### (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:

p(m, n, s(r)) → p(m, r, n)

p(m, s(n), 0) → p(0, n, m)

p(m, 0, 0) → m

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:

p(m, n, s(r)) → p(m, r, n)

p(m, s(n), 0) → p(0, n, m)

p(m, 0, 0) → m

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:

s0(0) → 0

00() → 0

p0(0, 0, 0) → 1

p1(0, 0, 0) → 1

01() → 2

p1(2, 0, 0) → 1

p1(2, 0, 2) → 1

0 → 1

2 → 1

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