### (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(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(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(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(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:

proper(true) → ok(true)
top(ok(X)) → top(active(X))
proper(c) → ok(c)
proper(false) → ok(false)
f(mark(X)) → mark(f(X))
f(ok(X)) → ok(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))

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:

proper(true) → ok(true)
top(ok(X)) → top(active(X))
proper(c) → ok(c)
proper(false) → ok(false)
f(mark(X)) → mark(f(X))
f(ok(X)) → ok(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))

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]
transitions:
true0() → 0
ok0(0) → 0
active0(0) → 0
c0() → 0
false0() → 0
mark0(0) → 0
proper0(0) → 1
top0(0) → 2
f0(0) → 3
if0(0, 0, 0) → 4
true1() → 5
ok1(5) → 1
active1(0) → 6
top1(6) → 2
c1() → 7
ok1(7) → 1
false1() → 8
ok1(8) → 1
f1(0) → 9
mark1(9) → 3
f1(0) → 10
ok1(10) → 3
if1(0, 0, 0) → 11
mark1(11) → 4
if1(0, 0, 0) → 12
ok1(12) → 4
proper1(0) → 13
top1(13) → 2
ok1(5) → 13
ok1(7) → 13
ok1(8) → 13
mark1(9) → 9
mark1(9) → 10
ok1(10) → 9
ok1(10) → 10
mark1(11) → 11
mark1(11) → 12
ok1(12) → 11
ok1(12) → 12
active2(5) → 14
top2(14) → 2
active2(7) → 14
active2(8) → 14