let R be the TRS under consideration

a(a(a(a(_1)))) -> a(c(a(c(c(_1))))) is in elim_R(R)
let l0 be the left-hand side of this rule
p0 = 0.0.0 is a position in l0
we have l0|p0 = a(_1)
a(a(a(a(_2)))) -> a(c(a(c(c(_2))))) is in R
let r'0 be the right-hand side of this rule
theta0 = {_1/c(a(c(c(_2))))} is a mgu of l0|p0 and r'0

==> a(a(a(a(a(a(a(_1))))))) -> a(c(a(c(c(c(a(c(c(_1))))))))) is in EU_R^1
let l1 be the left-hand side of this rule
p1 = 0 is a position in l1
we have l1|p1 = a(a(a(a(a(a(_1))))))
c(c(c(_2))) -> a(a(a(_2))) is in R
let r'1 be the right-hand side of this rule
theta1 = {_2/a(a(a(_1)))} is a mgu of l1|p1 and r'1

==> a(c(c(c(a(a(a(_1))))))) -> a(c(a(c(c(c(a(c(c(_1))))))))) is in EU_R^2
let l2 be the left-hand side of this rule
p2 = 0.0.0.0.0.0 is a position in l2
we have l2|p2 = a(_1)
a(a(a(a(_2)))) -> a(c(a(c(c(_2))))) is in R
let r'2 be the right-hand side of this rule
theta2 = {_1/c(a(c(c(_2))))} is a mgu of l2|p2 and r'2

==> a(c(c(c(a(a(a(a(a(a(_1)))))))))) -> a(c(a(c(c(c(a(c(c(c(a(c(c(_1))))))))))))) is in EU_R^3
let l3 be the left-hand side of this rule
p3 = 0.0.0.0.0 is a position in l3
we have l3|p3 = a(a(a(a(a(_1)))))
c(c(c(_2))) -> a(a(a(_2))) is in R
let r'3 be the right-hand side of this rule
theta3 = {_2/a(a(_1))} is a mgu of l3|p3 and r'3

==> a(c(c(c(a(c(c(c(a(a(_1)))))))))) -> a(c(a(c(c(c(a(c(c(c(a(c(c(_1))))))))))))) is in EU_R^4
let l4 be the left-hand side of this rule
p4 = 0.0.0.0.0.0.0.0.0 is a position in l4
we have l4|p4 = a(_1)
a(a(a(a(_2)))) -> a(c(a(c(c(_2))))) is in R
let r'4 be the right-hand side of this rule
theta4 = {_1/c(a(c(c(_2))))} is a mgu of l4|p4 and r'4

==> a(c(c(c(a(c(c(c(a(a(a(a(a(_1))))))))))))) -> a(c(a(c(c(c(a(c(c(c(a(c(c(c(a(c(c(_1))))))))))))))))) is in EU_R^5
let l5 be the left-hand side of this rule
p5 = 0.0.0.0.0.0.0.0.0 is a position in l5
we have l5|p5 = a(a(a(a(_1))))
c(c(c(_2))) -> a(a(a(_2))) is in R
let r'5 be the right-hand side of this rule
theta5 = {_2/a(_1)} is a mgu of l5|p5 and r'5

==> a(c(c(c(a(c(c(c(a(c(c(c(a(_1))))))))))))) -> a(c(a(c(c(c(a(c(c(c(a(c(c(c(a(c(c(_1))))))))))))))))) is in EU_R^6
let l be the left-hand side and r be the right-hand side of this rule
let p = 0.0
let theta = {}
let theta' = {_1/c(c(_1))}
we have r|p = a(c(c(c(a(c(c(c(a(c(c(c(a(c(c(_1))))))))))))))) and
theta'(theta(l)) = theta(r|p)
so, theta(l) = a(c(c(c(a(c(c(c(a(c(c(c(a(_1))))))))))))) is non-terminating w.r.t. R

Termination disproved by the backward process
proof stopped at iteration i=6, depth k=17
1386 rule(s) generated