/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), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 618 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 485 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 101 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (26) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of cons: active, proper, cons
The following defined symbols can occur below the 1th argument of cons: active, proper, cons
The following defined symbols can occur below the 0th argument of top: active, proper, cons
The following defined symbols can occur below the 0th argument of proper: active, proper, cons
The following defined symbols can occur below the 0th argument of active: active, proper, cons

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0)
active(length(cons(N, L))) -> mark(U11(tt, L))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(tt) -> ok(tt)
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 5. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8]
transitions: 
zeros0() -> 0
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
tt0() -> 0
nil0() -> 0
active0(0) -> 1
cons0(0, 0) -> 2
U110(0, 0) -> 3
U120(0, 0) -> 4
s0(0) -> 5
length0(0) -> 6
proper0(0) -> 7
top0(0) -> 8
01() -> 10
zeros1() -> 11
cons1(10, 11) -> 9
mark1(9) -> 1
cons1(0, 0) -> 12
mark1(12) -> 2
U111(0, 0) -> 13
mark1(13) -> 3
U121(0, 0) -> 14
mark1(14) -> 4
s1(0) -> 15
mark1(15) -> 5
length1(0) -> 16
mark1(16) -> 6
zeros1() -> 17
ok1(17) -> 7
01() -> 18
ok1(18) -> 7
tt1() -> 19
ok1(19) -> 7
nil1() -> 20
ok1(20) -> 7
cons1(0, 0) -> 21
ok1(21) -> 2
U111(0, 0) -> 22
ok1(22) -> 3
U121(0, 0) -> 23
ok1(23) -> 4
s1(0) -> 24
ok1(24) -> 5
length1(0) -> 25
ok1(25) -> 6
proper1(0) -> 26
top1(26) -> 8
active1(0) -> 27
top1(27) -> 8
mark1(9) -> 27
mark1(12) -> 12
mark1(12) -> 21
mark1(13) -> 13
mark1(13) -> 22
mark1(14) -> 14
mark1(14) -> 23
mark1(15) -> 15
mark1(15) -> 24
mark1(16) -> 16
mark1(16) -> 25
ok1(17) -> 26
ok1(18) -> 26
ok1(19) -> 26
ok1(20) -> 26
ok1(21) -> 12
ok1(21) -> 21
ok1(22) -> 13
ok1(22) -> 22
ok1(23) -> 14
ok1(23) -> 23
ok1(24) -> 15
ok1(24) -> 24
ok1(25) -> 16
ok1(25) -> 25
proper2(9) -> 28
top2(28) -> 8
active2(17) -> 29
top2(29) -> 8
active2(18) -> 29
active2(19) -> 29
active2(20) -> 29
02() -> 31
zeros2() -> 32
cons2(31, 32) -> 30
mark2(30) -> 29
proper2(10) -> 33
proper2(11) -> 34
cons2(33, 34) -> 28
zeros2() -> 35
ok2(35) -> 34
02() -> 36
ok2(36) -> 33
proper3(30) -> 37
top3(37) -> 8
proper3(31) -> 38
proper3(32) -> 39
cons3(38, 39) -> 37
cons3(36, 35) -> 40
ok3(40) -> 28
zeros3() -> 41
ok3(41) -> 39
03() -> 42
ok3(42) -> 38
active3(40) -> 43
top3(43) -> 8
cons4(42, 41) -> 44
ok4(44) -> 37
active4(36) -> 45
cons4(45, 35) -> 43
active4(44) -> 46
top4(46) -> 8
active5(42) -> 47
cons5(47, 41) -> 46

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

(6)
BOUNDS(1, n^1)

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

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
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(zeros) -> mark(cons(0', zeros))
   active(U11(tt, L)) -> mark(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(length(nil)) -> mark(0')
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(10)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, U12, s, length, U11, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U11 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U11 < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, U12, s, length, U11, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U11 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U11 < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil: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).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, U12, s, length, U11, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U11 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U11 < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
U12, active, s, length, U11, proper, top

They will be analysed ascendingly in the following order:
U12 < active
s < active
length < active
U11 < active
active < top
U12 < proper
s < proper
length < proper
U11 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n874_0)

Induction Base:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n874_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil: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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n874_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, length, U11, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
U11 < active
active < top
s < proper
length < proper
U11 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2049_0))) -> *4_0, rt in Omega(n2049_0)

Induction Base:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2049_0, 1)))) ->_R^Omega(1)
mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2049_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).
----------------------------------------

(22)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n874_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2049_0))) -> *4_0, rt in Omega(n2049_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
length, active, U11, proper, top

They will be analysed ascendingly in the following order:
length < active
U11 < active
active < top
length < proper
U11 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2709_0))) -> *4_0, rt in Omega(n2709_0)

Induction Base:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2709_0, 1)))) ->_R^Omega(1)
mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2709_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).
----------------------------------------

(24)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n874_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2049_0))) -> *4_0, rt in Omega(n2049_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2709_0))) -> *4_0, rt in Omega(n2709_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
U11, active, proper, top

They will be analysed ascendingly in the following order:
U11 < active
active < top
U11 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3470_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3470_0)

Induction Base:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3470_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3470_0)), gen_zeros:0':mark:tt:nil: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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n874_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n874_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2049_0))) -> *4_0, rt in Omega(n2049_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2709_0))) -> *4_0, rt in Omega(n2709_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3470_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3470_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil: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