The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(0) → 0
i(i(z0)) → z0
i(+(z0, z1)) → +(i(z0), i(z1))
+(0, z0) → z0
+(z0, 0) → z0
+(i(z0), z0) → 0
+(z0, i(z0)) → 0
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(+(z0, i(z1)), z1) → z0
+(+(z0, z1), i(z1)) → z0

Tuples:

I(+(z0, z1)) → c2(+'(i(z0), i(z1)), I(z0), I(z1))
+'(z0, +(z1, z2)) → c7(+'(+(z0, z1), z2), +'(z0, z1))

S tuples:

I(+(z0, z1)) → c2(+'(i(z0), i(z1)), I(z0), I(z1))
+'(z0, +(z1, z2)) → c7(+'(+(z0, z1), z2), +'(z0, z1))

K tuples:none
Defined Rule Symbols:

i, +

Defined Pair Symbols:

I, +'

Compound Symbols:

c2, c7

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

I(+(z0, z1)) → c2(+'(i(z0), i(z1)), I(z0), I(z1))
+'(z0, +(z1, z2)) → c7(+'(+(z0, z1), z2), +'(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

i(0) → 0
i(i(z0)) → z0
i(+(z0, z1)) → +(i(z0), i(z1))
+(0, z0) → z0
+(z0, 0) → z0
+(i(z0), z0) → 0
+(z0, i(z0)) → 0
+(z0, +(z1, z2)) → +(+(z0, z1), z2)
+(+(z0, i(z1)), z1) → z0
+(+(z0, z1), i(z1)) → z0

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

i, +

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))