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

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

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

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
not(not(x)) → x
not(or(x, y)) → and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) → or(not(not(not(x))), not(not(not(y))))

Types:
not :: or:and → or:and
or :: or:and → or:and → or:and
and :: or:and → or:and → or:and
hole_or:and1_0 :: or:and
gen_or:and2_0 :: Nat → or:and

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
not

(6) Obligation:

Innermost TRS:
Rules:
not(not(x)) → x
not(or(x, y)) → and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) → or(not(not(not(x))), not(not(not(y))))

Types:
not :: or:and → or:and
or :: or:and → or:and → or:and
and :: or:and → or:and → or:and
hole_or:and1_0 :: or:and
gen_or:and2_0 :: Nat → or:and

Generator Equations:
gen_or:and2_0(0) ⇔ hole_or:and1_0
gen_or:and2_0(+(x, 1)) ⇔ or(hole_or:and1_0, gen_or:and2_0(x))

The following defined symbols remain to be analysed:
not

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
not(gen_or:and2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
not(gen_or:and2_0(0))

Induction Step:
not(gen_or:and2_0(+(n4_0, 1))) →_R^Ω(1)
and(not(not(not(hole_or:and1_0))), not(not(not(gen_or:and2_0(n4_0))))) →_R^Ω(1)
and(not(hole_or:and1_0), not(not(not(gen_or:and2_0(n4_0))))) →_IH
and(not(hole_or:and1_0), not(not(*3_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
not(not(x)) → x
not(or(x, y)) → and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) → or(not(not(not(x))), not(not(not(y))))

Types:
not :: or:and → or:and
or :: or:and → or:and → or:and
and :: or:and → or:and → or:and
hole_or:and1_0 :: or:and
gen_or:and2_0 :: Nat → or:and

Lemmas:
not(gen_or:and2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_or:and2_0(0) ⇔ hole_or:and1_0
gen_or:and2_0(+(x, 1)) ⇔ or(hole_or:and1_0, gen_or:and2_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
not(gen_or:and2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
not(not(x)) → x
not(or(x, y)) → and(not(not(not(x))), not(not(not(y))))
not(and(x, y)) → or(not(not(not(x))), not(not(not(y))))

Types:
not :: or:and → or:and
or :: or:and → or:and → or:and
and :: or:and → or:and → or:and
hole_or:and1_0 :: or:and
gen_or:and2_0 :: Nat → or:and

Lemmas:
not(gen_or:and2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_or:and2_0(0) ⇔ hole_or:and1_0
gen_or:and2_0(+(x, 1)) ⇔ or(hole_or:and1_0, gen_or:and2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
not(gen_or:and2_0(n4_0)) → *3_0, rt ∈ Ω(n4₀)

(14) BOUNDS(n^1, INF)