Runtime Complexity (innermost) proof of /tmp/tmpBrUt5Z/Ex1_2_AEL03

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

0 CpxTRS

↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CpxTRS

↳3 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CpxWeightedTrs

↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CpxTypedWeightedTrs

↳7 CompletionProof (UPPER BOUND(ID), 1 ms)

↳8 CpxTypedWeightedCompleteTrs

↳9 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳10 CpxTypedWeightedCompleteTrs

↳11 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)

↳12 CpxRNTS

↳13 InliningProof (UPPER BOUND(ID), 159 ms)

↳14 CpxRNTS

↳15 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CpxRNTS

↳17 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 5 ms)

↳18 CpxRNTS

↳19 IntTrsBoundProof (UPPER BOUND(ID), 213 ms)

↳20 CpxRNTS

↳21 IntTrsBoundProof (UPPER BOUND(ID), 10 ms)

↳22 CpxRNTS

↳23 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳24 CpxRNTS

↳25 IntTrsBoundProof (UPPER BOUND(ID), 104 ms)

↳26 CpxRNTS

↳27 IntTrsBoundProof (UPPER BOUND(ID), 133 ms)

↳28 CpxRNTS

↳29 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳30 CpxRNTS

↳31 IntTrsBoundProof (UPPER BOUND(ID), 145 ms)

↳32 CpxRNTS

↳33 IntTrsBoundProof (UPPER BOUND(ID), 57 ms)

↳34 CpxRNTS

↳35 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳36 CpxRNTS

↳37 IntTrsBoundProof (UPPER BOUND(ID), 131 ms)

↳38 CpxRNTS

↳39 IntTrsBoundProof (UPPER BOUND(ID), 14 ms)

↳40 CpxRNTS

↳41 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳42 CpxRNTS

↳43 IntTrsBoundProof (UPPER BOUND(ID), 93 ms)

↳44 CpxRNTS

↳45 IntTrsBoundProof (UPPER BOUND(ID), 46 ms)

↳46 CpxRNTS

↳47 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳48 CpxRNTS

↳49 IntTrsBoundProof (UPPER BOUND(ID), 121 ms)

↳50 CpxRNTS

↳51 IntTrsBoundProof (UPPER BOUND(ID), 15 ms)

↳52 CpxRNTS

↳53 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳54 CpxRNTS

↳55 IntTrsBoundProof (UPPER BOUND(ID), 100 ms)

↳56 CpxRNTS

↳57 IntTrsBoundProof (UPPER BOUND(ID), 45 ms)

↳58 CpxRNTS

↳59 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳60 CpxRNTS

↳61 IntTrsBoundProof (UPPER BOUND(ID), 122 ms)

↳62 CpxRNTS

↳63 IntTrsBoundProof (UPPER BOUND(ID), 76 ms)

↳64 CpxRNTS

↳65 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳66 CpxRNTS

↳67 IntTrsBoundProof (UPPER BOUND(ID), 4008 ms)

↳68 CpxRNTS

↳69 IntTrsBoundProof (UPPER BOUND(ID), 288 ms)

↳70 CpxRNTS

↳71 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳72 CpxRNTS

↳73 IntTrsBoundProof (UPPER BOUND(ID), 154 ms)

↳74 CpxRNTS

↳75 IntTrsBoundProof (UPPER BOUND(ID), 54 ms)

↳76 CpxRNTS

↳77 FinalProof (⇔, 0 ms)

↳78 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)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 1th argument of 2ndspos: cons, from
The following defined symbols can occur below the 0th argument of from: s, cons, from, activate
The following defined symbols can occur below the 0th argument of s: s, cons, from, activate
The following defined symbols can occur below the 0th argument of cons: s, cons, from, activate
The following defined symbols can occur below the 1th argument of cons: s, cons, from, activate

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
plus(s(X), Y) → s(plus(X, Y))
times(s(X), Y) → plus(Y, times(X, Y))

(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:

2ndspos(0, Z) → rnil
plus(0, Y) → Y
activate(X) → X
times(0, Y) → 0
cons(X1, X2) → n__cons(X1, X2)
activate(n__s(X)) → s(activate(X))
square(X) → times(X, X)
activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
pi(X) → 2ndspos(X, from(0))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
2ndsneg(0, Z) → rnil

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:

2ndspos(0, Z) → rnil [1]
plus(0, Y) → Y [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
cons(X1, X2) → n__cons(X1, X2) [1]
activate(n__s(X)) → s(activate(X)) [1]
square(X) → times(X, X) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
pi(X) → 2ndspos(X, from(0)) [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1]
2ndsneg(0, Z) → rnil [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

2ndspos(0, Z) → rnil [1]
plus(0, Y) → Y [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
cons(X1, X2) → n__cons(X1, X2) [1]
activate(n__s(X)) → s(activate(X)) [1]
square(X) → times(X, X) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
pi(X) → 2ndspos(X, from(0)) [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1]
2ndsneg(0, Z) → rnil [1]

The TRS has the following type information:

2ndspos :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from → rnil
0 :: 0:n__cons:n__s:n__from
rnil :: rnil
plus :: 0:n__cons:n__s:n__from → plus → plus
activate :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
times :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
cons :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
n__cons :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
n__s :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
s :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
square :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
n__from :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
from :: 0:n__cons:n__s:n__from → 0:n__cons:n__s:n__from
pi :: 0:n__cons:n__s:n__from → rnil
2ndsneg :: 0:n__cons:n__s:n__from → a → rnil

Rewrite Strategy: INNERMOST

(7) 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:

2ndspos
plus
times
square
pi
2ndsneg

activate
from
s
cons

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(8) 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:

2ndspos(0, Z) → rnil [1]
plus(0, Y) → Y [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
cons(X1, X2) → n__cons(X1, X2) [1]
activate(n__s(X)) → s(activate(X)) [1]
square(X) → times(X, X) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
pi(X) → 2ndspos(X, from(0)) [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1]
2ndsneg(0, Z) → rnil [1]

The TRS has the following type information:

Rewrite Strategy: INNERMOST

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

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

(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:

2ndspos(0, Z) → rnil [1]
plus(0, Y) → Y [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
cons(X1, X2) → n__cons(X1, X2) [1]
activate(n__s(X)) → s(X) [2]
activate(n__s(n__s(X'))) → s(s(activate(X'))) [2]
activate(n__s(n__from(X''))) → s(from(activate(X''))) [2]
activate(n__s(n__cons(X1', X2'))) → s(cons(activate(X1'), X2')) [2]
square(X) → times(X, X) [1]
activate(n__from(X)) → from(X) [2]
activate(n__from(n__s(X3))) → from(s(activate(X3))) [2]
activate(n__from(n__from(X4))) → from(from(activate(X4))) [2]
activate(n__from(n__cons(X1'', X2''))) → from(cons(activate(X1''), X2'')) [2]
from(X) → cons(X, n__from(n__s(X))) [1]
pi(X) → 2ndspos(X, cons(0, n__from(n__s(0)))) [2]
pi(X) → 2ndspos(X, n__from(0)) [2]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(n__cons(X1, X2)) → cons(X1, X2) [2]
activate(n__cons(n__s(X5), X2)) → cons(s(activate(X5)), X2) [2]
activate(n__cons(n__from(X6), X2)) → cons(from(activate(X6)), X2) [2]
activate(n__cons(n__cons(X11, X21), X2)) → cons(cons(activate(X11), X21), X2) [2]
2ndsneg(0, Z) → rnil [1]

The TRS has the following type information:

Rewrite Strategy: INNERMOST

(11) 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
rnil => 0
const => 0
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ s(s(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ s(from(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ s(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ from(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ from(s(activate(X3))) :|: z = 1 + (1 + X3), X3 >= 0
activate(z) -{ 2 }→ from(from(activate(X4))) :|: z = 1 + (1 + X4), X4 >= 0
activate(z) -{ 2 }→ from(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
cons(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ cons(X, 1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
pi(z) -{ 2 }→ 2ndspos(X, cons(0, 1 + (1 + 0))) :|: X >= 0, z = X
pi(z) -{ 2 }→ 2ndspos(X, 1 + 0) :|: X >= 0, z = X
plus(z, z') -{ 1 }→ Y :|: z' = Y, Y >= 0, z = 0
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
square(z) -{ 1 }→ times(X, X) :|: X >= 0, z = X
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0

(13) InliningProof (UPPER BOUND(ID) transformation)

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

2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0
cons(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ cons(X, 1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(14) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(s(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ s(from(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ s(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ from(s(activate(X3))) :|: z = 1 + (1 + X3), X3 >= 0
activate(z) -{ 2 }→ from(from(activate(X4))) :|: z = 1 + (1 + X4), X4 >= 0
activate(z) -{ 2 }→ from(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(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) -{ 2 }→ 1 + X1 + X2 :|: X >= 0, z = X, X1 >= 0, X2 >= 0, X = X1, 1 + (1 + X) = X2
pi(z) -{ 3 }→ 2ndspos(X, 1 + X1 + X2) :|: X >= 0, z = X, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: X >= 0, z = X, Z >= 0, 1 + 0 = Z, X = 0
plus(z, z') -{ 1 }→ Y :|: z' = Y, Y >= 0, z = 0
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
square(z) -{ 2 }→ 0 :|: X >= 0, z = X, X = Y, Y >= 0, X = 0
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

Found the following analysis order by SCC decomposition:

{ from }
{ 2ndsneg }
{ cons }
{ 2ndspos }
{ s }
{ times }
{ square }
{ plus }
{ activate }
{ pi }

(18) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndsneg}, {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}

(19) 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

(20) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndsneg}, {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: ?, size: O(n¹) [3 + 2·z]

(21) 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: 2

(22) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: 2ndsneg
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(26) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: 2ndsneg
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(28) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: ?, size: O(n¹) [1 + z + z']

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

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: 2ndspos
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(38) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']
2ndspos: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: 2ndspos
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(40) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 3 }→ 2ndspos(z, 1 + X1 + X2) :|: z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']
2ndspos: runtime: O(1) [1], size: O(1) [0]

(41) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(42) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']
2ndspos: runtime: O(1) [1], size: O(1) [0]

(43) 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

(44) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(45) 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

(46) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(47) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(48) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(50) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(52) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(53) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(54) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(55) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(56) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}, {pi}
Previous analysis results are:

(57) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(58) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}
Previous analysis results are:

from: runtime: O(1) [2], size: O(n¹) [3 + 2·z]
2ndsneg: runtime: O(1) [1], size: O(1) [0]
cons: runtime: O(1) [1], size: O(n¹) [1 + z + z']
2ndspos: runtime: O(1) [1], size: O(1) [0]
s: runtime: O(1) [1], size: O(n¹) [1 + z]
times: runtime: O(1) [1], size: O(1) [0]
square: runtime: O(1) [2], size: O(1) [0]

(59) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(60) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}
Previous analysis results are:

(61) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(62) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}
Previous analysis results are:

(63) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(64) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}
Previous analysis results are:

(65) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(66) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}
Previous analysis results are:

(67) 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: ?

(68) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}
Previous analysis results are:

(69) 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: 11 + 37·z

(70) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 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(cons(activate(X1''), X2'')) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 2 }→ cons(s(activate(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(from(activate(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}
Previous analysis results are:

(71) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(72) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 15 + 37·X5 }→ s11 :|: s9 >= 0, s9 <= inf1, s10 >= 0, s10 <= 1 * s9 + 1, s11 >= 0, s11 <= 1 * s10 + 1 * X2 + 1, X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ -57 + 37·z }→ s14 :|: s12 >= 0, s12 <= inf2, s13 >= 0, s13 <= 2 * s12 + 3, s14 >= 0, s14 <= 2 * s13 + 3, z - 2 >= 0
activate(z) -{ 16 + 37·X1'' }→ s17 :|: s15 >= 0, s15 <= inf3, s16 >= 0, s16 <= 1 * s15 + 1 * X2'' + 1, s17 >= 0, s17 <= 2 * s16 + 3, X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 15 + 37·X11 }→ s2 :|: s'' >= 0, s'' <= inf, s1 >= 0, s1 <= 1 * s'' + 1 * X21 + 1, s2 >= 0, s2 <= 1 * s1 + 1 * X2 + 1, z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 16 + 37·X6 }→ s20 :|: s18 >= 0, s18 <= inf4, s19 >= 0, s19 <= 2 * s18 + 3, s20 >= 0, s20 <= 1 * s19 + 1 * X2 + 1, X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ -58 + 37·z }→ s23 :|: s21 >= 0, s21 <= inf5, s22 >= 0, s22 <= 2 * s21 + 3, s23 >= 0, s23 <= 1 * s22 + 1, z - 2 >= 0
activate(z) -{ -58 + 37·z }→ s26 :|: s24 >= 0, s24 <= inf6, s25 >= 0, s25 <= 1 * s24 + 1, s26 >= 0, s26 <= 2 * s25 + 3, z - 2 >= 0
activate(z) -{ -59 + 37·z }→ s5 :|: s3 >= 0, s3 <= inf', s4 >= 0, s4 <= 1 * s3 + 1, s5 >= 0, s5 <= 1 * s4 + 1, z - 2 >= 0
activate(z) -{ 15 + 37·X1' }→ s8 :|: s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= 1 * s6 + 1 * X2' + 1, s8 >= 0, s8 <= 1 * s7 + 1, z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}
Previous analysis results are:

(73) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: pi
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(74) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 15 + 37·X5 }→ s11 :|: s9 >= 0, s9 <= inf1, s10 >= 0, s10 <= 1 * s9 + 1, s11 >= 0, s11 <= 1 * s10 + 1 * X2 + 1, X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ -57 + 37·z }→ s14 :|: s12 >= 0, s12 <= inf2, s13 >= 0, s13 <= 2 * s12 + 3, s14 >= 0, s14 <= 2 * s13 + 3, z - 2 >= 0
activate(z) -{ 16 + 37·X1'' }→ s17 :|: s15 >= 0, s15 <= inf3, s16 >= 0, s16 <= 1 * s15 + 1 * X2'' + 1, s17 >= 0, s17 <= 2 * s16 + 3, X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 15 + 37·X11 }→ s2 :|: s'' >= 0, s'' <= inf, s1 >= 0, s1 <= 1 * s'' + 1 * X21 + 1, s2 >= 0, s2 <= 1 * s1 + 1 * X2 + 1, z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 16 + 37·X6 }→ s20 :|: s18 >= 0, s18 <= inf4, s19 >= 0, s19 <= 2 * s18 + 3, s20 >= 0, s20 <= 1 * s19 + 1 * X2 + 1, X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ -58 + 37·z }→ s23 :|: s21 >= 0, s21 <= inf5, s22 >= 0, s22 <= 2 * s21 + 3, s23 >= 0, s23 <= 1 * s22 + 1, z - 2 >= 0
activate(z) -{ -58 + 37·z }→ s26 :|: s24 >= 0, s24 <= inf6, s25 >= 0, s25 <= 1 * s24 + 1, s26 >= 0, s26 <= 2 * s25 + 3, z - 2 >= 0
activate(z) -{ -59 + 37·z }→ s5 :|: s3 >= 0, s3 <= inf', s4 >= 0, s4 <= 1 * s3 + 1, s5 >= 0, s5 <= 1 * s4 + 1, z - 2 >= 0
activate(z) -{ 15 + 37·X1' }→ s8 :|: s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= 1 * s6 + 1 * X2' + 1, s8 >= 0, s8 <= 1 * s7 + 1, z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}
Previous analysis results are:

(75) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: pi
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 4

(76) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 15 + 37·X5 }→ s11 :|: s9 >= 0, s9 <= inf1, s10 >= 0, s10 <= 1 * s9 + 1, s11 >= 0, s11 <= 1 * s10 + 1 * X2 + 1, X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0
activate(z) -{ -57 + 37·z }→ s14 :|: s12 >= 0, s12 <= inf2, s13 >= 0, s13 <= 2 * s12 + 3, s14 >= 0, s14 <= 2 * s13 + 3, z - 2 >= 0
activate(z) -{ 16 + 37·X1'' }→ s17 :|: s15 >= 0, s15 <= inf3, s16 >= 0, s16 <= 1 * s15 + 1 * X2'' + 1, s17 >= 0, s17 <= 2 * s16 + 3, X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0
activate(z) -{ 15 + 37·X11 }→ s2 :|: s'' >= 0, s'' <= inf, s1 >= 0, s1 <= 1 * s'' + 1 * X21 + 1, s2 >= 0, s2 <= 1 * s1 + 1 * X2 + 1, z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0
activate(z) -{ 16 + 37·X6 }→ s20 :|: s18 >= 0, s18 <= inf4, s19 >= 0, s19 <= 2 * s18 + 3, s20 >= 0, s20 <= 1 * s19 + 1 * X2 + 1, X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0
activate(z) -{ -58 + 37·z }→ s23 :|: s21 >= 0, s21 <= inf5, s22 >= 0, s22 <= 2 * s21 + 3, s23 >= 0, s23 <= 1 * s22 + 1, z - 2 >= 0
activate(z) -{ -58 + 37·z }→ s26 :|: s24 >= 0, s24 <= inf6, s25 >= 0, s25 <= 1 * s24 + 1, s26 >= 0, s26 <= 2 * s25 + 3, z - 2 >= 0
activate(z) -{ -59 + 37·z }→ s5 :|: s3 >= 0, s3 <= inf', s4 >= 0, s4 <= 1 * s3 + 1, s5 >= 0, s5 <= 1 * s4 + 1, z - 2 >= 0
activate(z) -{ 15 + 37·X1' }→ s8 :|: s6 >= 0, s6 <= inf'', s7 >= 0, s7 <= 1 * s6 + 1 * X2' + 1, s8 >= 0, s8 <= 1 * s7 + 1, z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
pi(z) -{ 4 }→ s' :|: s' >= 0, s' <= 0, z >= 0, X1 >= 0, X2 >= 0, 0 = X1, 1 + (1 + 0) = X2
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
Previous analysis results are:

(77) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CompletionProof (UPPER BOUND(ID) transformation)

(8) Obligation:

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

(10) Obligation:

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

(12) Obligation:

(13) InliningProof (UPPER BOUND(ID) transformation)

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(20) Obligation:

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

(22) Obligation:

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

(24) Obligation:

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(26) Obligation:

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

(28) Obligation:

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

(30) Obligation:

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(32) Obligation:

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

(34) Obligation:

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

(36) Obligation:

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(38) Obligation:

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

(40) Obligation:

(41) ResultPropagationProof (UPPER BOUND(ID) transformation)

(42) Obligation:

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(44) Obligation:

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

(46) Obligation:

(47) ResultPropagationProof (UPPER BOUND(ID) transformation)

(48) Obligation:

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(50) Obligation:

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(52) Obligation:

(53) ResultPropagationProof (UPPER BOUND(ID) transformation)

(54) Obligation:

(55) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(56) Obligation:

(57) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(58) Obligation:

(59) ResultPropagationProof (UPPER BOUND(ID) transformation)

(60) Obligation:

(61) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(62) Obligation:

(63) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(64) Obligation:

(65) ResultPropagationProof (UPPER BOUND(ID) transformation)

(66) Obligation:

(67) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(68) Obligation:

(69) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(70) Obligation:

(71) ResultPropagationProof (UPPER BOUND(ID) transformation)

(72) Obligation:

(73) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(74) Obligation:

(75) IntTrsBoundProof (UPPER BOUND(ID) transformation)

(76) Obligation:

(77) FinalProof (EQUIVALENT transformation)

(78) BOUNDS(1, n^1)