The Runtime Complexity (innermost) 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, from(s(X)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, Z))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, 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)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

from(X) → cons(X, from(s(X)))
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, Z))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, Z))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, 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)

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
from/0
cons/0
cons2/0
rcons/0
posrecip/0
negrecip/0

(4) Obligation:

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

from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

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

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

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

(8) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

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

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

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol from.

(10) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_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

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)

Induction Base:
plus(gen_0':s6_0(0), gen_0':s6_0(b)) →_R^Ω(1)
gen_0':s6_0(b)

Induction Step:
plus(gen_0':s6_0(+(n11_0, 1)), gen_0':s6_0(b)) →_R^Ω(1)
s(plus(gen_0':s6_0(n11_0), gen_0':s6_0(b))) →_IH
s(gen_0':s6_0(+(b, c12_0)))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

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

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

Induction Base:
times(gen_0':s6_0(0), gen_0':s6_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s6_0(+(n1026_0, 1)), gen_0':s6_0(b)) →_R^Ω(1)
plus(gen_0':s6_0(b), times(gen_0':s6_0(n1026_0), gen_0':s6_0(b))) →_IH
plus(gen_0':s6_0(b), gen_0':s6_0(*(c1027_0, b))) →_L^{Ω(1 + b)}
gen_0':s6_0(+(b, *(n1026_0, b)))

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

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

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

(18) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

The following defined symbols remain to be analysed:
2ndspos

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

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s6_0(n1026_0), gen_0':s6_0(b)) → gen_0':s6_0(*(n1026_0, b)), rt ∈ Ω(1 + b·n1026₀ + n1026₀)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
Rules:
from → cons(from)
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(Z)) → 2ndspos(s(N), cons2(Z))
2ndspos(s(N), cons2(cons(Z))) → rcons(2ndsneg(N, Z))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(Z)) → 2ndsneg(s(N), cons2(Z))
2ndsneg(s(N), cons2(cons(Z))) → rcons(2ndspos(N, Z))
pi(X) → 2ndspos(X, from)
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)

Types:
from :: cons:cons2
cons :: cons:cons2 → cons:cons2
2ndspos :: 0':s → cons:cons2 → rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s → 0':s
cons2 :: cons:cons2 → cons:cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: 0':s → cons:cons2 → rnil:rcons
pi :: 0':s → rnil:rcons
plus :: 0':s → 0':s → 0':s
times :: 0':s → 0':s → 0':s
square :: 0':s → 0':s
hole_cons:cons21_0 :: cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_cons:cons24_0 :: Nat → cons:cons2
gen_rnil:rcons5_0 :: Nat → rnil:rcons
gen_0':s6_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)

Generator Equations:
gen_cons:cons24_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons24_0(+(x, 1)) ⇔ cons(gen_cons:cons24_0(x))
gen_rnil:rcons5_0(0) ⇔ rnil
gen_rnil:rcons5_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) ⇔ 0'
gen_0':s6_0(+(x, 1)) ⇔ s(gen_0':s6_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s6_0(n11_0), gen_0':s6_0(b)) → gen_0':s6_0(+(n11_0, b)), rt ∈ Ω(1 + n11₀)

(28) BOUNDS(n^1, INF)