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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) 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), 315 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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__length(nil) -> 0
   a__length(cons(X, Y)) -> s(a__length1(Y))
   a__length1(X) -> a__length(X)
   mark(from(X)) -> a__from(mark(X))
   mark(length(X)) -> a__length(X)
   mark(length1(X)) -> a__length1(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(0) -> 0
   a__from(X) -> from(X)
   a__length(X) -> length(X)
   a__length1(X) -> length1(X)

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

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__from(X) -> cons(mark(X), from(s(X)))
   a__length(nil) -> 0'
   a__length(cons(X, Y)) -> s(a__length1(Y))
   a__length1(X) -> a__length(X)
   mark(from(X)) -> a__from(mark(X))
   mark(length(X)) -> a__length(X)
   mark(length1(X)) -> a__length1(X)
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(0') -> 0'
   a__from(X) -> from(X)
   a__length(X) -> length(X)
   a__length1(X) -> length1(X)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
a__from(X) -> cons(mark(X), from(s(X)))
a__length(nil) -> 0'
a__length(cons(X, Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0') -> 0'
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length1(X) -> length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat -> s:from:cons:nil:0':length:length1

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__from, mark, a__length, a__length1

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

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

(6)
Obligation:
Innermost TRS:
Rules:
a__from(X) -> cons(mark(X), from(s(X)))
a__length(nil) -> 0'
a__length(cons(X, Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0') -> 0'
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length1(X) -> length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat -> s:from:cons:nil:0':length:length1


Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) <=> nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)


The following defined symbols remain to be analysed:
a__length1, a__from, mark, a__length

They will be analysed ascendingly in the following order:
a__from = mark
a__length < mark
a__length1 < mark
a__length = a__length1

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) -> gen_s:from:cons:nil:0':length:length12_0(n58_0), rt in Omega(1 + n58_0)

Induction Base:
mark(gen_s:from:cons:nil:0':length:length12_0(0)) ->_R^Omega(1)
nil

Induction Step:
mark(gen_s:from:cons:nil:0':length:length12_0(+(n58_0, 1))) ->_R^Omega(1)
cons(mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)), nil) ->_IH
cons(gen_s:from:cons:nil:0':length:length12_0(c59_0), nil)

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:

Innermost TRS:
Rules:
a__from(X) -> cons(mark(X), from(s(X)))
a__length(nil) -> 0'
a__length(cons(X, Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0') -> 0'
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length1(X) -> length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat -> s:from:cons:nil:0':length:length1


Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) <=> nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)


The following defined symbols remain to be analysed:
mark, a__from

They will be analysed ascendingly in the following order:
a__from = mark

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
a__from(X) -> cons(mark(X), from(s(X)))
a__length(nil) -> 0'
a__length(cons(X, Y)) -> s(a__length1(Y))
a__length1(X) -> a__length(X)
mark(from(X)) -> a__from(mark(X))
mark(length(X)) -> a__length(X)
mark(length1(X)) -> a__length1(X)
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(0') -> 0'
a__from(X) -> from(X)
a__length(X) -> length(X)
a__length1(X) -> length1(X)

Types:
a__from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
cons :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
mark :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
from :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
s :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
a__length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
nil :: s:from:cons:nil:0':length:length1
0' :: s:from:cons:nil:0':length:length1
a__length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
length1 :: s:from:cons:nil:0':length:length1 -> s:from:cons:nil:0':length:length1
hole_s:from:cons:nil:0':length:length11_0 :: s:from:cons:nil:0':length:length1
gen_s:from:cons:nil:0':length:length12_0 :: Nat -> s:from:cons:nil:0':length:length1


Lemmas:
mark(gen_s:from:cons:nil:0':length:length12_0(n58_0)) -> gen_s:from:cons:nil:0':length:length12_0(n58_0), rt in Omega(1 + n58_0)


Generator Equations:
gen_s:from:cons:nil:0':length:length12_0(0) <=> nil
gen_s:from:cons:nil:0':length:length12_0(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length12_0(x), nil)


The following defined symbols remain to be analysed:
a__from

They will be analysed ascendingly in the following order:
a__from = mark