YES(O(1), O(n^2)) 2.73/1.13 YES(O(1), O(n^2)) 2.73/1.16 2.73/1.16 2.73/1.16 2.73/1.16 2.73/1.16 2.73/1.16 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.73/1.16 2.73/1.16 2.73/1.16
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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
2.73/1.16 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.73/1.16 
↳2 CdtProblem
2.73/1.16 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.73/1.16 
↳4 CdtProblem
2.73/1.16 
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
2.73/1.16 
↳6 CdtProblem
2.73/1.16 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.73/1.16 
↳8 CdtProblem
2.73/1.16 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
2.73/1.16 
↳10 CdtProblem
2.73/1.16 
↳11 CdtKnowledgeProof (⇔)
2.73/1.16 
↳12 BOUNDS(O(1), O(1))
2.73/1.16
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2.73/1.16

(0) Obligation:

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

from(X) → cons(X, n__from(s(X))) 2.73/1.16
first(0, Z) → nil 2.73/1.16
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) 2.73/1.16
sel(0, cons(X, Z)) → X 2.73/1.16
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) 2.73/1.16
from(X) → n__from(X) 2.73/1.16
first(X1, X2) → n__first(X1, X2) 2.73/1.16
activate(n__from(X)) → from(X) 2.73/1.16
activate(n__first(X1, X2)) → first(X1, X2) 2.73/1.16
activate(X) → X

Rewrite Strategy: INNERMOST
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2.73/1.16

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

Converted CpxTRS to CDT
2.73/1.16
2.73/1.16

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0) 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
sel(0, cons(z0, z1)) → z0 2.73/1.16
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.73/1.16
activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7(FROM(z0)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7(FROM(z0)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c7, c8

2.73/1.16
2.73/1.16

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

Removed 1 trailing tuple parts
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2.73/1.16

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0) 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
sel(0, cons(z0, z1)) → z0 2.73/1.16
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.73/1.16
activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8, c7

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2.73/1.16

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

Removed 1 trailing nodes:

ACTIVATE(n__from(z0)) → c7
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2.73/1.16

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0) 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
sel(0, cons(z0, z1)) → z0 2.73/1.16
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.73/1.16
activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8, c7

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2.73/1.16

(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.

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
We considered the (Usable) Rules:

activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0)
And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
The order we found is given by the following interpretation:
Polynomial interpretation : 2.73/1.16

POL(0) = [3]    2.73/1.16
POL(ACTIVATE(x1)) = [4]    2.73/1.16
POL(FIRST(x1, x2)) = [4]    2.73/1.16
POL(SEL(x1, x2)) = [4]x1    2.73/1.16
POL(activate(x1)) = [4]x1    2.73/1.16
POL(c3(x1)) = x1    2.73/1.16
POL(c6(x1, x2)) = x1 + x2    2.73/1.16
POL(c7) = 0    2.73/1.16
POL(c8(x1)) = x1    2.73/1.16
POL(cons(x1, x2)) = [4]    2.73/1.16
POL(first(x1, x2)) = [2] + [2]x2    2.73/1.16
POL(from(x1)) = [4] + [2]x1    2.73/1.16
POL(n__first(x1, x2)) = [2] + x2    2.73/1.16
POL(n__from(x1)) = [2] + x1    2.73/1.16
POL(nil) = [1]    2.73/1.16
POL(s(x1)) = [2] + x1   
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2.73/1.16

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0) 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
sel(0, cons(z0, z1)) → z0 2.73/1.16
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.73/1.16
activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8, c7

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2.73/1.16

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
We considered the (Usable) Rules:

activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0)
And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
The order we found is given by the following interpretation:
Polynomial interpretation : 2.73/1.16

POL(0) = [3]    2.73/1.16
POL(ACTIVATE(x1)) = [2]x1    2.73/1.16
POL(FIRST(x1, x2)) = [2]x2    2.73/1.16
POL(SEL(x1, x2)) = x1·x2    2.73/1.16
POL(activate(x1)) = [1] + x1    2.73/1.16
POL(c3(x1)) = x1    2.73/1.16
POL(c6(x1, x2)) = x1 + x2    2.73/1.16
POL(c7) = 0    2.73/1.16
POL(c8(x1)) = x1    2.73/1.16
POL(cons(x1, x2)) = [1] + x2    2.73/1.16
POL(first(x1, x2)) = [1] + x2    2.73/1.16
POL(from(x1)) = [2]    2.73/1.16
POL(n__first(x1, x2)) = x2    2.73/1.16
POL(n__from(x1)) = [1]    2.73/1.16
POL(nil) = 0    2.73/1.16
POL(s(x1)) = [2] + x1   
2.73/1.16
2.73/1.16

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.73/1.16
from(z0) → n__from(z0) 2.73/1.16
first(0, z0) → nil 2.73/1.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 2.73/1.16
first(z0, z1) → n__first(z0, z1) 2.73/1.16
sel(0, cons(z0, z1)) → z0 2.73/1.16
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.73/1.16
activate(n__from(z0)) → from(z0) 2.73/1.16
activate(n__first(z0, z1)) → first(z0, z1) 2.73/1.16
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2)) 2.73/1.16
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7
S tuples:

ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))
K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.73/1.16
ACTIVATE(n__from(z0)) → c7 2.73/1.16
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8, c7

2.73/1.16
2.73/1.16

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1)) 2.73/1.16
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
Now S is empty
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2.73/1.16

(12) BOUNDS(O(1), O(1))

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3.08/1.30 EOF