### (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:

+(x, 0) → x
+(x, i(x)) → 0
+(+(x, y), z) → +(x, +(y, z))
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(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))
*(x, +(y, z)) → +(*(x, y), *(x, z))
*(+(x, y), z) → +(*(x, z), *(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:

+(x, i(x)) → 0
+(x, 0) → 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:

+(x, i(x)) → 0
+(x, 0) → x

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, i(z0)) → 0
+(z0, 0) → z0
Tuples:

+'(z0, i(z0)) → c
+'(z0, 0) → c1
S tuples:

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

+

Defined Pair Symbols:

+'

Compound Symbols:

c, c1

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

Removed 2 trailing nodes:

+'(z0, 0) → c1
+'(z0, i(z0)) → c

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

+(z0, i(z0)) → 0
+(z0, 0) → 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