### (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:

*(i(x), x) → 1
*(1, y) → y
*(x, 0) → 0
*(*(x, y), z) → *(x, *(y, z))

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:
*(*(x, y), z) → *(x, *(y, z))

### (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:

*(i(x), x) → 1
*(x, 0) → 0
*(1, y) → y

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:

*(i(x), x) → 1
*(x, 0) → 0
*(1, y) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:

*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
S tuples:

*'(i(z0), z0) → c
*'(z0, 0) → c1
*'(1, z0) → c2
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c, c1, c2

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

Removed 3 trailing nodes:

*'(1, z0) → c2
*'(z0, 0) → c1
*'(i(z0), z0) → c

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(i(z0), z0) → 1
*(z0, 0) → 0
*(1, z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:none

Compound Symbols:none

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

The set S is empty