### (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), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(X)

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, 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(s(X), X) → f(X, a(X))
f(X, c(X)) → f(s(X), X)
f(X, X) → c(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(s(z0), z0) → f(z0, a(z0))
f(z0, c(z0)) → f(s(z0), z0)
f(z0, z0) → c(z0)
Tuples:

F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
F(z0, z0) → c3
S tuples:

F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
F(z0, z0) → c3
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:

F

Compound Symbols:

c1, c2, c3

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

Removed 3 trailing nodes:

F(s(z0), z0) → c1(F(z0, a(z0)))
F(z0, c(z0)) → c2(F(s(z0), z0))
F(z0, z0) → c3

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(s(z0), z0) → f(z0, a(z0))
f(z0, c(z0)) → f(s(z0), z0)
f(z0, z0) → c(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty