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

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), conf(Xs)).
conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X1, [], X2) :- ','(!, failure(a)).
del(H, X, T) :- ','(head(X, H), tail(X, T)).
del(X, Y, .(H, T2)) :- ','(head(Y, H), ','(tail(Y, T1), del(X, T1, T2))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
failure(b).
eq(X, X).

Queries:

goal(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), conf(Xs)).
conf(X) :- ','(del2(X, Z), ','(del(U, Y, Z), conf(Y))).
del2(X, Y) :- ','(del(U, X, Z), del(V, Z, Y)).
del(X1, [], X2) :- failure(a).
del(H, X, T) :- ','(head(X, H), tail(X, T)).
del(X, Y, .(H, T2)) :- ','(head(Y, H), ','(tail(Y, T1), del(X, T1, T2))).
s2l(0, L) :- eq(L, []).
s2l(X, .(X3, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X4).
head(.(H, X5), H).
tail([], []).
tail(.(X6, Xs), Xs).
p(0, 0).
p(s(X), X).
failure(b).
eq(X, X).

Queries:

goal(g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2l_in: (b,f)
conf_in: (b) (f)
del2_in: (b,f) (f,f)
del_in: (f,b,f) (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U14_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U15_GA(X, X3, Xs, p_out_ga(X, P)) → U16_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, conf_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → CONF_IN_G(Xs)
CONF_IN_G(X) → U3_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U6_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X1, [], X2) → U8_AGA(X1, X2, failure_in_g(a))
DEL_IN_AGA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AGA(H, X, T) → U9_AGA(H, X, T, head_in_ga(X, H))
DEL_IN_AGA(H, X, T) → HEAD_IN_GA(X, H)
U9_AGA(H, X, T, head_out_ga(X, H)) → U10_AGA(H, X, T, tail_in_ga(X, T))
U9_AGA(H, X, T, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
DEL_IN_AGA(X, Y, .(H, T2)) → HEAD_IN_GA(Y, H)
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T1)
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_AGA(X, Y, H, T2, del_in_aga(X, T1, T2))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)
U6_GA(X, Y, del_out_aga(U, X, Z)) → U7_GA(X, Y, del_in_aaa(V, Z, Y))
U6_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
DEL_IN_AAA(X1, [], X2) → U8_AAA(X1, X2, failure_in_g(a))
DEL_IN_AAA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AAA(H, X, T) → U9_AAA(H, X, T, head_in_aa(X, H))
DEL_IN_AAA(H, X, T) → HEAD_IN_AA(X, H)
U9_AAA(H, X, T, head_out_aa(X, H)) → U10_AAA(H, X, T, tail_in_aa(X, T))
U9_AAA(H, X, T, head_out_aa(X, H)) → TAIL_IN_AA(X, T)
DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
DEL_IN_AAA(X, Y, .(H, T2)) → HEAD_IN_AA(Y, H)
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → TAIL_IN_AA(Y, T1)
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_AAA(X, Y, H, T2, del_in_aaa(X, T1, T2))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)
U3_G(X, del2_out_ga(X, Z)) → U4_G(X, del_in_aaa(U, Y, Z))
U3_G(X, del2_out_ga(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_G(X, del_out_aaa(U, Y, Z)) → U5_G(X, conf_in_a(Y))
U4_G(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U6_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
U6_AA(X, Y, del_out_aaa(U, X, Z)) → U7_AA(X, Y, del_in_aaa(V, Z, Y))
U6_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U3_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_A(X, del_out_aaa(U, Y, Z)) → U5_A(X, conf_in_a(Y))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U14_GA(x1, x2)  =  U14_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
CONF_IN_G(x1)  =  CONF_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
DEL2_IN_GA(x1, x2)  =  DEL2_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x5)
U13_AGA(x1, x2, x3, x4, x5)  =  U13_AGA(x2, x5)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U8_AAA(x1, x2, x3)  =  U8_AAA(x3)
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U10_AAA(x1, x2, x3, x4)  =  U10_AAA(x4)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
DEL2_IN_AA(x1, x2)  =  DEL2_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2)  =  U5_A(x2)

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

(6) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U14_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U15_GA(X, X3, Xs, p_out_ga(X, P)) → U16_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, conf_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → CONF_IN_G(Xs)
CONF_IN_G(X) → U3_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U6_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X1, [], X2) → U8_AGA(X1, X2, failure_in_g(a))
DEL_IN_AGA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AGA(H, X, T) → U9_AGA(H, X, T, head_in_ga(X, H))
DEL_IN_AGA(H, X, T) → HEAD_IN_GA(X, H)
U9_AGA(H, X, T, head_out_ga(X, H)) → U10_AGA(H, X, T, tail_in_ga(X, T))
U9_AGA(H, X, T, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
DEL_IN_AGA(X, Y, .(H, T2)) → HEAD_IN_GA(Y, H)
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T1)
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_AGA(X, Y, H, T2, del_in_aga(X, T1, T2))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)
U6_GA(X, Y, del_out_aga(U, X, Z)) → U7_GA(X, Y, del_in_aaa(V, Z, Y))
U6_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
DEL_IN_AAA(X1, [], X2) → U8_AAA(X1, X2, failure_in_g(a))
DEL_IN_AAA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AAA(H, X, T) → U9_AAA(H, X, T, head_in_aa(X, H))
DEL_IN_AAA(H, X, T) → HEAD_IN_AA(X, H)
U9_AAA(H, X, T, head_out_aa(X, H)) → U10_AAA(H, X, T, tail_in_aa(X, T))
U9_AAA(H, X, T, head_out_aa(X, H)) → TAIL_IN_AA(X, T)
DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
DEL_IN_AAA(X, Y, .(H, T2)) → HEAD_IN_AA(Y, H)
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → TAIL_IN_AA(Y, T1)
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_AAA(X, Y, H, T2, del_in_aaa(X, T1, T2))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)
U3_G(X, del2_out_ga(X, Z)) → U4_G(X, del_in_aaa(U, Y, Z))
U3_G(X, del2_out_ga(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_G(X, del_out_aaa(U, Y, Z)) → U5_G(X, conf_in_a(Y))
U4_G(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U6_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
U6_AA(X, Y, del_out_aaa(U, X, Z)) → U7_AA(X, Y, del_in_aaa(V, Z, Y))
U6_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U3_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_A(X, del_out_aaa(U, Y, Z)) → U5_A(X, conf_in_a(Y))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U14_GA(x1, x2)  =  U14_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
CONF_IN_G(x1)  =  CONF_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
DEL2_IN_GA(x1, x2)  =  DEL2_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x2, x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x5)
U13_AGA(x1, x2, x3, x4, x5)  =  U13_AGA(x2, x5)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U8_AAA(x1, x2, x3)  =  U8_AAA(x3)
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U10_AAA(x1, x2, x3, x4)  =  U10_AAA(x4)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U4_G(x1, x2)  =  U4_G(x1, x2)
U5_G(x1, x2)  =  U5_G(x1, x2)
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
DEL2_IN_AA(x1, x2)  =  DEL2_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2)  =  U5_A(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 43 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)

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:

DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)

The TRS R consists of the following rules:

head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)

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:

DEL_IN_AAA → U11_AAA(head_in_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(14) Rewriting (EQUIVALENT transformation)

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

DEL_IN_AAA → U11_AAA(head_out_aa)

(15) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(16) 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.

(17) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(18) 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].

head_in_aa

(19) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(20) Rewriting (EQUIVALENT transformation)

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

U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

(21) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

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

tail_in_aa

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

(24) QReductionProof (EQUIVALENT transformation)

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

tail_in_aa

(25) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

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

(26) 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 = DEL_IN_AAA evaluates to t =DEL_IN_AAA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

DEL_IN_AAA → U11_AAA(head_out_aa)
with rule DEL_IN_AAA → U11_AAA(head_out_aa) at position [] and matcher [ ]

U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)
with rule U11_AAA(head_out_aa) → U12_AAA(tail_out_aa) at position [] and matcher [ ]

U12_AAA(tail_out_aa) → DEL_IN_AAA
with rule U12_AAA(tail_out_aa) → DEL_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.

(27) FALSE

(28) Obligation:

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

CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
U4_A(x1, x2)  =  U4_A(x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
failure_in_g(x1)  =  failure_in_g(x1)
a  =  a
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
U4_A(x1, x2)  =  U4_A(x2)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

CONF_IN_A → U3_A(del2_in_aa)
U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U8_aaa(failure_in_g(a))
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(33) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

del_in_aaa → U8_aaa(failure_in_g(a))

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 1   
POL(U10_aaa(x₁)) = 1 + x₁   
POL(U11_aaa(x₁)) = 1 + x₁   
POL(U12_aaa(x₁)) = 1 + x₁   
POL(U13_aaa(x₁)) = x₁   
POL(U3_A(x₁)) = x₁   
POL(U4_A(x₁)) = x₁   
POL(U6_aa(x₁)) = x₁   
POL(U7_aa(x₁)) = x₁   
POL(U8_aaa(x₁)) = x₁   
POL(U9_aaa(x₁)) = 1 + x₁   
POL(a) = 0   
POL(del2_in_aa) = 1   
POL(del2_out_aa) = 1   
POL(del_in_aaa) = 1   
POL(del_out_aaa) = 1   
POL(failure_in_g(x₁)) = x₁   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(34) Obligation:

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

CONF_IN_A → U3_A(del2_in_aa)
U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A(del2_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A(U6_aa(del_in_aaa))

(36) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(37) 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.

(38) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(39) 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].

del2_in_aa

(40) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(41) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_A(del2_out_aa) → U4_A(del_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))

(42) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(43) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))

(44) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(45) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))

(46) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(47) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa))

(48) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(49) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa))

(50) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(51) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(del_out_aaa)

(52) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(del_out_aaa)

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(53) RootLabelingFC2Proof (SOUND transformation)

We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

(54) Obligation:

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

U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_in_aaa}(del_in_aaa) → U4_A_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U3_A_{del_in_aaa}(del_in_aaa) → U3_A_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U4_A_{del_in_aaa}(del_in_aaa) → U4_A_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U3_A_{del_in_aaa}(del_in_aaa) → U3_A_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U4_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U4_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U4_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U4_A_{del2_out_aa}(del2_out_aa)
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U4_A_{head_in_aa}(head_in_aa) → U4_A_{head_out_aa}(head_out_aa)
U3_A_{head_in_aa}(head_in_aa) → U3_A_{head_out_aa}(head_out_aa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U3_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U3_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{tail_in_aa}(tail_in_aa) → U4_A_{tail_out_aa}(tail_out_aa)
U3_A_{tail_in_aa}(tail_in_aa) → U3_A_{tail_out_aa}(tail_out_aa)
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U3_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U3_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U3_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U3_A_{del_out_aaa}(del_out_aaa)
U4_A_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U3_A_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U3_A_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{del_out_aaa}(del_out_aaa)
U3_A_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U3_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

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

(55) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(56) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

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

(57) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U10_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U10_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U6_aa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U7_aa_1}(x₁)) = x₁   
POL(U10_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U10_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U10_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U10_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U10_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U10_aaa_{head_out_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U11_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U11_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U11_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U11_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U11_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U12_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U12_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U12_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U12_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U12_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U12_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U13_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U13_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U13_aaa_{head_out_aa}(x₁)) = x₁   
POL(U13_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U13_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U6_aa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U6_aa_{del2_out_aa}(x₁)) = x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U6_aa_{head_out_aa}(x₁)) = x₁   
POL(U6_aa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U6_aa_{tail_out_aa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U7_aa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U7_aa_{del2_out_aa}(x₁)) = x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U7_aa_{head_out_aa}(x₁)) = x₁   
POL(U7_aa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U7_aa_{tail_out_aa}(x₁)) = x₁   
POL(U9_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U9_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U9_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U9_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(U9_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U9_aaa_{tail_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(58) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(59) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(60) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(61) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa)) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))

(62) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(63) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(64) Obligation:

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

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(65) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(66) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(67) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(68) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(69) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(70) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(71) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(72) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(73) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(74) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(75) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(76) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(77) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(78) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(79) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(80) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(81) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(82) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(83) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(84) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(85) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(86) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(87) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(88) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(89) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(90) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(91) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(92) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(93) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))

(94) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(95) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))

(96) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(97) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

(98) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(99) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = 3 + x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = 3 + x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(100) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(101) Obligation:

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

DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x5)

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

(102) UsableRulesProof (EQUIVALENT transformation)

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

(103) Obligation:

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

DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)

The TRS R consists of the following rules:

head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x2, x5)

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

(104) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
U15_ga(x1, x2, x3, x4)  =  U15_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)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x1, x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x1, x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g(x1)
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga(x2)
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga(x1)
U10_aga(x1, x2, x3, x4)  =  U10_aga(x2, x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x1, x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x2, x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x2, x5)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
U5_g(x1, x2)  =  U5_g(x1, x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g(x1)
goal_out_g(x1)  =  goal_out_g(x1)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)

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

(105) UsableRulesProof (EQUIVALENT transformation)

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

(106) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

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
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x1, x4)

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

(107) 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)
s2l_in: (b,f)
conf_in: (b) (f)
del2_in: (b,f) (f,f)
del_in: (f,b,f) (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(108) Obligation:

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

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g

(109) DependencyPairsProof (EQUIVALENT transformation)

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U14_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U15_GA(X, X3, Xs, p_out_ga(X, P)) → U16_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, conf_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → CONF_IN_G(Xs)
CONF_IN_G(X) → U3_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U6_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X1, [], X2) → U8_AGA(X1, X2, failure_in_g(a))
DEL_IN_AGA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AGA(H, X, T) → U9_AGA(H, X, T, head_in_ga(X, H))
DEL_IN_AGA(H, X, T) → HEAD_IN_GA(X, H)
U9_AGA(H, X, T, head_out_ga(X, H)) → U10_AGA(H, X, T, tail_in_ga(X, T))
U9_AGA(H, X, T, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
DEL_IN_AGA(X, Y, .(H, T2)) → HEAD_IN_GA(Y, H)
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T1)
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_AGA(X, Y, H, T2, del_in_aga(X, T1, T2))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)
U6_GA(X, Y, del_out_aga(U, X, Z)) → U7_GA(X, Y, del_in_aaa(V, Z, Y))
U6_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
DEL_IN_AAA(X1, [], X2) → U8_AAA(X1, X2, failure_in_g(a))
DEL_IN_AAA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AAA(H, X, T) → U9_AAA(H, X, T, head_in_aa(X, H))
DEL_IN_AAA(H, X, T) → HEAD_IN_AA(X, H)
U9_AAA(H, X, T, head_out_aa(X, H)) → U10_AAA(H, X, T, tail_in_aa(X, T))
U9_AAA(H, X, T, head_out_aa(X, H)) → TAIL_IN_AA(X, T)
DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
DEL_IN_AAA(X, Y, .(H, T2)) → HEAD_IN_AA(Y, H)
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → TAIL_IN_AA(Y, T1)
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_AAA(X, Y, H, T2, del_in_aaa(X, T1, T2))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)
U3_G(X, del2_out_ga(X, Z)) → U4_G(X, del_in_aaa(U, Y, Z))
U3_G(X, del2_out_ga(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_G(X, del_out_aaa(U, Y, Z)) → U5_G(X, conf_in_a(Y))
U4_G(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U6_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
U6_AA(X, Y, del_out_aaa(U, X, Z)) → U7_AA(X, Y, del_in_aaa(V, Z, Y))
U6_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U3_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_A(X, del_out_aaa(U, Y, Z)) → U5_A(X, conf_in_a(Y))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U14_GA(x1, x2)  =  U14_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
CONF_IN_G(x1)  =  CONF_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
DEL2_IN_GA(x1, x2)  =  DEL2_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x5)
U13_AGA(x1, x2, x3, x4, x5)  =  U13_AGA(x5)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U8_AAA(x1, x2, x3)  =  U8_AAA(x3)
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U10_AAA(x1, x2, x3, x4)  =  U10_AAA(x4)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
DEL2_IN_AA(x1, x2)  =  DEL2_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2)  =  U5_A(x2)

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

(110) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2l_in_ga(X, Xs))
GOAL_IN_G(X) → S2L_IN_GA(X, Xs)
S2L_IN_GA(0, L) → U14_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X3, Xs)) → P_IN_GA(X, P)
U15_GA(X, X3, Xs, p_out_ga(X, P)) → U16_GA(X, X3, Xs, s2l_in_ga(P, Xs))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_G(X, s2l_out_ga(X, Xs)) → U2_G(X, conf_in_g(Xs))
U1_G(X, s2l_out_ga(X, Xs)) → CONF_IN_G(Xs)
CONF_IN_G(X) → U3_G(X, del2_in_ga(X, Z))
CONF_IN_G(X) → DEL2_IN_GA(X, Z)
DEL2_IN_GA(X, Y) → U6_GA(X, Y, del_in_aga(U, X, Z))
DEL2_IN_GA(X, Y) → DEL_IN_AGA(U, X, Z)
DEL_IN_AGA(X1, [], X2) → U8_AGA(X1, X2, failure_in_g(a))
DEL_IN_AGA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AGA(H, X, T) → U9_AGA(H, X, T, head_in_ga(X, H))
DEL_IN_AGA(H, X, T) → HEAD_IN_GA(X, H)
U9_AGA(H, X, T, head_out_ga(X, H)) → U10_AGA(H, X, T, tail_in_ga(X, T))
U9_AGA(H, X, T, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
DEL_IN_AGA(X, Y, .(H, T2)) → HEAD_IN_GA(Y, H)
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → TAIL_IN_GA(Y, T1)
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_AGA(X, Y, H, T2, del_in_aga(X, T1, T2))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)
U6_GA(X, Y, del_out_aga(U, X, Z)) → U7_GA(X, Y, del_in_aaa(V, Z, Y))
U6_GA(X, Y, del_out_aga(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
DEL_IN_AAA(X1, [], X2) → U8_AAA(X1, X2, failure_in_g(a))
DEL_IN_AAA(X1, [], X2) → FAILURE_IN_G(a)
DEL_IN_AAA(H, X, T) → U9_AAA(H, X, T, head_in_aa(X, H))
DEL_IN_AAA(H, X, T) → HEAD_IN_AA(X, H)
U9_AAA(H, X, T, head_out_aa(X, H)) → U10_AAA(H, X, T, tail_in_aa(X, T))
U9_AAA(H, X, T, head_out_aa(X, H)) → TAIL_IN_AA(X, T)
DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
DEL_IN_AAA(X, Y, .(H, T2)) → HEAD_IN_AA(Y, H)
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → TAIL_IN_AA(Y, T1)
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_AAA(X, Y, H, T2, del_in_aaa(X, T1, T2))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)
U3_G(X, del2_out_ga(X, Z)) → U4_G(X, del_in_aaa(U, Y, Z))
U3_G(X, del2_out_ga(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_G(X, del_out_aaa(U, Y, Z)) → U5_G(X, conf_in_a(Y))
U4_G(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)
CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
CONF_IN_A(X) → DEL2_IN_AA(X, Z)
DEL2_IN_AA(X, Y) → U6_AA(X, Y, del_in_aaa(U, X, Z))
DEL2_IN_AA(X, Y) → DEL_IN_AAA(U, X, Z)
U6_AA(X, Y, del_out_aaa(U, X, Z)) → U7_AA(X, Y, del_in_aaa(V, Z, Y))
U6_AA(X, Y, del_out_aaa(U, X, Z)) → DEL_IN_AAA(V, Z, Y)
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U3_A(X, del2_out_aa(X, Z)) → DEL_IN_AAA(U, Y, Z)
U4_A(X, del_out_aaa(U, Y, Z)) → U5_A(X, conf_in_a(Y))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U14_GA(x1, x2)  =  U14_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U16_GA(x1, x2, x3, x4)  =  U16_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
CONF_IN_G(x1)  =  CONF_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
DEL2_IN_GA(x1, x2)  =  DEL2_IN_GA(x1)
U6_GA(x1, x2, x3)  =  U6_GA(x3)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U8_AGA(x1, x2, x3)  =  U8_AGA(x3)
FAILURE_IN_G(x1)  =  FAILURE_IN_G(x1)
U9_AGA(x1, x2, x3, x4)  =  U9_AGA(x2, x4)
HEAD_IN_GA(x1, x2)  =  HEAD_IN_GA(x1)
U10_AGA(x1, x2, x3, x4)  =  U10_AGA(x4)
TAIL_IN_GA(x1, x2)  =  TAIL_IN_GA(x1)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x5)
U13_AGA(x1, x2, x3, x4, x5)  =  U13_AGA(x5)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U8_AAA(x1, x2, x3)  =  U8_AAA(x3)
U9_AAA(x1, x2, x3, x4)  =  U9_AAA(x4)
HEAD_IN_AA(x1, x2)  =  HEAD_IN_AA
U10_AAA(x1, x2, x3, x4)  =  U10_AAA(x4)
TAIL_IN_AA(x1, x2)  =  TAIL_IN_AA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)
U13_AAA(x1, x2, x3, x4, x5)  =  U13_AAA(x5)
U4_G(x1, x2)  =  U4_G(x2)
U5_G(x1, x2)  =  U5_G(x2)
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
DEL2_IN_AA(x1, x2)  =  DEL2_IN_AA
U6_AA(x1, x2, x3)  =  U6_AA(x3)
U7_AA(x1, x2, x3)  =  U7_AA(x3)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2)  =  U5_A(x2)

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

(111) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 43 less nodes.

(112) Complex Obligation (AND)

(113) Obligation:

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

DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)

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

(114) UsableRulesProof (EQUIVALENT transformation)

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

(115) Obligation:

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

DEL_IN_AAA(X, Y, .(H, T2)) → U11_AAA(X, Y, H, T2, head_in_aa(Y, H))
U11_AAA(X, Y, H, T2, head_out_aa(Y, H)) → U12_AAA(X, Y, H, T2, tail_in_aa(Y, T1))
U12_AAA(X, Y, H, T2, tail_out_aa(Y, T1)) → DEL_IN_AAA(X, T1, T2)

The TRS R consists of the following rules:

head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
DEL_IN_AAA(x1, x2, x3)  =  DEL_IN_AAA
U11_AAA(x1, x2, x3, x4, x5)  =  U11_AAA(x5)
U12_AAA(x1, x2, x3, x4, x5)  =  U12_AAA(x5)

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

(116) PiDPToQDPProof (SOUND transformation)

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

(117) Obligation:

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

DEL_IN_AAA → U11_AAA(head_in_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(118) Rewriting (EQUIVALENT transformation)

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

DEL_IN_AAA → U11_AAA(head_out_aa)

(119) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

head_in_aa → head_out_aa
tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(120) 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.

(121) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

head_in_aa
tail_in_aa

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

(122) 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].

head_in_aa

(123) Obligation:

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

U11_AAA(head_out_aa) → U12_AAA(tail_in_aa)
U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(124) Rewriting (EQUIVALENT transformation)

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

U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

(125) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

The TRS R consists of the following rules:

tail_in_aa → tail_out_aa

The set Q consists of the following terms:

tail_in_aa

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

(126) 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.

(127) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

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

tail_in_aa

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

(128) 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].

tail_in_aa

(129) Obligation:

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

U12_AAA(tail_out_aa) → DEL_IN_AAA
DEL_IN_AAA → U11_AAA(head_out_aa)
U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)

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

(130) 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 = DEL_IN_AAA evaluates to t =DEL_IN_AAA

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

DEL_IN_AAA → U11_AAA(head_out_aa)
with rule DEL_IN_AAA → U11_AAA(head_out_aa) at position [] and matcher [ ]

U11_AAA(head_out_aa) → U12_AAA(tail_out_aa)
with rule U11_AAA(head_out_aa) → U12_AAA(tail_out_aa) at position [] and matcher [ ]

U12_AAA(tail_out_aa) → DEL_IN_AAA
with rule U12_AAA(tail_out_aa) → DEL_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.

(131) FALSE

(132) Obligation:

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

CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
U4_A(x1, x2)  =  U4_A(x2)

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

(133) UsableRulesProof (EQUIVALENT transformation)

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

(134) Obligation:

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

CONF_IN_A(X) → U3_A(X, del2_in_aa(X, Z))
U3_A(X, del2_out_aa(X, Z)) → U4_A(X, del_in_aaa(U, Y, Z))
U4_A(X, del_out_aaa(U, Y, Z)) → CONF_IN_A(Y)

The TRS R consists of the following rules:

del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
failure_in_g(x1)  =  failure_in_g(x1)
a  =  a
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
CONF_IN_A(x1)  =  CONF_IN_A
U3_A(x1, x2)  =  U3_A(x2)
U4_A(x1, x2)  =  U4_A(x2)

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

(135) PiDPToQDPProof (SOUND transformation)

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

(136) Obligation:

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

CONF_IN_A → U3_A(del2_in_aa)
U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U8_aaa(failure_in_g(a))
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(137) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

del_in_aaa → U8_aaa(failure_in_g(a))

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 1   
POL(U10_aaa(x₁)) = 1 + x₁   
POL(U11_aaa(x₁)) = 1 + x₁   
POL(U12_aaa(x₁)) = 1 + x₁   
POL(U13_aaa(x₁)) = x₁   
POL(U3_A(x₁)) = x₁   
POL(U4_A(x₁)) = x₁   
POL(U6_aa(x₁)) = x₁   
POL(U7_aa(x₁)) = x₁   
POL(U8_aaa(x₁)) = x₁   
POL(U9_aaa(x₁)) = 1 + x₁   
POL(a) = 0   
POL(del2_in_aa) = 1   
POL(del2_out_aa) = 1   
POL(del_in_aaa) = 1   
POL(del_out_aaa) = 1   
POL(failure_in_g(x₁)) = x₁   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(138) Obligation:

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

CONF_IN_A → U3_A(del2_in_aa)
U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(139) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A(del2_in_aa) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A(U6_aa(del_in_aaa))

(140) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del2_in_aa → U6_aa(del_in_aaa)
del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U10_aaa(tail_out_aa) → del_out_aaa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
tail_in_aa → tail_out_aa
U13_aaa(del_out_aaa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(141) 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.

(142) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del2_in_aa
del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(143) 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].

del2_in_aa

(144) Obligation:

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

U3_A(del2_out_aa) → U4_A(del_in_aaa)
U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(145) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U3_A(del2_out_aa) → U4_A(del_in_aaa) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))

(146) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(147) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U9_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))

(148) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(149) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U11_aaa(head_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))

(150) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(151) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U9_aaa(head_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa))

(152) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(153) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U10_aaa(tail_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa))

(154) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa))

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(155) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U3_A(del2_out_aa) → U4_A(U10_aaa(tail_out_aa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A(del2_out_aa) → U4_A(del_out_aaa)

(156) Obligation:

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

U4_A(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A(U6_aa(del_in_aaa))
U3_A(del2_out_aa) → U4_A(U11_aaa(head_out_aa))
U3_A(del2_out_aa) → U4_A(del_out_aaa)

The TRS R consists of the following rules:

del_in_aaa → U9_aaa(head_in_aa)
del_in_aaa → U11_aaa(head_in_aa)
U6_aa(del_out_aaa) → U7_aa(del_in_aaa)
U7_aa(del_out_aaa) → del2_out_aa
head_in_aa → head_out_aa
U11_aaa(head_out_aa) → U12_aaa(tail_in_aa)
tail_in_aa → tail_out_aa
U12_aaa(tail_out_aa) → U13_aaa(del_in_aaa)
U13_aaa(del_out_aaa) → del_out_aaa
U9_aaa(head_out_aa) → U10_aaa(tail_in_aa)
U10_aaa(tail_out_aa) → del_out_aaa

The set Q consists of the following terms:

del_in_aaa
U6_aa(x0)
U9_aaa(x0)
U11_aaa(x0)
U7_aa(x0)
head_in_aa
U10_aaa(x0)
U12_aaa(x0)
tail_in_aa
U13_aaa(x0)

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

(157) RootLabelingFC2Proof (SOUND transformation)

We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

(158) Obligation:

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

U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_in_aaa}(del_in_aaa) → U4_A_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U3_A_{del_in_aaa}(del_in_aaa) → U3_A_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U4_A_{del_in_aaa}(del_in_aaa) → U4_A_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U3_A_{del_in_aaa}(del_in_aaa) → U3_A_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U4_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U4_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U4_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U4_A_{del2_out_aa}(del2_out_aa)
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U4_A_{head_in_aa}(head_in_aa) → U4_A_{head_out_aa}(head_out_aa)
U3_A_{head_in_aa}(head_in_aa) → U3_A_{head_out_aa}(head_out_aa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U3_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U3_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{tail_in_aa}(tail_in_aa) → U4_A_{tail_out_aa}(tail_out_aa)
U3_A_{tail_in_aa}(tail_in_aa) → U3_A_{tail_out_aa}(tail_out_aa)
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U3_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U3_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U3_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U3_A_{del_out_aaa}(del_out_aaa)
U4_A_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U3_A_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U3_A_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{del_out_aaa}(del_out_aaa)
U3_A_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U3_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

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

(159) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 17 less nodes.

(160) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

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

(161) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{del_in_aaa}(del_in_aaa) → U9_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U11_aaa_{del_in_aaa}(del_in_aaa) → U11_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U12_aaa_{del_in_aaa}(del_in_aaa) → U12_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{del_in_aaa}(del_in_aaa) → U10_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U6_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U7_aa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del2_out_aa}(del2_out_aa)
U11_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del2_out_aa}(del2_out_aa)
U7_aa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del2_out_aa}(del2_out_aa)
U12_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del2_out_aa}(del2_out_aa)
U13_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del2_out_aa}(del2_out_aa)
U6_aa_{head_in_aa}(head_in_aa) → U6_aa_{head_out_aa}(head_out_aa)
U7_aa_{head_in_aa}(head_in_aa) → U7_aa_{head_out_aa}(head_out_aa)
U12_aaa_{head_in_aa}(head_in_aa) → U12_aaa_{head_out_aa}(head_out_aa)
U13_aaa_{head_in_aa}(head_in_aa) → U13_aaa_{head_out_aa}(head_out_aa)
U10_aaa_{head_in_aa}(head_in_aa) → U10_aaa_{head_out_aa}(head_out_aa)
U9_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{tail_in_aa}(tail_in_aa) → U9_aaa_{tail_out_aa}(tail_out_aa)
U11_aaa_{tail_in_aa}(tail_in_aa) → U11_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{tail_in_aa}(tail_in_aa) → U6_aa_{tail_out_aa}(tail_out_aa)
U7_aa_{tail_in_aa}(tail_in_aa) → U7_aa_{tail_out_aa}(tail_out_aa)
U13_aaa_{tail_in_aa}(tail_in_aa) → U13_aaa_{tail_out_aa}(tail_out_aa)
U9_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U11_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U12_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U10_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U9_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del_out_aaa}(del_out_aaa)
U9_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U9_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U11_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U11_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U12_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U10_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U10_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U9_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U9_aaa_{del_out_aaa}(del_out_aaa)
U11_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U11_aaa_{del_out_aaa}(del_out_aaa)
U12_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U12_aaa_{del_out_aaa}(del_out_aaa)
U10_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U10_aaa_{del_out_aaa}(del_out_aaa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U10_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U10_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U6_aa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{U7_aa_1}(x₁)) = x₁   
POL(U10_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U10_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U10_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U10_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U10_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U10_aaa_{head_out_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U11_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U11_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U11_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U11_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U11_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U11_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U11_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U12_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U12_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U12_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U12_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U12_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U12_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U12_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U12_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U13_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U13_aaa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U13_aaa_{head_out_aa}(x₁)) = x₁   
POL(U13_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U13_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U6_aa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U6_aa_{del2_out_aa}(x₁)) = x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U6_aa_{head_out_aa}(x₁)) = x₁   
POL(U6_aa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U6_aa_{tail_out_aa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U7_aa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U7_aa_{del2_out_aa}(x₁)) = x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{head_in_aa}(x₁)) = 1 + x₁   
POL(U7_aa_{head_out_aa}(x₁)) = x₁   
POL(U7_aa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U7_aa_{tail_out_aa}(x₁)) = x₁   
POL(U9_aaa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U11_aaa_1}(x₁)) = 3 + x₁   
POL(U9_aaa_{U12_aaa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{U13_aaa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U6_aa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U9_aaa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U9_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U9_aaa_{del_in_aaa}(x₁)) = 4 + x₁   
POL(U9_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(U9_aaa_{tail_in_aa}(x₁)) = 1 + x₁   
POL(U9_aaa_{tail_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(162) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(163) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U10_aaa_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U10_aaa_{del2_out_aa}(del2_out_aa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{U7_aa_1}(x₁)) = 1 + x₁   
POL(U10_aaa_{del2_out_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(164) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(165) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa)) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))

(166) Obligation:

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

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(167) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(168) Obligation:

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

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa))
U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(169) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(170) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(171) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa)) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

(172) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(173) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(174) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(175) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(176) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(177) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(178) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(179) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(180) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(181) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

(182) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(183) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

(184) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(185) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(186) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(187) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(188) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(189) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

(190) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(191) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(192) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(193) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(194) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(195) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

(196) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(197) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))

(198) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(199) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))

(200) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(201) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa))) at position [0] we obtained the following new rules [LPAR04]:

U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

(202) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(203) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U6_aa_{del_in_aaa}(del_in_aaa) → U6_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U7_aa_{del_in_aaa}(del_in_aaa) → U7_aa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U6_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U6_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{del_out_aaa}(del_out_aaa)

Used ordering: Polynomial interpretation [POLO]:

POL(CONF_IN_A) = 0   
POL(U10_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U10_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U11_aaa_{head_in_aa}(x₁)) = x₁   
POL(U11_aaa_{head_out_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_in_aa}(x₁)) = x₁   
POL(U12_aaa_{tail_out_aa}(x₁)) = x₁   
POL(U13_aaa_{U10_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U11_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U12_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U13_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{U9_aaa_1}(x₁)) = x₁   
POL(U13_aaa_{del_in_aaa}(x₁)) = x₁   
POL(U13_aaa_{del_out_aaa}(x₁)) = x₁   
POL(U3_A_{U6_aa_1}(x₁)) = x₁   
POL(U3_A_{U7_aa_1}(x₁)) = x₁   
POL(U3_A_{del2_out_aa}(x₁)) = x₁   
POL(U4_A_{U11_aaa_1}(x₁)) = x₁   
POL(U4_A_{U12_aaa_1}(x₁)) = x₁   
POL(U4_A_{U13_aaa_1}(x₁)) = x₁   
POL(U4_A_{del_out_aaa}(x₁)) = x₁   
POL(U6_aa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U6_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U6_aa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U6_aa_{del_in_aaa}(x₁)) = 3 + x₁   
POL(U6_aa_{del_out_aaa}(x₁)) = x₁   
POL(U7_aa_{U10_aaa_1}(x₁)) = 1 + x₁   
POL(U7_aa_{U11_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U12_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U13_aaa_1}(x₁)) = x₁   
POL(U7_aa_{U9_aaa_1}(x₁)) = 2 + x₁   
POL(U7_aa_{del_in_aaa}(x₁)) = 3 + x₁   
POL(U7_aa_{del_out_aaa}(x₁)) = x₁   
POL(U9_aaa_{head_in_aa}(x₁)) = x₁   
POL(U9_aaa_{head_out_aa}(x₁)) = x₁   
POL(del2_out_aa) = 0   
POL(del_in_aaa) = 0   
POL(del_out_aaa) = 0   
POL(head_in_aa) = 0   
POL(head_out_aa) = 0   
POL(tail_in_aa) = 0   
POL(tail_out_aa) = 0   

(204) Obligation:

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

U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{del2_out_aa}(del2_out_aa)
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa))
U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{del_out_aaa}(del_out_aaa) → CONF_IN_A
U3_A_{del2_out_aa}(del2_out_aa) → U4_A_{del_out_aaa}(del_out_aaa)
U4_A_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)))
CONF_IN_A → U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa))
U3_A_{U6_aa_1}(U6_aa_{del_out_aaa}(del_out_aaa)) → U3_A_{U7_aa_1}(U7_aa_{del_out_aaa}(del_out_aaa))
U4_A_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U4_A_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa))

The TRS R consists of the following rules:

U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U9_aaa_1}(U9_aaa_{head_in_aa}(head_in_aa))
U13_aaa_{del_in_aaa}(del_in_aaa) → U13_aaa_{U11_aaa_1}(U11_aaa_{head_in_aa}(head_in_aa))
U9_aaa_{head_in_aa}(head_in_aa) → U9_aaa_{head_out_aa}(head_out_aa)
U11_aaa_{head_in_aa}(head_in_aa) → U11_aaa_{head_out_aa}(head_out_aa)
U6_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U6_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U7_aa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U7_aa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U11_aaa_1}(U11_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U12_aaa_1}(U12_aaa_{tail_in_aa}(tail_in_aa))
U12_aaa_{tail_in_aa}(tail_in_aa) → U12_aaa_{tail_out_aa}(tail_out_aa)
U10_aaa_{tail_in_aa}(tail_in_aa) → U10_aaa_{tail_out_aa}(tail_out_aa)
U6_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U6_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U7_aa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U7_aa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U13_aaa_{U12_aaa_1}(U12_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{U13_aaa_1}(U13_aaa_{del_in_aaa}(del_in_aaa))
U6_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U6_aa_{del_out_aaa}(del_out_aaa)
U7_aa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U7_aa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U13_aaa_1}(U13_aaa_{del_out_aaa}(del_out_aaa)) → U13_aaa_{del_out_aaa}(del_out_aaa)
U13_aaa_{U9_aaa_1}(U9_aaa_{head_out_aa}(head_out_aa)) → U13_aaa_{U10_aaa_1}(U10_aaa_{tail_in_aa}(tail_in_aa))
U13_aaa_{U10_aaa_1}(U10_aaa_{tail_out_aa}(tail_out_aa)) → U13_aaa_{del_out_aaa}(del_out_aaa)

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

(205) Obligation:

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

DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x5)

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

(206) UsableRulesProof (EQUIVALENT transformation)

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

(207) Obligation:

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

DEL_IN_AGA(X, Y, .(H, T2)) → U11_AGA(X, Y, H, T2, head_in_ga(Y, H))
U11_AGA(X, Y, H, T2, head_out_ga(Y, H)) → U12_AGA(X, Y, H, T2, tail_in_ga(Y, T1))
U12_AGA(X, Y, H, T2, tail_out_ga(Y, T1)) → DEL_IN_AGA(X, T1, T2)

The TRS R consists of the following rules:

head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x2)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
DEL_IN_AGA(x1, x2, x3)  =  DEL_IN_AGA(x2)
U11_AGA(x1, x2, x3, x4, x5)  =  U11_AGA(x2, x5)
U12_AGA(x1, x2, x3, x4, x5)  =  U12_AGA(x5)

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

(208) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2l_in_ga(X, Xs))
s2l_in_ga(0, L) → U14_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U14_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X3, Xs)) → U15_ga(X, X3, Xs, 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)
U15_ga(X, X3, Xs, p_out_ga(X, P)) → U16_ga(X, X3, Xs, s2l_in_ga(P, Xs))
U16_ga(X, X3, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X3, Xs))
U1_g(X, s2l_out_ga(X, Xs)) → U2_g(X, conf_in_g(Xs))
conf_in_g(X) → U3_g(X, del2_in_ga(X, Z))
del2_in_ga(X, Y) → U6_ga(X, Y, del_in_aga(U, X, Z))
del_in_aga(X1, [], X2) → U8_aga(X1, X2, failure_in_g(a))
failure_in_g(b) → failure_out_g(b)
U8_aga(X1, X2, failure_out_g(a)) → del_out_aga(X1, [], X2)
del_in_aga(H, X, T) → U9_aga(H, X, T, head_in_ga(X, H))
head_in_ga([], X4) → head_out_ga([], X4)
head_in_ga(.(H, X5), H) → head_out_ga(.(H, X5), H)
U9_aga(H, X, T, head_out_ga(X, H)) → U10_aga(H, X, T, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X6, Xs), Xs) → tail_out_ga(.(X6, Xs), Xs)
U10_aga(H, X, T, tail_out_ga(X, T)) → del_out_aga(H, X, T)
del_in_aga(X, Y, .(H, T2)) → U11_aga(X, Y, H, T2, head_in_ga(Y, H))
U11_aga(X, Y, H, T2, head_out_ga(Y, H)) → U12_aga(X, Y, H, T2, tail_in_ga(Y, T1))
U12_aga(X, Y, H, T2, tail_out_ga(Y, T1)) → U13_aga(X, Y, H, T2, del_in_aga(X, T1, T2))
U13_aga(X, Y, H, T2, del_out_aga(X, T1, T2)) → del_out_aga(X, Y, .(H, T2))
U6_ga(X, Y, del_out_aga(U, X, Z)) → U7_ga(X, Y, del_in_aaa(V, Z, Y))
del_in_aaa(X1, [], X2) → U8_aaa(X1, X2, failure_in_g(a))
U8_aaa(X1, X2, failure_out_g(a)) → del_out_aaa(X1, [], X2)
del_in_aaa(H, X, T) → U9_aaa(H, X, T, head_in_aa(X, H))
head_in_aa([], X4) → head_out_aa([], X4)
head_in_aa(.(H, X5), H) → head_out_aa(.(H, X5), H)
U9_aaa(H, X, T, head_out_aa(X, H)) → U10_aaa(H, X, T, tail_in_aa(X, T))
tail_in_aa([], []) → tail_out_aa([], [])
tail_in_aa(.(X6, Xs), Xs) → tail_out_aa(.(X6, Xs), Xs)
U10_aaa(H, X, T, tail_out_aa(X, T)) → del_out_aaa(H, X, T)
del_in_aaa(X, Y, .(H, T2)) → U11_aaa(X, Y, H, T2, head_in_aa(Y, H))
U11_aaa(X, Y, H, T2, head_out_aa(Y, H)) → U12_aaa(X, Y, H, T2, tail_in_aa(Y, T1))
U12_aaa(X, Y, H, T2, tail_out_aa(Y, T1)) → U13_aaa(X, Y, H, T2, del_in_aaa(X, T1, T2))
U13_aaa(X, Y, H, T2, del_out_aaa(X, T1, T2)) → del_out_aaa(X, Y, .(H, T2))
U7_ga(X, Y, del_out_aaa(V, Z, Y)) → del2_out_ga(X, Y)
U3_g(X, del2_out_ga(X, Z)) → U4_g(X, del_in_aaa(U, Y, Z))
U4_g(X, del_out_aaa(U, Y, Z)) → U5_g(X, conf_in_a(Y))
conf_in_a(X) → U3_a(X, del2_in_aa(X, Z))
del2_in_aa(X, Y) → U6_aa(X, Y, del_in_aaa(U, X, Z))
U6_aa(X, Y, del_out_aaa(U, X, Z)) → U7_aa(X, Y, del_in_aaa(V, Z, Y))
U7_aa(X, Y, del_out_aaa(V, Z, Y)) → del2_out_aa(X, Y)
U3_a(X, del2_out_aa(X, Z)) → U4_a(X, del_in_aaa(U, Y, Z))
U4_a(X, del_out_aaa(U, Y, Z)) → U5_a(X, conf_in_a(Y))
U5_a(X, conf_out_a(Y)) → conf_out_a(X)
U5_g(X, conf_out_a(Y)) → conf_out_g(X)
U2_g(X, conf_out_g(Xs)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
0  =  0
U14_ga(x1, x2)  =  U14_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
[]  =  []
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x4)
.(x1, x2)  =  .(x2)
U2_g(x1, x2)  =  U2_g(x2)
conf_in_g(x1)  =  conf_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
del2_in_ga(x1, x2)  =  del2_in_ga(x1)
U6_ga(x1, x2, x3)  =  U6_ga(x3)
del_in_aga(x1, x2, x3)  =  del_in_aga(x2)
U8_aga(x1, x2, x3)  =  U8_aga(x3)
failure_in_g(x1)  =  failure_in_g(x1)
b  =  b
failure_out_g(x1)  =  failure_out_g
a  =  a
del_out_aga(x1, x2, x3)  =  del_out_aga
U9_aga(x1, x2, x3, x4)  =  U9_aga(x2, x4)
head_in_ga(x1, x2)  =  head_in_ga(x1)
head_out_ga(x1, x2)  =  head_out_ga
U10_aga(x1, x2, x3, x4)  =  U10_aga(x4)
tail_in_ga(x1, x2)  =  tail_in_ga(x1)
tail_out_ga(x1, x2)  =  tail_out_ga(x2)
U11_aga(x1, x2, x3, x4, x5)  =  U11_aga(x2, x5)
U12_aga(x1, x2, x3, x4, x5)  =  U12_aga(x5)
U13_aga(x1, x2, x3, x4, x5)  =  U13_aga(x5)
U7_ga(x1, x2, x3)  =  U7_ga(x3)
del_in_aaa(x1, x2, x3)  =  del_in_aaa
U8_aaa(x1, x2, x3)  =  U8_aaa(x3)
del_out_aaa(x1, x2, x3)  =  del_out_aaa
U9_aaa(x1, x2, x3, x4)  =  U9_aaa(x4)
head_in_aa(x1, x2)  =  head_in_aa
head_out_aa(x1, x2)  =  head_out_aa
U10_aaa(x1, x2, x3, x4)  =  U10_aaa(x4)
tail_in_aa(x1, x2)  =  tail_in_aa
tail_out_aa(x1, x2)  =  tail_out_aa
U11_aaa(x1, x2, x3, x4, x5)  =  U11_aaa(x5)
U12_aaa(x1, x2, x3, x4, x5)  =  U12_aaa(x5)
U13_aaa(x1, x2, x3, x4, x5)  =  U13_aaa(x5)
del2_out_ga(x1, x2)  =  del2_out_ga
U4_g(x1, x2)  =  U4_g(x2)
U5_g(x1, x2)  =  U5_g(x2)
conf_in_a(x1)  =  conf_in_a
U3_a(x1, x2)  =  U3_a(x2)
del2_in_aa(x1, x2)  =  del2_in_aa
U6_aa(x1, x2, x3)  =  U6_aa(x3)
U7_aa(x1, x2, x3)  =  U7_aa(x3)
del2_out_aa(x1, x2)  =  del2_out_aa
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2)  =  U5_a(x2)
conf_out_a(x1)  =  conf_out_a
conf_out_g(x1)  =  conf_out_g
goal_out_g(x1)  =  goal_out_g
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x4)

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

(209) UsableRulesProof (EQUIVALENT transformation)

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

(210) Obligation:

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

S2L_IN_GA(X, .(X3, Xs)) → U15_GA(X, X3, Xs, p_in_ga(X, P))
U15_GA(X, X3, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)

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
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U15_GA(x1, x2, x3, x4)  =  U15_GA(x4)

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