### (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(f(x)) → f(c(f(x)))

f(f(x)) → f(d(f(x)))

g(c(x)) → x

g(d(x)) → x

g(c(h(0))) → g(d(1))

g(c(1)) → g(d(h(0)))

g(h(x)) → g(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(c(f(x)))

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(h(0))) → g(d(1))

g(d(x)) → x

g(h(x)) → g(x)

g(c(x)) → x

g(c(1)) → g(d(h(0)))

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(h(0))) → g(d(1))

g(d(x)) → x

g(h(x)) → g(x)

g(c(x)) → x

g(c(1)) → g(d(h(0)))

Rewrite Strategy: INNERMOST

### (5) CpxTrsMatchBoundsProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2.

The certificate found is represented by the following graph.

Start state: 3

Accept states: [4]

Transitions:

3→4[g_1|0, h_1|1, c_1|1, 0|1, d_1|1, 1|1, g_1|1, 1|2]

3→5[g_1|1]

3→7[g_1|1]

3→9[h_1|2]

4→4[c_1|0, h_1|0, 0|0, d_1|0, 1|0]

5→6[d_1|1]

6→4[1|1]

7→8[d_1|1]

8→9[h_1|1]

9→4[0|1]

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