| import Mathlib |
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| namespace OAI |
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| section |
| open scoped BigOperators |
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| open scoped BigOperators |
| namespace ExactQuantumFactoring |
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| /-- The telescoping estimate for any distinct integers at least two. -/ |
| theorem prod_le_card_mul_pred_prod (s : Finset ℕ) (hs : ∀ p ∈ s, 2 ≤ p) : |
| (∏ p ∈ s, p) ≤ (s.card + 1) * ∏ p ∈ s, (p - 1) := by |
| induction s using Finset.induction_on_max with |
| | empty => simp |
| | insert a s hmax ih => |
| have ha : a ∉ s := fun h => (lt_irrefl a) (hmax a h) |
| have ha2 : 2 ≤ a := hs a (Finset.mem_insert_self _ _) |
| have hs2 : ∀ p ∈ s, 2 ≤ p := fun p hp => hs p (Finset.mem_insert_of_mem hp) |
| have hc : s.card + 2 ≤ a := by |
| have hass : s ⊆ Finset.Ico 2 a := fun p hp => Finset.mem_Ico.mpr ⟨hs2 p hp, hmax p hp⟩ |
| have := Finset.card_le_card hass |
| rw [Nat.card_Ico] at this |
| omega |
| have hstep : a * (s.card + 1) ≤ (s.card + 2) * (a - 1) := by |
| have : a - 1 + 1 = a := by omega |
| nlinarith |
| simp only [Finset.prod_insert ha, Finset.card_insert_of_notMem ha] |
| calc |
| a * ∏ p ∈ s, p ≤ a * ((s.card + 1) * ∏ p ∈ s, (p - 1)) := |
| Nat.mul_le_mul_left _ (ih hs2) |
| _ ≤ (s.card + 1 + 1) * (a - 1) * ∏ p ∈ s, (p - 1) := by |
| simpa [Nat.mul_assoc] using Nat.mul_le_mul_right (∏ p ∈ s, (p - 1)) hstep |
| _ = (s.card + 1 + 1) * ((a - 1) * ∏ p ∈ s, (p - 1)) := by ring |
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| /-- The number of distinct prime factors is less than a strict binary cutoff. -/ |
| theorem primeFactors_card_lt {d n : ℕ} (hd : 0 < d) (hb : d < 2 ^ n) : |
| d.primeFactors.card < n := by |
| have hp : 2 ^ d.primeFactors.card ≤ ∏ p ∈ d.primeFactors, p := by |
| rw [← Finset.prod_const] |
| exact Finset.prod_le_prod |
| (fun prime hprime => (Nat.prime_of_mem_primeFactors hprime).two_le) |
| have hp' : (∏ p ∈ d.primeFactors, p) ≤ d := |
| Nat.le_of_dvd hd (Nat.prod_primeFactors_dvd d) |
| exact (Nat.pow_lt_pow_iff_right (by decide : 1 < 2)).mp (hp.trans_lt (hp'.trans_lt hb)) |
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| /-- Integer form of (5.2), retaining the original global binary cutoff. -/ |
| theorem le_succ_mul_totient {d n : ℕ} (hd : 0 < d) (hb : d < 2 ^ n) : |
| d ≤ (n + 1) * Nat.totient d := by |
| have hp := prod_le_card_mul_pred_prod d.primeFactors |
| (fun p hp => (Nat.prime_of_mem_primeFactors hp).two_le) |
| have hprodpos : 0 < ∏ p ∈ d.primeFactors, (p - 1) := by |
| apply Finset.prod_pos |
| intro p hp |
| have := (Nat.prime_of_mem_primeFactors hp).two_le |
| omega |
| have hx := Nat.mul_le_mul_left (Nat.totient d) hp |
| rw [Nat.totient_mul_prod_primeFactors] at hx |
| have hbase : d ≤ (d.primeFactors.card + 1) * Nat.totient d := by |
| apply Nat.le_of_mul_le_mul_right (c := ∏ p ∈ d.primeFactors, (p - 1)) _ hprodpos |
| simpa [Nat.mul_assoc, Nat.mul_left_comm, Nat.mul_comm] using hx |
| exact hbase.trans (Nat.mul_le_mul_right _ (by have := primeFactors_card_lt hd hb; omega)) |
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| /-- Real ratio version, ready for the exact transition-probability bound. -/ |
| theorem totient_ratio_lower {d n : ℕ} (hd : 0 < d) (hb : d < 2 ^ n) : |
| 1 / ((n : ℝ) + 1) ≤ (Nat.totient d : ℝ) / d := by |
| have hn : 0 < (n : ℝ) + 1 := by positivity |
| have hd' : 0 < (d : ℝ) := by exact_mod_cast hd |
| rw [div_le_div_iff₀ hn hd'] |
| norm_num only [one_mul] |
| exact_mod_cast (le_succ_mul_totient hd hb).trans_eq (Nat.mul_comm _ _) |
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| end ExactQuantumFactoring |
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| end |
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| end OAI |
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