(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(opp(x)) → opp(D(x))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(s(x), s(y)) → s(s(b(x, y)))
b(b(x, y), z) → b(x, b(y, z))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
D(
x1) =
D(
x1)
t =
t
s(
x1) =
x1
h =
h
constant =
constant
b(
x1,
x2) =
b(
x1,
x2)
c(
x1,
x2) =
c(
x1,
x2)
m(
x1,
x2) =
m(
x1,
x2)
opp(
x1) =
opp(
x1)
div(
x1,
x2) =
div(
x1,
x2)
pow(
x1,
x2) =
pow(
x1,
x2)
2 =
2
ln(
x1) =
ln(
x1)
1 =
1
Recursive path order with status [RPO].
Quasi-Precedence:
[D1, opp1, 1] > [h, constant]
[D1, opp1, 1] > [b2, c2, m2, pow2, ln1] > div2
[D1, opp1, 1] > 2
Status:
D1: multiset
t: multiset
h: multiset
constant: multiset
b2: [1,2]
c2: multiset
m2: multiset
opp1: multiset
div2: multiset
pow2: multiset
2: multiset
ln1: multiset
1: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
D(t) → s(h)
D(constant) → h
D(b(x, y)) → b(D(x), D(y))
D(c(x, y)) → b(c(y, D(x)), c(x, D(y)))
D(m(x, y)) → m(D(x), D(y))
D(div(x, y)) → m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
b(h, x) → x
b(x, h) → x
b(b(x, y), z) → b(x, b(y, z))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
D(opp(x)) → opp(D(x))
b(s(x), s(y)) → s(s(b(x, y)))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
b2 > s1 > D1 > opp1
and weight map:
D_1=2
opp_1=1
s_1=1
b_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
D(opp(x)) → opp(D(x))
b(s(x), s(y)) → s(s(b(x, y)))
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES