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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 9289 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__nats -> cons(0, incr(nats))
   a__pairs -> cons(0, incr(odds))
   a__odds -> a__incr(a__pairs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__head(cons(X, XS)) -> mark(X)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(nats) -> a__nats
   mark(pairs) -> a__pairs
   mark(odds) -> a__odds
   mark(incr(X)) -> a__incr(mark(X))
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__nats -> nats
   a__pairs -> pairs
   a__odds -> odds
   a__incr(X) -> incr(X)
   a__head(X) -> head(X)
   a__tail(X) -> tail(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__nats -> cons(0', incr(nats))
   a__pairs -> cons(0', incr(odds))
   a__odds -> a__incr(a__pairs)
   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
   a__head(cons(X, XS)) -> mark(X)
   a__tail(cons(X, XS)) -> mark(XS)
   mark(nats) -> a__nats
   mark(pairs) -> a__pairs
   mark(odds) -> a__odds
   mark(incr(X)) -> a__incr(mark(X))
   mark(head(X)) -> a__head(mark(X))
   mark(tail(X)) -> a__tail(mark(X))
   mark(0') -> 0'
   mark(s(X)) -> s(mark(X))
   mark(nil) -> nil
   mark(cons(X1, X2)) -> cons(mark(X1), X2)
   a__nats -> nats
   a__pairs -> pairs
   a__odds -> odds
   a__incr(X) -> incr(X)
   a__head(X) -> head(X)
   a__tail(X) -> tail(X)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
a__nats -> cons(0', incr(nats))
a__pairs -> cons(0', incr(odds))
a__odds -> a__incr(a__pairs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) -> mark(X)
a__tail(cons(X, XS)) -> mark(XS)
mark(nats) -> a__nats
mark(pairs) -> a__pairs
mark(odds) -> a__odds
mark(incr(X)) -> a__incr(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(tail(X)) -> a__tail(mark(X))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a__nats -> nats
a__pairs -> pairs
a__odds -> odds
a__incr(X) -> incr(X)
a__head(X) -> head(X)
a__tail(X) -> tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
s :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nil :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
hole_0':nats:incr:cons:odds:s:pairs:head:tail:nil1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0 :: Nat -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil

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

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

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = a__tail

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

(6)
Obligation:
Innermost TRS:
Rules:
a__nats -> cons(0', incr(nats))
a__pairs -> cons(0', incr(odds))
a__odds -> a__incr(a__pairs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) -> mark(X)
a__tail(cons(X, XS)) -> mark(XS)
mark(nats) -> a__nats
mark(pairs) -> a__pairs
mark(odds) -> a__odds
mark(incr(X)) -> a__incr(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(tail(X)) -> a__tail(mark(X))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a__nats -> nats
a__pairs -> pairs
a__odds -> odds
a__incr(X) -> incr(X)
a__head(X) -> head(X)
a__tail(X) -> tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
s :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nil :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
hole_0':nats:incr:cons:odds:s:pairs:head:tail:nil1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0 :: Nat -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil


Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(0) <=> 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(+(x, 1)) <=> cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(x), 0')


The following defined symbols remain to be analysed:
a__incr, a__odds, mark, a__head, a__tail

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = a__tail

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(n252277_0)) -> gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(n252277_0), rt in Omega(1 + n252277_0)

Induction Base:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(+(n252277_0, 1))) ->_R^Omega(1)
cons(mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(n252277_0)), 0') ->_IH
cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(c252278_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__nats -> cons(0', incr(nats))
a__pairs -> cons(0', incr(odds))
a__odds -> a__incr(a__pairs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) -> mark(X)
a__tail(cons(X, XS)) -> mark(XS)
mark(nats) -> a__nats
mark(pairs) -> a__pairs
mark(odds) -> a__odds
mark(incr(X)) -> a__incr(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(tail(X)) -> a__tail(mark(X))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a__nats -> nats
a__pairs -> pairs
a__odds -> odds
a__incr(X) -> incr(X)
a__head(X) -> head(X)
a__tail(X) -> tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
s :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nil :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
hole_0':nats:incr:cons:odds:s:pairs:head:tail:nil1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0 :: Nat -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil


Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(0) <=> 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(+(x, 1)) <=> cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(x), 0')


The following defined symbols remain to be analysed:
mark, a__odds, a__head, a__tail

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = a__tail

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
a__nats -> cons(0', incr(nats))
a__pairs -> cons(0', incr(odds))
a__odds -> a__incr(a__pairs)
a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
a__head(cons(X, XS)) -> mark(X)
a__tail(cons(X, XS)) -> mark(XS)
mark(nats) -> a__nats
mark(pairs) -> a__pairs
mark(odds) -> a__odds
mark(incr(X)) -> a__incr(mark(X))
mark(head(X)) -> a__head(mark(X))
mark(tail(X)) -> a__tail(mark(X))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(mark(X1), X2)
a__nats -> nats
a__pairs -> pairs
a__odds -> odds
a__incr(X) -> incr(X)
a__head(X) -> head(X)
a__tail(X) -> tail(X)

Types:
a__nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
cons :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
0' :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nats :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__odds :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__incr :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
s :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
mark :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
a__tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
pairs :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
head :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
tail :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil
nil :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
hole_0':nats:incr:cons:odds:s:pairs:head:tail:nil1_0 :: 0':nats:incr:cons:odds:s:pairs:head:tail:nil
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0 :: Nat -> 0':nats:incr:cons:odds:s:pairs:head:tail:nil


Lemmas:
mark(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(n252277_0)) -> gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(n252277_0), rt in Omega(1 + n252277_0)


Generator Equations:
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(0) <=> 0'
gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(+(x, 1)) <=> cons(gen_0':nats:incr:cons:odds:s:pairs:head:tail:nil2_0(x), 0')


The following defined symbols remain to be analysed:
a__odds, a__incr, a__head, a__tail

They will be analysed ascendingly in the following order:
a__odds = a__incr
a__odds = mark
a__odds = a__head
a__odds = a__tail
a__incr = mark
a__incr = a__head
a__incr = a__tail
mark = a__head
mark = a__tail
a__head = a__tail