### (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)) → f(g(x, x))
g(0, 1) → s(0)
01

Rewrite Strategy: FULL

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

The following defined symbols can occur below the 0th argument of f: g
The following defined symbols can occur below the 0th argument of g: g
The following defined symbols can occur below the 1th argument of g: g

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
g(0, 1) → s(0)

### (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:

01
f(s(x)) → f(g(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:

01
f(s(x)) → f(g(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:

01
f(s(z0)) → f(g(z0, z0))
Tuples:

0'c
F(s(z0)) → c1(F(g(z0, z0)))
S tuples:

0'c
F(s(z0)) → c1(F(g(z0, z0)))
K tuples:none
Defined Rule Symbols:

0, f

Defined Pair Symbols:

0', F

Compound Symbols:

c, c1

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

Removed 2 trailing nodes:

F(s(z0)) → c1(F(g(z0, z0)))
0'c

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

01
f(s(z0)) → f(g(z0, z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

0, f

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty