The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__primes → a__sieve(a__from(s(s(0))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__primes, a__sieve, a__from, mark, a__head, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(6) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__sieve, a__primes, a__from, mark, a__head, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sieve.

(8) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
mark, a__primes, a__from, a__head, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Induction Base:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(n11_0, 1))) →_R^Ω(1)
s(mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0))) →_IH
s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(c12_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__primes, a__sieve, a__from, a__head, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__primes.

(13) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__from, a__sieve, a__head, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__from.

(15) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__head, a__sieve, a__tail, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__head.

(17) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__tail, a__sieve, a__if

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__tail.

(19) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__if, a__sieve

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__if.

(21) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

The following defined symbols remain to be analysed:
a__sieve

They will be analysed ascendingly in the following order:
a__primes = a__sieve
a__primes = a__from
a__primes = mark
a__primes = a__head
a__primes = a__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__head
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__head
a__from = a__tail
a__from = a__if
mark = a__head
mark = a__tail
mark = a__if
a__head = a__tail
a__head = a__if
a__tail = a__if

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sieve.

(23) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
a__primes → a__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, Y)) → mark(X)
a__tail(cons(X, Y)) → mark(Y)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
a__filter(s(s(X)), cons(Y, Z)) → a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
a__sieve(cons(X, Y)) → cons(mark(X), filter(X, sieve(Y)))
mark(primes) → a__primes
mark(sieve(X)) → a__sieve(mark(X))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(tail(X)) → a__tail(mark(X))
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(filter(X1, X2)) → a__filter(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(true) → true
mark(false) → false
mark(divides(X1, X2)) → divides(mark(X1), mark(X2))
a__primes → primes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__head(X) → head(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:
a__primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
s :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
0' :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
cons :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
mark :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
from :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
true :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
false :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
a__filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
divides :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
filter :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
sieve :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
primes :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
head :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
tail :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
if :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
hole_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if1_0 :: 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0 :: Nat → 0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if

Lemmas:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(0) ⇔ 0'
gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(+(x, 1)) ⇔ s(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0)) → gen_0':s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if2_0(n11_0), rt ∈ Ω(1 + n11₀)

(28) BOUNDS(n^1, INF)