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

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(x), not(y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [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(and(x, y)) → or(not(x), not(y))
not(or(x, y)) → and(not(x), not(y))
and(x, or(y, z)) → or(and(x, y), and(x, z))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

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

(7) 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

(8) Obligation:

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

Types:
not :: or → or
and :: or → or → or
or :: 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

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

Proved the following rewrite lemma:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
and(gen_or2_0(a), gen_or2_0(+(1, 0)))

Induction Step:
and(gen_or2_0(a), gen_or2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
or(and(gen_or2_0(a), hole_or1_0), and(gen_or2_0(a), gen_or2_0(+(1, n4_0)))) →_IH
or(and(gen_or2_0(a), 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:

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

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

Lemmas:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

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

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
not(gen_or2_0(+(1, n3887_0))) → *3_0, rt ∈ Ω(n3887₀)

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

Induction Step:
not(gen_or2_0(+(1, +(n3887_0, 1)))) →_R^Ω(1)
and(not(hole_or1_0), not(gen_or2_0(+(1, n3887_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).

(13) Complex Obligation (BEST)

(14) Obligation:

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

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

Lemmas:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
not(gen_or2_0(+(1, n3887_0))) → *3_0, rt ∈ Ω(n3887₀)

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:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

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

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

Lemmas:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)
not(gen_or2_0(+(1, n3887_0))) → *3_0, rt ∈ Ω(n3887₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

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

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

Lemmas:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
and(gen_or2_0(a), gen_or2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(22) BOUNDS(n^1, INF)