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

(0) Obligation:

Runtime Complexity TRS:
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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
plus(s(X), Y) →⁺ s(plus(X, Y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / s(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
2ndspos, 2ndsneg, plus, times

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg
plus < times

(8) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

The following defined symbols remain to be analysed:
plus, 2ndspos, 2ndsneg, times

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg
plus < times

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Induction Base:
plus(gen_s:n__from:0':n__cons4_0(0), gen_s:n__from:0':n__cons4_0(b)) →_R^Ω(1)
gen_s:n__from:0':n__cons4_0(b)

Induction Step:
plus(gen_s:n__from:0':n__cons4_0(+(n7_0, 1)), gen_s:n__from:0':n__cons4_0(b)) →_R^Ω(1)
s(plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b))) →_IH
s(gen_s:n__from:0':n__cons4_0(+(b, c8_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

The following defined symbols remain to be analysed:
times, 2ndspos, 2ndsneg

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

Induction Base:
times(gen_s:n__from:0':n__cons4_0(0), gen_s:n__from:0':n__cons4_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_s:n__from:0':n__cons4_0(+(n724_0, 1)), gen_s:n__from:0':n__cons4_0(b)) →_R^Ω(1)
plus(gen_s:n__from:0':n__cons4_0(b), times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b))) →_IH
plus(gen_s:n__from:0':n__cons4_0(b), gen_s:n__from:0':n__cons4_0(*(c725_0, b))) →_L^{Ω(1 + b)}
gen_s:n__from:0':n__cons4_0(+(b, *(n724_0, b)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

The following defined symbols remain to be analysed:
2ndsneg, 2ndspos

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol 2ndsneg.

(16) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

The following defined symbols remain to be analysed:
2ndspos

They will be analysed ascendingly in the following order:
2ndspos = 2ndsneg

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol 2ndspos.

(18) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

(20) BOUNDS(n^2, INF)

(21) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt ∈ Ω(1 + b·n724₀ + n724₀)

(23) BOUNDS(n^2, INF)

(24) Obligation:

TRS:
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

Types:
from :: s:n__from:0':n__cons → s:n__from:0':n__cons
cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons → s:n__from:0':n__cons
s :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:n__from:0':n__cons → posrecip:negrecip
activate :: s:n__from:0':n__cons → s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons → s:n__from:0':n__cons → rnil:rcons
negrecip :: s:n__from:0':n__cons → posrecip:negrecip
pi :: s:n__from:0':n__cons → rnil:rcons
plus :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
times :: s:n__from:0':n__cons → s:n__from:0':n__cons → s:n__from:0':n__cons
square :: s:n__from:0':n__cons → s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat → s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) ⇔ s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons5_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) → gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

(26) BOUNDS(n^1, INF)