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

Infered types.

(4) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

(5) 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

(6) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_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

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(8) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_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:0'6_0(n60_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n60_0, b)), rt ∈ Ω(1 + n60₀)

Induction Base:
plus(gen_s:0'6_0(0), gen_s:0'6_0(b)) →_R^Ω(1)
gen_s:0'6_0(b)

Induction Step:
plus(gen_s:0'6_0(+(n60_0, 1)), gen_s:0'6_0(b)) →_R^Ω(1)
s(plus(gen_s:0'6_0(n60_0), gen_s:0'6_0(b))) →_IH
s(gen_s:0'6_0(+(b, c61_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:0'6_0(n60_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n60_0, b)), rt ∈ Ω(1 + n60₀)

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_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:0'6_0(n975_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n975_0, b)), rt ∈ Ω(1 + b·n975₀ + n975₀)

Induction Base:
times(gen_s:0'6_0(0), gen_s:0'6_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_s:0'6_0(+(n975_0, 1)), gen_s:0'6_0(b)) →_R^Ω(1)
plus(gen_s:0'6_0(b), times(gen_s:0'6_0(n975_0), gen_s:0'6_0(b))) →_IH
plus(gen_s:0'6_0(b), gen_s:0'6_0(*(c976_0, b))) →_L^{Ω(1 + b)}
gen_s:0'6_0(+(b, *(n975_0, b)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:0'6_0(n60_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n60_0, b)), rt ∈ Ω(1 + n60₀)
times(gen_s:0'6_0(n975_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n975_0, b)), rt ∈ Ω(1 + b·n975₀ + n975₀)

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_0(x))

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

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

(15) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:0'6_0(n60_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n60_0, b)), rt ∈ Ω(1 + n60₀)
times(gen_s:0'6_0(n975_0), gen_s:0'6_0(b)) → gen_s:0'6_0(*(n975_0, b)), rt ∈ Ω(1 + b·n975₀ + n975₀)

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_0(x))

No more defined symbols left to analyse.

(16) Obligation:

Innermost TRS:
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)

Types:
from :: s:0' → cons:cons2
cons :: s:0' → cons:cons2 → cons:cons2
s :: s:0' → s:0'
2ndspos :: s:0' → cons:cons2 → rnil:rcons
0' :: s:0'
rnil :: rnil:rcons
cons2 :: s:0' → cons:cons2 → cons:cons2
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: s:0' → posrecip:negrecip
2ndsneg :: s:0' → cons:cons2 → rnil:rcons
negrecip :: s:0' → posrecip:negrecip
pi :: s:0' → rnil:rcons
plus :: s:0' → s:0' → s:0'
times :: s:0' → s:0' → s:0'
square :: s:0' → s:0'
hole_cons:cons21_0 :: cons:cons2
hole_s:0'2_0 :: s:0'
hole_rnil:rcons3_0 :: rnil:rcons
hole_posrecip:negrecip4_0 :: posrecip:negrecip
gen_cons:cons25_0 :: Nat → cons:cons2
gen_s:0'6_0 :: Nat → s:0'
gen_rnil:rcons7_0 :: Nat → rnil:rcons

Lemmas:
plus(gen_s:0'6_0(n60_0), gen_s:0'6_0(b)) → gen_s:0'6_0(+(n60_0, b)), rt ∈ Ω(1 + n60₀)

Generator Equations:
gen_cons:cons25_0(0) ⇔ hole_cons:cons21_0
gen_cons:cons25_0(+(x, 1)) ⇔ cons(0', gen_cons:cons25_0(x))
gen_s:0'6_0(0) ⇔ 0'
gen_s:0'6_0(+(x, 1)) ⇔ s(gen_s:0'6_0(x))
gen_rnil:rcons7_0(0) ⇔ rnil
gen_rnil:rcons7_0(+(x, 1)) ⇔ rcons(posrecip(0'), gen_rnil:rcons7_0(x))

No more defined symbols left to analyse.