/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 80 ms]
        (16) BEST
            (17) proven lower bound
                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
                (19) BOUNDS(n^2, INF)
            (20) 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, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, X)
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   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, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
   2ndsneg(0', Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
   pi(X) -> 2ndspos(X, from(0'))
   plus(0', Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0', Y) -> 0'
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, X)
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
from/0
cons/0
n__from/0
cons2/0
rcons/0
posrecip/0
negrecip/0

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

(4)
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 -> cons(n__from)
   2ndspos(0', Z) -> rnil
   2ndspos(s(N), cons(Z)) -> 2ndspos(s(N), cons2(activate(Z)))
   2ndspos(s(N), cons2(cons(Z))) -> rcons(2ndsneg(N, activate(Z)))
   2ndsneg(0', Z) -> rnil
   2ndsneg(s(N), cons(Z)) -> 2ndsneg(s(N), cons2(activate(Z)))
   2ndsneg(s(N), cons2(cons(Z))) -> rcons(2ndspos(N, activate(Z)))
   pi(X) -> 2ndspos(X, from)
   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 -> n__from
   activate(n__from) -> from
   activate(X) -> X

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

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

(6)
Obligation:
TRS:
Rules:
from -> cons(n__from)
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(Z)) -> 2ndspos(s(N), cons2(activate(Z)))
2ndspos(s(N), cons2(cons(Z))) -> rcons(2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(Z)) -> 2ndsneg(s(N), cons2(activate(Z)))
2ndsneg(s(N), cons2(cons(Z))) -> rcons(2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from)
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 -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s

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

(7) 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

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

(8)
Obligation:
TRS:
Rules:
from -> cons(n__from)
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(Z)) -> 2ndspos(s(N), cons2(activate(Z)))
2ndspos(s(N), cons2(cons(Z))) -> rcons(2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(Z)) -> 2ndsneg(s(N), cons2(activate(Z)))
2ndsneg(s(N), cons2(cons(Z))) -> rcons(2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from)
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 -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons:cons24_0(0) <=> n__from
gen_n__from:cons:cons24_0(+(x, 1)) <=> cons(gen_n__from:cons:cons24_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_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

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s6_0(n8_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n8_0, b)), rt in Omega(1 + n8_0)

Induction Base:
plus(gen_0':s6_0(0), gen_0':s6_0(b)) ->_R^Omega(1)
gen_0':s6_0(b)

Induction Step:
plus(gen_0':s6_0(+(n8_0, 1)), gen_0':s6_0(b)) ->_R^Omega(1)
s(plus(gen_0':s6_0(n8_0), gen_0':s6_0(b))) ->_IH
s(gen_0':s6_0(+(b, c9_0)))

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

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

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons:cons24_0(0) <=> n__from
gen_n__from:cons:cons24_0(+(x, 1)) <=> cons(gen_n__from:cons:cons24_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


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

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

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
from -> cons(n__from)
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(Z)) -> 2ndspos(s(N), cons2(activate(Z)))
2ndspos(s(N), cons2(cons(Z))) -> rcons(2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(Z)) -> 2ndsneg(s(N), cons2(activate(Z)))
2ndsneg(s(N), cons2(cons(Z))) -> rcons(2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from)
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 -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(n8_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n8_0, b)), rt in Omega(1 + n8_0)


Generator Equations:
gen_n__from:cons:cons24_0(0) <=> n__from
gen_n__from:cons:cons24_0(+(x, 1)) <=> cons(gen_n__from:cons:cons24_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


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

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s6_0(n1027_0), gen_0':s6_0(b)) -> gen_0':s6_0(*(n1027_0, b)), rt in Omega(1 + b*n1027_0 + n1027_0)

Induction Base:
times(gen_0':s6_0(0), gen_0':s6_0(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s6_0(+(n1027_0, 1)), gen_0':s6_0(b)) ->_R^Omega(1)
plus(gen_0':s6_0(b), times(gen_0':s6_0(n1027_0), gen_0':s6_0(b))) ->_IH
plus(gen_0':s6_0(b), gen_0':s6_0(*(c1028_0, b))) ->_L^Omega(1 + b)
gen_0':s6_0(+(b, *(n1027_0, b)))

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^2 for the following obligation:

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

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(n8_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n8_0, b)), rt in Omega(1 + n8_0)


Generator Equations:
gen_n__from:cons:cons24_0(0) <=> n__from
gen_n__from:cons:cons24_0(+(x, 1)) <=> cons(gen_n__from:cons:cons24_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


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

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

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

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

(19)
BOUNDS(n^2, INF)

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

(20)
Obligation:
TRS:
Rules:
from -> cons(n__from)
2ndspos(0', Z) -> rnil
2ndspos(s(N), cons(Z)) -> 2ndspos(s(N), cons2(activate(Z)))
2ndspos(s(N), cons2(cons(Z))) -> rcons(2ndsneg(N, activate(Z)))
2ndsneg(0', Z) -> rnil
2ndsneg(s(N), cons(Z)) -> 2ndsneg(s(N), cons2(activate(Z)))
2ndsneg(s(N), cons2(cons(Z))) -> rcons(2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from)
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 -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons:cons2
cons :: n__from:cons:cons2 -> n__from:cons:cons2
n__from :: n__from:cons:cons2
2ndspos :: 0':s -> n__from:cons:cons2 -> rnil:rcons
0' :: 0':s
rnil :: rnil:rcons
s :: 0':s -> 0':s
cons2 :: n__from:cons:cons2 -> n__from:cons:cons2
activate :: n__from:cons:cons2 -> n__from:cons:cons2
rcons :: rnil:rcons -> rnil:rcons
2ndsneg :: 0':s -> n__from:cons:cons2 -> rnil:rcons
pi :: 0':s -> rnil:rcons
plus :: 0':s -> 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
square :: 0':s -> 0':s
hole_n__from:cons:cons21_0 :: n__from:cons:cons2
hole_rnil:rcons2_0 :: rnil:rcons
hole_0':s3_0 :: 0':s
gen_n__from:cons:cons24_0 :: Nat -> n__from:cons:cons2
gen_rnil:rcons5_0 :: Nat -> rnil:rcons
gen_0':s6_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s6_0(n8_0), gen_0':s6_0(b)) -> gen_0':s6_0(+(n8_0, b)), rt in Omega(1 + n8_0)
times(gen_0':s6_0(n1027_0), gen_0':s6_0(b)) -> gen_0':s6_0(*(n1027_0, b)), rt in Omega(1 + b*n1027_0 + n1027_0)


Generator Equations:
gen_n__from:cons:cons24_0(0) <=> n__from
gen_n__from:cons:cons24_0(+(x, 1)) <=> cons(gen_n__from:cons:cons24_0(x))
gen_rnil:rcons5_0(0) <=> rnil
gen_rnil:rcons5_0(+(x, 1)) <=> rcons(gen_rnil:rcons5_0(x))
gen_0':s6_0(0) <=> 0'
gen_0':s6_0(+(x, 1)) <=> s(gen_0':s6_0(x))


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

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