### (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:

norm(nil) → 0

norm(g(x, y)) → s(norm(x))

f(x, nil) → g(nil, x)

f(x, g(y, z)) → g(f(x, y), z)

rem(nil, y) → nil

rem(g(x, y), 0) → g(x, y)

rem(g(x, y), s(z)) → rem(x, z)

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:

norm(nil) → 0

norm(g(x, y)) → s(norm(x))

f(x, nil) → g(nil, x)

f(x, g(y, z)) → g(f(x, y), z)

rem(nil, y) → nil

rem(g(x, y), 0) → g(x, y)

rem(g(x, y), s(z)) → rem(x, z)

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:

nil0() → 0

00() → 0

g0(0, 0) → 0

s0(0) → 0

norm0(0) → 1

f0(0, 0) → 2

rem0(0, 0) → 3

01() → 1

norm1(0) → 4

s1(4) → 1

nil1() → 5

g1(5, 0) → 2

f1(0, 0) → 6

g1(6, 0) → 2

nil1() → 3

g1(0, 0) → 3

rem1(0, 0) → 3

01() → 4

s1(4) → 4

g1(5, 0) → 6

g1(6, 0) → 6

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