The Runtime Complexity (full) 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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
not(or(x, y)) →⁺ and(not(not(not(x))), not(not(not(y))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [x / or(x, y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) 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: FULL

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

Infered types.

(6) Obligation:

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

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
not

(8) Obligation:

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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) Obligation:

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.

(15) 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)

(16) BOUNDS(n^1, INF)