(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:
Lexicographic path order with status [LPO].
Precedence:
prime1 > s1 > true > =2
prime1 > s1 > and2 > =2
prime1 > s1 > not1 > =2
prime1 > s1 > divp2 > 0 > false > =2
prime1 > s1 > divp2 > rem2 > =2
prime1 > prime12 > true > =2
prime1 > prime12 > and2 > =2
prime1 > prime12 > not1 > =2
prime1 > prime12 > divp2 > 0 > false > =2
prime1 > prime12 > divp2 > rem2 > =2
Status:
prime1: [1]
0: []
false: []
s1: [1]
prime12: [1,2]
true: []
and2: [1,2]
not1: [1]
divp2: [1,2]
=2: [1,2]
rem2: [1,2]
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) TRUE