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

(0) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tree(T)).
tree(nil) :- !.
tree(X) :- ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R)))).
s2t(0, L) :- ','(!, eq(L, nil)).
s2t(X, node(T, X1, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X2, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X3, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, nil)).
left(nil, nil).
left(node(L, X5, X6), L).
right(nil, nil).
right(node(X7, X8, R), R).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tree(T)).
tree(nil) :- true.
tree(X) :- ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R)))).
s2t(0, L) :- eq(L, nil).
s2t(X, node(T, X1, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X2, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X3, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, nil)).
left(nil, nil).
left(node(L, X5, X6), L).
right(nil, nil).
right(node(X7, X8, R), R).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(3) UndefinedPredicateHandlerProof (SOUND transformation)

Added facts for all undefined predicates [PROLOG].

(4) Obligation:

Clauses:

goal(X) :- ','(s2t(X, T), tree(T)).
tree(nil) :- true.
tree(X) :- ','(left(T, L), ','(right(T, R), ','(tree(L), tree(R)))).
s2t(0, L) :- eq(L, nil).
s2t(X, node(T, X1, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X2, T)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(T, X3, nil)) :- ','(p(X, P), s2t(P, T)).
s2t(X, node(nil, X4, nil)).
left(nil, nil).
left(node(L, X5, X6), L).
right(nil, nil).
right(node(X7, X8, R), R).
p(0, 0).
p(s(X), X).
eq(X, X).
true.

Queries:

goal(g).

(5) PrologToPiTRSProof (SOUND transformation)

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)

(7) DependencyPairsProof (EQUIVALENT transformation)

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U8_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X1, T)) → P_IN_GA(X, P)
U9_GA(X, T, X1, p_out_ga(X, P)) → U10_GA(X, T, X1, s2t_in_ga(P, T))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X2, T)) → P_IN_GA(X, P)
U11_GA(X, X2, T, p_out_ga(X, P)) → U12_GA(X, X2, T, s2t_in_ga(P, T))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, nil)) → P_IN_GA(X, P)
U13_GA(X, T, X3, p_out_ga(X, P)) → U14_GA(X, T, X3, s2t_in_ga(P, T))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tree_in_g(T))
U1_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(nil) → U3_G(true_in_)
TREE_IN_G(nil) → TRUE_IN_
TREE_IN_G(X) → U4_G(X, left_in_aa(T, L))
TREE_IN_G(X) → LEFT_IN_AA(T, L)
U4_G(X, left_out_aa(T, L)) → U5_G(X, T, L, right_in_aa(T, R))
U4_G(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_G(X, T, L, right_out_aa(T, R)) → U6_G(X, R, tree_in_a(L))
U5_G(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
TREE_IN_A(nil) → U3_A(true_in_)
TREE_IN_A(nil) → TRUE_IN_
TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
TREE_IN_A(X) → LEFT_IN_AA(T, L)
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U4_A(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
U6_A(X, R, tree_out_a(L)) → U7_A(X, tree_in_a(R))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U6_G(X, R, tree_out_a(L)) → U7_G(X, tree_in_a(R))
U6_G(X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U8_GA(x1, x2)  =  U8_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U3_G(x1)  =  U3_G(x1)
TRUE_IN_  =  TRUE_IN_
U4_G(x1, x2)  =  U4_G(x1, x2)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x4)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_G(x1, x2, x3)  =  U6_G(x1, x3)
TREE_IN_A(x1)  =  TREE_IN_A
U3_A(x1)  =  U3_A(x1)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)
U7_A(x1, x2)  =  U7_A(x2)
U7_G(x1, x2)  =  U7_G(x1, x2)

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

(8) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U8_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X1, T)) → P_IN_GA(X, P)
U9_GA(X, T, X1, p_out_ga(X, P)) → U10_GA(X, T, X1, s2t_in_ga(P, T))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X2, T)) → P_IN_GA(X, P)
U11_GA(X, X2, T, p_out_ga(X, P)) → U12_GA(X, X2, T, s2t_in_ga(P, T))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, nil)) → P_IN_GA(X, P)
U13_GA(X, T, X3, p_out_ga(X, P)) → U14_GA(X, T, X3, s2t_in_ga(P, T))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tree_in_g(T))
U1_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(nil) → U3_G(true_in_)
TREE_IN_G(nil) → TRUE_IN_
TREE_IN_G(X) → U4_G(X, left_in_aa(T, L))
TREE_IN_G(X) → LEFT_IN_AA(T, L)
U4_G(X, left_out_aa(T, L)) → U5_G(X, T, L, right_in_aa(T, R))
U4_G(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_G(X, T, L, right_out_aa(T, R)) → U6_G(X, R, tree_in_a(L))
U5_G(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
TREE_IN_A(nil) → U3_A(true_in_)
TREE_IN_A(nil) → TRUE_IN_
TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
TREE_IN_A(X) → LEFT_IN_AA(T, L)
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U4_A(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
U6_A(X, R, tree_out_a(L)) → U7_A(X, tree_in_a(R))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U6_G(X, R, tree_out_a(L)) → U7_G(X, tree_in_a(R))
U6_G(X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U8_GA(x1, x2)  =  U8_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x1, x4)
U2_G(x1, x2)  =  U2_G(x1, x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U3_G(x1)  =  U3_G(x1)
TRUE_IN_  =  TRUE_IN_
U4_G(x1, x2)  =  U4_G(x1, x2)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x4)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_G(x1, x2, x3)  =  U6_G(x1, x3)
TREE_IN_A(x1)  =  TREE_IN_A
U3_A(x1)  =  U3_A(x1)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)
U7_A(x1, x2)  =  U7_A(x2)
U7_G(x1, x2)  =  U7_G(x1, x2)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 27 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)
TREE_IN_A(x1)  =  TREE_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)

The TRS R consists of the following rules:

left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
tree_in_a(nil) → U3_a(true_in_)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U3_a(true_out_) → tree_out_a(nil)
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
true_in_ → true_out_
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)

The argument filtering Pi contains the following mapping:
nil  =  nil
node(x1, x2, x3)  =  node(x1, x3)
true_in_  =  true_in_
true_out_  =  true_out_
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
TREE_IN_A(x1)  =  TREE_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

TREE_IN_A → U4_A(left_in_aa)
U4_A(left_out_aa) → U5_A(right_in_aa)
U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(16) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TREE_IN_A → U4_A(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TREE_IN_A → U4_A(left_out_aa)

(17) Obligation:

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

U4_A(left_out_aa) → U5_A(right_in_aa)
U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(18) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U4_A(left_out_aa) → U5_A(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U4_A(left_out_aa) → U5_A(right_out_aa)

(19) Obligation:

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

U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(20) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(right_out_aa) → U6_A(tree_in_a) at position [0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U3_a(true_in_))
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))

(21) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U3_a(true_in_))
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(22) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U3_a(true_in_)) at position [0,0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U3_a(true_out_))

(23) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))
U5_A(right_out_aa) → U6_A(U3_a(true_out_))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(24) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U4_a(left_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))

(25) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U3_a(true_out_))
U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(26) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U3_a(true_out_)) at position [0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(tree_out_a)

(27) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))
U5_A(right_out_aa) → U6_A(tree_out_a)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(28) 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 = TREE_IN_A evaluates to t =TREE_IN_A

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

TREE_IN_A → U4_A(left_out_aa)
with rule TREE_IN_A → U4_A(left_out_aa) at position [] and matcher [ ]

U4_A(left_out_aa) → U5_A(right_out_aa)
with rule U4_A(left_out_aa) → U5_A(right_out_aa) at position [] and matcher [ ]

U5_A(right_out_aa) → TREE_IN_A
with rule U5_A(right_out_aa) → TREE_IN_A

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.

(29) FALSE

(30) Obligation:

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

S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x1, x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1, x2)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x1, x2)
U9_ga(x1, x2, x3, x4)  =  U9_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)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x1, x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x1, x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x1, x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g(x1)
U4_g(x1, x2)  =  U4_g(x1, x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x1, x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x1, x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x1, x2)
goal_out_g(x1)  =  goal_out_g(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
0  =  0
nil  =  nil
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x1, x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x1, x4)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

S2T_IN_GA(X) → U9_GA(X, p_in_ga(X))
U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U11_GA(X, p_in_ga(X))
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(X, p_in_ga(X))
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)

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

(35) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U9_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))

(36) Obligation:

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

U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U11_GA(X, p_in_ga(X))
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(X, p_in_ga(X))
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)

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

(37) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U11_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))

(38) Obligation:

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

U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(X, p_in_ga(X))
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)

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

(39) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U13_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))

(40) Obligation:

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

U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)

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

(41) UsableRulesProof (EQUIVALENT transformation)

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

(42) Obligation:

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

U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))

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

p_in_ga(x0)

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

(43) QReductionProof (EQUIVALENT transformation)

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

p_in_ga(x0)

(44) Obligation:

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

U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(46) Obligation:

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

U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

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

(47) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U11_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U11_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U11_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(48) Obligation:

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

U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U11_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U11_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

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

(49) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_GA(X, p_out_ga(X, P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U13_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

(50) Obligation:

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

S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U11_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U11_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
U13_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

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

(52) Complex Obligation (AND)

(53) Obligation:

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

U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
S2T_IN_GA(0) → U11_GA(0, p_out_ga(0, 0))
U11_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
S2T_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
U13_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)

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

(54) NonTerminationProof (EQUIVALENT transformation)

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

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

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0))
with rule S2T_IN_GA(0) → U9_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)
with rule U9_GA(0, p_out_ga(0, 0)) → S2T_IN_GA(0)

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


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

(55) FALSE

(56) Obligation:

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

S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
U11_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)

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

(57) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

• U9_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• S2T_IN_GA(s(x0)) → U9_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

• U11_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2T_IN_GA(z0)
  The graph contains the following edges 1 > 1, 2 > 1

• S2T_IN_GA(s(x0)) → U11_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

• S2T_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
  The graph contains the following edges 1 >= 1

(58) TRUE

(59) PrologToPiTRSProof (SOUND transformation)

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(60) Obligation:

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

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g

(61) DependencyPairsProof (EQUIVALENT transformation)

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U8_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X1, T)) → P_IN_GA(X, P)
U9_GA(X, T, X1, p_out_ga(X, P)) → U10_GA(X, T, X1, s2t_in_ga(P, T))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X2, T)) → P_IN_GA(X, P)
U11_GA(X, X2, T, p_out_ga(X, P)) → U12_GA(X, X2, T, s2t_in_ga(P, T))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, nil)) → P_IN_GA(X, P)
U13_GA(X, T, X3, p_out_ga(X, P)) → U14_GA(X, T, X3, s2t_in_ga(P, T))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tree_in_g(T))
U1_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(nil) → U3_G(true_in_)
TREE_IN_G(nil) → TRUE_IN_
TREE_IN_G(X) → U4_G(X, left_in_aa(T, L))
TREE_IN_G(X) → LEFT_IN_AA(T, L)
U4_G(X, left_out_aa(T, L)) → U5_G(X, T, L, right_in_aa(T, R))
U4_G(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_G(X, T, L, right_out_aa(T, R)) → U6_G(X, R, tree_in_a(L))
U5_G(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
TREE_IN_A(nil) → U3_A(true_in_)
TREE_IN_A(nil) → TRUE_IN_
TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
TREE_IN_A(X) → LEFT_IN_AA(T, L)
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U4_A(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
U6_A(X, R, tree_out_a(L)) → U7_A(X, tree_in_a(R))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U6_G(X, R, tree_out_a(L)) → U7_G(X, tree_in_a(R))
U6_G(X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U8_GA(x1, x2)  =  U8_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U3_G(x1)  =  U3_G(x1)
TRUE_IN_  =  TRUE_IN_
U4_G(x1, x2)  =  U4_G(x2)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_G(x1, x2, x3, x4)  =  U5_G(x4)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_G(x1, x2, x3)  =  U6_G(x3)
TREE_IN_A(x1)  =  TREE_IN_A
U3_A(x1)  =  U3_A(x1)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)
U7_A(x1, x2)  =  U7_A(x2)
U7_G(x1, x2)  =  U7_G(x2)

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

(62) Obligation:

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

GOAL_IN_G(X) → U1_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(0, L) → U8_GA(L, eq_in_ag(L, nil))
S2T_IN_GA(0, L) → EQ_IN_AG(L, nil)
S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X1, T)) → P_IN_GA(X, P)
U9_GA(X, T, X1, p_out_ga(X, P)) → U10_GA(X, T, X1, s2t_in_ga(P, T))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
S2T_IN_GA(X, node(nil, X2, T)) → P_IN_GA(X, P)
U11_GA(X, X2, T, p_out_ga(X, P)) → U12_GA(X, X2, T, s2t_in_ga(P, T))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
S2T_IN_GA(X, node(T, X3, nil)) → P_IN_GA(X, P)
U13_GA(X, T, X3, p_out_ga(X, P)) → U14_GA(X, T, X3, s2t_in_ga(P, T))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)
U1_G(X, s2t_out_ga(X, T)) → U2_G(X, tree_in_g(T))
U1_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(nil) → U3_G(true_in_)
TREE_IN_G(nil) → TRUE_IN_
TREE_IN_G(X) → U4_G(X, left_in_aa(T, L))
TREE_IN_G(X) → LEFT_IN_AA(T, L)
U4_G(X, left_out_aa(T, L)) → U5_G(X, T, L, right_in_aa(T, R))
U4_G(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_G(X, T, L, right_out_aa(T, R)) → U6_G(X, R, tree_in_a(L))
U5_G(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
TREE_IN_A(nil) → U3_A(true_in_)
TREE_IN_A(nil) → TRUE_IN_
TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
TREE_IN_A(X) → LEFT_IN_AA(T, L)
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U4_A(X, left_out_aa(T, L)) → RIGHT_IN_AA(T, R)
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)
U6_A(X, R, tree_out_a(L)) → U7_A(X, tree_in_a(R))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U6_G(X, R, tree_out_a(L)) → U7_G(X, tree_in_a(R))
U6_G(X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U8_GA(x1, x2)  =  U8_GA(x2)
EQ_IN_AG(x1, x2)  =  EQ_IN_AG(x2)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)
U14_GA(x1, x2, x3, x4)  =  U14_GA(x4)
U2_G(x1, x2)  =  U2_G(x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U3_G(x1)  =  U3_G(x1)
TRUE_IN_  =  TRUE_IN_
U4_G(x1, x2)  =  U4_G(x2)
LEFT_IN_AA(x1, x2)  =  LEFT_IN_AA
U5_G(x1, x2, x3, x4)  =  U5_G(x4)
RIGHT_IN_AA(x1, x2)  =  RIGHT_IN_AA
U6_G(x1, x2, x3)  =  U6_G(x3)
TREE_IN_A(x1)  =  TREE_IN_A
U3_A(x1)  =  U3_A(x1)
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)
U7_A(x1, x2)  =  U7_A(x2)
U7_G(x1, x2)  =  U7_G(x2)

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

(63) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 27 less nodes.

(64) Complex Obligation (AND)

(65) Obligation:

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

TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g
TREE_IN_A(x1)  =  TREE_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)

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

(66) UsableRulesProof (EQUIVALENT transformation)

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

(67) Obligation:

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

TREE_IN_A(X) → U4_A(X, left_in_aa(T, L))
U4_A(X, left_out_aa(T, L)) → U5_A(X, T, L, right_in_aa(T, R))
U5_A(X, T, L, right_out_aa(T, R)) → U6_A(X, R, tree_in_a(L))
U6_A(X, R, tree_out_a(L)) → TREE_IN_A(R)
U5_A(X, T, L, right_out_aa(T, R)) → TREE_IN_A(L)

The TRS R consists of the following rules:

left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
tree_in_a(nil) → U3_a(true_in_)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U3_a(true_out_) → tree_out_a(nil)
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
true_in_ → true_out_
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)

The argument filtering Pi contains the following mapping:
nil  =  nil
node(x1, x2, x3)  =  node(x1, x3)
true_in_  =  true_in_
true_out_  =  true_out_
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
TREE_IN_A(x1)  =  TREE_IN_A
U4_A(x1, x2)  =  U4_A(x2)
U5_A(x1, x2, x3, x4)  =  U5_A(x4)
U6_A(x1, x2, x3)  =  U6_A(x3)

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

(68) PiDPToQDPProof (SOUND transformation)

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

(69) Obligation:

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

TREE_IN_A → U4_A(left_in_aa)
U4_A(left_out_aa) → U5_A(right_in_aa)
U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(70) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule TREE_IN_A → U4_A(left_in_aa) at position [0] we obtained the following new rules [LPAR04]:

TREE_IN_A → U4_A(left_out_aa)

(71) Obligation:

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

U4_A(left_out_aa) → U5_A(right_in_aa)
U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(72) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U4_A(left_out_aa) → U5_A(right_in_aa) at position [0] we obtained the following new rules [LPAR04]:

U4_A(left_out_aa) → U5_A(right_out_aa)

(73) Obligation:

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

U5_A(right_out_aa) → U6_A(tree_in_a)
U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(74) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U5_A(right_out_aa) → U6_A(tree_in_a) at position [0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U3_a(true_in_))
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))

(75) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U3_a(true_in_))
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(76) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U3_a(true_in_)) at position [0,0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U3_a(true_out_))

(77) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U4_a(left_in_aa))
U5_A(right_out_aa) → U6_A(U3_a(true_out_))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(78) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U4_a(left_in_aa)) at position [0,0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))

(79) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U3_a(true_out_))
U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(80) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U5_A(right_out_aa) → U6_A(U3_a(true_out_)) at position [0] we obtained the following new rules [LPAR04]:

U5_A(right_out_aa) → U6_A(tree_out_a)

(81) Obligation:

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

U6_A(tree_out_a) → TREE_IN_A
U5_A(right_out_aa) → TREE_IN_A
TREE_IN_A → U4_A(left_out_aa)
U4_A(left_out_aa) → U5_A(right_out_aa)
U5_A(right_out_aa) → U6_A(U4_a(left_out_aa))
U5_A(right_out_aa) → U6_A(tree_out_a)

The TRS R consists of the following rules:

left_in_aa → left_out_aa
right_in_aa → right_out_aa
tree_in_a → U3_a(true_in_)
tree_in_a → U4_a(left_in_aa)
U3_a(true_out_) → tree_out_a
U4_a(left_out_aa) → U5_a(right_in_aa)
true_in_ → true_out_
U5_a(right_out_aa) → U6_a(tree_in_a)
U6_a(tree_out_a) → U7_a(tree_in_a)
U7_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

left_in_aa
right_in_aa
tree_in_a
U3_a(x0)
U4_a(x0)
true_in_
U5_a(x0)
U6_a(x0)
U7_a(x0)

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

(82) 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 = TREE_IN_A evaluates to t =TREE_IN_A

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

TREE_IN_A → U4_A(left_out_aa)
with rule TREE_IN_A → U4_A(left_out_aa) at position [] and matcher [ ]

U4_A(left_out_aa) → U5_A(right_out_aa)
with rule U4_A(left_out_aa) → U5_A(right_out_aa) at position [] and matcher [ ]

U5_A(right_out_aa) → TREE_IN_A
with rule U5_A(right_out_aa) → TREE_IN_A

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.

(83) FALSE

(84) Obligation:

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

S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

goal_in_g(X) → U1_g(X, s2t_in_ga(X, T))
s2t_in_ga(0, L) → U8_ga(L, eq_in_ag(L, nil))
eq_in_ag(X, X) → eq_out_ag(X, X)
U8_ga(L, eq_out_ag(L, nil)) → s2t_out_ga(0, L)
s2t_in_ga(X, node(T, X1, T)) → U9_ga(X, T, X1, 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)
U9_ga(X, T, X1, p_out_ga(X, P)) → U10_ga(X, T, X1, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X2, T)) → U11_ga(X, X2, T, p_in_ga(X, P))
U11_ga(X, X2, T, p_out_ga(X, P)) → U12_ga(X, X2, T, s2t_in_ga(P, T))
s2t_in_ga(X, node(T, X3, nil)) → U13_ga(X, T, X3, p_in_ga(X, P))
U13_ga(X, T, X3, p_out_ga(X, P)) → U14_ga(X, T, X3, s2t_in_ga(P, T))
s2t_in_ga(X, node(nil, X4, nil)) → s2t_out_ga(X, node(nil, X4, nil))
U14_ga(X, T, X3, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X3, nil))
U12_ga(X, X2, T, s2t_out_ga(P, T)) → s2t_out_ga(X, node(nil, X2, T))
U10_ga(X, T, X1, s2t_out_ga(P, T)) → s2t_out_ga(X, node(T, X1, T))
U1_g(X, s2t_out_ga(X, T)) → U2_g(X, tree_in_g(T))
tree_in_g(nil) → U3_g(true_in_)
true_in_ → true_out_
U3_g(true_out_) → tree_out_g(nil)
tree_in_g(X) → U4_g(X, left_in_aa(T, L))
left_in_aa(nil, nil) → left_out_aa(nil, nil)
left_in_aa(node(L, X5, X6), L) → left_out_aa(node(L, X5, X6), L)
U4_g(X, left_out_aa(T, L)) → U5_g(X, T, L, right_in_aa(T, R))
right_in_aa(nil, nil) → right_out_aa(nil, nil)
right_in_aa(node(X7, X8, R), R) → right_out_aa(node(X7, X8, R), R)
U5_g(X, T, L, right_out_aa(T, R)) → U6_g(X, R, tree_in_a(L))
tree_in_a(nil) → U3_a(true_in_)
U3_a(true_out_) → tree_out_a(nil)
tree_in_a(X) → U4_a(X, left_in_aa(T, L))
U4_a(X, left_out_aa(T, L)) → U5_a(X, T, L, right_in_aa(T, R))
U5_a(X, T, L, right_out_aa(T, R)) → U6_a(X, R, tree_in_a(L))
U6_a(X, R, tree_out_a(L)) → U7_a(X, tree_in_a(R))
U7_a(X, tree_out_a(R)) → tree_out_a(X)
U6_g(X, R, tree_out_a(L)) → U7_g(X, tree_in_a(R))
U7_g(X, tree_out_a(R)) → tree_out_g(X)
U2_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
0  =  0
U8_ga(x1, x2)  =  U8_ga(x2)
eq_in_ag(x1, x2)  =  eq_in_ag(x2)
eq_out_ag(x1, x2)  =  eq_out_ag(x1)
nil  =  nil
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
U9_ga(x1, x2, x3, x4)  =  U9_ga(x4)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)
U11_ga(x1, x2, x3, x4)  =  U11_ga(x4)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x4)
U14_ga(x1, x2, x3, x4)  =  U14_ga(x4)
node(x1, x2, x3)  =  node(x1, x3)
U2_g(x1, x2)  =  U2_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
U3_g(x1)  =  U3_g(x1)
true_in_  =  true_in_
true_out_  =  true_out_
tree_out_g(x1)  =  tree_out_g
U4_g(x1, x2)  =  U4_g(x2)
left_in_aa(x1, x2)  =  left_in_aa
left_out_aa(x1, x2)  =  left_out_aa
U5_g(x1, x2, x3, x4)  =  U5_g(x4)
right_in_aa(x1, x2)  =  right_in_aa
right_out_aa(x1, x2)  =  right_out_aa
U6_g(x1, x2, x3)  =  U6_g(x3)
tree_in_a(x1)  =  tree_in_a
U3_a(x1)  =  U3_a(x1)
tree_out_a(x1)  =  tree_out_a
U4_a(x1, x2)  =  U4_a(x2)
U5_a(x1, x2, x3, x4)  =  U5_a(x4)
U6_a(x1, x2, x3)  =  U6_a(x3)
U7_a(x1, x2)  =  U7_a(x2)
U7_g(x1, x2)  =  U7_g(x2)
goal_out_g(x1)  =  goal_out_g
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)

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

(85) UsableRulesProof (EQUIVALENT transformation)

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

(86) Obligation:

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

S2T_IN_GA(X, node(T, X1, T)) → U9_GA(X, T, X1, p_in_ga(X, P))
U9_GA(X, T, X1, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(nil, X2, T)) → U11_GA(X, X2, T, p_in_ga(X, P))
U11_GA(X, X2, T, p_out_ga(X, P)) → S2T_IN_GA(P, T)
S2T_IN_GA(X, node(T, X3, nil)) → U13_GA(X, T, X3, p_in_ga(X, P))
U13_GA(X, T, X3, p_out_ga(X, P)) → S2T_IN_GA(P, T)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
0  =  0
nil  =  nil
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x4)
U11_GA(x1, x2, x3, x4)  =  U11_GA(x4)
U13_GA(x1, x2, x3, x4)  =  U13_GA(x4)

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

(87) PiDPToQDPProof (SOUND transformation)

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

(88) Obligation:

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

S2T_IN_GA(X) → U9_GA(p_in_ga(X))
U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U11_GA(p_in_ga(X))
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)

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

(89) UsableRulesReductionPairsProof (EQUIVALENT transformation)

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

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0   
POL(S2T_IN_GA(x₁)) = x₁   
POL(U11_GA(x₁)) = x₁   
POL(U13_GA(x₁)) = x₁   
POL(U9_GA(x₁)) = x₁   
POL(p_in_ga(x₁)) = x₁   
POL(p_out_ga(x₁)) = 2·x₁   
POL(s(x₁)) = 2 + 2·x₁   

(90) Obligation:

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

S2T_IN_GA(X) → U9_GA(p_in_ga(X))
U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U11_GA(p_in_ga(X))
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(91) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U9_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U9_GA(p_out_ga(0))

(92) Obligation:

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

U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U11_GA(p_in_ga(X))
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(93) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U11_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U11_GA(p_out_ga(0))

(94) Obligation:

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

U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(95) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule S2T_IN_GA(X) → U13_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

S2T_IN_GA(0) → U13_GA(p_out_ga(0))

(96) Obligation:

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

U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(97) 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.

(98) Obligation:

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

U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))

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

p_in_ga(x0)

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

(99) QReductionProof (EQUIVALENT transformation)

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

p_in_ga(x0)

(100) Obligation:

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

U9_GA(p_out_ga(P)) → S2T_IN_GA(P)
U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))

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

(101) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U9_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U9_GA(p_out_ga(0)) → S2T_IN_GA(0)

(102) Obligation:

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

U11_GA(p_out_ga(P)) → S2T_IN_GA(P)
U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))
U9_GA(p_out_ga(0)) → S2T_IN_GA(0)

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

(103) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U11_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U11_GA(p_out_ga(0)) → S2T_IN_GA(0)

(104) Obligation:

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

U13_GA(p_out_ga(P)) → S2T_IN_GA(P)
S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))
U9_GA(p_out_ga(0)) → S2T_IN_GA(0)
U11_GA(p_out_ga(0)) → S2T_IN_GA(0)

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

(105) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U13_GA(p_out_ga(P)) → S2T_IN_GA(P) we obtained the following new rules [LPAR04]:

U13_GA(p_out_ga(0)) → S2T_IN_GA(0)

(106) Obligation:

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

S2T_IN_GA(0) → U9_GA(p_out_ga(0))
S2T_IN_GA(0) → U11_GA(p_out_ga(0))
S2T_IN_GA(0) → U13_GA(p_out_ga(0))
U9_GA(p_out_ga(0)) → S2T_IN_GA(0)
U11_GA(p_out_ga(0)) → S2T_IN_GA(0)
U13_GA(p_out_ga(0)) → S2T_IN_GA(0)

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

(107) 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 = U9_GA(p_out_ga(0)) evaluates to t =U9_GA(p_out_ga(0))

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

U9_GA(p_out_ga(0)) → S2T_IN_GA(0)
with rule U9_GA(p_out_ga(0)) → S2T_IN_GA(0) at position [] and matcher [ ]

S2T_IN_GA(0) → U9_GA(p_out_ga(0))
with rule S2T_IN_GA(0) → U9_GA(p_out_ga(0))

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


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

(108) FALSE