### (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:

g(c(x, s(y))) → g(c(s(x), y))

f(c(s(x), y)) → f(c(x, s(y)))

f(f(x)) → f(d(f(x)))

f(x) → x

Rewrite Strategy: FULL

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

The TRS does not nest defined symbols.

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:

f(f(x)) → f(d(f(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:

g(c(x, s(y))) → g(c(s(x), y))

f(x) → x

f(c(s(x), y)) → f(c(x, s(y)))

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:

g(c(x, s(y))) → g(c(s(x), y))

f(x) → x

f(c(s(x), y)) → f(c(x, s(y)))

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]

transitions:

c0(0, 0) → 0

s0(0) → 0

g0(0) → 1

f0(0) → 2

s1(0) → 4

c1(4, 0) → 3

g1(3) → 1

s1(0) → 6

c1(0, 6) → 5

f1(5) → 2

s1(4) → 4

s1(6) → 6

0 → 2

5 → 2

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