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

(0) Obligation:

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

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__from(X)) →⁺ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

The rewrite sequence
activate(n__from(X)) →⁺ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].

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

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

Types:
from :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
cons :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
n__from :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
n__s :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
2ndspos :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → rnil:rcons
0' :: n__s:n__from:0':n__cons
rnil :: rnil:rcons
s :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
n__cons :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
rcons :: posrecip:negrecip → rnil:rcons → rnil:rcons
posrecip :: n__s:n__from:0':n__cons → posrecip:negrecip
activate :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
2ndsneg :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → rnil:rcons
negrecip :: n__s:n__from:0':n__cons → posrecip:negrecip
pi :: n__s:n__from:0':n__cons → rnil:rcons
plus :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
times :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
square :: n__s:n__from:0':n__cons → n__s:n__from:0':n__cons
hole_n__s:n__from:0':n__cons1_0 :: n__s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_n__s:n__from:0':n__cons4_0 :: Nat → n__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, activate, 2ndsneg, plus, times

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

(8) Obligation:

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

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

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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:
activate, 2ndspos, 2ndsneg, plus, times

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

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

Proved the following rewrite lemma:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Induction Base:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, 0)))

Induction Step:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, +(n7_0, 1)))) →_R^Ω(1)
from(activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0)))) →_IH
from(*6_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(n__s(X)))
2ndspos(0', Z) → rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) → rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) → rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) → rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0'))
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0', Y) → 0'
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
cons(X1, X2) → n__cons(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(X) → X

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

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

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

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

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

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

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

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

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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

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

Lemmas:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_n__s:n__from:0':n__cons4_0(0) ⇔ 0'
gen_n__s:n__from:0':n__cons4_0(+(x, 1)) ⇔ n__from(gen_n__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.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__s:n__from:0':n__cons4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

(24) BOUNDS(n^1, INF)