### (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(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(sel(0, cons(X, Z))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
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(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(sel(0, cons(X, Z))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

### (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))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
proper(0) → ok(0)
first(X1, mark(X2)) → mark(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
sel(X1, mark(X2)) → mark(sel(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))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
proper(0) → ok(0)
first(X1, mark(X2)) → mark(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
sel(X1, mark(X2)) → mark(sel(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
cons0(0, 0) → 4
sel0(0, 0) → 5
first0(0, 0) → 6
s0(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
cons1(0, 0) → 12
ok1(12) → 4
sel1(0, 0) → 13
mark1(13) → 5
first1(0, 0) → 14
ok1(14) → 6
first1(0, 0) → 15
mark1(15) → 6
s1(0) → 16
ok1(16) → 7
s1(0) → 17
mark1(17) → 7
01() → 18
ok1(18) → 2
sel1(0, 0) → 19
ok1(19) → 5
cons1(0, 0) → 20
mark1(20) → 4
proper1(0) → 21
top1(21) → 1
ok1(9) → 21
ok1(10) → 10
ok1(10) → 11
mark1(11) → 10
mark1(11) → 11
ok1(12) → 12
ok1(12) → 20
mark1(13) → 13
mark1(13) → 19
ok1(14) → 14
ok1(14) → 15
mark1(15) → 14
mark1(15) → 15
ok1(16) → 16
ok1(16) → 17
mark1(17) → 16
mark1(17) → 17
ok1(18) → 21
ok1(19) → 13
ok1(19) → 19
mark1(20) → 12
mark1(20) → 20
active2(9) → 22
top2(22) → 1
active2(18) → 22