### (0) Obligation:

The Runtime Complexity (full) of the given

*CpxTRS* could be proven to be

BOUNDS(1, 1).

The TRS R consists of the following rules:

f(s(x)) → f(g(x, x))

g(0, 1) → s(0)

0 → 1

Rewrite Strategy: FULL

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

The following defined symbols can occur below the 0th argument of f: g

The following defined symbols can occur below the 0th argument of g: g

The following defined symbols can occur below the 1th argument of g: g

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

g(0, 1) → s(0)

### (2) Obligation:

The Runtime Complexity (full) of the given

*CpxTRS* could be proven to be

BOUNDS(1, 1).

The TRS R consists of the following rules:

0 → 1

f(s(x)) → f(g(x, x))

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, 1).

The TRS R consists of the following rules:

0 → 1

f(s(x)) → f(g(x, x))

Rewrite Strategy: INNERMOST

### (5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT
### (6) Obligation:

Complexity Dependency Tuples Problem

Rules:

0 → 1

f(s(z0)) → f(g(z0, z0))

Tuples:

0' → c

F(s(z0)) → c1(F(g(z0, z0)))

S tuples:

0' → c

F(s(z0)) → c1(F(g(z0, z0)))

K tuples:none

Defined Rule Symbols:

0, f

Defined Pair Symbols:

0', F

Compound Symbols:

c, c1

### (7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing nodes:

F(s(z0)) → c1(F(g(z0, z0)))

0' → c

### (8) Obligation:

Complexity Dependency Tuples Problem

Rules:

0 → 1

f(s(z0)) → f(g(z0, z0))

Tuples:none

S tuples:none

K tuples:none

Defined Rule Symbols:

0, f

Defined Pair Symbols:none

Compound Symbols:none

### (9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

### (10) BOUNDS(1, 1)