### (0) Obligation:

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

a__primesa__sieve(a__from(s(s(0))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(sieve(from(X133734_4))) →+ a__sieve(cons(mark(mark(X133734_4)), from(s(mark(X133734_4)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X133734_4 / sieve(from(X133734_4))].
The result substitution is [ ].

The rewrite sequence
mark(sieve(from(X133734_4))) →+ a__sieve(cons(mark(mark(X133734_4)), from(s(mark(X133734_4)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X133734_4 / sieve(from(X133734_4))].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

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

a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(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: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

### (7) 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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

### (8) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

### (9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (10) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

### (11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:
0'

Induction Step:

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

### (13) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

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

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

### (15) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

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

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

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

### (17) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

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

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

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

### (19) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

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

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

### (21) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

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

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

### (23) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

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__tail
a__primes = a__if
a__sieve = a__from
a__sieve = mark
a__sieve = a__tail
a__sieve = a__if
a__from = mark
a__from = a__tail
a__from = a__if
mark = a__tail
mark = a__if
a__tail = a__if

### (24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (25) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:

### (28) Obligation:

TRS:
Rules:
a__primesa__sieve(a__from(s(s(0'))))
a__from(X) → cons(mark(X), from(s(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(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__primesprimes
a__sieve(X) → sieve(X)
a__from(X) → from(X)
a__tail(X) → tail(X)
a__if(X1, X2, X3) → if(X1, X2, X3)
a__filter(X1, X2) → filter(X1, X2)

Types:

Lemmas: