YES(O(1), O(n^1)) 0.00/0.82 YES(O(1), O(n^1)) 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 0.00/0.85 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.85 0.00/0.85 0.00/0.85
0.00/0.85
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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))
0.00/0.85 
↳2 CdtProblem
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↳3 CdtUnreachableProof (⇔)
0.00/0.85 
↳4 CdtProblem
0.00/0.85 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳6 CdtProblem
0.00/0.85 
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
0.00/0.85 
↳8 CdtProblem
0.00/0.85 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.85 
↳10 CdtProblem
0.00/0.85 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
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↳12 CdtProblem
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↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
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↳14 BOUNDS(O(1), O(1))
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0.00/0.85 0.00/0.85
0.00/0.85
0.00/0.85

(0) Obligation:

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

from(X) → cons(X, n__from(n__s(X))) 0.00/0.85
2ndspos(0, Z) → rnil 0.00/0.85
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z))) 0.00/0.85
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z))) 0.00/0.85
2ndsneg(0, Z) → rnil 0.00/0.85
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z))) 0.00/0.85
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z))) 0.00/0.85
pi(X) → 2ndspos(X, from(0)) 0.00/0.85
plus(0, Y) → Y 0.00/0.85
plus(s(X), Y) → s(plus(X, Y)) 0.00/0.85
times(0, Y) → 0 0.00/0.85
times(s(X), Y) → plus(Y, times(X, Y)) 0.00/0.85
square(X) → times(X, X) 0.00/0.85
from(X) → n__from(X) 0.00/0.85
s(X) → n__s(X) 0.00/0.85
activate(n__from(X)) → from(activate(X)) 0.00/0.85
activate(n__s(X)) → s(activate(X)) 0.00/0.85
activate(X) → X

Rewrite Strategy: INNERMOST
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0.00/0.85

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

Converted CpxTRS to CDT
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0.00/0.85

(2) 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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 0.00/0.85
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1)) 0.00/0.85
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 0.00/0.85
SQUARE(z0) → c13(TIMES(z0, z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 0.00/0.85
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1)) 0.00/0.85
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 0.00/0.85
SQUARE(z0) → c13(TIMES(z0, z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PI, PLUS, TIMES, SQUARE, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c8, c10, c12, c13, c15, c16

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0.00/0.85

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2)) 0.00/0.85
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 0.00/0.85
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1)) 0.00/0.85
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
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0.00/0.85

(4) 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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 0.00/0.85
SQUARE(z0) → c13(TIMES(z0, z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 0.00/0.85
SQUARE(z0) → c13(TIMES(z0, z0)) 0.00/0.85
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

PI, SQUARE, ACTIVATE

Compound Symbols:

c8, c13, c15, c16

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0.00/0.85

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

Removed 5 trailing tuple parts
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0.00/0.85

(6) 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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

PI(z0) → c8 0.00/0.85
SQUARE(z0) → c13 0.00/0.85
ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

PI(z0) → c8 0.00/0.85
SQUARE(z0) → c13 0.00/0.85
ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

PI, SQUARE, ACTIVATE

Compound Symbols:

c8, c13, c15, c16

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0.00/0.85

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

Removed 2 trailing nodes:

PI(z0) → c8 0.00/0.85
SQUARE(z0) → c13
0.00/0.85
0.00/0.85

(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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

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0.00/0.85

(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)) → c15(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
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(c15(x1)) = x1    0.00/0.85
POL(c16(x1)) = x1    0.00/0.85
POL(n__from(x1)) = [1] + x1    0.00/0.85
POL(n__s(x1)) = x1   
0.00/0.85
0.00/0.85

(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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

0.00/0.85
0.00/0.85

(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__s(z0)) → c16(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.85

POL(ACTIVATE(x1)) = [3]x1    0.00/0.85
POL(c15(x1)) = x1    0.00/0.85
POL(c16(x1)) = x1    0.00/0.85
POL(n__from(x1)) = x1    0.00/0.85
POL(n__s(x1)) = [1] + x1   
0.00/0.85
0.00/0.85

(12) 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
2ndspos(0, z0) → rnil 0.00/0.85
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 0.00/0.85
2ndsneg(0, z0) → rnil 0.00/0.85
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 0.00/0.85
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 0.00/0.85
pi(z0) → 2ndspos(z0, from(0)) 0.00/0.85
plus(0, z0) → z0 0.00/0.85
plus(s(z0), z1) → s(plus(z0, z1)) 0.00/0.85
times(0, z0) → 0 0.00/0.85
times(s(z0), z1) → plus(z1, times(z0, z1)) 0.00/0.85
square(z0) → times(z0, z0) 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(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0)) 0.00/0.85
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

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0.00/0.85

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

The set S is empty
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0.00/0.85

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

0.00/0.85
0.00/0.85
0.00/0.90 EOF