The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

from(X) → cons(X, n__from(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)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

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

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

2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z))) [1]
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z))) [1]

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

cons(X1, X2) → n__cons(X1, X2) [1]

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

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

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

The TRS has the following type information:

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

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
2ndsneg
pi
square
activate

(c) The following functions are completely defined:

times
from
plus
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: from(X) → cons(X, n__from(s(X))) [1] 2ndspos(0, Z) → rnil [1] 2ndsneg(0, Z) → rnil [1] pi(X) → 2ndspos(X, from(0)) [1] plus(0, Y) → Y [1] plus(s(X), Y) → s(plus(X, Y)) [1] times(0, Y) → 0 [1] times(s(X), Y) → plus(Y, times(X, Y)) [1] square(X) → times(X, X) [1] from(X) → n__from(X) [1] cons(X1, X2) → n__cons(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(n__cons(X1, X2)) → cons(X1, X2) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:n__cons cons :: s:0 → n__from:n__cons → n__from:n__cons n__from :: s:0 → n__from:n__cons s :: s:0 → s:0 2ndspos :: s:0 → n__from:n__cons → rnil 0 :: s:0 rnil :: rnil 2ndsneg :: s:0 → a → rnil pi :: s:0 → rnil plus :: s:0 → s:0 → s:0 times :: s:0 → s:0 → s:0 square :: s:0 → s:0 n__cons :: s:0 → n__from:n__cons → n__from:n__cons activate :: n__from:n__cons → n__from:n__cons const :: n__from:n__cons const1 :: a 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: from(X) → cons(X, n__from(s(X))) [1] 2ndspos(0, Z) → rnil [1] 2ndsneg(0, Z) → rnil [1] pi(X) → 2ndspos(X, cons(0, n__from(s(0)))) [2] pi(X) → 2ndspos(X, n__from(0)) [2] plus(0, Y) → Y [1] plus(s(X), Y) → s(plus(X, Y)) [1] times(0, Y) → 0 [1] times(s(0), Y) → plus(Y, 0) [2] times(s(s(X')), Y) → plus(Y, plus(Y, times(X', Y))) [2] square(X) → times(X, X) [1] from(X) → n__from(X) [1] cons(X1, X2) → n__cons(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(n__cons(X1, X2)) → cons(X1, X2) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:n__cons cons :: s:0 → n__from:n__cons → n__from:n__cons n__from :: s:0 → n__from:n__cons s :: s:0 → s:0 2ndspos :: s:0 → n__from:n__cons → rnil 0 :: s:0 rnil :: rnil 2ndsneg :: s:0 → a → rnil pi :: s:0 → rnil plus :: s:0 → s:0 → s:0 times :: s:0 → s:0 → s:0 square :: s:0 → s:0 n__cons :: s:0 → n__from:n__cons → n__from:n__cons activate :: n__from:n__cons → n__from:n__cons const :: n__from:n__cons const1 :: a 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) -{ 1 }→ from(X) :|: z = 1 + X, X >= 0
activate(z) -{ 1 }→ cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
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
plus(z, z') -{ 1 }→ 1 + plus(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
square(z) -{ 1 }→ times(X, X) :|: X >= 0, z = X
times(z, z') -{ 2 }→ plus(Y, plus(Y, times(X', Y))) :|: z' = Y, Y >= 0, X' >= 0, z = 1 + (1 + X')
times(z, z') -{ 2 }→ plus(Y, 0) :|: z' = Y, Y >= 0, z = 1 + 0
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
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

(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 }→ cons(X', 1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
square(z) -{ 1 }→ times(X, X) :|: X >= 0, z = X
times(z, z') -{ 2 }→ plus(Y, plus(Y, times(X', Y))) :|: z' = Y, Y >= 0, X' >= 0, z = 1 + (1 + X')
times(z, z') -{ 2 }→ plus(Y, 0) :|: z' = Y, Y >= 0, z = 1 + 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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 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 } { plus } { activate } { pi } { times } { square }

(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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

(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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndsneg}, {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {cons}, {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {plus}, {activate}, {pi}, {times}, {square}
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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}, {times}, {square}
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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}, {times}, {square}
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: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z + 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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}, {pi}, {times}, {square}
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] plus: runtime: ?, size: O(n1) [z + z']

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

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

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 2 }→ plus(z', 0) :|: z' >= 0, z = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z']

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

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}, {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: ?, size: O(n1) [1 + 2·z]

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

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 8

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z]

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z]

(55) 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

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: ?, size: O(1) [0]

(57) 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

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], 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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0]

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

Computed SIZE bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2·z·z' + 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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0] times: runtime: ?, size: O(n2) [2·z·z' + z']

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

Computed RUNTIME bound using PUBS for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + 4·z + 2·z·z' + z'

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 1 }→ times(z, z) :|: z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ 2 }→ plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0] times: runtime: O(n2) [3 + 4·z + 2·z·z' + z'], size: O(n2) [2·z·z' + z']

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 4 + 5·z + 2·z² }→ s5 :|: s5 >= 0, s5 <= 1 * z + 2 * (z * z), z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ -1 + 4·z + 2·z·z' + -1·z' }→ s4 :|: s2 >= 0, s2 <= 1 * z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= 1 * z' + 1 * s2, s4 >= 0, s4 <= 1 * z' + 1 * s3, z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0] times: runtime: O(n2) [3 + 4·z + 2·z·z' + z'], size: O(n2) [2·z·z' + z']

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

Computed SIZE bound using KoAT for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + 2·z²

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 4 + 5·z + 2·z² }→ s5 :|: s5 >= 0, s5 <= 1 * z + 2 * (z * z), z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ -1 + 4·z + 2·z·z' + -1·z' }→ s4 :|: s2 >= 0, s2 <= 1 * z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= 1 * z' + 1 * s2, s4 >= 0, s4 <= 1 * z' + 1 * s3, z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0] times: runtime: O(n2) [3 + 4·z + 2·z·z' + z'], size: O(n2) [2·z·z' + z'] square: runtime: ?, size: O(n2) [z + 2·z2]

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

Computed RUNTIME bound using KoAT for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 4 + 5·z + 2·z²

(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) -{ 3 }→ 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 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
plus(z, z') -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * z', z' >= 0, z - 1 >= 0
square(z) -{ 4 + 5·z + 2·z² }→ s5 :|: s5 >= 0, s5 <= 1 * z + 2 * (z * z), z >= 0
times(z, z') -{ 3 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * z' + 1 * 0, z' >= 0, z = 1 + 0
times(z, z') -{ -1 + 4·z + 2·z·z' + -1·z' }→ s4 :|: s2 >= 0, s2 <= 1 * z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= 1 * z' + 1 * s2, s4 >= 0, s4 <= 1 * z' + 1 * s3, z' >= 0, z - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
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] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] activate: runtime: O(1) [8], size: O(n1) [1 + 2·z] pi: runtime: O(1) [4], size: O(1) [0] times: runtime: O(n2) [3 + 4·z + 2·z·z' + z'], size: O(n2) [2·z·z' + z'] square: runtime: O(n2) [4 + 5·z + 2·z2], size: O(n2) [z + 2·z2]

(71) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(72) BOUNDS(1, n^2)