YES(O(1), O(n^2)) 2.40/1.10 YES(O(1), O(n^2)) 2.81/1.17 2.81/1.17 2.81/1.17 2.81/1.17 2.81/1.17 2.81/1.17 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 2.81/1.17 2.81/1.17 2.81/1.17
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2.81/1.17
2.81/1.17
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
2.81/1.17 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
2.81/1.17 
↳2 CdtProblem
2.81/1.17 
↳3 CdtUnreachableProof (⇔)
2.81/1.17 
↳4 CdtProblem
2.81/1.17 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
2.81/1.17 
↳6 CdtProblem
2.81/1.17 
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)
2.81/1.17 
↳8 CdtProblem
2.81/1.17 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
2.81/1.17 
↳10 CdtProblem
2.81/1.17 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
2.81/1.17 
↳12 CdtProblem
2.81/1.17 
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))
2.81/1.17 
↳14 BOUNDS(O(1), O(1))
2.81/1.17
2.81/1.17 2.81/1.17
2.81/1.17
2.81/1.17

(0) Obligation:

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

from(X) → cons(X, n__from(s(X))) 2.81/1.17
2ndspos(0, Z) → rnil 2.81/1.17
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z))) 2.81/1.17
2ndsneg(0, Z) → rnil 2.81/1.17
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) 2.81/1.17
pi(X) → 2ndspos(X, from(0)) 2.81/1.17
plus(0, Y) → Y 2.81/1.17
plus(s(X), Y) → s(plus(X, Y)) 2.81/1.17
times(0, Y) → 0 2.81/1.17
times(s(X), Y) → plus(Y, times(X, Y)) 2.81/1.17
square(X) → times(X, X) 2.81/1.17
from(X) → n__from(X) 2.81/1.17
cons(X1, X2) → n__cons(X1, X2) 2.81/1.17
activate(n__from(X)) → from(X) 2.81/1.17
activate(n__cons(X1, X2)) → cons(X1, X2) 2.81/1.17
activate(X) → X

Rewrite Strategy: INNERMOST
2.81/1.17
2.81/1.17

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

Converted CpxTRS to CDT
2.81/1.17
2.81/1.17

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0)))) 2.81/1.17
2NDSPOS(s(z0), cons(z1, n__cons(z2, z3))) → c3(ACTIVATE(z2), 2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 2.81/1.17
2NDSNEG(s(z0), cons(z1, n__cons(z2, z3))) → c5(ACTIVATE(z2), 2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 2.81/1.17
PI(z0) → c6(2NDSPOS(z0, from(0)), FROM(0)) 2.81/1.17
PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14(CONS(z0, z1))
S tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0)))) 2.81/1.17
2NDSPOS(s(z0), cons(z1, n__cons(z2, z3))) → c3(ACTIVATE(z2), 2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 2.81/1.17
2NDSNEG(s(z0), cons(z1, n__cons(z2, z3))) → c5(ACTIVATE(z2), 2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 2.81/1.17
PI(z0) → c6(2NDSPOS(z0, from(0)), FROM(0)) 2.81/1.17
PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14(CONS(z0, z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c, c3, c5, c6, c8, c10, c11, c13, c14

2.81/1.17
2.81/1.17

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

2NDSPOS(s(z0), cons(z1, n__cons(z2, z3))) → c3(ACTIVATE(z2), 2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 2.81/1.17
2NDSNEG(s(z0), cons(z1, n__cons(z2, z3))) → c5(ACTIVATE(z2), 2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
2.81/1.17
2.81/1.17

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0)))) 2.81/1.17
PI(z0) → c6(2NDSPOS(z0, from(0)), FROM(0)) 2.81/1.17
PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14(CONS(z0, z1))
S tuples:

FROM(z0) → c(CONS(z0, n__from(s(z0)))) 2.81/1.17
PI(z0) → c6(2NDSPOS(z0, from(0)), FROM(0)) 2.81/1.17
PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14(CONS(z0, z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

FROM, PI, PLUS, TIMES, SQUARE, ACTIVATE

Compound Symbols:

c, c6, c8, c10, c11, c13, c14

2.81/1.17
2.81/1.17

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

Removed 3 trailing tuple parts
2.81/1.17
2.81/1.17

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
FROM(z0) → c 2.81/1.17
PI(z0) → c6(FROM(0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14
S tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
SQUARE(z0) → c11(TIMES(z0, z0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0)) 2.81/1.17
FROM(z0) → c 2.81/1.17
PI(z0) → c6(FROM(0)) 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

PLUS, TIMES, SQUARE, ACTIVATE, FROM, PI

Compound Symbols:

c8, c10, c11, c13, c, c6, c14

2.81/1.17
2.81/1.17

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

SQUARE(z0) → c11(TIMES(z0, z0))
Removed 4 trailing nodes:

FROM(z0) → c 2.81/1.17
ACTIVATE(n__cons(z0, z1)) → c14 2.81/1.17
PI(z0) → c6(FROM(0)) 2.81/1.17
ACTIVATE(n__from(z0)) → c13(FROM(z0))
2.81/1.17
2.81/1.17

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
S tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

PLUS, TIMES

Compound Symbols:

c8, c10

2.81/1.17
2.81/1.17

(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.

TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
We considered the (Usable) Rules:

times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1))
And the Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.81/1.17

POL(0) = [3]    2.81/1.17
POL(PLUS(x1, x2)) = [1]    2.81/1.17
POL(TIMES(x1, x2)) = [4]x1    2.81/1.17
POL(c10(x1, x2)) = x1 + x2    2.81/1.17
POL(c8(x1)) = x1    2.81/1.17
POL(plus(x1, x2)) = [3] + [5]x2    2.81/1.17
POL(s(x1)) = [1] + x1    2.81/1.17
POL(times(x1, x2)) = 0   
2.81/1.17
2.81/1.17

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
S tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1))
K tuples:

TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
Defined Rule Symbols:

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

Defined Pair Symbols:

PLUS, TIMES

Compound Symbols:

c8, c10

2.81/1.17
2.81/1.17

(11) 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.

PLUS(s(z0), z1) → c8(PLUS(z0, z1))
We considered the (Usable) Rules:

times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1))
And the Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 2.81/1.17

POL(0) = 0    2.81/1.17
POL(PLUS(x1, x2)) = [2]x1    2.81/1.17
POL(TIMES(x1, x2)) = x1·x2    2.81/1.17
POL(c10(x1, x2)) = x1 + x2    2.81/1.17
POL(c8(x1)) = x1    2.81/1.17
POL(plus(x1, x2)) = [2] + [2]x1 + x2    2.81/1.17
POL(s(x1)) = [2] + x1    2.81/1.17
POL(times(x1, x2)) = x1·x2 + [2]x12   
2.81/1.17
2.81/1.17

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 2.81/1.17
from(z0) → n__from(z0) 2.81/1.17
2ndspos(0, z0) → rnil 2.81/1.17
2ndspos(s(z0), cons(z1, n__cons(z2, z3))) → rcons(posrecip(activate(z2)), 2ndsneg(z0, activate(z3))) 2.81/1.17
2ndsneg(0, z0) → rnil 2.81/1.17
2ndsneg(s(z0), cons(z1, n__cons(z2, z3))) → rcons(negrecip(activate(z2)), 2ndspos(z0, activate(z3))) 2.81/1.17
pi(z0) → 2ndspos(z0, from(0)) 2.81/1.17
plus(0, z0) → z0 2.81/1.17
plus(s(z0), z1) → s(plus(z0, z1)) 2.81/1.17
times(0, z0) → 0 2.81/1.17
times(s(z0), z1) → plus(z1, times(z0, z1)) 2.81/1.17
square(z0) → times(z0, z0) 2.81/1.17
cons(z0, z1) → n__cons(z0, z1) 2.81/1.17
activate(n__from(z0)) → from(z0) 2.81/1.17
activate(n__cons(z0, z1)) → cons(z0, z1) 2.81/1.17
activate(z0) → z0
Tuples:

PLUS(s(z0), z1) → c8(PLUS(z0, z1)) 2.81/1.17
TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
S tuples:none
K tuples:

TIMES(s(z0), z1) → c10(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 2.81/1.17
PLUS(s(z0), z1) → c8(PLUS(z0, z1))
Defined Rule Symbols:

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

Defined Pair Symbols:

PLUS, TIMES

Compound Symbols:

c8, c10

2.81/1.17
2.81/1.17

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

The set S is empty
2.81/1.17
2.81/1.17

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

2.81/1.17
2.81/1.17
3.10/1.22 EOF