Left Termination proof of /tmp/tmpXz4gU9/in-fb.pl

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

in₂(X, tree₃(X, underscore, underscore1)).
in₂(X, tree₃(Y, Left, underscore2)) :- less₂(X, Y), in₂(X, Left).
in₂(X, tree₃(Y, underscore3, Right)) :- less₂(Y, X), in₂(X, Right).
less₂(0₀, s₁(underscore4)).
less₂(s₁(X), s₁(Y)) :- less₂(X, Y).

With regard to the inferred argument filtering the predicates were used in the following modes:
in₂: (f,b) (b,b)
less₂: (f,b) (f,f) (b,b) (b,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

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:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_AG₂(X, Y)
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)
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_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IF_IN_2_IN_2_AG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IF_IN_2_IN_2_GG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GG₂(Y, X)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IF_IN_2_IN_4_GG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GA₂(Y, X)
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_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IF_IN_2_IN_4_AG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IN_2_IN_GG₂(X, Right)

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

The argument filtering Pi contains the following mapping:
in_2_in_ag₂(x1, x2) = in_2_in_ag₁(x2)
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
in_2_out_ag₂(x1, x2) = in_2_out_ag₁(x1)
if_in_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_1_ag₂(x3, x5)
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_in_2_in_2_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_2_ag₂(x1, x5)
in_2_in_gg₂(x1, x2) = in_2_in_gg₂(x1, x2)
in_2_out_gg₂(x1, x2) = in_2_out_gg
if_in_2_in_1_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_1_gg₃(x1, x3, x5)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_in_2_in_2_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_2_gg₁(x5)
if_in_2_in_3_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_3_gg₃(x1, x4, x5)
if_in_2_in_4_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_4_gg₁(x5)
if_in_2_in_3_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_3_ag₂(x4, x5)
less_2_in_ga₂(x1, x2) = less_2_in_ga₁(x1)
less_2_out_ga₂(x1, x2) = less_2_out_ga₁(x2)
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_less_2_in_1_ga₃(x1, x2, x3) = if_less_2_in_1_ga₁(x3)
if_in_2_in_4_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_4_ag₂(x1, x5)
IN_2_IN_AG₂(x1, x2) = IN_2_IN_AG₁(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_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_GG₃(x1, x4, x5)
IF_IN_2_IN_3_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_AG₂(x4, x5)
IN_2_IN_GG₂(x1, x2) = IN_2_IN_GG₂(x1, x2)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_GG₃(x1, x3, x5)
IF_IN_2_IN_2_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_GG₁(x5)
IF_IN_2_IN_2_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_AG₂(x1, x5)
IF_IN_2_IN_4_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_AG₂(x1, x5)
IF_IN_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_AG₂(x3, x5)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
IF_LESS_2_IN_1_GG₃(x1, x2, x3) = IF_LESS_2_IN_1_GG₁(x3)
IF_IN_2_IN_4_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_GG₁(x5)
LESS_2_IN_GG₂(x1, x2) = LESS_2_IN_GG₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_AG₂(X, Y)
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)
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_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IF_IN_2_IN_2_AG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IF_IN_2_IN_2_GG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GG₂(Y, X)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IF_IN_2_IN_4_GG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GA₂(Y, X)
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_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IF_IN_2_IN_4_AG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IN_2_IN_GG₂(X, Right)

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

↳ 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:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

↳ 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:

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
less_2_in_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
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_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
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_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_GG₃(x1, x4, x5)
IN_2_IN_GG₂(x1, x2) = IN_2_IN_GG₂(x1, x2)
IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_GG₃(x1, x3, x5)

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:

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₃(X, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₃(X, Left, less_2_out_gg) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₃(X, Right, less_2_out_gg) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₃(X, Left, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1) -> less_2_out_gg
less_2_in_gg₂(s_1, s_1) -> if_less_2_in_1_gg₁(less_2_in_aa)
if_less_2_in_1_gg₁(less_2_out_aa₂(X, Y)) -> less_2_out_gg
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_gg₂(x0, x1)
if_less_2_in_1_gg₁(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_IN_2_IN_3_GG₃, IN_2_IN_GG₂, IF_IN_2_IN_1_GG₃}.
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:

IF_IN_2_IN_3_GG₃(X, Right, less_2_out_gg) -> IN_2_IN_GG₂(X, Right)
The graph contains the following edges 1 >= 1, 2 >= 2
IF_IN_2_IN_1_GG₃(X, Left, less_2_out_gg) -> IN_2_IN_GG₂(X, Left)
The graph contains the following edges 1 >= 1, 2 >= 2
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₃(X, Left, less_2_in_gg₂(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₃(X, Right, less_2_in_gg₂(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

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:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_AG₂(X, Y)
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)
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_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IF_IN_2_IN_2_AG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IF_IN_2_IN_2_GG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GG₂(Y, X)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IF_IN_2_IN_4_GG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GA₂(Y, X)
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_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IF_IN_2_IN_4_AG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IN_2_IN_GG₂(X, Right)

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

The argument filtering Pi contains the following mapping:
in_2_in_ag₂(x1, x2) = in_2_in_ag₁(x2)
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
in_2_out_ag₂(x1, x2) = in_2_out_ag₂(x1, x2)
if_in_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_1_ag₄(x2, x3, x4, x5)
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_in_2_in_2_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_2_ag₅(x1, x2, x3, x4, x5)
in_2_in_gg₂(x1, x2) = in_2_in_gg₂(x1, x2)
in_2_out_gg₂(x1, x2) = in_2_out_gg₂(x1, x2)
if_in_2_in_1_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_1_gg₅(x1, x2, x3, x4, x5)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg₂(x1, x2)
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_in_2_in_2_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_2_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_3_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_3_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_4_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_4_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_3_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_3_ag₄(x2, x3, x4, x5)
less_2_in_ga₂(x1, x2) = less_2_in_ga₁(x1)
less_2_out_ga₂(x1, x2) = less_2_out_ga₂(x1, x2)
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_less_2_in_1_ga₃(x1, x2, x3) = if_less_2_in_1_ga₁(x3)
if_in_2_in_4_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_4_ag₅(x1, x2, x3, x4, x5)
IN_2_IN_AG₂(x1, x2) = IN_2_IN_AG₁(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_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_3_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_AG₄(x2, x3, x4, x5)
IN_2_IN_GG₂(x1, x2) = IN_2_IN_GG₂(x1, x2)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_2_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_2_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_AG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_4_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_AG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_AG₄(x2, x3, x4, x5)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
IF_LESS_2_IN_1_GG₃(x1, x2, x3) = IF_LESS_2_IN_1_GG₁(x3)
IF_IN_2_IN_4_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_GG₅(x1, x2, x3, x4, x5)
LESS_2_IN_GG₂(x1, x2) = LESS_2_IN_GG₂(x1, x2)

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:

IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
IN_2_IN_AG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_AG₂(X, Y)
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)
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_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IF_IN_2_IN_2_AG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_AG₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> LESS_2_IN_GG₂(X, Y)
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> IF_LESS_2_IN_1_GG₃(X, Y, less_2_in_aa₂(X, Y))
LESS_2_IN_GG₂(s_1₁(X), s_1₁(Y)) -> LESS_2_IN_AA₂(X, Y)
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IF_IN_2_IN_2_GG₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GG₂(Y, X)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IF_IN_2_IN_4_GG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
IN_2_IN_AG₂(X, tree_3₃(Y, underscore3, Right)) -> LESS_2_IN_GA₂(Y, X)
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_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IF_IN_2_IN_4_AG₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
IF_IN_2_IN_3_AG₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> IN_2_IN_GG₂(X, Right)

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

The argument filtering Pi contains the following mapping:
in_2_in_ag₂(x1, x2) = in_2_in_ag₁(x2)
tree_3₃(x1, x2, x3) = tree_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
in_2_out_ag₂(x1, x2) = in_2_out_ag₂(x1, x2)
if_in_2_in_1_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_1_ag₄(x2, x3, x4, x5)
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_in_2_in_2_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_2_ag₅(x1, x2, x3, x4, x5)
in_2_in_gg₂(x1, x2) = in_2_in_gg₂(x1, x2)
in_2_out_gg₂(x1, x2) = in_2_out_gg₂(x1, x2)
if_in_2_in_1_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_1_gg₅(x1, x2, x3, x4, x5)
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg₂(x1, x2)
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
if_in_2_in_2_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_2_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_3_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_3_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_4_gg₅(x1, x2, x3, x4, x5) = if_in_2_in_4_gg₅(x1, x2, x3, x4, x5)
if_in_2_in_3_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_3_ag₄(x2, x3, x4, x5)
less_2_in_ga₂(x1, x2) = less_2_in_ga₁(x1)
less_2_out_ga₂(x1, x2) = less_2_out_ga₂(x1, x2)
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_less_2_in_1_ga₃(x1, x2, x3) = if_less_2_in_1_ga₁(x3)
if_in_2_in_4_ag₅(x1, x2, x3, x4, x5) = if_in_2_in_4_ag₅(x1, x2, x3, x4, x5)
IN_2_IN_AG₂(x1, x2) = IN_2_IN_AG₁(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_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_3_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_AG₄(x2, x3, x4, x5)
IN_2_IN_GG₂(x1, x2) = IN_2_IN_GG₂(x1, x2)
IF_LESS_2_IN_1_AA₃(x1, x2, x3) = IF_LESS_2_IN_1_AA₁(x3)
IF_LESS_2_IN_1_GA₃(x1, x2, x3) = IF_LESS_2_IN_1_GA₁(x3)
IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_2_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_GG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_2_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_2_AG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_4_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_AG₅(x1, x2, x3, x4, x5)
IF_IN_2_IN_1_AG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_AG₄(x2, x3, x4, x5)
LESS_2_IN_AA₂(x1, x2) = LESS_2_IN_AA
IF_LESS_2_IN_1_GG₃(x1, x2, x3) = IF_LESS_2_IN_1_GG₁(x3)
IF_IN_2_IN_4_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_4_GG₅(x1, x2, x3, x4, x5)
LESS_2_IN_GG₂(x1, x2) = LESS_2_IN_GG₂(x1, x2)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

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

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

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

LESS_2_IN_AA -> LESS_2_IN_AA

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

in_2_in_ag₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_ag₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_ag₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_in_ag₂(X, Y))
less_2_in_ag₂(0_0, s_1₁(underscore4)) -> less_2_out_ag₂(0_0, s_1₁(underscore4))
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_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
less_2_in_aa₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_aa₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_aa₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1₁(X), s_1₁(Y))
if_less_2_in_1_ag₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_ag₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_ag₅(X, Y, Left, underscore2, less_2_out_ag₂(X, Y)) -> if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(X, underscore, underscore1)) -> in_2_out_gg₂(X, tree_3₃(X, underscore, underscore1))
in_2_in_gg₂(X, tree_3₃(Y, Left, underscore2)) -> if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
if_in_2_in_1_gg₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_in_gg₂(X, Left))
in_2_in_gg₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
if_in_2_in_3_gg₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_gg₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_gg₂(X, tree_3₃(Y, underscore3, Right))
if_in_2_in_2_gg₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_gg₂(X, tree_3₃(Y, Left, underscore2))
if_in_2_in_2_ag₅(X, Y, Left, underscore2, in_2_out_gg₂(X, Left)) -> in_2_out_ag₂(X, tree_3₃(Y, Left, underscore2))
in_2_in_ag₂(X, tree_3₃(Y, underscore3, Right)) -> if_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_in_ga₂(Y, X))
less_2_in_ga₂(0_0, s_1₁(underscore4)) -> less_2_out_ga₂(0_0, s_1₁(underscore4))
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_in_2_in_3_ag₅(X, Y, underscore3, Right, less_2_out_ga₂(Y, X)) -> if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_in_gg₂(X, Right))
if_in_2_in_4_ag₅(X, Y, underscore3, Right, in_2_out_gg₂(X, Right)) -> in_2_out_ag₂(X, tree_3₃(Y, underscore3, Right))

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1₁(underscore4)) -> less_2_out_gg₂(0_0, s_1₁(underscore4))
less_2_in_gg₂(s_1₁(X), s_1₁(Y)) -> if_less_2_in_1_gg₃(X, Y, less_2_in_aa₂(X, Y))
if_less_2_in_1_gg₃(X, Y, less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1₁(X), s_1₁(Y))
less_2_in_aa₂(0_0, s_1₁(underscore4)) -> less_2_out_aa₂(0_0, s_1₁(underscore4))
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_3₃(x1, x2, x3)
0_0 = 0_0
s_1₁(x1) = s_1
less_2_in_gg₂(x1, x2) = less_2_in_gg₂(x1, x2)
less_2_out_gg₂(x1, x2) = less_2_out_gg₂(x1, x2)
if_less_2_in_1_gg₃(x1, x2, x3) = if_less_2_in_1_gg₁(x3)
less_2_in_aa₂(x1, x2) = less_2_in_aa
less_2_out_aa₂(x1, x2) = less_2_out_aa₂(x1, x2)
if_less_2_in_1_aa₃(x1, x2, x3) = if_less_2_in_1_aa₁(x3)
IF_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_3_GG₅(x1, x2, x3, x4, x5)
IN_2_IN_GG₂(x1, x2) = IN_2_IN_GG₂(x1, x2)
IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5) = IF_IN_2_IN_1_GG₅(x1, x2, x3, x4, x5)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))

The TRS R consists of the following rules:

less_2_in_gg₂(0_0, s_1) -> less_2_out_gg₂(0_0, s_1)
less_2_in_gg₂(s_1, s_1) -> if_less_2_in_1_gg₁(less_2_in_aa)
if_less_2_in_1_gg₁(less_2_out_aa₂(X, Y)) -> less_2_out_gg₂(s_1, s_1)
less_2_in_aa -> less_2_out_aa₂(0_0, s_1)
less_2_in_aa -> if_less_2_in_1_aa₁(less_2_in_aa)
if_less_2_in_1_aa₁(less_2_out_aa₂(X, Y)) -> less_2_out_aa₂(s_1, s_1)

The set Q consists of the following terms:

less_2_in_gg₂(x0, x1)
if_less_2_in_1_gg₁(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_IN_2_IN_3_GG₅, IN_2_IN_GG₂, IF_IN_2_IN_1_GG₅}.
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:

IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_out_gg₂(Y, X)) -> IN_2_IN_GG₂(X, Right)
The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2
IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_out_gg₂(X, Y)) -> IN_2_IN_GG₂(X, Left)
The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2
IN_2_IN_GG₂(X, tree_3₃(Y, Left, underscore2)) -> IF_IN_2_IN_1_GG₅(X, Y, Left, underscore2, less_2_in_gg₂(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
IN_2_IN_GG₂(X, tree_3₃(Y, underscore3, Right)) -> IF_IN_2_IN_3_GG₅(X, Y, underscore3, Right, less_2_in_gg₂(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4