Runtime Complexity (innermost) proof of /tmp/tmpe1Fji9/ExIntrod_GM99

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 0 ms)

↳8 CdtProblem

↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 91 ms)

↳10 CdtProblem

↳11 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

↳12 CdtProblem

↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0))))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
head(cons(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
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))))
filter(z0, z1) → n__filter(z0, z1)
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1))))
sieve(z0) → n__sieve(z0)
s(z0) → n__s(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__sieve(z0)) → sieve(activate(z0))
activate(z0) → z0

Tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c2
HEAD(cons(z0, z1)) → c3
TAIL(cons(z0, z1)) → c4(ACTIVATE(z1))
IF(true, z0, z1) → c5(ACTIVATE(z0))
IF(false, z0, z1) → c6(ACTIVATE(z1))
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))
FILTER(z0, z1) → c8
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1))
SIEVE(z0) → c10
S(z0) → c11
CONS(z0, z1) → c12
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c18

S tuples:

PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
FROM(z0) → c2
HEAD(cons(z0, z1)) → c3
TAIL(cons(z0, z1)) → c4(ACTIVATE(z1))
IF(true, z0, z1) → c5(ACTIVATE(z0))
IF(false, z0, z1) → c6(ACTIVATE(z1))
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))
FILTER(z0, z1) → c8
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1))
SIEVE(z0) → c10
S(z0) → c11
CONS(z0, z1) → c12
ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c18

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

PRIMES, FROM, HEAD, TAIL, IF, FILTER, SIEVE, S, CONS, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 5 leading nodes:

TAIL(cons(z0, z1)) → c4(ACTIVATE(z1))
IF(true, z0, z1) → c5(ACTIVATE(z0))
IF(false, z0, z1) → c6(ACTIVATE(z1))
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))
SIEVE(cons(z0, z1)) → c9(CONS(z0, n__filter(z0, n__sieve(activate(z1)))), ACTIVATE(z1))

Removed 9 trailing nodes:

S(z0) → c11
SIEVE(z0) → c10
PRIMES → c(SIEVE(from(s(s(0)))), FROM(s(s(0))), S(s(0)), S(0))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
ACTIVATE(z0) → c18
FROM(z0) → c2
HEAD(cons(z0, z1)) → c3
CONS(z0, z1) → c12
FILTER(z0, z1) → c8

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0))))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
head(cons(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
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))))
filter(z0, z1) → n__filter(z0, z1)
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1))))
sieve(z0) → n__sieve(z0)
s(z0) → n__s(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__sieve(z0)) → sieve(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(SIEVE(activate(z0)), ACTIVATE(z0))

S tuples:

ACTIVATE(n__from(z0)) → c13(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(FILTER(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(CONS(activate(z0), z1), ACTIVATE(z0))
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:

ACTIVATE

Compound Symbols:

c13, c14, c15, c16, c17

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

Removed 5 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

primes → sieve(from(s(s(0))))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
head(cons(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
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))))
filter(z0, z1) → n__filter(z0, z1)
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1))))
sieve(z0) → n__sieve(z0)
s(z0) → n__s(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__sieve(z0)) → sieve(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

S tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
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

Compound Symbols:

c13, c14, c15, c16, c17

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

primes → sieve(from(s(s(0))))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
head(cons(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
if(true, z0, z1) → activate(z0)
if(false, z0, z1) → activate(z1)
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))))
filter(z0, z1) → n__filter(z0, z1)
sieve(cons(z0, z1)) → cons(z0, n__filter(z0, n__sieve(activate(z1))))
sieve(z0) → n__sieve(z0)
s(z0) → n__s(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__filter(z0, z1)) → filter(activate(z0), activate(z1))
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__sieve(z0)) → sieve(activate(z0))
activate(z0) → z0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

S tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14, c15, c16, c17

(9) CdtRuleRemovalProof (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(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = [2] + [2]x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(n__cons(x₁, x₂)) = [4] + x₁
POL(n__filter(x₁, x₂)) = [2] + x₁ + x₂
POL(n__from(x₁)) = [2] + x₁
POL(n__s(x₁)) = [2] + x₁
POL(n__sieve(x₁)) = x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

S tuples:

ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

K tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14, c15, c16, c17

(11) CdtRuleRemovalProof (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__sieve(z0)) → c17(ACTIVATE(z0))

We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x₁)) = [3] + x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(n__cons(x₁, x₂)) = x₁
POL(n__filter(x₁, x₂)) = [4] + x₁ + x₂
POL(n__from(x₁)) = x₁
POL(n__s(x₁)) = x₁
POL(n__sieve(x₁)) = [1] + x₁

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c13(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
ACTIVATE(n__filter(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__cons(z0, z1)) → c16(ACTIVATE(z0))
ACTIVATE(n__sieve(z0)) → c17(ACTIVATE(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c13, c14, c15, c16, c17

(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

(7) CdtUsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(10) Obligation:

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

(12) Obligation:

(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

(14) BOUNDS(O(1), O(1))