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:

minus(minus(x)) → x
minux(+(x, y)) → +(minus(y), minus(x))
+(minus(x), +(x, y)) → y
+(+(x, y), minus(y)) → x

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(minus(z0)) → z0
minux(+(z0, z1)) → +(minus(z1), minus(z0))
+(minus(z0), +(z0, z1)) → z1
+(+(z0, z1), minus(z1)) → z0

Tuples:

MINUX(+(z0, z1)) → c1(+'(minus(z1), minus(z0)), MINUS(z1), MINUS(z0))

S tuples:

MINUX(+(z0, z1)) → c1(+'(minus(z1), minus(z0)), MINUS(z1), MINUS(z0))

K tuples:none
Defined Rule Symbols:

minus, minux, +

Defined Pair Symbols:

MINUX

Compound Symbols:

c1

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

MINUX(+(z0, z1)) → c1(+'(minus(z1), minus(z0)), MINUS(z1), MINUS(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

minus(minus(z0)) → z0
minux(+(z0, z1)) → +(minus(z1), minus(z0))
+(minus(z0), +(z0, z1)) → z1
+(+(z0, z1), minus(z1)) → z0

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

minus, minux, +

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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