/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 485 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), 111 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 56 ms]
        (18) 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(adx(zeros))
   active(zeros) -> mark(cons(0, zeros))
   active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
   active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
   active(hd(cons(X, Y))) -> mark(X)
   active(tl(cons(X, Y))) -> mark(Y)
   active(adx(X)) -> adx(active(X))
   active(incr(X)) -> incr(active(X))
   active(hd(X)) -> hd(active(X))
   active(tl(X)) -> tl(active(X))
   adx(mark(X)) -> mark(adx(X))
   incr(mark(X)) -> mark(incr(X))
   hd(mark(X)) -> mark(hd(X))
   tl(mark(X)) -> mark(tl(X))
   proper(nats) -> ok(nats)
   proper(adx(X)) -> adx(proper(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(incr(X)) -> incr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(hd(X)) -> hd(proper(X))
   proper(tl(X)) -> tl(proper(X))
   adx(ok(X)) -> ok(adx(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   hd(ok(X)) -> ok(hd(X))
   tl(ok(X)) -> ok(tl(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(adx(zeros))
   active(zeros) -> mark(cons(0', zeros))
   active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
   active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
   active(hd(cons(X, Y))) -> mark(X)
   active(tl(cons(X, Y))) -> mark(Y)
   active(adx(X)) -> adx(active(X))
   active(incr(X)) -> incr(active(X))
   active(hd(X)) -> hd(active(X))
   active(tl(X)) -> tl(active(X))
   adx(mark(X)) -> mark(adx(X))
   incr(mark(X)) -> mark(incr(X))
   hd(mark(X)) -> mark(hd(X))
   tl(mark(X)) -> mark(tl(X))
   proper(nats) -> ok(nats)
   proper(adx(X)) -> adx(proper(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(incr(X)) -> incr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(hd(X)) -> hd(proper(X))
   proper(tl(X)) -> tl(proper(X))
   adx(ok(X)) -> ok(adx(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   incr(ok(X)) -> ok(incr(X))
   s(ok(X)) -> ok(s(X))
   hd(ok(X)) -> ok(hd(X))
   tl(ok(X)) -> ok(tl(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(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok

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

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

They will be analysed ascendingly in the following order:
adx < active
cons < active
s < active
incr < active
hd < active
tl < active
active < top
adx < proper
cons < proper
s < proper
incr < proper
hd < proper
tl < proper
proper < top

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

(6)
Obligation:
TRS:
Rules:
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


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


The following defined symbols remain to be analysed:
adx, active, cons, s, incr, hd, tl, proper, top

They will be analysed ascendingly in the following order:
adx < active
cons < active
s < active
incr < active
hd < active
tl < active
active < top
adx < proper
cons < proper
s < proper
incr < proper
hd < proper
tl < proper
proper < top

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

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

Induction Base:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
mark(adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_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).
----------------------------------------

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


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


The following defined symbols remain to be analysed:
adx, active, cons, s, incr, hd, tl, proper, top

They will be analysed ascendingly in the following order:
adx < active
cons < active
s < active
incr < active
hd < active
tl < active
active < top
adx < proper
cons < proper
s < proper
incr < proper
hd < proper
tl < proper
proper < top

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


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


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n382_0))) -> *4_0, rt in Omega(n382_0)

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

Induction Step:
incr(gen_nats:zeros:mark:0':ok3_0(+(1, +(n382_0, 1)))) ->_R^Omega(1)
mark(incr(gen_nats:zeros:mark:0':ok3_0(+(1, n382_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(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n382_0))) -> *4_0, rt in Omega(n382_0)


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


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

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n843_0))) -> *4_0, rt in Omega(n843_0)

Induction Base:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
hd(gen_nats:zeros:mark:0':ok3_0(+(1, +(n843_0, 1)))) ->_R^Omega(1)
mark(hd(gen_nats:zeros:mark:0':ok3_0(+(1, n843_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(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n382_0))) -> *4_0, rt in Omega(n382_0)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n843_0))) -> *4_0, rt in Omega(n843_0)


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


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1405_0))) -> *4_0, rt in Omega(n1405_0)

Induction Base:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, 0)))

Induction Step:
tl(gen_nats:zeros:mark:0':ok3_0(+(1, +(n1405_0, 1)))) ->_R^Omega(1)
mark(tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1405_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(adx(zeros))
active(zeros) -> mark(cons(0', zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
incr(mark(X)) -> mark(incr(X))
hd(mark(X)) -> mark(hd(X))
tl(mark(X)) -> mark(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(ok(X)) -> ok(incr(X))
s(ok(X)) -> ok(s(X))
hd(ok(X)) -> ok(hd(X))
tl(ok(X)) -> ok(tl(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
nats :: nats:zeros:mark:0':ok
mark :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
adx :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
zeros :: nats:zeros:mark:0':ok
cons :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
0' :: nats:zeros:mark:0':ok
incr :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
s :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
hd :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
tl :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
proper :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
ok :: nats:zeros:mark:0':ok -> nats:zeros:mark:0':ok
top :: nats:zeros:mark:0':ok -> top
hole_nats:zeros:mark:0':ok1_0 :: nats:zeros:mark:0':ok
hole_top2_0 :: top
gen_nats:zeros:mark:0':ok3_0 :: Nat -> nats:zeros:mark:0':ok


Lemmas:
adx(gen_nats:zeros:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
incr(gen_nats:zeros:mark:0':ok3_0(+(1, n382_0))) -> *4_0, rt in Omega(n382_0)
hd(gen_nats:zeros:mark:0':ok3_0(+(1, n843_0))) -> *4_0, rt in Omega(n843_0)
tl(gen_nats:zeros:mark:0':ok3_0(+(1, n1405_0))) -> *4_0, rt in Omega(n1405_0)


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