### (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(g(X), Y)) → mark(f(X, f(g(X), Y)))

active(f(X1, X2)) → f(active(X1), X2)

active(g(X)) → g(active(X))

f(mark(X1), X2) → mark(f(X1, X2))

g(mark(X)) → mark(g(X))

proper(f(X1, X2)) → f(proper(X1), proper(X2))

proper(g(X)) → g(proper(X))

f(ok(X1), ok(X2)) → ok(f(X1, X2))

g(ok(X)) → ok(g(X))

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(g(X), Y)) → mark(f(X, f(g(X), Y)))

active(f(X1, X2)) → f(active(X1), X2)

active(g(X)) → g(active(X))

proper(f(X1, X2)) → f(proper(X1), proper(X2))

proper(g(X)) → g(proper(X))

### (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:

f(mark(X1), X2) → mark(f(X1, X2))

g(ok(X)) → ok(g(X))

top(ok(X)) → top(active(X))

f(ok(X1), ok(X2)) → ok(f(X1, X2))

g(mark(X)) → mark(g(X))

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 does not nest defined symbols, 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:

f(mark(X1), X2) → mark(f(X1, X2))

g(ok(X)) → ok(g(X))

top(ok(X)) → top(active(X))

f(ok(X1), ok(X2)) → ok(f(X1, X2))

g(mark(X)) → mark(g(X))

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 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:

final states : [1, 2, 3]

transitions:

mark0(0) → 0

ok0(0) → 0

active0(0) → 0

proper0(0) → 0

f0(0, 0) → 1

g0(0) → 2

top0(0) → 3

f1(0, 0) → 4

mark1(4) → 1

g1(0) → 5

ok1(5) → 2

active1(0) → 6

top1(6) → 3

f1(0, 0) → 7

ok1(7) → 1

g1(0) → 8

mark1(8) → 2

proper1(0) → 9

top1(9) → 3

mark1(4) → 4

mark1(4) → 7

ok1(5) → 5

ok1(5) → 8

ok1(7) → 4

ok1(7) → 7

mark1(8) → 5

mark1(8) → 8

### (6) BOUNDS(1, n^1)