| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
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|
| /-! Bounded binary-root search required by the appendix's perfect-power check. |
| The search uses only comparisons and powers with exponent at most the input |
| bit bound; no unbounded Nat.nthRoot primitive is used by the algorithm. -/ |
| namespace ExactQuantumFactoring.Primality |
|
|
| def rootSearch (m e : ℕ) : (k guess : ℕ) → ℕ |
| | 0, guess => guess |
| | k+1, guess => |
| let next := guess+2^k |
| rootSearch m e k (if next^e ≤ m then next else guess) |
|
|
| lemma rootSearch_spec {m e : ℕ} (_he : 0 < e) (k guess : ℕ) |
| (hlo : guess^e ≤ m) (hhi : m < (guess+2^k)^e) : |
| (rootSearch m e k guess)^e ≤ m ∧ m < (rootSearch m e k guess+1)^e := by |
| induction k generalizing guess with |
| | zero => simpa only [rootSearch,pow_zero] using And.intro hlo hhi |
| | succ k ih => |
| simp only [rootSearch] |
| by_cases hnext : (guess+2^k)^e ≤ m |
| · rw [ite_eq_left hnext] |
| apply ih _ hnext |
| have hh : guess+2^k+2^k=guess+2^(k+1) := by rw [pow_succ]; omega |
| rw [hh] |
| exact hhi |
| · rw [ite_eq_right hnext] |
| exact ih _ hlo (by omega) |
|
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| lemma rootSearch_eq {m e : ℕ} (he : 0 < e) (k guess b : ℕ) |
| (hlo : guess^e ≤ m) (hhi : m < (guess+2^k)^e) (hb : b^e=m) : |
| rootSearch m e k guess=b := by |
| obtain ⟨hl,hh⟩ := rootSearch_spec he k guess hlo hhi |
| have hl' : rootSearch m e k guess ≤ b := |
| (Nat.pow_le_pow_iff_left he.ne').mp (hl.trans_eq hb.symm) |
| have hh' : b < rootSearch m e k guess+1 := |
| (Nat.pow_lt_pow_iff_left he.ne').mp (hb ▸ hh) |
| omega |
|
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| def boundedRoot (m e n : ℕ) : ℕ := rootSearch m e n 0 |
|
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| lemma boundedRoot_spec {m e n : ℕ} (he : 0 < e) (hm : m < 2^n) : |
| (boundedRoot m e n)^e ≤ m ∧ m < (boundedRoot m e n+1)^e := by |
| apply rootSearch_spec he n 0 |
| · simp [Nat.ne_of_gt he] |
| · simp only [zero_add] |
| exact hm.trans_le (Nat.le_self_pow he.ne' (2^n)) |
|
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| lemma boundedRoot_pow_eq {m e n b : ℕ} (he : 0 < e) (hm : m < 2^n) (hb : b^e=m) : |
| boundedRoot m e n=b := by |
| apply rootSearch_eq he n 0 b |
| · simp [Nat.ne_of_gt he] |
| · simp only [zero_add] |
| exact hm.trans_le (Nat.le_self_pow he.ne' (2^n)) |
| · exact hb |
|
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| lemma perfectPower_exponent_lt {m n b e : ℕ} (hm : m < 2^n) (hb : 2 ≤ b) |
| (h : b^e=m) : e < n := by |
| have hh : 2^e < 2^n := (Nat.pow_le_pow_left hb e).trans_lt (h ▸ hm) |
| exact (Nat.pow_lt_pow_iff_right (by decide : 1 < 2)).mp hh |
|
|
| end ExactQuantumFactoring.Primality |
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| end |
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| end OAI |
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|
