YES(O(1), O(n^1)) 0.00/0.90 YES(O(1), O(n^1)) 0.00/0.94 0.00/0.94 0.00/0.94 0.00/0.94 0.00/0.94 0.00/0.94 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.94 0.00/0.94 0.00/0.94
0.00/0.94
0.00/0.94
0.00/0.94
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))
0.00/0.94 
↳2 CdtProblem
0.00/0.94 
↳3 CdtUnreachableProof (⇔)
0.00/0.94 
↳4 CdtProblem
0.00/0.94 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.94 
↳6 CdtProblem
0.00/0.94 
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)
0.00/0.94 
↳8 CdtProblem
0.00/0.94 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.94 
↳10 CdtProblem
0.00/0.94 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.94 
↳12 CdtProblem
0.00/0.94 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.94 
↳14 CdtProblem
0.00/0.94 
↳15 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.94 
↳16 BOUNDS(O(1), O(1))
0.00/0.94
0.00/0.94 0.00/0.94
0.00/0.94
0.00/0.94

(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X))) 0.00/0.94
head(cons(X, XS)) → X 0.00/0.94
2nd(cons(X, XS)) → head(activate(XS)) 0.00/0.94
take(0, XS) → nil 0.00/0.94
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) 0.00/0.94
sel(0, cons(X, XS)) → X 0.00/0.94
sel(s(N), cons(X, XS)) → sel(N, activate(XS)) 0.00/0.94
from(X) → n__from(X) 0.00/0.94
s(X) → n__s(X) 0.00/0.94
take(X1, X2) → n__take(X1, X2) 0.00/0.94
activate(n__from(X)) → from(activate(X)) 0.00/0.94
activate(n__s(X)) → s(activate(X)) 0.00/0.94
activate(n__take(X1, X2)) → take(activate(X1), activate(X2)) 0.00/0.94
activate(X) → X

Rewrite Strategy: INNERMOST
0.00/0.94
0.00/0.94

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

Converted CpxTRS to CDT
0.00/0.94
0.00/0.94

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 0.00/0.94
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 0.00/0.94
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(S(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 0.00/0.94
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 0.00/0.94
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(S(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

2ND, TAKE, SEL, ACTIVATE

Compound Symbols:

c3, c5, c8, c10, c11, c12

0.00/0.94
0.00/0.94

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 0.00/0.94
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2))
0.00/0.94
0.00/0.94

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(S(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(S(activate(z0)), ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(TAKE(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c3, c10, c11, c12

0.00/0.94
0.00/0.94

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

Removed 4 trailing tuple parts
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0.00/0.94

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

2ND(cons(z0, z1)) → c3(ACTIVATE(z1)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

2ND(cons(z0, z1)) → c3(ACTIVATE(z1)) 0.00/0.94
ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c3, c10, c11, c12

0.00/0.94
0.00/0.94

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

2ND(cons(z0, z1)) → c3(ACTIVATE(z1))
0.00/0.94
0.00/0.94

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c10, c11, c12

0.00/0.94
0.00/0.94

(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__from(z0)) → c10(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.94

POL(ACTIVATE(x1)) = x1    0.00/0.94
POL(c10(x1)) = x1    0.00/0.94
POL(c11(x1)) = x1    0.00/0.94
POL(c12(x1, x2)) = x1 + x2    0.00/0.94
POL(n__from(x1)) = [4] + x1    0.00/0.94
POL(n__s(x1)) = x1    0.00/0.94
POL(n__take(x1, x2)) = x1 + x2   
0.00/0.94
0.00/0.94

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0))
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c10, c11, c12

0.00/0.94
0.00/0.94

(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__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.94

POL(ACTIVATE(x1)) = [1] + [2]x1    0.00/0.94
POL(c10(x1)) = x1    0.00/0.94
POL(c11(x1)) = x1    0.00/0.94
POL(c12(x1, x2)) = x1 + x2    0.00/0.94
POL(n__from(x1)) = [3] + x1    0.00/0.94
POL(n__s(x1)) = x1    0.00/0.94
POL(n__take(x1, x2)) = [2] + x1 + x2   
0.00/0.94
0.00/0.94

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0))
K tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c10, c11, c12

0.00/0.94
0.00/0.94

(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)) → c11(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.94

POL(ACTIVATE(x1)) = [5]x1    0.00/0.94
POL(c10(x1)) = x1    0.00/0.94
POL(c11(x1)) = x1    0.00/0.94
POL(c12(x1, x2)) = x1 + x2    0.00/0.94
POL(n__from(x1)) = x1    0.00/0.94
POL(n__s(x1)) = [1] + x1    0.00/0.94
POL(n__take(x1, x2)) = [5] + x1 + x2   
0.00/0.94
0.00/0.94

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.94
from(z0) → n__from(z0) 0.00/0.94
head(cons(z0, z1)) → z0 0.00/0.94
2nd(cons(z0, z1)) → head(activate(z1)) 0.00/0.94
take(0, z0) → nil 0.00/0.94
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 0.00/0.94
take(z0, z1) → n__take(z0, z1) 0.00/0.94
sel(0, cons(z0, z1)) → z0 0.00/0.94
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 0.00/0.94
s(z0) → n__s(z0) 0.00/0.94
activate(n__from(z0)) → from(activate(z0)) 0.00/0.94
activate(n__s(z0)) → s(activate(z0)) 0.00/0.94
activate(n__take(z0, z1)) → take(activate(z0), activate(z1)) 0.00/0.94
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c10(ACTIVATE(z0)) 0.00/0.94
ACTIVATE(n__take(z0, z1)) → c12(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.94
ACTIVATE(n__s(z0)) → c11(ACTIVATE(z0))
Defined Rule Symbols:

from, head, 2nd, take, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c10, c11, c12

0.00/0.94
0.00/0.94

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

The set S is empty
0.00/0.94
0.00/0.94

(16) BOUNDS(O(1), O(1))

0.00/0.94
0.00/0.94
0.00/1.00 EOF