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

(0) Obligation:

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

or(x, x) → x
and(x, x) → x
not(not(x)) → x
not(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

or(z0, z0) → z0
and(z0, z0) → z0
not(not(z0)) → z0
not(and(z0, z1)) → or(not(z0), not(z1))
not(or(z0, z1)) → and(not(z0), not(z1))

Tuples:

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

S tuples:

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

K tuples:none
Defined Rule Symbols:

or, and, not

Defined Pair Symbols:

NOT

Compound Symbols:

c3, c4

(3) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

We considered the (Usable) Rules:

not(not(z0)) → z0
not(and(z0, z1)) → or(not(z0), not(z1))
not(or(z0, z1)) → and(not(z0), not(z1))
and(z0, z0) → z0
or(z0, z0) → z0

And the Tuples:

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0   
POL(NOT(x₁)) = [2]x₁   
POL(OR(x₁, x₂)) = [1] + x₂   
POL(and(x₁, x₂)) = [4] + [3]x₁ + [5]x₂   
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(not(x₁)) = x₁   
POL(or(x₁, x₂)) = [4] + [3]x₁ + [5]x₂   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

or(z0, z0) → z0
and(z0, z0) → z0
not(not(z0)) → z0
not(and(z0, z1)) → or(not(z0), not(z1))
not(or(z0, z1)) → and(not(z0), not(z1))

Tuples:

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

S tuples:none
K tuples:

NOT(and(z0, z1)) → c3(OR(not(z0), not(z1)), NOT(z0), NOT(z1))
NOT(or(z0, z1)) → c4(AND(not(z0), not(z1)), NOT(z0), NOT(z1))

Defined Rule Symbols:

or, and, not

Defined Pair Symbols:

NOT

Compound Symbols:

c3, c4

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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