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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, 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), 271 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   minus(X, 0) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
   from(X) -> n__from(X)
   zWquot(X1, X2) -> n__zWquot(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__zWquot(X1, X2)) -> zWquot(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^1, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   sel(0', cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   minus(X, 0') -> 0'
   minus(s(X), s(Y)) -> minus(X, Y)
   quot(0', s(Y)) -> 0'
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
   zWquot(XS, nil) -> nil
   zWquot(nil, XS) -> nil
   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
   from(X) -> n__from(X)
   zWquot(X1, X2) -> n__zWquot(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__zWquot(X1, X2)) -> zWquot(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)))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0') -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) -> n__from(X)
zWquot(X1, X2) -> n__zWquot(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X

Types:
from :: s:0' -> n__from:cons:nil:n__zWquot
cons :: s:0' -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
n__from :: s:0' -> n__from:cons:nil:n__zWquot
s :: s:0' -> s:0'
sel :: s:0' -> n__from:cons:nil:n__zWquot -> s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
minus :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat -> n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat -> s:0'

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sel, activate, minus, quot

They will be analysed ascendingly in the following order:
activate < sel
quot < activate
minus < quot

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

(6)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0') -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) -> n__from(X)
zWquot(X1, X2) -> n__zWquot(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X

Types:
from :: s:0' -> n__from:cons:nil:n__zWquot
cons :: s:0' -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
n__from :: s:0' -> n__from:cons:nil:n__zWquot
s :: s:0' -> s:0'
sel :: s:0' -> n__from:cons:nil:n__zWquot -> s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
minus :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat -> n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) <=> n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) <=> cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
minus, sel, activate, quot

They will be analysed ascendingly in the following order:
activate < sel
quot < activate
minus < quot

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> gen_s:0'4_0(0), rt in Omega(1 + n6_0)

Induction Base:
minus(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
gen_s:0'4_0(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)))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0') -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) -> n__from(X)
zWquot(X1, X2) -> n__zWquot(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X

Types:
from :: s:0' -> n__from:cons:nil:n__zWquot
cons :: s:0' -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
n__from :: s:0' -> n__from:cons:nil:n__zWquot
s :: s:0' -> s:0'
sel :: s:0' -> n__from:cons:nil:n__zWquot -> s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
minus :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat -> n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) <=> n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) <=> cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
minus, sel, activate, quot

They will be analysed ascendingly in the following order:
activate < sel
quot < activate
minus < quot

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
from(X) -> cons(X, n__from(s(X)))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0') -> 0'
minus(s(X), s(Y)) -> minus(X, Y)
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
from(X) -> n__from(X)
zWquot(X1, X2) -> n__zWquot(X1, X2)
activate(n__from(X)) -> from(X)
activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X

Types:
from :: s:0' -> n__from:cons:nil:n__zWquot
cons :: s:0' -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
n__from :: s:0' -> n__from:cons:nil:n__zWquot
s :: s:0' -> s:0'
sel :: s:0' -> n__from:cons:nil:n__zWquot -> s:0'
0' :: s:0'
activate :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
minus :: s:0' -> s:0' -> s:0'
quot :: s:0' -> s:0' -> s:0'
zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
nil :: n__from:cons:nil:n__zWquot
n__zWquot :: n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot -> n__from:cons:nil:n__zWquot
hole_n__from:cons:nil:n__zWquot1_0 :: n__from:cons:nil:n__zWquot
hole_s:0'2_0 :: s:0'
gen_n__from:cons:nil:n__zWquot3_0 :: Nat -> n__from:cons:nil:n__zWquot
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
minus(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> gen_s:0'4_0(0), rt in Omega(1 + n6_0)


Generator Equations:
gen_n__from:cons:nil:n__zWquot3_0(0) <=> n__from(0')
gen_n__from:cons:nil:n__zWquot3_0(+(x, 1)) <=> cons(0', gen_n__from:cons:nil:n__zWquot3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
quot, sel, activate

They will be analysed ascendingly in the following order:
activate < sel
quot < activate