/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 480 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 71 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
        (20) 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:

   active(nats) -> mark(cons(0, incr(nats)))
   active(pairs) -> mark(cons(0, incr(odds)))
   active(odds) -> mark(incr(pairs))
   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
   active(head(cons(X, XS))) -> mark(X)
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(incr(X)) -> incr(active(X))
   active(s(X)) -> s(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   incr(mark(X)) -> mark(incr(X))
   s(mark(X)) -> mark(s(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(nats) -> ok(nats)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(incr(X)) -> incr(proper(X))
   proper(pairs) -> ok(pairs)
   proper(odds) -> ok(odds)
   proper(s(X)) -> s(proper(X))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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:

   active(nats) -> mark(cons(0', incr(nats)))
   active(pairs) -> mark(cons(0', incr(odds)))
   active(odds) -> mark(incr(pairs))
   active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
   active(head(cons(X, XS))) -> mark(X)
   active(tail(cons(X, XS))) -> mark(XS)
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(incr(X)) -> incr(active(X))
   active(s(X)) -> s(active(X))
   active(head(X)) -> head(active(X))
   active(tail(X)) -> tail(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   incr(mark(X)) -> mark(incr(X))
   s(mark(X)) -> mark(s(X))
   head(mark(X)) -> mark(head(X))
   tail(mark(X)) -> mark(tail(X))
   proper(nats) -> ok(nats)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(incr(X)) -> incr(proper(X))
   proper(pairs) -> ok(pairs)
   proper(odds) -> ok(odds)
   proper(s(X)) -> s(proper(X))
   proper(head(X)) -> head(proper(X))
   proper(tail(X)) -> tail(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   head(ok(X)) -> ok(head(X))
   tail(ok(X)) -> ok(tail(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(4)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, incr, s, head, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
head < active
tail < active
active < top
cons < proper
incr < proper
s < proper
head < proper
tail < proper
proper < top

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

(6)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, incr, s, head, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
head < active
tail < active
active < top
cons < proper
incr < proper
s < proper
head < proper
tail < proper
proper < top

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)), gen_nats:0':mark:pairs:odds:ok3_0(b))

Induction Step:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n5_0, 1))), gen_nats:0':mark:pairs:odds:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b))) ->_IH
mark(*4_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:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, incr, s, head, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
head < active
tail < active
active < top
cons < proper
incr < proper
s < proper
head < proper
tail < proper
proper < top

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
incr, active, s, head, tail, proper, top

They will be analysed ascendingly in the following order:
incr < active
s < active
head < active
tail < active
active < top
incr < proper
s < proper
head < proper
tail < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0))) -> *4_0, rt in Omega(n888_0)

Induction Base:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n888_0, 1)))) ->_R^Omega(1)
mark(incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0))) -> *4_0, rt in Omega(n888_0)


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, head, tail, proper, top

They will be analysed ascendingly in the following order:
s < active
head < active
tail < active
active < top
s < proper
head < proper
tail < proper
proper < top

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1401_0))) -> *4_0, rt in Omega(n1401_0)

Induction Base:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n1401_0, 1)))) ->_R^Omega(1)
mark(s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1401_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0))) -> *4_0, rt in Omega(n888_0)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1401_0))) -> *4_0, rt in Omega(n1401_0)


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
head, active, tail, proper, top

They will be analysed ascendingly in the following order:
head < active
tail < active
active < top
head < proper
tail < proper
proper < top

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2015_0))) -> *4_0, rt in Omega(n2015_0)

Induction Base:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n2015_0, 1)))) ->_R^Omega(1)
mark(head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2015_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0))) -> *4_0, rt in Omega(n888_0)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1401_0))) -> *4_0, rt in Omega(n1401_0)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2015_0))) -> *4_0, rt in Omega(n2015_0)


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
tail, active, proper, top

They will be analysed ascendingly in the following order:
tail < active
active < top
tail < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2730_0))) -> *4_0, rt in Omega(n2730_0)

Induction Base:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, 0)))

Induction Step:
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, +(n2730_0, 1)))) ->_R^Omega(1)
mark(tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2730_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(nats) -> mark(cons(0', incr(nats)))
active(pairs) -> mark(cons(0', incr(odds)))
active(odds) -> mark(incr(pairs))
active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) -> mark(X)
active(tail(cons(X, XS))) -> mark(XS)
active(cons(X1, X2)) -> cons(active(X1), X2)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
s(mark(X)) -> mark(s(X))
head(mark(X)) -> mark(head(X))
tail(mark(X)) -> mark(tail(X))
proper(nats) -> ok(nats)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(pairs) -> ok(pairs)
proper(odds) -> ok(odds)
proper(s(X)) -> s(proper(X))
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
head(ok(X)) -> ok(head(X))
tail(ok(X)) -> ok(tail(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
nats :: nats:0':mark:pairs:odds:ok
mark :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
cons :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
0' :: nats:0':mark:pairs:odds:ok
incr :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
pairs :: nats:0':mark:pairs:odds:ok
odds :: nats:0':mark:pairs:odds:ok
s :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
head :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
tail :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
proper :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
ok :: nats:0':mark:pairs:odds:ok -> nats:0':mark:pairs:odds:ok
top :: nats:0':mark:pairs:odds:ok -> top
hole_nats:0':mark:pairs:odds:ok1_0 :: nats:0':mark:pairs:odds:ok
hole_top2_0 :: top
gen_nats:0':mark:pairs:odds:ok3_0 :: Nat -> nats:0':mark:pairs:odds:ok


Lemmas:
cons(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n5_0)), gen_nats:0':mark:pairs:odds:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n888_0))) -> *4_0, rt in Omega(n888_0)
s(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n1401_0))) -> *4_0, rt in Omega(n1401_0)
head(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2015_0))) -> *4_0, rt in Omega(n2015_0)
tail(gen_nats:0':mark:pairs:odds:ok3_0(+(1, n2730_0))) -> *4_0, rt in Omega(n2730_0)


Generator Equations:
gen_nats:0':mark:pairs:odds:ok3_0(0) <=> nats
gen_nats:0':mark:pairs:odds:ok3_0(+(x, 1)) <=> mark(gen_nats:0':mark:pairs:odds:ok3_0(x))


The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top