/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^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 13.4 s]
(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__pairNs -> cons(0, incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0, XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(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__pairNs -> cons(0', incr(oddNs))
   a__oddNs -> a__incr(a__pairNs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__take(0', XS) -> nil
   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
   a__zip(nil, XS) -> nil
   a__zip(X, nil) -> nil
   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
   a__tail(cons(X, XS)) -> mark(XS)
   a__repItems(nil) -> nil
   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
   mark(pairNs) -> a__pairNs
   mark(incr(X)) -> a__incr(mark(X))
   mark(oddNs) -> a__oddNs
   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
   mark(tail(X)) -> a__tail(mark(X))
   mark(repItems(X)) -> a__repItems(mark(X))
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   mark(0') -> 0'
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
   a__pairNs -> pairNs
   a__incr(X) -> incr(X)
   a__oddNs -> oddNs
   a__take(X1, X2) -> take(X1, X2)
   a__zip(X1, X2) -> zip(X1, X2)
   a__tail(X) -> tail(X)
   a__repItems(X) -> repItems(X)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
a__pairNs -> cons(0', incr(oddNs))
a__oddNs -> a__incr(a__pairNs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__take(0', XS) -> nil
a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a__zip(nil, XS) -> nil
a__zip(X, nil) -> nil
a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) -> mark(XS)
a__repItems(nil) -> nil
a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) -> a__pairNs
mark(incr(X)) -> a__incr(mark(X))
mark(oddNs) -> a__oddNs
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(repItems(X)) -> a__repItems(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
a__pairNs -> pairNs
a__incr(X) -> incr(X)
a__oddNs -> oddNs
a__take(X1, X2) -> take(X1, X2)
a__zip(X1, X2) -> zip(X1, X2)
a__tail(X) -> tail(X)
a__repItems(X) -> repItems(X)

Types:
a__pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
cons :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
0' :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
s :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
mark :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
nil :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pair :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
hole_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail1_0 :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0 :: Nat -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__oddNs, a__incr, mark, a__tail, a__repItems

They will be analysed ascendingly in the following order:
a__oddNs = a__incr
a__oddNs = mark
a__oddNs = a__tail
a__oddNs = a__repItems
a__incr = mark
a__incr = a__tail
a__incr = a__repItems
mark = a__tail
mark = a__repItems
a__tail = a__repItems

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

(6)
Obligation:
Innermost TRS:
Rules:
a__pairNs -> cons(0', incr(oddNs))
a__oddNs -> a__incr(a__pairNs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__take(0', XS) -> nil
a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a__zip(nil, XS) -> nil
a__zip(X, nil) -> nil
a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) -> mark(XS)
a__repItems(nil) -> nil
a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) -> a__pairNs
mark(incr(X)) -> a__incr(mark(X))
mark(oddNs) -> a__oddNs
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(repItems(X)) -> a__repItems(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
a__pairNs -> pairNs
a__incr(X) -> incr(X)
a__oddNs -> oddNs
a__take(X1, X2) -> take(X1, X2)
a__zip(X1, X2) -> zip(X1, X2)
a__tail(X) -> tail(X)
a__repItems(X) -> repItems(X)

Types:
a__pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
cons :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
0' :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
s :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
mark :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
nil :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pair :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
hole_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail1_0 :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0 :: Nat -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail


Generator Equations:
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(0) <=> 0'
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(+(x, 1)) <=> cons(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(x), 0')


The following defined symbols remain to be analysed:
a__incr, a__oddNs, mark, a__tail, a__repItems

They will be analysed ascendingly in the following order:
a__oddNs = a__incr
a__oddNs = mark
a__oddNs = a__tail
a__oddNs = a__repItems
a__incr = mark
a__incr = a__tail
a__incr = a__repItems
mark = a__tail
mark = a__repItems
a__tail = a__repItems

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(n1256293_0)) -> gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(n1256293_0), rt in Omega(1 + n1256293_0)

Induction Base:
mark(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(+(n1256293_0, 1))) ->_R^Omega(1)
cons(mark(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(n1256293_0)), 0') ->_IH
cons(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(c1256294_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:

Innermost TRS:
Rules:
a__pairNs -> cons(0', incr(oddNs))
a__oddNs -> a__incr(a__pairNs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__take(0', XS) -> nil
a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a__zip(nil, XS) -> nil
a__zip(X, nil) -> nil
a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) -> mark(XS)
a__repItems(nil) -> nil
a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) -> a__pairNs
mark(incr(X)) -> a__incr(mark(X))
mark(oddNs) -> a__oddNs
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(repItems(X)) -> a__repItems(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
a__pairNs -> pairNs
a__incr(X) -> incr(X)
a__oddNs -> oddNs
a__take(X1, X2) -> take(X1, X2)
a__zip(X1, X2) -> zip(X1, X2)
a__tail(X) -> tail(X)
a__repItems(X) -> repItems(X)

Types:
a__pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
cons :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
0' :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
s :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
mark :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
nil :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pair :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
hole_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail1_0 :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0 :: Nat -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail


Generator Equations:
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(0) <=> 0'
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(+(x, 1)) <=> cons(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(x), 0')


The following defined symbols remain to be analysed:
mark, a__oddNs, a__tail, a__repItems

They will be analysed ascendingly in the following order:
a__oddNs = a__incr
a__oddNs = mark
a__oddNs = a__tail
a__oddNs = a__repItems
a__incr = mark
a__incr = a__tail
a__incr = a__repItems
mark = a__tail
mark = a__repItems
a__tail = a__repItems

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
a__pairNs -> cons(0', incr(oddNs))
a__oddNs -> a__incr(a__pairNs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__take(0', XS) -> nil
a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
a__zip(nil, XS) -> nil
a__zip(X, nil) -> nil
a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
a__tail(cons(X, XS)) -> mark(XS)
a__repItems(nil) -> nil
a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
mark(pairNs) -> a__pairNs
mark(incr(X)) -> a__incr(mark(X))
mark(oddNs) -> a__oddNs
mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
mark(tail(X)) -> a__tail(mark(X))
mark(repItems(X)) -> a__repItems(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
a__pairNs -> pairNs
a__incr(X) -> incr(X)
a__oddNs -> oddNs
a__take(X1, X2) -> take(X1, X2)
a__zip(X1, X2) -> zip(X1, X2)
a__tail(X) -> tail(X)
a__repItems(X) -> repItems(X)

Types:
a__pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
cons :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
0' :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__oddNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__incr :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
s :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
mark :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
nil :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
take :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pair :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
zip :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
a__repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
repItems :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
pairNs :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
tail :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
hole_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail1_0 :: 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0 :: Nat -> 0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail


Lemmas:
mark(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(n1256293_0)) -> gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(n1256293_0), rt in Omega(1 + n1256293_0)


Generator Equations:
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(0) <=> 0'
gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(+(x, 1)) <=> cons(gen_0':oddNs:incr:cons:s:nil:take:pair:zip:repItems:pairNs:tail2_0(x), 0')


The following defined symbols remain to be analysed:
a__oddNs, a__incr, a__tail, a__repItems

They will be analysed ascendingly in the following order:
a__oddNs = a__incr
a__oddNs = mark
a__oddNs = a__tail
a__oddNs = a__repItems
a__incr = mark
a__incr = a__tail
a__incr = a__repItems
mark = a__tail
mark = a__repItems
a__tail = a__repItems