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

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

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

(6) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

(9) 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

(10) Obligation:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_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

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_0(x))

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

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 times.

(16) Obligation:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_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:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_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:

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

Types:
from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__from :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
n__s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
2ndspos :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
0' :: n__s:n__from:cons:0':cons2
rnil :: rnil:rcons
s :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
cons2 :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
activate :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
rcons :: rnil:rcons → rnil:rcons
2ndsneg :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → rnil:rcons
pi :: n__s:n__from:cons:0':cons2 → rnil:rcons
plus :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
times :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
square :: n__s:n__from:cons:0':cons2 → n__s:n__from:cons:0':cons2
hole_n__s:n__from:cons:0':cons21_0 :: n__s:n__from:cons:0':cons2
hole_rnil:rcons2_0 :: rnil:rcons
gen_n__s:n__from:cons:0':cons23_0 :: Nat → n__s:n__from:cons:0':cons2
gen_rnil:rcons4_0 :: Nat → rnil:rcons

Generator Equations:
gen_n__s:n__from:cons:0':cons23_0(0) ⇔ 0'
gen_n__s:n__from:cons:0':cons23_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0':cons23_0(x))
gen_rnil:rcons4_0(0) ⇔ rnil
gen_rnil:rcons4_0(+(x, 1)) ⇔ rcons(gen_rnil:rcons4_0(x))

No more defined symbols left to analyse.