### (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:

f(s(X)) → f(X)

g(cons(0, Y)) → g(Y)

g(cons(s(X), Y)) → s(X)

h(cons(X, Y)) → h(g(cons(X, Y)))

Rewrite Strategy: FULL

### (1) 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.

### (2) 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(s(X)) → f(X)

g(cons(0, Y)) → g(Y)

g(cons(s(X), Y)) → s(X)

h(cons(X, Y)) → h(g(cons(X, Y)))

Rewrite Strategy: INNERMOST

### (3) 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:

s0(0) → 0

cons0(0, 0) → 0

00() → 0

f0(0) → 1

g0(0) → 2

h0(0) → 3

f1(0) → 1

g1(0) → 2

s1(0) → 2

cons1(0, 0) → 5

g1(5) → 4

h1(4) → 3

g1(0) → 4

s1(0) → 4

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