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:

p(s(x)) → x
s(p(x)) → x
+(0, y) → y
+(s(x), y) → s(+(x, y))
+(p(x), y) → p(+(x, y))
minus(0) → 0
minus(s(x)) → p(minus(x))
minus(p(x)) → s(minus(x))
*(0, y) → 0
*(s(x), y) → +(*(x, y), y)
*(p(x), y) → +(*(x, y), minus(y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
s(p(z0)) → z0
+(0, z0) → z0
+(s(z0), z1) → s(+(z0, z1))
+(p(z0), z1) → p(+(z0, z1))
minus(0) → 0
minus(s(z0)) → p(minus(z0))
minus(p(z0)) → s(minus(z0))
*(0, z0) → 0
*(s(z0), z1) → +(*(z0, z1), z1)
*(p(z0), z1) → +(*(z0, z1), minus(z1))

Tuples:

+'(s(z0), z1) → c3(S(+(z0, z1)), +'(z0, z1))
+'(p(z0), z1) → c4(P(+(z0, z1)), +'(z0, z1))
MINUS(s(z0)) → c6(P(minus(z0)), MINUS(z0))
MINUS(p(z0)) → c7(S(minus(z0)), MINUS(z0))
*'(s(z0), z1) → c9(+'(*(z0, z1), z1), *'(z0, z1))
*'(p(z0), z1) → c10(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))

S tuples:

+'(s(z0), z1) → c3(S(+(z0, z1)), +'(z0, z1))
+'(p(z0), z1) → c4(P(+(z0, z1)), +'(z0, z1))
MINUS(s(z0)) → c6(P(minus(z0)), MINUS(z0))
MINUS(p(z0)) → c7(S(minus(z0)), MINUS(z0))
*'(s(z0), z1) → c9(+'(*(z0, z1), z1), *'(z0, z1))
*'(p(z0), z1) → c10(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))

K tuples:none
Defined Rule Symbols:

p, s, +, minus, *

Defined Pair Symbols:

+', MINUS, *'

Compound Symbols:

c3, c4, c6, c7, c9, c10

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

+'(s(z0), z1) → c3(S(+(z0, z1)), +'(z0, z1))
+'(p(z0), z1) → c4(P(+(z0, z1)), +'(z0, z1))
MINUS(s(z0)) → c6(P(minus(z0)), MINUS(z0))
MINUS(p(z0)) → c7(S(minus(z0)), MINUS(z0))
*'(s(z0), z1) → c9(+'(*(z0, z1), z1), *'(z0, z1))
*'(p(z0), z1) → c10(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
s(p(z0)) → z0
+(0, z0) → z0
+(s(z0), z1) → s(+(z0, z1))
+(p(z0), z1) → p(+(z0, z1))
minus(0) → 0
minus(s(z0)) → p(minus(z0))
minus(p(z0)) → s(minus(z0))
*(0, z0) → 0
*(s(z0), z1) → +(*(z0, z1), z1)
*(p(z0), z1) → +(*(z0, z1), minus(z1))

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

p, s, +, minus, *

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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