### (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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(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))
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
unquote1(mark(X)) → mark(unquote1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
unquote1(ok(X)) → ok(unquote1(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
unquote(ok(X)) → ok(unquote(X))
unquote(mark(X)) → mark(unquote(X))
first1(X1, mark(X2)) → mark(first1(X1, X2))
proper(01) → ok(01)
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote1(ok(X)) → ok(quote1(X))
first(mark(X1), X2) → mark(first(X1, X2))
proper(nil1) → ok(nil1)
cons1(mark(X1), X2) → mark(cons1(X1, X2))
quote(ok(X)) → ok(quote(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s1(ok(X)) → ok(s1(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
proper(0) → ok(0)
first1(mark(X1), X2) → mark(first1(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
s1(mark(X)) → mark(s1(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))
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
unquote1(mark(X)) → mark(unquote1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
unquote1(ok(X)) → ok(unquote1(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
unquote(ok(X)) → ok(unquote(X))
unquote(mark(X)) → mark(unquote(X))
first1(X1, mark(X2)) → mark(first1(X1, X2))
proper(01) → ok(01)
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote1(ok(X)) → ok(quote1(X))
first(mark(X1), X2) → mark(first(X1, X2))
proper(nil1) → ok(nil1)
cons1(mark(X1), X2) → mark(cons1(X1, X2))
quote(ok(X)) → ok(quote(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s1(ok(X)) → ok(s1(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
proper(0) → ok(0)
first1(mark(X1), X2) → mark(first1(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))
s1(mark(X)) → mark(s1(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, 8, 9, 10, 11, 12, 13, 14, 15, 16]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
010() → 0
nil0() → 0
nil10() → 0
00() → 0
top0(0) → 1
from0(0) → 2
unquote10(0) → 3
sel10(0, 0) → 4
cons0(0, 0) → 5
first0(0, 0) → 6
unquote0(0) → 7
first10(0, 0) → 8
proper0(0) → 9
fcons0(0, 0) → 10
sel0(0, 0) → 11
cons10(0, 0) → 12
quote10(0) → 13
quote0(0) → 14
s10(0) → 15
s0(0) → 16
active1(0) → 17
top1(17) → 1
from1(0) → 18
ok1(18) → 2
from1(0) → 19
mark1(19) → 2
unquote11(0) → 20
mark1(20) → 3
sel11(0, 0) → 21
mark1(21) → 4
cons1(0, 0) → 22
ok1(22) → 5
unquote11(0) → 23
ok1(23) → 3
first1(0, 0) → 24
ok1(24) → 6
unquote1(0) → 25
ok1(25) → 7
unquote1(0) → 26
mark1(26) → 7
first11(0, 0) → 27
mark1(27) → 8
011() → 28
ok1(28) → 9
fcons1(0, 0) → 29
mark1(29) → 10
first1(0, 0) → 30
mark1(30) → 6
sel1(0, 0) → 31
ok1(31) → 11
sel1(0, 0) → 32
mark1(32) → 11
cons11(0, 0) → 33
mark1(33) → 12
fcons1(0, 0) → 34
ok1(34) → 10
first11(0, 0) → 35
ok1(35) → 8
nil1() → 36
ok1(36) → 9
cons11(0, 0) → 37
ok1(37) → 12
quote11(0) → 38
ok1(38) → 13
nil11() → 39
ok1(39) → 9
quote1(0) → 40
ok1(40) → 14
sel11(0, 0) → 41
ok1(41) → 4
s11(0) → 42
ok1(42) → 15
s1(0) → 43
ok1(43) → 16
s1(0) → 44
mark1(44) → 16
01() → 45
ok1(45) → 9
cons1(0, 0) → 46
mark1(46) → 5
proper1(0) → 47
top1(47) → 1
s11(0) → 48
mark1(48) → 15
ok1(18) → 18
ok1(18) → 19
mark1(19) → 18
mark1(19) → 19
mark1(20) → 20
mark1(20) → 23
mark1(21) → 21
mark1(21) → 41
ok1(22) → 22
ok1(22) → 46
ok1(23) → 20
ok1(23) → 23
ok1(24) → 24
ok1(24) → 30
ok1(25) → 25
ok1(25) → 26
mark1(26) → 25
mark1(26) → 26
mark1(27) → 27
mark1(27) → 35
ok1(28) → 47
mark1(29) → 29
mark1(29) → 34
mark1(30) → 24
mark1(30) → 30
ok1(31) → 31
ok1(31) → 32
mark1(32) → 31
mark1(32) → 32
mark1(33) → 33
mark1(33) → 37
ok1(34) → 29
ok1(34) → 34
ok1(35) → 27
ok1(35) → 35
ok1(36) → 47
ok1(37) → 33
ok1(37) → 37
ok1(38) → 38
ok1(39) → 47
ok1(40) → 40
ok1(41) → 21
ok1(41) → 41
ok1(42) → 42
ok1(42) → 48
ok1(43) → 43
ok1(43) → 44
mark1(44) → 43
mark1(44) → 44
ok1(45) → 47
mark1(46) → 22
mark1(46) → 46
mark1(48) → 42
mark1(48) → 48
active2(28) → 49
top2(49) → 1
active2(36) → 49
active2(39) → 49
active2(45) → 49