### (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, X, c)) → mark(f(X, c, X))
active(c) → mark(b)
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))
proper(b) → ok(b)
proper(c) → ok(c)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
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(f(b, X, c)) → mark(f(X, c, X))
active(f(X1, X2, X3)) → f(X1, active(X2), X3)
proper(f(X1, X2, X3)) → f(proper(X1), proper(X2), proper(X3))

### (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(b) → ok(b)
proper(c) → ok(c)
active(c) → mark(b)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
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(b) → ok(b)
proper(c) → ok(c)
active(c) → mark(b)
f(ok(X1), ok(X2), ok(X3)) → ok(f(X1, X2, X3))
f(X1, mark(X2), X3) → mark(f(X1, X2, X3))
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:
ok0(0) → 0
b0() → 0
c0() → 0
mark0(0) → 0
top0(0) → 1
proper0(0) → 2
active0(0) → 3
f0(0, 0, 0) → 4
active1(0) → 5
top1(5) → 1
b1() → 6
ok1(6) → 2
c1() → 7
ok1(7) → 2
b1() → 8
mark1(8) → 3
f1(0, 0, 0) → 9
ok1(9) → 4
f1(0, 0, 0) → 10
mark1(10) → 4
proper1(0) → 11
top1(11) → 1
ok1(6) → 11
ok1(7) → 11
mark1(8) → 5
ok1(9) → 9
ok1(9) → 10
mark1(10) → 9
mark1(10) → 10
active2(6) → 12
top2(12) → 1
active2(7) → 12
proper2(8) → 13
top2(13) → 1
b2() → 14
ok2(14) → 13
b2() → 15
mark2(15) → 12
active3(14) → 16
top3(16) → 1
proper3(15) → 17
top3(17) → 1
b3() → 18
ok3(18) → 17
active4(18) → 19
top4(19) → 1