/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 302 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

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


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0'))
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0', Y) -> 0'
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
from(X) -> n__from(X)
cons(X1, X2) -> n__cons(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__cons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
2ndspos, 2ndsneg, plus, times

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

----------------------------------------

(6)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0'))
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0', Y) -> 0'
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
from(X) -> n__from(X)
cons(X1, X2) -> n__cons(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__cons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons


Generator Equations:
gen_s:n__from:0':n__cons4_0(0) <=> 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) <=> s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(posrecip(0'), gen_rnil:rcons5_0(x))


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

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

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt in Omega(1 + n7_0)

Induction Base:
plus(gen_s:n__from:0':n__cons4_0(0), gen_s:n__from:0':n__cons4_0(b)) ->_R^Omega(1)
gen_s:n__from:0':n__cons4_0(b)

Induction Step:
plus(gen_s:n__from:0':n__cons4_0(+(n7_0, 1)), gen_s:n__from:0':n__cons4_0(b)) ->_R^Omega(1)
s(plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b))) ->_IH
s(gen_s:n__from:0':n__cons4_0(+(b, c8_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons


Generator Equations:
gen_s:n__from:0':n__cons4_0(0) <=> 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) <=> s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(posrecip(0'), gen_rnil:rcons5_0(x))


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

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

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0'))
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0', Y) -> 0'
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
from(X) -> n__from(X)
cons(X1, X2) -> n__cons(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__cons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons


Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt in Omega(1 + n7_0)


Generator Equations:
gen_s:n__from:0':n__cons4_0(0) <=> 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) <=> s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(posrecip(0'), gen_rnil:rcons5_0(x))


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

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

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt in Omega(1 + b*n724_0 + n724_0)

Induction Base:
times(gen_s:n__from:0':n__cons4_0(0), gen_s:n__from:0':n__cons4_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_s:n__from:0':n__cons4_0(+(n724_0, 1)), gen_s:n__from:0':n__cons4_0(b)) ->_R^Omega(1)
plus(gen_s:n__from:0':n__cons4_0(b), times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b))) ->_IH
plus(gen_s:n__from:0':n__cons4_0(b), gen_s:n__from:0':n__cons4_0(*(c725_0, b))) ->_L^Omega(1 + b)
gen_s:n__from:0':n__cons4_0(+(b, *(n724_0, b)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(14)
Complex Obligation (BEST)

----------------------------------------

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

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

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons


Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt in Omega(1 + n7_0)


Generator Equations:
gen_s:n__from:0':n__cons4_0(0) <=> 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) <=> s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(posrecip(0'), gen_rnil:rcons5_0(x))


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

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

----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^2, INF)

----------------------------------------

(18)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(X, n__cons(Y, Z))) -> rcons(posrecip(activate(Y)), 2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(X, n__cons(Y, Z))) -> rcons(negrecip(activate(Y)), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0'))
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0', Y) -> 0'
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
from(X) -> n__from(X)
cons(X1, X2) -> n__cons(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__cons(X1, X2)) -> cons(X1, X2)
activate(X) -> X

Types:
from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
n__from :: s:n__from:0':n__cons -> s:n__from:0':n__cons
s :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndspos :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
0' :: s:n__from:0':n__cons
rnil :: rnil:rcons
n__cons :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:n__from:0':n__cons -> posrecip:negrecip
activate :: s:n__from:0':n__cons -> s:n__from:0':n__cons
2ndsneg :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> rnil:rcons
negrecip :: s:n__from:0':n__cons -> posrecip:negrecip
pi :: s:n__from:0':n__cons -> rnil:rcons
plus :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
times :: s:n__from:0':n__cons -> s:n__from:0':n__cons -> s:n__from:0':n__cons
square :: s:n__from:0':n__cons -> s:n__from:0':n__cons
hole_s:n__from:0':n__cons1_0 :: s:n__from:0':n__cons
hole_rnil:rcons2_0 :: rnil:rcons
hole_posrecip:negrecip3_0 :: posrecip:negrecip
gen_s:n__from:0':n__cons4_0 :: Nat -> s:n__from:0':n__cons
gen_rnil:rcons5_0 :: Nat -> rnil:rcons


Lemmas:
plus(gen_s:n__from:0':n__cons4_0(n7_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(+(n7_0, b)), rt in Omega(1 + n7_0)
times(gen_s:n__from:0':n__cons4_0(n724_0), gen_s:n__from:0':n__cons4_0(b)) -> gen_s:n__from:0':n__cons4_0(*(n724_0, b)), rt in Omega(1 + b*n724_0 + n724_0)


Generator Equations:
gen_s:n__from:0':n__cons4_0(0) <=> 0'
gen_s:n__from:0':n__cons4_0(+(x, 1)) <=> s(gen_s:n__from:0':n__cons4_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(posrecip(0'), gen_rnil:rcons5_0(x))


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

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