### (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(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
an__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

### (2) 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:

f(n__f(n__a)) → f(n__g(f(n__a)))
f(X) → n__f(X)
an__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
an__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:

F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
F(z0) → c1
Ac2
G(z0) → c3
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
ACTIVATE(z0) → c7
S tuples:

F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
F(z0) → c1
Ac2
G(z0) → c3
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(n__a) → c5(A)
ACTIVATE(n__g(z0)) → c6(G(z0))
ACTIVATE(z0) → c7
K tuples:none
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:

F, A, G, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

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

Removed 8 trailing nodes:

ACTIVATE(n__g(z0)) → c6(G(z0))
F(n__f(n__a)) → c(F(n__g(f(n__a))), F(n__a))
G(z0) → c3
ACTIVATE(n__f(z0)) → c4(F(z0))
ACTIVATE(z0) → c7
Ac2
F(z0) → c1
ACTIVATE(n__a) → c5(A)

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(n__f(n__a)) → f(n__g(f(n__a)))
f(z0) → n__f(z0)
an__a
g(z0) → n__g(z0)
activate(n__f(z0)) → f(z0)
activate(n__a) → a
activate(n__g(z0)) → g(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, a, g, activate

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty