)
if_insert_3_in_1_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_aga₃(x3, x4, 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_insert_3_in_2_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_aga₃(x3, x4, x6)
insert_3_in_gga₃(x1, x2, x3) = insert_3_in_gga₂(x1, x2)
insert_3_out_gga₃(x1, x2, x3) = insert_3_out_gga₃(x1, x2, x3)
if_insert_3_in_1_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_1_gga₄(x1, x3, x4, 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)
if_insert_3_in_2_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_2_gga₄(x1, x3, x4, x6)
if_insert_3_in_3_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_gga₄(x1, x3, x4, x6)
less_2_in_ag₂(x1, x2) = less_2_in_ag₁(x2)
less_2_out_ag₂(x1, x2) = less_2_out_ag₂(x1, x2)
if_less_2_in_1_ag₃(x1, x2, x3) = if_less_2_in_1_ag₁(x3)
if_insert_3_in_4_gga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_gga₄(x1, x3, x4, x6)
if_insert_3_in_3_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_3_aga₃(x3, x4, x6)
if_insert_3_in_4_aga₆(x1, x2, x3, x4, x5, x6) = if_insert_3_in_4_aga₃(x3, x4, x6)
IF_INSERT_3_IN_2_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_AGA₃(x3, x4, x6)
INSERT_3_IN_GGA₃(x1, x2, x3) = INSERT_3_IN_GGA₂(x1, x2)
LESS_2_IN_GA₂(x1, x2) = LESS_2_IN_GA₁(x1)
LESS_2_IN_AG₂(x1, x2) = LESS_2_IN_AG₁(x2)
IF_LESS_2_IN_1_AG₃(x1, x2, x3) = IF_LESS_2_IN_1_AG₁(x3)
IF_INSERT_3_IN_1_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_2_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_2_GGA₄(x1, x3, x4, x6)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_INSERT_3_IN_3_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_AGA₃(x3, x4, x6)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_INSERT_3_IN_4_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_GGA₄(x1, x3, x4, x6)
IF_INSERT_3_IN_3_GGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_3_GGA₄(x1, x3, x4, x6)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
INSERT_3_IN_AGA₃(x1, x2, x3) = INSERT_3_IN_AGA₁(x2)
IF_INSERT_3_IN_4_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_4_AGA₃(x3, x4, x6)
IF_INSERT_3_IN_1_AGA₆(x1, x2, x3, x4, x5, x6) = IF_INSERT_3_IN_1_AGA₃(x3, x4, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph contains 2 SCCs with 17 less nodes.

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LESS_2_IN_AA₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

              ↳ PiDP

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

LESS_2_IN_AA₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)

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

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

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

IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> INSERT_3_IN_GGA₃(X, Right, Right1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
IF_INSERT_3_IN_1_GGA₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> INSERT_3_IN_GGA₃(X, Left, Left1)
INSERT_3_IN_GGA₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> IF_INSERT_3_IN_3_GGA₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))

The TRS R consists of the following rules:

insert_3_in_aga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_aga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_aga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_in_aa₂(X, Y))
less_2_in_aa₂(0_0, s_1₁(underscore)) -> less_2_out_aa₂(0_0, s_1₁(underscore))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_aga₆(X, Y, Left, Right, Left1, less_2_out_aa₂(X, Y)) -> if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, void_0, tree_3₃(X, void_0, void_0)) -> insert_3_out_gga₃(X, void_0, tree_3₃(X, void_0, void_0))
insert_3_in_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right)) -> insert_3_out_gga₃(X, tree_3₃(X, Left, Right), tree_3₃(X, Left, Right))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right)) -> if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_in_ga₂(X, Y))
less_2_in_ga₂(0_0, s_1₁(underscore)) -> less_2_out_ga₂(0_0, s_1₁(underscore))
less_2_in_ga₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ga₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ga₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ga₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_1_gga₆(X, Y, Left, Right, Left1, less_2_out_ga₂(X, Y)) -> if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_in_gga₃(X, Left, Left1))
insert_3_in_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_in_ag₂(Y, X))
less_2_in_ag₂(0_0, s_1₁(underscore)) -> less_2_out_ag₂(0_0, s_1₁(underscore))
less_2_in_ag₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_ag₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_insert_3_in_3_gga₆(X, Y, Left, Right, Right1, less_2_out_ag₂(Y, X)) -> if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_gga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))
if_insert_3_in_2_gga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_gga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
if_insert_3_in_2_aga₆(X, Y, Left, Right, Left1, insert_3_out_gga₃(X, Left, Left1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left1, Right))
insert_3_in_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1)) -> if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_in_aa₂(Y, X))
if_insert_3_in_3_aga₆(X, Y, Left, Right, Right1, less_2_out_aa₂(Y, X)) -> if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_in_gga₃(X, Right, Right1))
if_insert_3_in_4_aga₆(X, Y, Left, Right, Right1, insert_3_out_gga₃(X, Right, Right1)) -> insert_3_out_aga₃(X, tree_3₃(Y, Left, Right), tree_3₃(Y, Left, Right1))