### (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(f(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(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(a)) → f(g(n__f(a)))

### (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:

activate(n__f(X)) → f(X)
f(X) → n__f(X)
activate(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:

activate(n__f(X)) → f(X)
f(X) → n__f(X)
activate(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:

activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:

ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
S tuples:

ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
K tuples:none
Defined Rule Symbols:

activate, f

Defined Pair Symbols:

ACTIVATE, F

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

F(z0) → c2
ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

activate, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty