The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

0 CpxTRS
↳1 DependencyGraphProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxWeightedTrs
↳5 InnermostUnusableRulesProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CpxWeightedTrs
↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CpxTypedWeightedTrs
↳9 CompletionProof (UPPER BOUND(ID), 0 ms)
↳10 CpxTypedWeightedCompleteTrs
↳11 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 CpxTypedWeightedCompleteTrs
↳13 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳14 CpxRNTS
↳15 InliningProof (UPPER BOUND(ID), 133 ms)
↳16 CpxRNTS
↳17 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CpxRNTS
↳19 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳20 CpxRNTS
↳21 IntTrsBoundProof (UPPER BOUND(ID), 219 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 46 ms)
↳24 CpxRNTS
↳25 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳26 CpxRNTS
↳27 IntTrsBoundProof (UPPER BOUND(ID), 231 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 54 ms)
↳30 CpxRNTS
↳31 ResultPropagationProof (UPPER BOUND(ID), 2 ms)
↳32 CpxRNTS
↳33 IntTrsBoundProof (UPPER BOUND(ID), 157 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 6 ms)
↳36 CpxRNTS
↳37 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳38 CpxRNTS
↳39 IntTrsBoundProof (UPPER BOUND(ID), 112 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 26 ms)
↳42 CpxRNTS
↳43 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳44 CpxRNTS
↳45 IntTrsBoundProof (UPPER BOUND(ID), 8656 ms)
↳46 CpxRNTS
↳47 IntTrsBoundProof (UPPER BOUND(ID), 1612 ms)
↳48 CpxRNTS
↳49 FinalProof (⇔, 0 ms)
↳50 BOUNDS(1, n^1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

sel(0, cons(X, Z)) → X [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(activate(X)) [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) [1]
first(0, Z) → nil [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__s(X)) → s(activate(X)) [1]

Rewrite Strategy: INNERMOST

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

Removed the following rules with non-basic left-hand side, as they cannot be used in innermost rewriting:

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]

Due to the following rules that have to be used instead:

s(X) → n__s(X) [1]

(6) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

sel(0, cons(X, Z)) → X [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) [1]
first(0, Z) → nil [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__s(X)) → s(activate(X)) [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

sel(0, cons(X, Z)) → X [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) [1]
first(0, Z) → nil [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__s(X)) → s(activate(X)) [1]

The TRS has the following type information:
sel :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
0 :: 0:cons:n__first:n__from:n__s:nil
cons :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
activate :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
nil :: 0:cons:n__first:n__from:n__s:nil
s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil

Rewrite Strategy: INNERMOST

(9) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:


sel

(c) The following functions are completely defined:

activate
from
first
s

Due to the following rules being added:
none

And the following fresh constants: none

(10) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

sel(0, cons(X, Z)) → X [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) [1]
first(0, Z) → nil [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__s(X)) → s(activate(X)) [1]

The TRS has the following type information:
sel :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
0 :: 0:cons:n__first:n__from:n__s:nil
cons :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
activate :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
nil :: 0:cons:n__first:n__from:n__s:nil
s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil

Rewrite Strategy: INNERMOST

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

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(12) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

sel(0, cons(X, Z)) → X [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(n__from(X'))) → from(from(activate(X'))) [2]
activate(n__from(n__first(X1', X2'))) → from(first(activate(X1'), activate(X2'))) [2]
activate(n__from(X)) → from(X) [2]
activate(n__from(n__s(X''))) → from(s(activate(X''))) [2]
from(X) → cons(X, n__from(n__s(X))) [1]
activate(n__first(n__from(X3), n__from(X5))) → first(from(activate(X3)), from(activate(X5))) [3]
activate(n__first(n__from(X3), n__first(X11, X21))) → first(from(activate(X3)), first(activate(X11), activate(X21))) [3]
activate(n__first(n__from(X3), X2)) → first(from(activate(X3)), X2) [3]
activate(n__first(n__from(X3), n__s(X6))) → first(from(activate(X3)), s(activate(X6))) [3]
activate(n__first(n__first(X1'', X2''), n__from(X7))) → first(first(activate(X1''), activate(X2'')), from(activate(X7))) [3]
activate(n__first(n__first(X1'', X2''), n__first(X12, X22))) → first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) [3]
activate(n__first(n__first(X1'', X2''), X2)) → first(first(activate(X1''), activate(X2'')), X2) [3]
activate(n__first(n__first(X1'', X2''), n__s(X8))) → first(first(activate(X1''), activate(X2'')), s(activate(X8))) [3]
activate(n__first(X1, n__from(X9))) → first(X1, from(activate(X9))) [3]
activate(n__first(X1, n__first(X13, X23))) → first(X1, first(activate(X13), activate(X23))) [3]
activate(n__first(X1, X2)) → first(X1, X2) [3]
activate(n__first(X1, n__s(X10))) → first(X1, s(activate(X10))) [3]
activate(n__first(n__s(X4), n__from(X14))) → first(s(activate(X4)), from(activate(X14))) [3]
activate(n__first(n__s(X4), n__first(X15, X24))) → first(s(activate(X4)), first(activate(X15), activate(X24))) [3]
activate(n__first(n__s(X4), X2)) → first(s(activate(X4)), X2) [3]
activate(n__first(n__s(X4), n__s(X16))) → first(s(activate(X4)), s(activate(X16))) [3]
first(0, Z) → nil [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__s(n__from(X17))) → s(from(activate(X17))) [2]
activate(n__s(n__first(X18, X25))) → s(first(activate(X18), activate(X25))) [2]
activate(n__s(X)) → s(X) [2]
activate(n__s(n__s(X19))) → s(s(activate(X19))) [2]

The TRS has the following type information:
sel :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
0 :: 0:cons:n__first:n__from:n__s:nil
cons :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__first :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
activate :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
from :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
n__s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil
nil :: 0:cons:n__first:n__from:n__s:nil
s :: 0:cons:n__first:n__from:n__s:nil → 0:cons:n__first:n__from:n__s:nil

Rewrite Strategy: INNERMOST

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
nil => 1

(14) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ s(s(activate(X19))) :|: X19 >= 0, z = 1 + (1 + X19)
activate(z) -{ 2 }→ s(from(activate(X17))) :|: X17 >= 0, z = 1 + (1 + X17)
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ from(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ from(from(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
first(z, z') -{ 1 }→ 1 :|: Z >= 0, z' = Z, z = 0
first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

(15) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
first(z, z') -{ 1 }→ 1 :|: Z >= 0, z' = Z, z = 0
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(s(activate(X19))) :|: X19 >= 0, z = 1 + (1 + X19)
activate(z) -{ 2 }→ s(from(activate(X17))) :|: X17 >= 0, z = 1 + (1 + X17)
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ from(from(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: Z >= 0, z' = Z, z = 0
first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

(17) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ from }
{ first }
{ sel }
{ s }
{ activate }

(20) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {from}, {first}, {sel}, {s}, {activate}

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 3 + 2·z

(22) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {from}, {first}, {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: ?, size: O(n1) [3 + 2·z]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(24) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {first}, {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {first}, {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z + z'

(28) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {first}, {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: ?, size: O(n1) [1 + z + z']

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(30) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(32) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: sel
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z'

(34) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {sel}, {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: ?, size: O(n1) [z']

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: sel
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(36) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(38) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: s
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z

(40) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {s}, {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']
s: runtime: ?, size: O(n1) [1 + z]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: s
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(42) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']
s: runtime: O(1) [1], size: O(n1) [1 + z]

(43) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(44) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']
s: runtime: O(1) [1], size: O(n1) [1 + z]

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(46) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed: {activate}
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']
s: runtime: O(1) [1], size: O(n1) [1 + z]
activate: runtime: ?, size: EXP

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 108 + 216·z

(48) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(first(activate(X18), activate(X25))) :|: X18 >= 0, z = 1 + (1 + X18 + X25), X25 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(first(activate(X1'), activate(X2'))) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ first(X1, s(activate(X10))) :|: X1 >= 0, X10 >= 0, z = 1 + X1 + (1 + X10)
activate(z) -{ 3 }→ first(X1, from(activate(X9))) :|: X1 >= 0, X9 >= 0, z = 1 + X1 + (1 + X9)
activate(z) -{ 3 }→ first(X1, first(activate(X13), activate(X23))) :|: X1 >= 0, z = 1 + X1 + (1 + X13 + X23), X13 >= 0, X23 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 3 }→ first(s(activate(X4)), s(activate(X16))) :|: X16 >= 0, z = 1 + (1 + X4) + (1 + X16), X4 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), from(activate(X14))) :|: X4 >= 0, z = 1 + (1 + X4) + (1 + X14), X14 >= 0
activate(z) -{ 3 }→ first(s(activate(X4)), first(activate(X15), activate(X24))) :|: z = 1 + (1 + X4) + (1 + X15 + X24), X4 >= 0, X15 >= 0, X24 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 3 }→ first(from(activate(X3)), s(activate(X6))) :|: X6 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X6)
activate(z) -{ 3 }→ first(from(activate(X3)), from(activate(X5))) :|: X5 >= 0, X3 >= 0, z = 1 + (1 + X3) + (1 + X5)
activate(z) -{ 3 }→ first(from(activate(X3)), first(activate(X11), activate(X21))) :|: X11 >= 0, X21 >= 0, z = 1 + (1 + X3) + (1 + X11 + X21), X3 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), s(activate(X8))) :|: X1'' >= 0, X8 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X8), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), from(activate(X7))) :|: X1'' >= 0, X7 >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X7), X2'' >= 0
activate(z) -{ 3 }→ first(first(activate(X1''), activate(X2'')), first(activate(X12), activate(X22))) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + (1 + X12 + X22), X12 >= 0, X22 >= 0, X2'' >= 0
activate(z) -{ 4 }→ 1 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, Z >= 0, X2 = Z, X1 = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 4 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
first(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z

Function symbols to be analyzed:
Previous analysis results are:
from: runtime: O(1) [1], size: O(n1) [3 + 2·z]
first: runtime: O(1) [1], size: O(n1) [1 + z + z']
sel: runtime: O(1) [1], size: O(n1) [z']
s: runtime: O(1) [1], size: O(n1) [1 + z]
activate: runtime: O(n1) [108 + 216·z], size: EXP

(49) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(50) BOUNDS(1, n^1)