(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, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(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(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) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(from(X)) →+ cons(mark(mark(X)), from(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X / from(X)].
The result substitution is [ ].
The rewrite sequence
mark(from(X)) →+ cons(mark(mark(X)), from(s(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / from(X)].
The result substitution is [ ].
(2) BOUNDS(2^n, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(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, 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(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
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0', Z) → rnil
a__2ndspos(s(N), cons(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, 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(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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
(7) 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__squareThey 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
(8) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Generator Equations:
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
mark(
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(
n4_0)) →
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(
n4_0), rt ∈ Ω(1 + n4
0)
Induction Base:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0)) →RΩ(1)
0'
Induction Step:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(n4_0, 1))) →RΩ(1)
cons(mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)), 0') →IH
cons(gen_s:from:cons:0':rnil: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 ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__from.
(13) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__2ndspos.
(15) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__2ndsneg.
(17) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__pi.
(19) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__plus.
(21) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__times.
(23) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil: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
(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__square.
(25) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')
No more defined symbols left to analyse.
(26) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n40)
(27) BOUNDS(n^1, INF)
(28) Obligation:
TRS:
Rules:
a__from(
X) →
cons(
mark(
X),
from(
s(
X)))
a__2ndspos(
0',
Z) →
rnila__2ndspos(
s(
N),
cons(
X,
cons(
Y,
Z))) →
rcons(
posrecip(
mark(
Y)),
a__2ndsneg(
mark(
N),
mark(
Z)))
a__2ndsneg(
0',
Z) →
rnila__2ndsneg(
s(
N),
cons(
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) →
nilmark(
cons(
X1,
X2)) →
cons(
mark(
X1),
X2)
mark(
rnil) →
rnilmark(
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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
cons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
mark :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
from :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
s :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
0' :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rnil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
rcons :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
posrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
negrecip :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
a__square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndspos :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
2ndsneg :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
pi :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
plus :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
times :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
square :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
nil :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
hole_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil1_0 :: s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0 :: Nat → s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil
Lemmas:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil: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:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(0) ⇔ 0'
gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(+(x, 1)) ⇔ cons(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(x), 0')
No more defined symbols left to analyse.
(29) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0)) → gen_s:from:cons:0':rnil:posrecip:rcons:negrecip:2ndspos:2ndsneg:pi:plus:times:square:nil2_0(n4_0), rt ∈ Ω(1 + n40)
(30) BOUNDS(n^1, INF)