 143  simp [Nat.factorization_zero]
 144
 145/-- For a prime `p`, `c(p) = J(p)`. -/
 146theorem costSpectrumValue_prime {p : ℕ} (hp : Nat.Prime p) :
 147    costSpectrumValue p = primeCost p := by
 148  unfold costSpectrumValue
 149  rw [Nat.Prime.factorization hp]
 150  simp [Finsupp.sum_single_index]
 151
 152/-- For a prime power `p^k`, `c(p^k) = k · J(p)`. -/
 153theorem costSpectrumValue_pow {p k : ℕ} (hp : Nat.Prime p) :
 154    costSpectrumValue (p ^ k) = (k : ℝ) * primeCost p := by
 155  unfold costSpectrumValue
 156  rw [Nat.Prime.factorization_pow hp]
 157  simp [Finsupp.sum_single_index]
 158
 159/-- The cost is completely additive over coprime products.
 160    For arbitrary products with positive factors, the same identity holds
 161    because `Nat.factorization` is additive on positive multiplications. -/
 162theorem costSpectrumValue_mul {m n : ℕ} (hm : m ≠ 0) (hn : n ≠ 0) :
 163    costSpectrumValue (m * n) = costSpectrumValue m + costSpectrumValue n := by
 164  unfold costSpectrumValue
 165  rw [Nat.factorization_mul hm hn]
 166  rw [Finsupp.sum_add_index']
 167  · intro p
 168    simp
 169  · intro p i j
 170    push_cast
 171    ring
 172
 173/-- The cost is nonnegative for any positive `n`.
 174    Each summand `k · J(p) ≥ 0` by primality of `p`, so the sum is ≥ 0. -/
 175theorem costSpectrumValue_nonneg (n : ℕ) :
 176    0 ≤ costSpectrumValue n := by

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.