### (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(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(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(a(b(f(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:

f(a(g(X))) → b(X)
b(X) → a(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:

f(a(g(X))) → b(X)
b(X) → a(X)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:

F(a(g(z0))) → c(B(z0))
B(z0) → c1
S tuples:

F(a(g(z0))) → c(B(z0))
B(z0) → c1
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:

F, B

Compound Symbols:

c, c1

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

Removed 2 trailing nodes:

F(a(g(z0))) → c(B(z0))
B(z0) → c1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, b

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty