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:

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

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, 1) → z0
*(1, z0) → z0
*(i(z0), z0) → 1
*(z0, i(z0)) → 1
*(z0, *(z1, z2)) → *(*(z0, z1), z2)
*(*(z0, z1), i(z1)) → z0
*(*(z0, i(z1)), z1) → z0
*(k(z0, z1), k(z1, z0)) → 1
*(*(i(z0), k(z1, z2)), z0) → k(*(*(i(z0), z1), z0), *(*(i(z0), z2), z0))
i(1) → 1
i(i(z0)) → z0
i(*(z0, z1)) → *(i(z1), i(z0))
k(z0, 1) → 1
k(z0, z0) → 1
k(*(z0, i(z1)), *(z1, i(z0))) → 1

Tuples:

*'(z0, *(z1, z2)) → c4(*'(*(z0, z1), z2), *'(z0, z1))
*'(*(i(z0), k(z1, z2)), z0) → c8(K(*(*(i(z0), z1), z0), *(*(i(z0), z2), z0)), *'(*(i(z0), z1), z0), *'(i(z0), z1), I(z0), *'(*(i(z0), z2), z0), *'(i(z0), z2), I(z0))
I(*(z0, z1)) → c11(*'(i(z1), i(z0)), I(z1), I(z0))

S tuples:

*'(z0, *(z1, z2)) → c4(*'(*(z0, z1), z2), *'(z0, z1))
*'(*(i(z0), k(z1, z2)), z0) → c8(K(*(*(i(z0), z1), z0), *(*(i(z0), z2), z0)), *'(*(i(z0), z1), z0), *'(i(z0), z1), I(z0), *'(*(i(z0), z2), z0), *'(i(z0), z2), I(z0))
I(*(z0, z1)) → c11(*'(i(z1), i(z0)), I(z1), I(z0))

K tuples:none
Defined Rule Symbols:

*, i, k

Defined Pair Symbols:

*', I

Compound Symbols:

c4, c8, c11

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

*'(z0, *(z1, z2)) → c4(*'(*(z0, z1), z2), *'(z0, z1))
*'(*(i(z0), k(z1, z2)), z0) → c8(K(*(*(i(z0), z1), z0), *(*(i(z0), z2), z0)), *'(*(i(z0), z1), z0), *'(i(z0), z1), I(z0), *'(*(i(z0), z2), z0), *'(i(z0), z2), I(z0))
I(*(z0, z1)) → c11(*'(i(z1), i(z0)), I(z1), I(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, 1) → z0
*(1, z0) → z0
*(i(z0), z0) → 1
*(z0, i(z0)) → 1
*(z0, *(z1, z2)) → *(*(z0, z1), z2)
*(*(z0, z1), i(z1)) → z0
*(*(z0, i(z1)), z1) → z0
*(k(z0, z1), k(z1, z0)) → 1
*(*(i(z0), k(z1, z2)), z0) → k(*(*(i(z0), z1), z0), *(*(i(z0), z2), z0))
i(1) → 1
i(i(z0)) → z0
i(*(z0, z1)) → *(i(z1), i(z0))
k(z0, 1) → 1
k(z0, z0) → 1
k(*(z0, i(z1)), *(z1, i(z0))) → 1

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

*, i, k

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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