(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
prime(0) → false
prime(s(0)) → false
prime(s(s(x))) → prime1(s(s(x)), s(x))
prime1(x, 0) → false
prime1(x, s(0)) → true
prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y)))
divp(x, y) → =(rem(x, y), 0)
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Combined order from the following AFS and order.
prime(
x1) =
prime(
x1)
0 =
0
false =
false
s(
x1) =
s(
x1)
prime1(
x1,
x2) =
prime1(
x1,
x2)
true =
true
and(
x1,
x2) =
and(
x1,
x2)
not(
x1) =
x1
divp(
x1,
x2) =
divp(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
rem(
x1,
x2) =
rem(
x1,
x2)
Recursive path order with status [RPO].
Quasi-Precedence:
prime1 > prime12 > false > [and2, =2, rem2]
prime1 > prime12 > s1 > [and2, =2, rem2]
prime1 > prime12 > true > [and2, =2, rem2]
prime1 > prime12 > divp2 > 0 > [and2, =2, rem2]
Status:
prime1: multiset
0: multiset
false: multiset
s1: multiset
prime12: [1,2]
true: multiset
and2: [1,2]
divp2: [2,1]
=2: multiset
rem2: multiset
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
prime(0) → false
prime(s(0)) → false
prime(s(s(x))) → prime1(s(s(x)), s(x))
prime1(x, 0) → false
prime1(x, s(0)) → true
prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y)))
divp(x, y) → =(rem(x, y), 0)
(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) YES