### (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(f(b, c, x)) → mark(f(x, x, x))
active(f(x, y, z)) → f(x, y, active(z))
active(d) → m(b)
f(x, y, mark(z)) → mark(f(x, y, z))
active(d) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
proper(d) → ok(d)
proper(f(x, y, z)) → f(proper(x), proper(y), proper(z))
f(ok(x), ok(y), ok(z)) → ok(f(x, y, z))
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: active, proper
The following defined symbols can occur below the 0th argument of active: active, proper
The following defined symbols can occur below the 0th argument of proper: active, proper

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(f(b, c, x)) → mark(f(x, x, x))
active(f(x, y, z)) → f(x, y, active(z))
proper(f(x, y, z)) → f(proper(x), proper(y), proper(z))

### (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(d) → m(b)
top(ok(x)) → top(active(x))
active(d) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(x, y, mark(z)) → mark(f(x, y, z))
proper(d) → ok(d)
f(ok(x), ok(y), ok(z)) → ok(f(x, y, z))
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:

active(d) → m(b)
top(ok(x)) → top(active(x))
active(d) → mark(c)
proper(b) → ok(b)
proper(c) → ok(c)
f(x, y, mark(z)) → mark(f(x, y, z))
proper(d) → ok(d)
f(ok(x), ok(y), ok(z)) → ok(f(x, y, z))
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 4.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
d0() → 0
m0(0) → 0
b0() → 0
ok0(0) → 0
mark0(0) → 0
c0() → 0
active0(0) → 1
top0(0) → 2
proper0(0) → 3
f0(0, 0, 0) → 4
b1() → 5
m1(5) → 1
active1(0) → 6
top1(6) → 2
c1() → 7
mark1(7) → 1
b1() → 8
ok1(8) → 3
c1() → 9
ok1(9) → 3
f1(0, 0, 0) → 10
mark1(10) → 4
d1() → 11
ok1(11) → 3
f1(0, 0, 0) → 12
ok1(12) → 4
proper1(0) → 13
top1(13) → 2
m1(5) → 6
mark1(7) → 6
ok1(8) → 13
ok1(9) → 13
mark1(10) → 10
mark1(10) → 12
ok1(11) → 13
ok1(12) → 10
ok1(12) → 12
active2(8) → 14
top2(14) → 2
active2(9) → 14
active2(11) → 14
proper2(7) → 15
top2(15) → 2
b2() → 16
m2(16) → 14
c2() → 17
mark2(17) → 14
c2() → 18
ok2(18) → 15
active3(18) → 19
top3(19) → 2
proper3(17) → 20
top3(20) → 2
c3() → 21
ok3(21) → 20
active4(21) → 22
top4(22) → 2