### (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:

merge(x, nil) → x

merge(nil, y) → y

merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))

merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Rewrite Strategy: FULL

### (1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, 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:

merge(x, nil) → x

merge(nil, y) → y

merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))

merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

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]

transitions:

nil0() → 0

++0(0, 0) → 0

u0() → 0

v0() → 0

merge0(0, 0) → 1

u1() → 4

v1() → 5

++1(4, 5) → 3

merge1(0, 3) → 2

++1(0, 2) → 1

++1(0, 0) → 7

v1() → 8

merge1(7, 8) → 6

++1(4, 6) → 1

++1(0, 2) → 2

++1(4, 6) → 2

0 → 1

3 → 2

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