### (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(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))

p(0) → g(0)

g(s(p(x))) → p(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(g(x), g(y)) → f(p(f(g(x), s(y))), g(s(p(x))))

g(s(p(x))) → p(x)

### (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:

p(0) → g(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, 1).

The TRS R consists of the following rules:

p(0) → g(0)

Rewrite Strategy: INNERMOST

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

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

Complexity Dependency Tuples Problem

Rules:

p(0) → g(0)

Tuples:

P(0) → c

S tuples:

P(0) → c

K tuples:none

Defined Rule Symbols:

p

Defined Pair Symbols:

P

Compound Symbols:

c

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

Removed 1 trailing nodes:

P(0) → c

### (8) Obligation:

Complexity Dependency Tuples Problem

Rules:

p(0) → g(0)

Tuples:none

S tuples:none

K tuples:none

Defined Rule Symbols:

p

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty

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