### (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(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))

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(c(f(x)))
f(f(x)) → f(d(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:

g(c(0)) → g(d(1))
g(d(x)) → x
g(c(x)) → x
g(c(1)) → g(d(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:

g(c(0)) → g(d(1))
g(d(x)) → x
g(c(x)) → x
g(c(1)) → g(d(0))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(c(0)) → g(d(1))
g(d(z0)) → z0
g(c(z0)) → z0
g(c(1)) → g(d(0))
Tuples:

G(c(0)) → c1(G(d(1)))
G(d(z0)) → c2
G(c(z0)) → c3
G(c(1)) → c4(G(d(0)))
S tuples:

G(c(0)) → c1(G(d(1)))
G(d(z0)) → c2
G(c(z0)) → c3
G(c(1)) → c4(G(d(0)))
K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:

G

Compound Symbols:

c1, c2, c3, c4

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

Removed 4 trailing nodes:

G(d(z0)) → c2
G(c(1)) → c4(G(d(0)))
G(c(0)) → c1(G(d(1)))
G(c(z0)) → c3

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

g(c(0)) → g(d(1))
g(d(z0)) → z0
g(c(z0)) → z0
g(c(1)) → g(d(0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

g

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty