The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Rewrite Strategy: FULL

(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:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__from, mark, a__2ndspos, a__2ndsneg, a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

(6) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
mark, a__from, a__2ndspos, a__2ndsneg, a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Induction Base:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(n4_0, 1))) →_R^Ω(1)
cons(mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)), 0') →_IH
cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(c5_0), 0')

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__from, a__2ndspos, a__2ndsneg, a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__2ndspos, a__2ndsneg, a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

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

Could not prove a rewrite lemma for the defined symbol a__2ndspos.

(13) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__2ndsneg, a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

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

Could not prove a rewrite lemma for the defined symbol a__2ndsneg.

(15) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__pi, a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

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

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

(17) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__plus, a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

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

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

(19) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__times, a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(21) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

The following defined symbols remain to be analysed:
a__square

They will be analysed ascendingly in the following order:
a__from = mark
a__from = a__2ndspos
a__from = a__2ndsneg
a__from = a__pi
a__from = a__plus
a__from = a__times
a__from = a__square
mark = a__2ndspos
mark = a__2ndsneg
mark = a__pi
mark = a__plus
mark = a__times
mark = a__square
a__2ndspos = a__2ndsneg
a__2ndspos = a__pi
a__2ndspos = a__plus
a__2ndspos = a__times
a__2ndspos = a__square
a__2ndsneg = a__pi
a__2ndsneg = a__plus
a__2ndsneg = a__times
a__2ndsneg = a__square
a__pi = a__plus
a__pi = a__times
a__pi = a__square
a__plus = a__times
a__plus = a__square
a__times = a__square

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0', Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0'))
a__plus(0', Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0', Y) → 0'
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Types:
a__from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons2 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil

Lemmas:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:cons2:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n4₀)

(28) BOUNDS(n^1, INF)