YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 0.00/0.84 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.84 0.00/0.84 0.00/0.84
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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))
0.00/0.84 
↳6 CdtProblem
0.00/0.84 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.84 
↳8 CdtProblem
0.00/0.84 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳10 CdtProblem
0.00/0.84 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳12 BOUNDS(O(1), O(1))
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0.00/0.84 0.00/0.84
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0.00/0.84

(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X))) 0.00/0.84
first(0, Z) → nil 0.00/0.84
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) 0.00/0.84
sel(0, cons(X, Z)) → X 0.00/0.84
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) 0.00/0.84
from(X) → n__from(X) 0.00/0.84
s(X) → n__s(X) 0.00/0.84
first(X1, X2) → n__first(X1, X2) 0.00/0.84
activate(n__from(X)) → from(activate(X)) 0.00/0.84
activate(n__s(X)) → s(activate(X)) 0.00/0.84
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) 0.00/0.84
activate(X) → X

Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.84
from(z0) → n__from(z0) 0.00/0.84
first(0, z0) → nil 0.00/0.84
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.84
first(z0, z1) → n__first(z0, z1) 0.00/0.84
sel(0, cons(z0, z1)) → z0 0.00/0.84
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__from(z0)) → from(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.84
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 0.00/0.84
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.84
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 0.00/0.84
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.84
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8, c9, c10

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(3) CdtUnreachableProof (EQUIVALENT transformation)

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

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 0.00/0.84
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
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0.00/0.84

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.84
from(z0) → n__from(z0) 0.00/0.84
first(0, z0) → nil 0.00/0.84
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.84
first(z0, z1) → n__first(z0, z1) 0.00/0.84
sel(0, cons(z0, z1)) → z0 0.00/0.84
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__from(z0)) → from(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.84
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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

Removed 3 trailing tuple parts
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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.84
from(z0) → n__from(z0) 0.00/0.84
first(0, z0) → nil 0.00/0.84
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.84
first(z0, z1) → n__first(z0, z1) 0.00/0.84
sel(0, cons(z0, z1)) → z0 0.00/0.84
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__from(z0)) → from(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.84
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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(7) 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)) → c8(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(ACTIVATE(x1)) = x1    0.00/0.85
POL(c10(x1, x2)) = x1 + x2    0.00/0.85
POL(c8(x1)) = x1    0.00/0.85
POL(c9(x1)) = x1    0.00/0.85
POL(n__first(x1, x2)) = x1 + x2    0.00/0.85
POL(n__from(x1)) = [4] + x1    0.00/0.85
POL(n__s(x1)) = x1   
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(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
first(0, z0) → nil 0.00/0.85
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.85
first(z0, z1) → n__first(z0, z1) 0.00/0.85
sel(0, cons(z0, z1)) → z0 0.00/0.85
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.85
s(z0) → n__s(z0) 0.00/0.85
activate(n__from(z0)) → from(activate(z0)) 0.00/0.85
activate(n__s(z0)) → s(activate(z0)) 0.00/0.85
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.85
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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(9) 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)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(ACTIVATE(x1)) = [2]x1    0.00/0.85
POL(c10(x1, x2)) = x1 + x2    0.00/0.85
POL(c8(x1)) = x1    0.00/0.85
POL(c9(x1)) = x1    0.00/0.85
POL(n__first(x1, x2)) = [1] + x1 + x2    0.00/0.85
POL(n__from(x1)) = x1    0.00/0.85
POL(n__s(x1)) = [1] + x1   
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(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.85
from(z0) → n__from(z0) 0.00/0.85
first(0, z0) → nil 0.00/0.85
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.85
first(z0, z1) → n__first(z0, z1) 0.00/0.85
sel(0, cons(z0, z1)) → z0 0.00/0.85
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.85
s(z0) → n__s(z0) 0.00/0.85
activate(n__from(z0)) → from(activate(z0)) 0.00/0.85
activate(n__s(z0)) → s(activate(z0)) 0.00/0.85
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.85
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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

The set S is empty
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(12) BOUNDS(O(1), O(1))

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0.00/0.89 EOF