(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Lexicographic path order with status [LPO].
Precedence:
a1 > c1 > d1
a1 > u1 > d1
a1 > b1 > d1
v1 > u1 > d1
v1 > b1 > d1
w1 > u1 > d1
w1 > b1 > d1
Status:
a1: [1]
c1: [1]
d1: [1]
u1: [1]
b1: [1]
v1: [1]
w1: [1]
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)
(2) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(3) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(4) TRUE