### (0) 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(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

### (1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(length1(X)) → length1(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:

top(ok(X)) → top(active(X))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(0) → ok(0)
length1(ok(X)) → ok(length1(X))
length(ok(X)) → ok(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))

Rewrite Strategy: FULL

### (3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

### (4) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
proper(0) → ok(0)
length1(ok(X)) → ok(length1(X))
length(ok(X)) → ok(length(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

### (5) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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 2.

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]
transitions:
ok0(0) → 0
active0(0) → 0
nil0() → 0
mark0(0) → 0
00() → 0
top0(0) → 1
proper0(0) → 2
from0(0) → 3
s0(0) → 4
cons0(0, 0) → 5
length10(0) → 6
length0(0) → 7
active1(0) → 8
top1(8) → 1
nil1() → 9
ok1(9) → 2
from1(0) → 10
ok1(10) → 3
from1(0) → 11
mark1(11) → 3
s1(0) → 12
ok1(12) → 4
s1(0) → 13
mark1(13) → 4
cons1(0, 0) → 14
ok1(14) → 5
01() → 15
ok1(15) → 2
length11(0) → 16
ok1(16) → 6
length1(0) → 17
ok1(17) → 7
cons1(0, 0) → 18
mark1(18) → 5
proper1(0) → 19
top1(19) → 1
ok1(9) → 19
ok1(10) → 10
ok1(10) → 11
mark1(11) → 10
mark1(11) → 11
ok1(12) → 12
ok1(12) → 13
mark1(13) → 12
mark1(13) → 13
ok1(14) → 14
ok1(14) → 18
ok1(15) → 19
ok1(16) → 16
ok1(17) → 17
mark1(18) → 14
mark1(18) → 18
active2(9) → 20
top2(20) → 1
active2(15) → 20