Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tappend(T, X1, X2)).
tappend(nil, Y, Z) :- ','(!, eq(Y, Z)).
tappend(T, T1, node(T1, X, T2)) :- ','(left(T, nil), ','(right(T, T2), value(T, X))).
tappend(T, T2, node(T1, X, T2)) :- ','(left(T, T1), ','(right(T, nil), value(T, X))).
tappend(T, T3, node(U, X, T2)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T1, T3, U)))).
tappend(T, T1, node(T1, X, U)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T2, T3, U)))).
s2t(0, L) :- ','(!, eq(L, nil)).
s2t(X, node(T, X3, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X5, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X6, nil)).
left(nil, nil).
left(node(L, X7, X8), L).
right(nil, nil).
right(node(X9, X10, R), R).
value(nil, nil).
value(node(X11, X, X12), X).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tappend(T, X1, X2)).
tappend(nil, Y, Z) :- eq(Y, Z).
tappend(T, T1, node(T1, X, T2)) :- ','(left(T, nil), ','(right(T, T2), value(T, X))).
tappend(T, T2, node(T1, X, T2)) :- ','(left(T, T1), ','(right(T, nil), value(T, X))).
tappend(T, T3, node(U, X, T2)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T1, T3, U)))).
tappend(T, T1, node(T1, X, U)) :- ','(left(T, T1), ','(right(T, T2), ','(value(T, X), tappend(T2, T3, U)))).
s2t(0, L) :- eq(L, nil).
s2t(X, node(T, X3, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X5, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X6, nil)).
left(nil, nil).
left(node(L, X7, X8), L).
right(nil, nil).
right(node(X9, X10, R), R).
value(nil, nil).
value(node(X11, X, X12), X).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2t_in: (b,f)
tappend_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U18_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, T)) → P_IN_GA(X, P)
U19_GA(X, T, X3, p_out_ga(X, P)) → U20_GA(X, T, X3, s2t_in_ga(P, T))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X4, T)) → P_IN_GA(X, P)
U21_GA(X, X4, T, p_out_ga(X, P)) → U22_GA(X, X4, T, s2t_in_ga(P, T))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, nil)) → P_IN_GA(X, P)
U23_GA(X, T, X5, p_out_ga(X, P)) → U24_GA(X, T, X5, s2t_in_ga(P, T))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tappend_in_aaa(T, X1, X2))
U1_G(X, s2t_out_ga(X, T)) → TAPPEND_IN_AAA(T, X1, X2)
TAPPEND_IN_AAA(nil, Y, Z) → U3_AAA(Y, Z, eq_in_aa(Y, Z))
TAPPEND_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → U4_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U5_AAA(T, T1, X, T2, right_in_aa(T, T2))
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U6_AAA(T, T1, X, T2, value_in_aa(T, X))
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U7_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U8_AAA(T, T2, T1, X, right_in_ag(T, nil))
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U9_AAA(T, T2, T1, X, value_in_aa(T, X))
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → LEFT_IN_AA(T, T1)
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U17_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U18_GA(x1, x2)  =  U18_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U3_AAA(x1, x2, x3)  =  U3_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)
U17_AAA(x1, x2, x3, x4, x5)  =  U17_AAA(x5)

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

(6) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U18_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, T)) → P_IN_GA(X, P)
U19_GA(X, T, X3, p_out_ga(X, P)) → U20_GA(X, T, X3, s2t_in_ga(P, T))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X4, T)) → P_IN_GA(X, P)
U21_GA(X, X4, T, p_out_ga(X, P)) → U22_GA(X, X4, T, s2t_in_ga(P, T))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, nil)) → P_IN_GA(X, P)
U23_GA(X, T, X5, p_out_ga(X, P)) → U24_GA(X, T, X5, s2t_in_ga(P, T))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tappend_in_aaa(T, X1, X2))
U1_G(X, s2t_out_ga(X, T)) → TAPPEND_IN_AAA(T, X1, X2)
TAPPEND_IN_AAA(nil, Y, Z) → U3_AAA(Y, Z, eq_in_aa(Y, Z))
TAPPEND_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → U4_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U5_AAA(T, T1, X, T2, right_in_aa(T, T2))
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U6_AAA(T, T1, X, T2, value_in_aa(T, X))
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U7_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U8_AAA(T, T2, T1, X, right_in_ag(T, nil))
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U9_AAA(T, T2, T1, X, value_in_aa(T, X))
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → LEFT_IN_AA(T, T1)
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U17_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U18_GA(x1, x2)  =  U18_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U3_AAA(x1, x2, x3)  =  U3_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)
U17_AAA(x1, x2, x3, x4, x5)  =  U17_AAA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 34 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)

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

(10) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(11) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)

The argument filtering Pi contains the following mapping:
nil  =  nil
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)

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

(12) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(13) Obligation:

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

TAPPEND_IN_AAA → U10_AAA(left_in_aa)
U10_AAA(left_out_aa) → U11_AAA(right_in_aa)
U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(14) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U10_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U10_AAA(left_out_aa)

(15) Obligation:

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

U10_AAA(left_out_aa) → U11_AAA(right_in_aa)
U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(16) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U10_AAA(left_out_aa) → U11_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U10_AAA(left_out_aa) → U11_AAA(right_out_aa)

(17) Obligation:

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

U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(18) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U11_AAA(right_out_aa) → U12_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U11_AAA(right_out_aa) → U12_AAA(value_out_aa)

(19) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(20) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U14_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U14_AAA(left_out_aa)

(21) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(22) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(23) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(24) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

left_in_aa

(25) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(26) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U14_AAA(left_out_aa) → U15_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

(27) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(28) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(29) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

right_in_aa

(31) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

value_in_aa

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

(32) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U15_AAA(right_out_aa) → U16_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

(33) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

value_in_aa

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

(34) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(35) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

R is empty.
The set Q consists of the following terms:

value_in_aa

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

(36) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

value_in_aa

(37) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(38) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = TAPPEND_IN_AAA evaluates to t =TAPPEND_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

TAPPEND_IN_AAA → U10_AAA(left_out_aa)
with rule TAPPEND_IN_AAA → U10_AAA(left_out_aa) at position [] and matcher [ ]

U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
with rule U10_AAA(left_out_aa) → U11_AAA(right_out_aa) at position [] and matcher [ ]

U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
with rule U11_AAA(right_out_aa) → U12_AAA(value_out_aa) at position [] and matcher [ ]

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
with rule U12_AAA(value_out_aa) → TAPPEND_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(39) FALSE

(40) Obligation:

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

S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga
U19_ga(x1, x2, x3, x4)  =  U19_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U2_g(x1, x2)  =  U2_g(x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)

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

(41) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(42) Obligation:

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

S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0  =  0
nil  =  nil
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)

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

(43) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(44) Obligation:

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

S2T_IN_GA(X) → U19_GA(p_in_ga(X))
U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U21_GA(p_in_ga(X))
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(45) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0   
POL(S2T_IN_GA(x₁)) = x₁   
POL(U19_GA(x₁)) = x₁   
POL(U21_GA(x₁)) = x₁   
POL(U23_GA(x₁)) = x₁   
POL(p_in_ga(x₁)) = x₁   
POL(p_out_ga(x₁)) = 2·x₁   
POL(s(x₁)) = 2 + 2·x₁   

(46) Obligation:

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

S2T_IN_GA(X) → U19_GA(p_in_ga(X))
U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U21_GA(p_in_ga(X))
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(47) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U19_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U19_GA(p_out_ga(0))

(48) Obligation:

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

U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U21_GA(p_in_ga(X))
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U21_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U21_GA(p_out_ga(0))

(50) Obligation:

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

U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(51) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U23_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U23_GA(p_out_ga(0))

(52) Obligation:

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

U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(53) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(54) Obligation:

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

U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(55) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(56) Obligation:

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

U19_GA(p_out_ga(P)) → S2T_IN_GA(P)
U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(57) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U19_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U19_GA(p_out_ga(0)) → S2T_IN_GA(0)

(58) Obligation:

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

U21_GA(p_out_ga(P)) → S2T_IN_GA(P)
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
U19_GA(p_out_ga(0)) → S2T_IN_GA(0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(59) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U21_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U21_GA(p_out_ga(0)) → S2T_IN_GA(0)

(60) Obligation:

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

U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
U19_GA(p_out_ga(0)) → S2T_IN_GA(0)
U21_GA(p_out_ga(0)) → S2T_IN_GA(0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(61) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U23_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U23_GA(p_out_ga(0)) → S2T_IN_GA(0)

(62) Obligation:

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

S2T_IN_GA(0) → U19_GA(p_out_ga(0))
S2T_IN_GA(0) → U21_GA(p_out_ga(0))
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
U19_GA(p_out_ga(0)) → S2T_IN_GA(0)
U21_GA(p_out_ga(0)) → S2T_IN_GA(0)
U23_GA(p_out_ga(0)) → S2T_IN_GA(0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(63) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U19_GA(p_out_ga(0)) evaluates to t =U19_GA(p_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U19_GA(p_out_ga(0)) → S2T_IN_GA(0)
with rule U19_GA(p_out_ga(0)) → S2T_IN_GA(0) at position [] and matcher [ ]

S2T_IN_GA(0) → U19_GA(p_out_ga(0))
with rule S2T_IN_GA(0) → U19_GA(p_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(64) FALSE

(65) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2t_in: (b,f)
tappend_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(66) Obligation:

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)

(67) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U18_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, T)) → P_IN_GA(X, P)
U19_GA(X, T, X3, p_out_ga(X, P)) → U20_GA(X, T, X3, s2t_in_ga(P, T))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X4, T)) → P_IN_GA(X, P)
U21_GA(X, X4, T, p_out_ga(X, P)) → U22_GA(X, X4, T, s2t_in_ga(P, T))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, nil)) → P_IN_GA(X, P)
U23_GA(X, T, X5, p_out_ga(X, P)) → U24_GA(X, T, X5, s2t_in_ga(P, T))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tappend_in_aaa(T, X1, X2))
U1_G(X, s2t_out_ga(X, T)) → TAPPEND_IN_AAA(T, X1, X2)
TAPPEND_IN_AAA(nil, Y, Z) → U3_AAA(Y, Z, eq_in_aa(Y, Z))
TAPPEND_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → U4_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U5_AAA(T, T1, X, T2, right_in_aa(T, T2))
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U6_AAA(T, T1, X, T2, value_in_aa(T, X))
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U7_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U8_AAA(T, T2, T1, X, right_in_ag(T, nil))
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U9_AAA(T, T2, T1, X, value_in_aa(T, X))
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → LEFT_IN_AA(T, T1)
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U17_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U18_GA(x1, x2)  =  U18_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U3_AAA(x1, x2, x3)  =  U3_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)
U17_AAA(x1, x2, x3, x4, x5)  =  U17_AAA(x5)

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

(68) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U18_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, T)) → P_IN_GA(X, P)
U19_GA(X, T, X3, p_out_ga(X, P)) → U20_GA(X, T, X3, s2t_in_ga(P, T))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X4, T)) → P_IN_GA(X, P)
U21_GA(X, X4, T, p_out_ga(X, P)) → U22_GA(X, X4, T, s2t_in_ga(P, T))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, nil)) → P_IN_GA(X, P)
U23_GA(X, T, X5, p_out_ga(X, P)) → U24_GA(X, T, X5, s2t_in_ga(P, T))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tappend_in_aaa(T, X1, X2))
U1_G(X, s2t_out_ga(X, T)) → TAPPEND_IN_AAA(T, X1, X2)
TAPPEND_IN_AAA(nil, Y, Z) → U3_AAA(Y, Z, eq_in_aa(Y, Z))
TAPPEND_IN_AAA(nil, Y, Z) → EQ_IN_AA(Y, Z)
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → U4_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U5_AAA(T, T1, X, T2, right_in_aa(T, T2))
U4_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U6_AAA(T, T1, X, T2, value_in_aa(T, X))
U5_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U7_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U8_AAA(T, T2, T1, X, right_in_ag(T, nil))
U7_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U9_AAA(T, T2, T1, X, value_in_aa(T, X))
U8_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → LEFT_IN_AA(T, T1)
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U17_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U18_GA(x1, x2)  =  U18_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U20_GA(x1, x2, x3, x4)  =  U20_GA(x1, x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x4)
U22_GA(x1, x2, x3, x4)  =  U22_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U3_AAA(x1, x2, x3)  =  U3_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U4_AAA(x1, x2, x3, x4, x5)  =  U4_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U5_AAA(x1, x2, x3, x4, x5)  =  U5_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_AAA(x1, x2, x3, x4, x5)  =  U6_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U7_AAA(x1, x2, x3, x4, x5)  =  U7_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U13_AAA(x1, x2, x3, x4, x5, x6)  =  U13_AAA(x6)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)
U17_AAA(x1, x2, x3, x4, x5)  =  U17_AAA(x5)

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

(69) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 34 less nodes.

(70) Complex Obligation (AND)

(71) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)

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

(72) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(73) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U10_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U10_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → U14_AAA(T, T1, X, U, left_in_aa(T, T1))
U14_AAA(T, T1, X, U, left_out_aa(T, T1)) → U15_AAA(T, T1, X, U, right_in_aa(T, T2))
U15_AAA(T, T1, X, U, right_out_aa(T, T2)) → U16_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U16_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)

The TRS R consists of the following rules:

left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)

The argument filtering Pi contains the following mapping:
nil  =  nil
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6)  =  U10_AAA(x6)
U11_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U11_AAA(x7)
U12_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_AAA(x7)
U14_AAA(x1, x2, x3, x4, x5)  =  U14_AAA(x5)
U15_AAA(x1, x2, x3, x4, x5)  =  U15_AAA(x5)
U16_AAA(x1, x2, x3, x4, x5, x6)  =  U16_AAA(x6)

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

(74) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(75) Obligation:

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

TAPPEND_IN_AAA → U10_AAA(left_in_aa)
U10_AAA(left_out_aa) → U11_AAA(right_in_aa)
U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(76) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U10_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U10_AAA(left_out_aa)

(77) Obligation:

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

U10_AAA(left_out_aa) → U11_AAA(right_in_aa)
U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(78) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U10_AAA(left_out_aa) → U11_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U10_AAA(left_out_aa) → U11_AAA(right_out_aa)

(79) Obligation:

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

U11_AAA(right_out_aa) → U12_AAA(value_in_aa)
U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(80) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U11_AAA(right_out_aa) → U12_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U11_AAA(right_out_aa) → U12_AAA(value_out_aa)

(81) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U14_AAA(left_in_aa)
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(82) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U14_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U14_AAA(left_out_aa)

(83) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
value_in_aa → value_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(84) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(85) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

left_in_aa
right_in_aa
value_in_aa

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

(86) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

left_in_aa

(87) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U14_AAA(left_out_aa) → U15_AAA(right_in_aa)
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(88) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U14_AAA(left_out_aa) → U15_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

(89) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa
right_in_aa → right_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(90) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(91) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

right_in_aa
value_in_aa

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

(92) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

right_in_aa

(93) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

value_in_aa

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

(94) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U15_AAA(right_out_aa) → U16_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

(95) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

The TRS R consists of the following rules:

value_in_aa → value_out_aa

The set Q consists of the following terms:

value_in_aa

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

(96) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(97) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

R is empty.
The set Q consists of the following terms:

value_in_aa

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

(98) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

value_in_aa

(99) Obligation:

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

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U10_AAA(left_out_aa)
U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
TAPPEND_IN_AAA → U14_AAA(left_out_aa)
U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
U15_AAA(right_out_aa) → U16_AAA(value_out_aa)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(100) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = TAPPEND_IN_AAA evaluates to t =TAPPEND_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

TAPPEND_IN_AAA → U10_AAA(left_out_aa)
with rule TAPPEND_IN_AAA → U10_AAA(left_out_aa) at position [] and matcher [ ]

U10_AAA(left_out_aa) → U11_AAA(right_out_aa)
with rule U10_AAA(left_out_aa) → U11_AAA(right_out_aa) at position [] and matcher [ ]

U11_AAA(right_out_aa) → U12_AAA(value_out_aa)
with rule U11_AAA(right_out_aa) → U12_AAA(value_out_aa) at position [] and matcher [ ]

U12_AAA(value_out_aa) → TAPPEND_IN_AAA
with rule U12_AAA(value_out_aa) → TAPPEND_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(101) FALSE

(102) Obligation:

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

S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U18_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U18_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X3, T)) → U19_ga(X, T, X3, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U19_ga(X, T, X3, p_out_ga(X, P)) → U20_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, T)) → U21_ga(X, X4, T, p_in_ga(X, P))
U21_ga(X, X4, T, p_out_ga(X, P)) → U22_ga(X, X4, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X5, nil)) → U23_ga(X, T, X5, p_in_ga(X, P))
U23_ga(X, T, X5, p_out_ga(X, P)) → U24_ga(X, T, X5, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X6, nil)) → s2t_out_ga(X, node(nil, X6, nil))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, nil))
U22_ga(X, X4, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X4, T))
U20_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tappend_in_aaa(T, X1, X2))
tappend_in_aaa(nil, Y, Z) → U3_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U3_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U4_aaa(T, T1, X, T2, left_in_ag(T, nil))
left_in_ag(nil, nil) → left_out_ag(nil, nil)
left_in_ag(node(L, X7, X8), L) → left_out_ag(node(L, X7, X8), L)
U4_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U5_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X9, X10, R), R) → right_out_aa(node(X9, X10, R), R)
U5_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U6_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X11, X, X12), X) → value_out_aa(node(X11, X, X12), X)
U6_aaa(T, T1, X, T2, value_out_aa(T, X)) → tappend_out_aaa(T, T1, node(T1, X, T2))
tappend_in_aaa(T, T2, node(T1, X, T2)) → U7_aaa(T, T2, T1, X, left_in_aa(T, T1))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X7, X8), L) → left_out_aa(node(L, X7, X8), L)
U7_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U8_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X9, X10, R), R) → right_out_ag(node(X9, X10, R), R)
U8_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U9_aaa(T, T2, T1, X, value_in_aa(T, X))
U9_aaa(T, T2, T1, X, value_out_aa(T, X)) → tappend_out_aaa(T, T2, node(T1, X, T2))
tappend_in_aaa(T, T3, node(U, X, T2)) → U10_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U10_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U11_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U11_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U12_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U12_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U13_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U14_aaa(T, T1, X, U, left_in_aa(T, T1))
U14_aaa(T, T1, X, U, left_out_aa(T, T1)) → U15_aaa(T, T1, X, U, right_in_aa(T, T2))
U15_aaa(T, T1, X, U, right_out_aa(T, T2)) → U16_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U16_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U17_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U17_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U13_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U2_g(X, tappend_out_aaa(T, X1, X2)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U18_ga(x1, x2)  =  U18_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1)
U19_ga(x1, x2, x3, x4)  =  U19_ga(x1, x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U20_ga(x1, x2, x3, x4)  =  U20_ga(x1, x4)
U21_ga(x1, x2, x3, x4)  =  U21_ga(x1, x4)
U22_ga(x1, x2, x3, x4)  =  U22_ga(x1, x4)
U23_ga(x1, x2, x3, x4)  =  U23_ga(x1, x4)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U2_g(x1, x2)  =  U2_g(x1, x2)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U3_aaa(x1, x2, x3)  =  U3_aaa(x3)
eq_in_aa(x1, x2)  =  eq_in_aa
eq_out_aa(x1, x2)  =  eq_out_aa
tappend_out_aaa(x1, x2, x3)  =  tappend_out_aaa
U4_aaa(x1, x2, x3, x4, x5)  =  U4_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U5_aaa(x1, x2, x3, x4, x5)  =  U5_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_aaa(x1, x2, x3, x4, x5)  =  U6_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U7_aaa(x1, x2, x3, x4, x5)  =  U7_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
U10_aaa(x1, x2, x3, x4, x5, x6)  =  U10_aaa(x6)
U11_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aaa(x7)
U12_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U12_aaa(x7)
U13_aaa(x1, x2, x3, x4, x5, x6)  =  U13_aaa(x6)
U14_aaa(x1, x2, x3, x4, x5)  =  U14_aaa(x5)
U15_aaa(x1, x2, x3, x4, x5)  =  U15_aaa(x5)
U16_aaa(x1, x2, x3, x4, x5, x6)  =  U16_aaa(x6)
U17_aaa(x1, x2, x3, x4, x5)  =  U17_aaa(x5)
goal_out_g(x1)  =  goal_out_g(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)

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

(103) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(104) Obligation:

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

S2T_IN_GA(X, node(T, X3, T)) → U19_GA(X, T, X3, p_in_ga(X, P))
U19_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X4, T)) → U21_GA(X, X4, T, p_in_ga(X, P))
U21_GA(X, X4, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X5, nil)) → U23_GA(X, T, X5, p_in_ga(X, P))
U23_GA(X, T, X5, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0  =  0
nil  =  nil
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U19_GA(x1, x2, x3, x4)  =  U19_GA(x1, x4)
U21_GA(x1, x2, x3, x4)  =  U21_GA(x1, x4)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)

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

(105) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(106) Obligation:

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

S2T_IN_GA(X) → U19_GA(X, p_in_ga(X))
U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U21_GA(X, p_in_ga(X))
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(X, p_in_ga(X))
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(107) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U19_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))

(108) Obligation:

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

U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U21_GA(X, p_in_ga(X))
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(X, p_in_ga(X))
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(109) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U21_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))

(110) Obligation:

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

U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U23_GA(X, p_in_ga(X))
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(111) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U23_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))

(112) Obligation:

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

U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(113) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(114) Obligation:

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

U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(115) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(116) Obligation:

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

U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(117) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U19_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(118) Obligation:

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

U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))
U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(119) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U21_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U21_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U21_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(120) Obligation:

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

U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))
U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U21_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U21_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(121) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U23_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U23_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(122) Obligation:

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

S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))
U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U21_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U21_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U23_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U23_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(123) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(124) Complex Obligation (AND)

(125) Obligation:

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

U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
S2T_IN_GA(0) → U21_GA(0, p_out_ga(0, 0))
U21_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
U23_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(126) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = S2T_IN_GA(0) evaluates to t =S2T_IN_GA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0))
with rule S2T_IN_GA(0) → U19_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
with rule U19_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.

(127) FALSE

(128) Obligation:

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

S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
U21_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))
U23_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(129) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• U19_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• S2T_IN_GA(s(x0)) → U19_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

• U21_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• U23_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• S2T_IN_GA(s(x0)) → U21_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

• S2T_IN_GA(s(x0)) → U23_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

(130) TRUE