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, *(minus(y), y)) → *(minus(*(y, y)), x)

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, *(minus(z1), z1)) → *(minus(*(z1, z1)), z0)

Tuples:

*'(z0, *(minus(z1), z1)) → c(*'(minus(*(z1, z1)), z0), *'(z1, z1))

S tuples:

*'(z0, *(minus(z1), z1)) → c(*'(minus(*(z1, z1)), z0), *'(z1, z1))

K tuples:none
Defined Rule Symbols:

*

Defined Pair Symbols:

*'

Compound Symbols:

c

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

*'(z0, *(minus(z1), z1)) → c(*'(minus(*(z1, z1)), z0), *'(z1, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

*(z0, *(minus(z1), z1)) → *(minus(*(z1, z1)), z0)

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

*

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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