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

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

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

Types:
not :: or → or
or :: or → or → or
and :: or → or → or
hole_or1_0 :: or
gen_or2_0 :: Nat → or

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

Heuristically decided to analyse the following defined symbols:
not, and

They will be analysed ascendingly in the following order:
and < not

(6) Obligation:

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

Types:
not :: or → or
or :: or → or → or
and :: or → or → or
hole_or1_0 :: or
gen_or2_0 :: Nat → or

Generator Equations:
gen_or2_0(0) ⇔ hole_or1_0
gen_or2_0(+(x, 1)) ⇔ or(hole_or1_0, gen_or2_0(x))

The following defined symbols remain to be analysed:
and, not

They will be analysed ascendingly in the following order:
and < not

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol and.

(8) Obligation:

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

Types:
not :: or → or
or :: or → or → or
and :: or → or → or
hole_or1_0 :: or
gen_or2_0 :: Nat → or

Generator Equations:
gen_or2_0(0) ⇔ hole_or1_0
gen_or2_0(+(x, 1)) ⇔ or(hole_or1_0, gen_or2_0(x))

The following defined symbols remain to be analysed:
not

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
not(gen_or2_0(+(1, n10843_0))) → *3_0, rt ∈ Ω(n10843₀)

Induction Base:
not(gen_or2_0(+(1, 0)))

Induction Step:
not(gen_or2_0(+(1, +(n10843_0, 1)))) →_R^Ω(1)
and(not(hole_or1_0), not(gen_or2_0(+(1, n10843_0)))) →_IH
and(not(hole_or1_0), *3_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

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

Types:
not :: or → or
or :: or → or → or
and :: or → or → or
hole_or1_0 :: or
gen_or2_0 :: Nat → or

Lemmas:
not(gen_or2_0(+(1, n10843_0))) → *3_0, rt ∈ Ω(n10843₀)

Generator Equations:
gen_or2_0(0) ⇔ hole_or1_0
gen_or2_0(+(x, 1)) ⇔ or(hole_or1_0, gen_or2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
not(gen_or2_0(+(1, n10843_0))) → *3_0, rt ∈ Ω(n10843₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

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

Types:
not :: or → or
or :: or → or → or
and :: or → or → or
hole_or1_0 :: or
gen_or2_0 :: Nat → or

Lemmas:
not(gen_or2_0(+(1, n10843_0))) → *3_0, rt ∈ Ω(n10843₀)

Generator Equations:
gen_or2_0(0) ⇔ hole_or1_0
gen_or2_0(+(x, 1)) ⇔ or(hole_or1_0, gen_or2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
not(gen_or2_0(+(1, n10843_0))) → *3_0, rt ∈ Ω(n10843₀)

(16) BOUNDS(n^1, INF)