YES(O(1), O(n^1)) 5.04/1.75 YES(O(1), O(n^1)) 5.42/1.86 5.42/1.86 5.42/1.86 5.42/1.86 5.42/1.86 5.42/1.86 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 5.42/1.86 5.42/1.86 5.42/1.86
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
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↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
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↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
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↳4 CdtProblem
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↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
5.42/1.86 
↳6 CdtProblem
5.42/1.86 
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)
5.42/1.86 
↳8 CdtProblem
5.42/1.86 
↳9 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
5.42/1.86 
↳10 CdtProblem
5.42/1.86 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
5.42/1.86 
↳12 CdtProblem
5.42/1.86 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
5.42/1.86 
↳14 CdtProblem
5.42/1.86 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
5.42/1.86 
↳16 CdtProblem
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↳17 CdtKnowledgeProof (⇔)
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↳18 BOUNDS(O(1), O(1))
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(0) Obligation:

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

primes → sieve(from(s(s(0)))) 5.42/1.86
from(X) → cons(X, n__from(n__s(X))) 5.42/1.86
head(cons(X, Y)) → X 5.42/1.86
tail(cons(X, Y)) → activate(Y) 5.42/1.86
if(true, X, Y) → activate(X) 5.42/1.86
if(false, X, Y) → activate(Y) 5.42/1.86
filter(s(s(X)), cons(Y, Z)) → if(divides(s(s(X)), Y), n__filter(n__s(n__s(X)), activate(Z)), n__cons(Y, n__filter(X, n__sieve(Y)))) 5.42/1.86
sieve(cons(X, Y)) → cons(X, n__filter(X, n__sieve(activate(Y)))) 5.42/1.86
from(X) → n__from(X) 5.42/1.86
s(X) → n__s(X) 5.42/1.86
filter(X1, X2) → n__filter(X1, X2) 5.42/1.86
cons(X1, X2) → n__cons(X1, X2) 5.42/1.86
sieve(X) → n__sieve(X) 5.42/1.86
activate(n__from(X)) → from(activate(X)) 5.42/1.86
activate(n__s(X)) → s(activate(X)) 5.42/1.86
activate(n__filter(X1, X2)) → filter(activate(X1), activate(X2)) 5.42/1.86
activate(n__cons(X1, X2)) → cons(activate(X1), X2) 5.42/1.86
activate(n__sieve(X)) → sieve(activate(X)) 5.42/1.86
activate(X) → X

Rewrite Strategy: INNERMOST
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5.42/1.86

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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5.42/1.86

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0)) 5.42/1.86
FROM(z0) → c1(CONS(z0, n__from(n__s(z0)))) 5.42/1.86
TAIL(cons(z0, z1)) → c4(ACTIVATE(z1)) 5.42/1.86
IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
FILTER(s(s(z0)), cons(z1, z2)) → c7(IF(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))), S(s(z0)), S(z0), ACTIVATE(z2)) 5.42/1.86
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
S tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0)) 5.42/1.86
FROM(z0) → c1(CONS(z0, n__from(n__s(z0)))) 5.42/1.86
TAIL(cons(z0, z1)) → c4(ACTIVATE(z1)) 5.42/1.86
IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
FILTER(s(s(z0)), cons(z1, z2)) → c7(IF(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))), S(s(z0)), S(z0), ACTIVATE(z2)) 5.42/1.86
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

PRIMES, FROM, TAIL, IF, FILTER, SIEVE, ACTIVATE

Compound Symbols:

c, c1, c4, c5, c6, c7, c9, c13, c14, c15, c16, c17

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5.42/1.86

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

TAIL(cons(z0, z1)) → c4(ACTIVATE(z1)) 5.42/1.86
FILTER(s(s(z0)), cons(z1, z2)) → c7(IF(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))), S(s(z0)), S(z0), ACTIVATE(z2)) 5.42/1.86
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1))
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5.42/1.86

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0)) 5.42/1.86
FROM(z0) → c1(CONS(z0, n__from(n__s(z0)))) 5.42/1.86
IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
S tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0)) 5.42/1.86
FROM(z0) → c1(CONS(z0, n__from(n__s(z0)))) 5.42/1.86
IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

PRIMES, FROM, IF, ACTIVATE

Compound Symbols:

c, c1, c5, c6, c13, c14, c15, c16, c17

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5.42/1.86

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 8 trailing tuple parts
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5.42/1.86

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

IF, ACTIVATE, PRIMES, FROM

Compound Symbols:

c5, c6, c13, c, c1, c14, c15, c16, c17

5.42/1.86
5.42/1.86

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

IF(true, z0, z1) → c5(ACTIVATE(z0)) 5.42/1.86
IF(false, z0, z1) → c6(ACTIVATE(z1))
Removed 2 trailing nodes:

PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1
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5.42/1.86

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

ACTIVATE, PRIMES, FROM

Compound Symbols:

c13, c, c1, c14, c15, c16, c17

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5.42/1.86

(9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

PRIMES → c(FROM(s(s(0))))
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5.42/1.86

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
K tuples:

PRIMES → c(FROM(s(s(0))))
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

ACTIVATE, PRIMES, FROM

Compound Symbols:

c13, c, c1, c14, c15, c16, c17

5.42/1.86
5.42/1.86

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
We considered the (Usable) Rules:

s(z0) → n__s(z0) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0)
And the Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.42/1.86

POL(0) = [1]    5.42/1.86
POL(ACTIVATE(x1)) = [4]x1    5.42/1.86
POL(FROM(x1)) = 0    5.42/1.86
POL(PRIMES) = [2]    5.42/1.86
POL(activate(x1)) = 0    5.42/1.86
POL(c(x1)) = x1    5.42/1.86
POL(c1) = 0    5.42/1.86
POL(c13(x1, x2)) = x1 + x2    5.42/1.86
POL(c14(x1)) = x1    5.42/1.86
POL(c15(x1, x2)) = x1 + x2    5.42/1.86
POL(c16(x1)) = x1    5.42/1.86
POL(c17(x1)) = x1    5.42/1.86
POL(cons(x1, x2)) = [3] + [3]x1 + [3]x2    5.42/1.86
POL(filter(x1, x2)) = [3]    5.42/1.86
POL(from(x1)) = [3]    5.42/1.86
POL(n__cons(x1, x2)) = [4] + x1    5.42/1.86
POL(n__filter(x1, x2)) = x1 + x2    5.42/1.86
POL(n__from(x1)) = x1    5.42/1.86
POL(n__s(x1)) = x1    5.42/1.86
POL(n__sieve(x1)) = x1    5.42/1.86
POL(s(x1)) = [2] + x1    5.42/1.86
POL(sieve(x1)) = [3]   
5.42/1.86
5.42/1.86

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
K tuples:

PRIMES → c(FROM(s(s(0)))) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

ACTIVATE, PRIMES, FROM

Compound Symbols:

c13, c, c1, c14, c15, c16, c17

5.42/1.86
5.42/1.86

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
We considered the (Usable) Rules:

s(z0) → n__s(z0) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0)
And the Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.42/1.86

POL(0) = [1]    5.42/1.86
POL(ACTIVATE(x1)) = [5] + [4]x1    5.42/1.86
POL(FROM(x1)) = 0    5.42/1.86
POL(PRIMES) = [4]    5.42/1.86
POL(activate(x1)) = [4] + [3]x1    5.42/1.86
POL(c(x1)) = x1    5.42/1.86
POL(c1) = 0    5.42/1.86
POL(c13(x1, x2)) = x1 + x2    5.42/1.86
POL(c14(x1)) = x1    5.42/1.86
POL(c15(x1, x2)) = x1 + x2    5.42/1.86
POL(c16(x1)) = x1    5.42/1.86
POL(c17(x1)) = x1    5.42/1.86
POL(cons(x1, x2)) = x1    5.42/1.86
POL(filter(x1, x2)) = [4] + x1 + x2    5.42/1.86
POL(from(x1)) = x1    5.42/1.86
POL(n__cons(x1, x2)) = x1    5.42/1.86
POL(n__filter(x1, x2)) = [4] + x1 + x2    5.42/1.86
POL(n__from(x1)) = x1    5.42/1.86
POL(n__s(x1)) = [1] + x1    5.42/1.86
POL(n__sieve(x1)) = [4] + x1    5.42/1.86
POL(s(x1)) = x1    5.42/1.86
POL(sieve(x1)) = [4] + x1   
5.42/1.86
5.42/1.86

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
FROM(z0) → c1
K tuples:

PRIMES → c(FROM(s(s(0)))) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

ACTIVATE, PRIMES, FROM

Compound Symbols:

c13, c, c1, c14, c15, c16, c17

5.42/1.86
5.42/1.86

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

s(z0) → n__s(z0) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0)
And the Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.42/1.86

POL(0) = [1]    5.42/1.86
POL(ACTIVATE(x1)) = [4]x1    5.42/1.86
POL(FROM(x1)) = 0    5.42/1.86
POL(PRIMES) = 0    5.42/1.86
POL(activate(x1)) = 0    5.42/1.86
POL(c(x1)) = x1    5.42/1.86
POL(c1) = 0    5.42/1.86
POL(c13(x1, x2)) = x1 + x2    5.42/1.86
POL(c14(x1)) = x1    5.42/1.86
POL(c15(x1, x2)) = x1 + x2    5.42/1.86
POL(c16(x1)) = x1    5.42/1.86
POL(c17(x1)) = x1    5.42/1.86
POL(cons(x1, x2)) = [3] + [3]x1 + [3]x2    5.42/1.86
POL(filter(x1, x2)) = [3]    5.42/1.86
POL(from(x1)) = [3]    5.42/1.86
POL(n__cons(x1, x2)) = x1    5.42/1.86
POL(n__filter(x1, x2)) = x1 + x2    5.42/1.86
POL(n__from(x1)) = [2] + x1    5.42/1.86
POL(n__s(x1)) = x1    5.42/1.86
POL(n__sieve(x1)) = x1    5.42/1.86
POL(s(x1)) = 0    5.42/1.86
POL(sieve(x1)) = [3]   
5.42/1.86
5.42/1.86

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0)))) 5.42/1.86
from(z0) → cons(z0, n__from(n__s(z0))) 5.42/1.86
from(z0) → n__from(z0) 5.42/1.86
head(cons(z0, z1)) → z0 5.42/1.86
tail(cons(z0, z1)) → activate(z1) 5.42/1.86
if(true, z0, z1) → activate(z0) 5.42/1.86
if(false, z0, z1) → activate(z1) 5.42/1.86
filter(s(s(z0)), cons(z1, z2)) → if(divides(s(s(z0)), z1), n__filter(n__s(n__s(z0)), activate(z2)), n__cons(z1, n__filter(z0, n__sieve(z1)))) 5.42/1.86
filter(z0, z1) → n__filter(z0, z1) 5.42/1.86
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1)))) 5.42/1.86
sieve(z0) → n__sieve(z0) 5.42/1.86
s(z0) → n__s(z0) 5.42/1.86
cons(z0, z1) → n__cons(z0, z1) 5.42/1.86
activate(n__from(z0)) → from(activate(z0)) 5.42/1.86
activate(n__s(z0)) → s(activate(z0)) 5.42/1.86
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1)) 5.42/1.86
activate(n__cons(z0, z1)) → cons(activate(z0), z1) 5.42/1.86
activate(n__sieve(z0)) → sieve(activate(z0)) 5.42/1.86
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0)) 5.42/1.86
PRIMES → c(FROM(s(s(0)))) 5.42/1.86
FROM(z0) → c1 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))
S tuples:

FROM(z0) → c1
K tuples:

PRIMES → c(FROM(s(s(0)))) 5.42/1.86
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1)) 5.42/1.86
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0)) 5.42/1.86
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

primes, from, head, tail, if, filter, sieve, s, cons, activate

Defined Pair Symbols:

ACTIVATE, PRIMES, FROM

Compound Symbols:

c13, c, c1, c14, c15, c16, c17

5.42/1.86
5.42/1.86

(17) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

FROM(z0) → c1
Now S is empty
5.42/1.86
5.42/1.86

(18) BOUNDS(O(1), O(1))

5.42/1.86
5.42/1.86
5.78/1.95 EOF