/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 12.0 s]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) 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__fst(0, Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(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__fst(0', Z) -> nil
   a__fst(s(X), cons(Y, Z)) -> cons(mark(Y), fst(X, Z))
   a__from(X) -> cons(mark(X), from(s(X)))
   a__add(0', X) -> mark(X)
   a__add(s(X), Y) -> s(add(X, Y))
   a__len(nil) -> 0'
   a__len(cons(X, Z)) -> s(len(Z))
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0') -> 0'
   mark(s(X)) -> s(X)
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
s/0
cons/1

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

(4)
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__fst(0', Z) -> nil
   a__fst(s, cons(Y)) -> cons(mark(Y))
   a__from(X) -> cons(mark(X))
   a__add(0', X) -> mark(X)
   a__add(s, Y) -> s
   a__len(nil) -> 0'
   a__len(cons(X)) -> s
   mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
   mark(from(X)) -> a__from(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(len(X)) -> a__len(mark(X))
   mark(0') -> 0'
   mark(s) -> s
   mark(nil) -> nil
   mark(cons(X1)) -> cons(mark(X1))
   a__fst(X1, X2) -> fst(X1, X2)
   a__from(X) -> from(X)
   a__add(X1, X2) -> add(X1, X2)
   a__len(X) -> len(X)

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

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

(6)
Obligation:
Innermost TRS:
Rules:
a__fst(0', Z) -> nil
a__fst(s, cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
a__add(0', X) -> mark(X)
a__add(s, Y) -> s
a__len(nil) -> 0'
a__len(cons(X)) -> s
mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(X1)) -> cons(mark(X1))
a__fst(X1, X2) -> fst(X1, X2)
a__from(X) -> from(X)
a__add(X1, X2) -> add(X1, X2)
a__len(X) -> len(X)

Types:
a__fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
0' :: 0':nil:s:cons:fst:from:add:len
nil :: 0':nil:s:cons:fst:from:add:len
s :: 0':nil:s:cons:fst:from:add:len
cons :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
mark :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
hole_0':nil:s:cons:fst:from:add:len1_0 :: 0':nil:s:cons:fst:from:add:len
gen_0':nil:s:cons:fst:from:add:len2_0 :: Nat -> 0':nil:s:cons:fst:from:add:len

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

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

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

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

(8)
Obligation:
Innermost TRS:
Rules:
a__fst(0', Z) -> nil
a__fst(s, cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
a__add(0', X) -> mark(X)
a__add(s, Y) -> s
a__len(nil) -> 0'
a__len(cons(X)) -> s
mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(X1)) -> cons(mark(X1))
a__fst(X1, X2) -> fst(X1, X2)
a__from(X) -> from(X)
a__add(X1, X2) -> add(X1, X2)
a__len(X) -> len(X)

Types:
a__fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
0' :: 0':nil:s:cons:fst:from:add:len
nil :: 0':nil:s:cons:fst:from:add:len
s :: 0':nil:s:cons:fst:from:add:len
cons :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
mark :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
hole_0':nil:s:cons:fst:from:add:len1_0 :: 0':nil:s:cons:fst:from:add:len
gen_0':nil:s:cons:fst:from:add:len2_0 :: Nat -> 0':nil:s:cons:fst:from:add:len


Generator Equations:
gen_0':nil:s:cons:fst:from:add:len2_0(0) <=> 0'
gen_0':nil:s:cons:fst:from:add:len2_0(+(x, 1)) <=> cons(gen_0':nil:s:cons:fst:from:add:len2_0(x))


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

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':nil:s:cons:fst:from:add:len2_0(n27352_0)) -> gen_0':nil:s:cons:fst:from:add:len2_0(n27352_0), rt in Omega(1 + n27352_0)

Induction Base:
mark(gen_0':nil:s:cons:fst:from:add:len2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':nil:s:cons:fst:from:add:len2_0(+(n27352_0, 1))) ->_R^Omega(1)
cons(mark(gen_0':nil:s:cons:fst:from:add:len2_0(n27352_0))) ->_IH
cons(gen_0':nil:s:cons:fst:from:add:len2_0(c27353_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:

Innermost TRS:
Rules:
a__fst(0', Z) -> nil
a__fst(s, cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
a__add(0', X) -> mark(X)
a__add(s, Y) -> s
a__len(nil) -> 0'
a__len(cons(X)) -> s
mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(X1)) -> cons(mark(X1))
a__fst(X1, X2) -> fst(X1, X2)
a__from(X) -> from(X)
a__add(X1, X2) -> add(X1, X2)
a__len(X) -> len(X)

Types:
a__fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
0' :: 0':nil:s:cons:fst:from:add:len
nil :: 0':nil:s:cons:fst:from:add:len
s :: 0':nil:s:cons:fst:from:add:len
cons :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
mark :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
hole_0':nil:s:cons:fst:from:add:len1_0 :: 0':nil:s:cons:fst:from:add:len
gen_0':nil:s:cons:fst:from:add:len2_0 :: Nat -> 0':nil:s:cons:fst:from:add:len


Generator Equations:
gen_0':nil:s:cons:fst:from:add:len2_0(0) <=> 0'
gen_0':nil:s:cons:fst:from:add:len2_0(+(x, 1)) <=> cons(gen_0':nil:s:cons:fst:from:add:len2_0(x))


The following defined symbols remain to be analysed:
mark

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

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Innermost TRS:
Rules:
a__fst(0', Z) -> nil
a__fst(s, cons(Y)) -> cons(mark(Y))
a__from(X) -> cons(mark(X))
a__add(0', X) -> mark(X)
a__add(s, Y) -> s
a__len(nil) -> 0'
a__len(cons(X)) -> s
mark(fst(X1, X2)) -> a__fst(mark(X1), mark(X2))
mark(from(X)) -> a__from(mark(X))
mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
mark(len(X)) -> a__len(mark(X))
mark(0') -> 0'
mark(s) -> s
mark(nil) -> nil
mark(cons(X1)) -> cons(mark(X1))
a__fst(X1, X2) -> fst(X1, X2)
a__from(X) -> from(X)
a__add(X1, X2) -> add(X1, X2)
a__len(X) -> len(X)

Types:
a__fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
0' :: 0':nil:s:cons:fst:from:add:len
nil :: 0':nil:s:cons:fst:from:add:len
s :: 0':nil:s:cons:fst:from:add:len
cons :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
mark :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
a__len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
fst :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
from :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
add :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
len :: 0':nil:s:cons:fst:from:add:len -> 0':nil:s:cons:fst:from:add:len
hole_0':nil:s:cons:fst:from:add:len1_0 :: 0':nil:s:cons:fst:from:add:len
gen_0':nil:s:cons:fst:from:add:len2_0 :: Nat -> 0':nil:s:cons:fst:from:add:len


Lemmas:
mark(gen_0':nil:s:cons:fst:from:add:len2_0(n27352_0)) -> gen_0':nil:s:cons:fst:from:add:len2_0(n27352_0), rt in Omega(1 + n27352_0)


Generator Equations:
gen_0':nil:s:cons:fst:from:add:len2_0(0) <=> 0'
gen_0':nil:s:cons:fst:from:add:len2_0(+(x, 1)) <=> cons(gen_0':nil:s:cons:fst:from:add:len2_0(x))


The following defined symbols remain to be analysed:
a__from

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