Left Termination of the query pattern goal_in_3(g, a, a) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

goal(A, B, C) :- ','(s2t(A, T), tapplast(T, B, C)).
tapplast(L, X, Last) :- ','(tappend(L, node(nil, X, nil), LX), tlast(Last, LX)).
tlast(X, node(nil, X, nil)).
tlast(X, node(L, X1, X2)) :- tlast(X, L).
tlast(X, node(X3, X4, R)) :- tlast(X, R).
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, X5, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X6, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X7, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X8, nil)).
left(nil, nil).
left(node(L, X9, X10), L).
right(nil, nil).
right(node(X11, X12, R), R).
value(nil, nil).
value(node(X13, X, X14), X).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g,a,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

goal(A, B, C) :- ','(s2t(A, T), tapplast(T, B, C)).
tapplast(L, X, Last) :- ','(tappend(L, node(nil, X, nil), LX), tlast(Last, LX)).
tlast(X, node(nil, X, nil)).
tlast(X, node(L, X1, X2)) :- tlast(X, L).
tlast(X, node(X3, X4, R)) :- tlast(X, R).
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, X5, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X6, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X7, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X8, nil)).
left(nil, nil).
left(node(L, X9, X10), L).
right(nil, nil).
right(node(X11, X12, R), R).
value(nil, nil).
value(node(X13, X, X14), X).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g,a,a).

(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,f,f)
s2t_in: (b,f)
tapplast_in: (f,f,f)
tappend_in: (f,f,f)
tlast_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa

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_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa

(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_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(0, L) → U22_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X5, T)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X6, T)) → P_IN_GA(X, P)
U25_GA(X, X6, T, p_out_ga(X, P)) → U26_GA(X, X6, T, s2t_in_ga(P, T))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X7, nil)) → P_IN_GA(X, P)
U27_GA(X, T, X7, p_out_ga(X, P)) → U28_GA(X, T, X7, s2t_in_ga(P, T))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(nil, Y, Z) → U7_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)) → U8_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U9_AAA(T, T1, X, T2, right_in_aa(T, T2))
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U10_AAA(T, T1, X, T2, value_in_aa(T, X))
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U11_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U12_AAA(T, T2, T1, X, right_in_ag(T, nil))
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U13_AAA(T, T2, T1, X, value_in_aa(T, X))
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_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)
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U21_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, X1, X2)) → U5_AA(X, L, X1, X2, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(X3, X4, R)) → U6_AA(X, X3, X4, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U22_GA(x1, x2)  =  U22_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x4)
U28_GA(x1, x2, x3, x4)  =  U28_GA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
TAPPLAST_IN_AAA(x1, x2, x3)  =  TAPPLAST_IN_AAA
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x4)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U7_AAA(x1, x2, x3)  =  U7_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U10_AAA(x1, x2, x3, x4, x5)  =  U10_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U17_AAA(x1, x2, x3, x4, x5, x6)  =  U17_AAA(x6)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)
U21_AAA(x1, x2, x3, x4, x5)  =  U21_AAA(x5)
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA
U5_AA(x1, x2, x3, x4, x5)  =  U5_AA(x5)
U6_AA(x1, x2, x3, x4, x5)  =  U6_AA(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_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(0, L) → U22_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X5, T)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X6, T)) → P_IN_GA(X, P)
U25_GA(X, X6, T, p_out_ga(X, P)) → U26_GA(X, X6, T, s2t_in_ga(P, T))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X7, nil)) → P_IN_GA(X, P)
U27_GA(X, T, X7, p_out_ga(X, P)) → U28_GA(X, T, X7, s2t_in_ga(P, T))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(nil, Y, Z) → U7_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)) → U8_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U9_AAA(T, T1, X, T2, right_in_aa(T, T2))
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U10_AAA(T, T1, X, T2, value_in_aa(T, X))
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U11_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U12_AAA(T, T2, T1, X, right_in_ag(T, nil))
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U13_AAA(T, T2, T1, X, value_in_aa(T, X))
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_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)
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U21_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, X1, X2)) → U5_AA(X, L, X1, X2, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(X3, X4, R)) → U6_AA(X, X3, X4, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U22_GA(x1, x2)  =  U22_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x4)
U28_GA(x1, x2, x3, x4)  =  U28_GA(x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x4)
TAPPLAST_IN_AAA(x1, x2, x3)  =  TAPPLAST_IN_AAA
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x4)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U7_AAA(x1, x2, x3)  =  U7_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U10_AAA(x1, x2, x3, x4, x5)  =  U10_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U17_AAA(x1, x2, x3, x4, x5, x6)  =  U17_AAA(x6)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)
U21_AAA(x1, x2, x3, x4, x5)  =  U21_AAA(x5)
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA
U5_AA(x1, x2, x3, x4, x5)  =  U5_AA(x5)
U6_AA(x1, x2, x3, x4, x5)  =  U6_AA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 40 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA

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:

TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA

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:

TLAST_IN_AA → TLAST_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = TLAST_IN_AA evaluates to t =TLAST_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TLAST_IN_AA to TLAST_IN_AA.

(15) FALSE

(16) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U20_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_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U20_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), 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
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

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 → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_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.

(21) 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)

(22) Obligation:

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

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 → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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.

(23) 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)

(24) Obligation:

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

U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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:

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.

(25) 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)

(26) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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:

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.

(27) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U18_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U18_AAA(left_out_aa)

(28) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(29) 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.

(30) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(31) 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

(32) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(33) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U18_AAA(left_out_aa) → U19_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U18_AAA(left_out_aa) → U19_AAA(right_out_aa)

(34) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(35) 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.

(36) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(37) 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

(38) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(39) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U19_AAA(right_out_aa) → U20_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U19_AAA(right_out_aa) → U20_AAA(value_out_aa)

(40) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_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.

(41) 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.

(42) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_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.

(43) 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

(44) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_AAA(value_out_aa)

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

(45) 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:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

TAPPEND_IN_AAA → U14_AAA(left_out_aa)
with rule TAPPEND_IN_AAA → U14_AAA(left_out_aa) at position [] and matcher [ ]

U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
with rule U14_AAA(left_out_aa) → U15_AAA(right_out_aa) at position [] and matcher [ ]

U15_AAA(right_out_aa) → U16_AAA(value_out_aa)
with rule U15_AAA(right_out_aa) → U16_AAA(value_out_aa) at position [] and matcher [ ]

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
with rule U16_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.

(46) FALSE

(47) Obligation:

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

S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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
U23_ga(x1, x2, x3, x4)  =  U23_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x4)

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

(48) UsableRulesProof (EQUIVALENT transformation)

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

(49) Obligation:

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

S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
U27_GA(X, T, X7, 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)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x4)

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

(50) PiDPToQDPProof (SOUND transformation)

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

(51) Obligation:

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

S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U25_GA(p_in_ga(X))
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(p_in_ga(X))
U27_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.

(52) 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(U23_GA(x₁)) = x₁   
POL(U25_GA(x₁)) = x₁   
POL(U27_GA(x₁)) = x₁   
POL(p_in_ga(x₁)) = x₁   
POL(p_out_ga(x₁)) = 2·x₁   
POL(s(x₁)) = 2 + 2·x₁   

(53) Obligation:

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

S2T_IN_GA(X) → U23_GA(p_in_ga(X))
U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U25_GA(p_in_ga(X))
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(p_in_ga(X))
U27_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.

(54) 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))

(55) 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(X) → U25_GA(p_in_ga(X))
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(p_in_ga(X))
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
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.

(56) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U25_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U25_GA(p_out_ga(0))

(57) Obligation:

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

U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(p_in_ga(X))
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_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.

(58) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U27_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U27_GA(p_out_ga(0))

(59) Obligation:

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

U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_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.

(60) 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.

(61) Obligation:

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

U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_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.

(62) 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)

(63) Obligation:

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

U23_GA(p_out_ga(P)) → S2T_IN_GA(P)
U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_GA(p_out_ga(0))

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

(64) 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)

(65) Obligation:

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

U25_GA(p_out_ga(P)) → S2T_IN_GA(P)
U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_GA(p_out_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.

(66) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U25_GA(p_out_ga(0)) → S2T_IN_GA(0)

(67) Obligation:

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

U27_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_GA(p_out_ga(0))
U23_GA(p_out_ga(0)) → S2T_IN_GA(0)
U25_GA(p_out_ga(0)) → S2T_IN_GA(0)

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

(68) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U27_GA(p_out_ga(0)) → S2T_IN_GA(0)

(69) Obligation:

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

S2T_IN_GA(0) → U23_GA(p_out_ga(0))
S2T_IN_GA(0) → U25_GA(p_out_ga(0))
S2T_IN_GA(0) → U27_GA(p_out_ga(0))
U23_GA(p_out_ga(0)) → S2T_IN_GA(0)
U25_GA(p_out_ga(0)) → S2T_IN_GA(0)
U27_GA(p_out_ga(0)) → S2T_IN_GA(0)

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

(70) 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 = U23_GA(p_out_ga(0)) evaluates to t =U23_GA(p_out_ga(0))

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U23_GA(p_out_ga(0)) → S2T_IN_GA(0)
with rule U23_GA(p_out_ga(0)) → S2T_IN_GA(0) at position [] and matcher [ ]

S2T_IN_GA(0) → U23_GA(p_out_ga(0))
with rule S2T_IN_GA(0) → U23_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.

(71) FALSE

(72) 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,f,f)
s2t_in: (b,f)
tapplast_in: (f,f,f)
tappend_in: (f,f,f)
tlast_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(73) Obligation:

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

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)

(74) 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_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(0, L) → U22_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X5, T)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X6, T)) → P_IN_GA(X, P)
U25_GA(X, X6, T, p_out_ga(X, P)) → U26_GA(X, X6, T, s2t_in_ga(P, T))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X7, nil)) → P_IN_GA(X, P)
U27_GA(X, T, X7, p_out_ga(X, P)) → U28_GA(X, T, X7, s2t_in_ga(P, T))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(nil, Y, Z) → U7_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)) → U8_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U9_AAA(T, T1, X, T2, right_in_aa(T, T2))
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U10_AAA(T, T1, X, T2, value_in_aa(T, X))
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U11_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U12_AAA(T, T2, T1, X, right_in_ag(T, nil))
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U13_AAA(T, T2, T1, X, value_in_aa(T, X))
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_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)
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U21_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, X1, X2)) → U5_AA(X, L, X1, X2, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(X3, X4, R)) → U6_AA(X, X3, X4, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U22_GA(x1, x2)  =  U22_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x1, x4)
U28_GA(x1, x2, x3, x4)  =  U28_GA(x1, x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
TAPPLAST_IN_AAA(x1, x2, x3)  =  TAPPLAST_IN_AAA
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x4)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U7_AAA(x1, x2, x3)  =  U7_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U10_AAA(x1, x2, x3, x4, x5)  =  U10_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U17_AAA(x1, x2, x3, x4, x5, x6)  =  U17_AAA(x6)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)
U21_AAA(x1, x2, x3, x4, x5)  =  U21_AAA(x5)
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA
U5_AA(x1, x2, x3, x4, x5)  =  U5_AA(x5)
U6_AA(x1, x2, x3, x4, x5)  =  U6_AA(x5)

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

(75) Obligation:

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

GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(0, L) → U22_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X5, T)) → U23_GA(X, T, X5, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X6, T)) → P_IN_GA(X, P)
U25_GA(X, X6, T, p_out_ga(X, P)) → U26_GA(X, X6, T, s2t_in_ga(P, T))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X7, nil)) → P_IN_GA(X, P)
U27_GA(X, T, X7, p_out_ga(X, P)) → U28_GA(X, T, X7, s2t_in_ga(P, T))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(nil, Y, Z) → U7_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)) → U8_AAA(T, T1, X, T2, left_in_ag(T, nil))
TAPPEND_IN_AAA(T, T1, node(T1, X, T2)) → LEFT_IN_AG(T, nil)
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → U9_AAA(T, T1, X, T2, right_in_aa(T, T2))
U8_AAA(T, T1, X, T2, left_out_ag(T, nil)) → RIGHT_IN_AA(T, T2)
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → U10_AAA(T, T1, X, T2, value_in_aa(T, X))
U9_AAA(T, T1, X, T2, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → U11_AAA(T, T2, T1, X, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T2, node(T1, X, T2)) → LEFT_IN_AA(T, T1)
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → U12_AAA(T, T2, T1, X, right_in_ag(T, nil))
U11_AAA(T, T2, T1, X, left_out_aa(T, T1)) → RIGHT_IN_AG(T, nil)
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → U13_AAA(T, T2, T1, X, value_in_aa(T, X))
U12_AAA(T, T2, T1, X, right_out_ag(T, nil)) → VALUE_IN_AA(T, X)
TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_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)
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U16_AAA(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_AAA(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
TAPPEND_IN_AAA(T, T1, node(T1, X, U)) → LEFT_IN_AA(T, T1)
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → RIGHT_IN_AA(T, T2)
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → VALUE_IN_AA(T, X)
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → U21_AAA(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U20_AAA(T, T1, X, U, T2, value_out_aa(T, X)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, X1, X2)) → U5_AA(X, L, X1, X2, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(X3, X4, R)) → U6_AA(X, X3, X4, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
GOAL_IN_GAA(x1, x2, x3)  =  GOAL_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4)  =  U1_GAA(x1, x4)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U22_GA(x1, x2)  =  U22_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U24_GA(x1, x2, x3, x4)  =  U24_GA(x1, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x4)
U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x1, x4)
U28_GA(x1, x2, x3, x4)  =  U28_GA(x1, x4)
U2_GAA(x1, x2, x3, x4)  =  U2_GAA(x1, x4)
TAPPLAST_IN_AAA(x1, x2, x3)  =  TAPPLAST_IN_AAA
U3_AAA(x1, x2, x3, x4)  =  U3_AAA(x4)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U7_AAA(x1, x2, x3)  =  U7_AAA(x3)
EQ_IN_AA(x1, x2)  =  EQ_IN_AA
U8_AAA(x1, x2, x3, x4, x5)  =  U8_AAA(x5)
LEFT_IN_AG(x1, x2)  =  LEFT_IN_AG(x2)
U9_AAA(x1, x2, x3, x4, x5)  =  U9_AAA(x5)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U10_AAA(x1, x2, x3, x4, x5)  =  U10_AAA(x5)
VALUE_IN_AA(x1, x2)  =  VALUE_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
RIGHT_IN_AG(x1, x2)  =  RIGHT_IN_AG(x2)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U17_AAA(x1, x2, x3, x4, x5, x6)  =  U17_AAA(x6)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)
U21_AAA(x1, x2, x3, x4, x5)  =  U21_AAA(x5)
U4_AAA(x1, x2, x3, x4)  =  U4_AAA(x4)
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA
U5_AA(x1, x2, x3, x4, x5)  =  U5_AA(x5)
U6_AA(x1, x2, x3, x4, x5)  =  U6_AA(x5)

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

(76) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 40 less nodes.

(77) Complex Obligation (AND)

(78) Obligation:

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

TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA

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

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

TLAST_IN_AA(X, node(X3, X4, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, X1, X2)) → TLAST_IN_AA(X, L)

R is empty.
The argument filtering Pi contains the following mapping:
TLAST_IN_AA(x1, x2)  =  TLAST_IN_AA

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

(81) PiDPToQDPProof (SOUND transformation)

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

(82) Obligation:

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

TLAST_IN_AA → TLAST_IN_AA

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

(83) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = TLAST_IN_AA evaluates to t =TLAST_IN_AA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TLAST_IN_AA to TLAST_IN_AA.

(84) FALSE

(85) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U20_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_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
TAPPEND_IN_AAA(x1, x2, x3)  =  TAPPEND_IN_AAA
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)

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

(86) UsableRulesProof (EQUIVALENT transformation)

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

(87) Obligation:

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

TAPPEND_IN_AAA(T, T3, node(U, X, T2)) → U14_AAA(T, T3, U, X, T2, left_in_aa(T, T1))
U14_AAA(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_AAA(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_AAA(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_AAA(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_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)) → U18_AAA(T, T1, X, U, left_in_aa(T, T1))
U18_AAA(T, T1, X, U, left_out_aa(T, T1)) → U19_AAA(T, T1, X, U, right_in_aa(T, T2))
U19_AAA(T, T1, X, U, right_out_aa(T, T2)) → U20_AAA(T, T1, X, U, T2, value_in_aa(T, X))
U20_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), 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
U14_AAA(x1, x2, x3, x4, x5, x6)  =  U14_AAA(x6)
U15_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U15_AAA(x7)
U16_AAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_AAA(x7)
U18_AAA(x1, x2, x3, x4, x5)  =  U18_AAA(x5)
U19_AAA(x1, x2, x3, x4, x5)  =  U19_AAA(x5)
U20_AAA(x1, x2, x3, x4, x5, x6)  =  U20_AAA(x6)

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

(88) PiDPToQDPProof (SOUND transformation)

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

(89) Obligation:

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

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 → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_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.

(90) 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)

(91) Obligation:

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

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 → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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.

(92) 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)

(93) Obligation:

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

U15_AAA(right_out_aa) → U16_AAA(value_in_aa)
U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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:

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.

(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:

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
TAPPEND_IN_AAA → U18_AAA(left_in_aa)
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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:

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.

(96) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TAPPEND_IN_AAA → U18_AAA(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TAPPEND_IN_AAA → U18_AAA(left_out_aa)

(97) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(98) 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.

(99) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(100) 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

(101) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U18_AAA(left_out_aa) → U19_AAA(right_in_aa)
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_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.

(102) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U18_AAA(left_out_aa) → U19_AAA(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U18_AAA(left_out_aa) → U19_AAA(right_out_aa)

(103) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(104) 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.

(105) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(106) 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

(107) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U19_AAA(right_out_aa) → U20_AAA(value_in_aa)
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_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.

(108) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U19_AAA(right_out_aa) → U20_AAA(value_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U19_AAA(right_out_aa) → U20_AAA(value_out_aa)

(109) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_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.

(110) 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.

(111) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_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.

(112) 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

(113) Obligation:

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

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
U20_AAA(value_out_aa) → TAPPEND_IN_AAA
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)
TAPPEND_IN_AAA → U18_AAA(left_out_aa)
U18_AAA(left_out_aa) → U19_AAA(right_out_aa)
U19_AAA(right_out_aa) → U20_AAA(value_out_aa)

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

(114) 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:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

TAPPEND_IN_AAA → U14_AAA(left_out_aa)
with rule TAPPEND_IN_AAA → U14_AAA(left_out_aa) at position [] and matcher [ ]

U14_AAA(left_out_aa) → U15_AAA(right_out_aa)
with rule U14_AAA(left_out_aa) → U15_AAA(right_out_aa) at position [] and matcher [ ]

U15_AAA(right_out_aa) → U16_AAA(value_out_aa)
with rule U15_AAA(right_out_aa) → U16_AAA(value_out_aa) at position [] and matcher [ ]

U16_AAA(value_out_aa) → TAPPEND_IN_AAA
with rule U16_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.

(115) FALSE

(116) Obligation:

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

S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
U27_GA(X, T, X7, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(0, L) → U22_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U22_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X5, T)) → U23_ga(X, T, X5, 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)
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, T)) → U25_ga(X, X6, T, p_in_ga(X, P))
U25_ga(X, X6, T, p_out_ga(X, P)) → U26_ga(X, X6, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X7, nil)) → U27_ga(X, T, X7, p_in_ga(X, P))
U27_ga(X, T, X7, p_out_ga(X, P)) → U28_ga(X, T, X7, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X8, nil)) → s2t_out_ga(X, node(nil, X8, nil))
U28_ga(X, T, X7, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X7, nil))
U26_ga(X, X6, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X6, T))
U24_ga(X, T, X5, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X5, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, Y, Z) → U7_aaa(Y, Z, eq_in_aa(Y, Z))
eq_in_aa(X, X) → eq_out_aa(X, X)
U7_aaa(Y, Z, eq_out_aa(Y, Z)) → tappend_out_aaa(nil, Y, Z)
tappend_in_aaa(T, T1, node(T1, X, T2)) → U8_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, X9, X10), L) → left_out_ag(node(L, X9, X10), L)
U8_aaa(T, T1, X, T2, left_out_ag(T, nil)) → U9_aaa(T, T1, X, T2, right_in_aa(T, T2))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X11, X12, R), R) → right_out_aa(node(X11, X12, R), R)
U9_aaa(T, T1, X, T2, right_out_aa(T, T2)) → U10_aaa(T, T1, X, T2, value_in_aa(T, X))
value_in_aa(nil, nil) → value_out_aa(nil, nil)
value_in_aa(node(X13, X, X14), X) → value_out_aa(node(X13, X, X14), X)
U10_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)) → U11_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, X9, X10), L) → left_out_aa(node(L, X9, X10), L)
U11_aaa(T, T2, T1, X, left_out_aa(T, T1)) → U12_aaa(T, T2, T1, X, right_in_ag(T, nil))
right_in_ag(nil, nil) → right_out_ag(nil, nil)
right_in_ag(node(X11, X12, R), R) → right_out_ag(node(X11, X12, R), R)
U12_aaa(T, T2, T1, X, right_out_ag(T, nil)) → U13_aaa(T, T2, T1, X, value_in_aa(T, X))
U13_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)) → U14_aaa(T, T3, U, X, T2, left_in_aa(T, T1))
U14_aaa(T, T3, U, X, T2, left_out_aa(T, T1)) → U15_aaa(T, T3, U, X, T2, T1, right_in_aa(T, T2))
U15_aaa(T, T3, U, X, T2, T1, right_out_aa(T, T2)) → U16_aaa(T, T3, U, X, T2, T1, value_in_aa(T, X))
U16_aaa(T, T3, U, X, T2, T1, value_out_aa(T, X)) → U17_aaa(T, T3, U, X, T2, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(T, T1, node(T1, X, U)) → U18_aaa(T, T1, X, U, left_in_aa(T, T1))
U18_aaa(T, T1, X, U, left_out_aa(T, T1)) → U19_aaa(T, T1, X, U, right_in_aa(T, T2))
U19_aaa(T, T1, X, U, right_out_aa(T, T2)) → U20_aaa(T, T1, X, U, T2, value_in_aa(T, X))
U20_aaa(T, T1, X, U, T2, value_out_aa(T, X)) → U21_aaa(T, T1, X, U, tappend_in_aaa(T2, T3, U))
U21_aaa(T, T1, X, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(T, T1, node(T1, X, U))
U17_aaa(T, T3, U, X, T2, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(T, T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, X1, X2)) → U5_aa(X, L, X1, X2, tlast_in_aa(X, L))
tlast_in_aa(X, node(X3, X4, R)) → U6_aa(X, X3, X4, R, tlast_in_aa(X, R))
U6_aa(X, X3, X4, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(X3, X4, R))
U5_aa(X, L, X1, X2, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, X1, X2))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)

The argument filtering Pi contains the following mapping:
goal_in_gaa(x1, x2, x3)  =  goal_in_gaa(x1)
U1_gaa(x1, x2, x3, x4)  =  U1_gaa(x1, x4)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U22_ga(x1, x2)  =  U22_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)
U23_ga(x1, x2, x3, x4)  =  U23_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)
U24_ga(x1, x2, x3, x4)  =  U24_ga(x1, x4)
U25_ga(x1, x2, x3, x4)  =  U25_ga(x1, x4)
U26_ga(x1, x2, x3, x4)  =  U26_ga(x1, x4)
U27_ga(x1, x2, x3, x4)  =  U27_ga(x1, x4)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x4)
U2_gaa(x1, x2, x3, x4)  =  U2_gaa(x1, x4)
tapplast_in_aaa(x1, x2, x3)  =  tapplast_in_aaa
U3_aaa(x1, x2, x3, x4)  =  U3_aaa(x4)
tappend_in_aaa(x1, x2, x3)  =  tappend_in_aaa
U7_aaa(x1, x2, x3)  =  U7_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
U8_aaa(x1, x2, x3, x4, x5)  =  U8_aaa(x5)
left_in_ag(x1, x2)  =  left_in_ag(x2)
left_out_ag(x1, x2)  =  left_out_ag(x2)
U9_aaa(x1, x2, x3, x4, x5)  =  U9_aaa(x5)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U10_aaa(x1, x2, x3, x4, x5)  =  U10_aaa(x5)
value_in_aa(x1, x2)  =  value_in_aa
value_out_aa(x1, x2)  =  value_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
right_in_ag(x1, x2)  =  right_in_ag(x2)
right_out_ag(x1, x2)  =  right_out_ag(x2)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
U14_aaa(x1, x2, x3, x4, x5, x6)  =  U14_aaa(x6)
U15_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U15_aaa(x7)
U16_aaa(x1, x2, x3, x4, x5, x6, x7)  =  U16_aaa(x7)
U17_aaa(x1, x2, x3, x4, x5, x6)  =  U17_aaa(x6)
U18_aaa(x1, x2, x3, x4, x5)  =  U18_aaa(x5)
U19_aaa(x1, x2, x3, x4, x5)  =  U19_aaa(x5)
U20_aaa(x1, x2, x3, x4, x5, x6)  =  U20_aaa(x6)
U21_aaa(x1, x2, x3, x4, x5)  =  U21_aaa(x5)
U4_aaa(x1, x2, x3, x4)  =  U4_aaa(x4)
tlast_in_aa(x1, x2)  =  tlast_in_aa
tlast_out_aa(x1, x2)  =  tlast_out_aa
U5_aa(x1, x2, x3, x4, x5)  =  U5_aa(x5)
U6_aa(x1, x2, x3, x4, x5)  =  U6_aa(x5)
tapplast_out_aaa(x1, x2, x3)  =  tapplast_out_aaa
goal_out_gaa(x1, x2, x3)  =  goal_out_gaa(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x1, x4)

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

(117) UsableRulesProof (EQUIVALENT transformation)

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

(118) Obligation:

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

S2T_IN_GA(X, node(T, X5, T)) → 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)
S2T_IN_GA(X, node(nil, X6, T)) → U25_GA(X, X6, T, p_in_ga(X, P))
U25_GA(X, X6, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X7, nil)) → U27_GA(X, T, X7, p_in_ga(X, P))
U27_GA(X, T, X7, 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)
U23_GA(x1, x2, x3, x4)  =  U23_GA(x1, x4)
U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x4)
U27_GA(x1, x2, x3, x4)  =  U27_GA(x1, x4)

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

(119) PiDPToQDPProof (SOUND transformation)

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

(120) Obligation:

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

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(X) → U25_GA(X, p_in_ga(X))
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(X, p_in_ga(X))
U27_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.

(121) 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))

(122) 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(X) → U25_GA(X, p_in_ga(X))
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(X, p_in_ga(X))
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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.

(123) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U25_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))

(124) Obligation:

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

U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U27_GA(X, p_in_ga(X))
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_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.

(125) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U27_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))

(126) Obligation:

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

U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_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.

(127) 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.

(128) Obligation:

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

U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_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.

(129) 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)

(130) Obligation:

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

U23_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))

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

(131) 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)

(132) Obligation:

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

U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))
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.

(133) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U25_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U25_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U25_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(134) Obligation:

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

U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))
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)
U25_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U25_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.

(135) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U27_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U27_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U27_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(136) Obligation:

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

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))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))
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)
U25_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U25_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U27_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U27_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.

(137) DependencyGraphProof (EQUIVALENT transformation)

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

(138) Complex Obligation (AND)

(139) Obligation:

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

U23_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0))
S2T_IN_GA(0) → U25_GA(0, p_out_ga(0, 0))
U25_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U27_GA(0, p_out_ga(0, 0))
U27_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.

(140) 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) → U23_GA(0, p_out_ga(0, 0))
with rule S2T_IN_GA(0) → U23_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U23_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
with rule U23_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.

(141) FALSE

(142) Obligation:

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

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)
S2T_IN_GA(s(x0)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
U25_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))
U27_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.

(143) 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:

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

• U25_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• U27_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)) → U25_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

• S2T_IN_GA(s(x0)) → U27_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

(144) TRUE