YES(O(1), O(n^1)) 0.00/0.75 YES(O(1), O(n^1)) 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 0.00/0.77 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 0.00/0.77 0.00/0.77 0.00/0.77
0.00/0.77
0.00/0.77
0.00/0.77
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
0.00/0.77 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
0.00/0.77 
↳2 CdtProblem
0.00/0.77 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
0.00/0.77 
↳4 CdtProblem
0.00/0.77 
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
0.00/0.77 
↳6 CdtProblem
0.00/0.77 
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.77 
↳8 CdtProblem
0.00/0.77 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
0.00/0.77 
↳10 CdtProblem
0.00/0.77 
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))
0.00/0.77 
↳12 BOUNDS(O(1), O(1))
0.00/0.77
0.00/0.77 0.00/0.77
0.00/0.77
0.00/0.77

(0) Obligation:

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

2nd(cons1(X, cons(Y, Z))) → Y 0.00/0.77
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1))) 0.00/0.77
from(X) → cons(X, n__from(n__s(X))) 0.00/0.77
from(X) → n__from(X) 0.00/0.77
s(X) → n__s(X) 0.00/0.77
activate(n__from(X)) → from(activate(X)) 0.00/0.77
activate(n__s(X)) → s(activate(X)) 0.00/0.77
activate(X) → X

Rewrite Strategy: INNERMOST
0.00/0.77
0.00/0.77

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

Converted CpxTRS to CDT
0.00/0.77
0.00/0.77

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1 0.00/0.77
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1))) 0.00/0.77
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1)) 0.00/0.77
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
S tuples:

2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1)) 0.00/0.77
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c1, c5, c6

0.00/0.77
0.00/0.77

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

Removed 3 trailing tuple parts
0.00/0.77
0.00/0.77

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1 0.00/0.77
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1))) 0.00/0.77
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1)) 0.00/0.77
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1)) 0.00/0.77
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c1, c5, c6

0.00/0.77
0.00/0.77

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1))
0.00/0.77
0.00/0.77

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1 0.00/0.77
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1))) 0.00/0.77
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

0.00/0.77
0.00/0.77

(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)) → c5(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.77

POL(ACTIVATE(x1)) = [2]x1    0.00/0.77
POL(c5(x1)) = x1    0.00/0.77
POL(c6(x1)) = x1    0.00/0.77
POL(n__from(x1)) = [1] + x1    0.00/0.77
POL(n__s(x1)) = x1   
0.00/0.77
0.00/0.77

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1 0.00/0.77
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1))) 0.00/0.77
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
K tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

0.00/0.77
0.00/0.77

(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)) → c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 0.00/0.77

POL(ACTIVATE(x1)) = [3]x1    0.00/0.77
POL(c5(x1)) = x1    0.00/0.77
POL(c6(x1)) = x1    0.00/0.77
POL(n__from(x1)) = x1    0.00/0.77
POL(n__s(x1)) = [1] + x1   
0.00/0.77
0.00/0.77

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1 0.00/0.77
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1))) 0.00/0.77
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.77
from(z0) → n__from(z0) 0.00/0.77
s(z0) → n__s(z0) 0.00/0.77
activate(n__from(z0)) → from(activate(z0)) 0.00/0.77
activate(n__s(z0)) → s(activate(z0)) 0.00/0.77
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0)) 0.00/0.77
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

0.00/0.77
0.00/0.77

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

The set S is empty
0.00/0.77
0.00/0.77

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

0.00/0.77
0.00/0.77
0.00/0.78 EOF