| import Mathlib.Algebra.Order.Floor.Semiring |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.Positivity |
| import Mathlib.Tactic.Ring |
| import OAI.NumberTheory.PiExponent.Analysis.DimensionLimits |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Filter |
| open scoped Topology |
|
|
| namespace PiExponent |
|
|
| def dimensionK (C : ℝ) (m : ℕ) : ℕ := ⌊C ^ m⌋₊ |
|
|
| def dimensionW (B : ℝ) (m : ℕ) : ℝ := (B ^ m)⁻¹ |
|
|
| def dimensionV (theta B C : ℝ) (m : ℕ) : ℝ := |
| 2 * (dimensionK C m : ℝ) * theta ^ m * dimensionW B m |
|
|
| theorem dimensionK_le (C : ℝ) (hC : 0 ≤ C) (m : ℕ) : |
| (dimensionK C m : ℝ) ≤ C ^ m := |
| Nat.floor_le (pow_nonneg hC m) |
|
|
| theorem dimensionK_one_le (C : ℝ) (hC : 1 ≤ C) (m : ℕ) : |
| 1 ≤ dimensionK C m := by |
| exact (Nat.one_le_floor_iff _).2 (one_le_pow₀ hC) |
|
|
| theorem dimensionW_pos (B : ℝ) (hB : 0 < B) (m : ℕ) : |
| 0 < dimensionW B m := by |
| exact inv_pos.mpr (pow_pos hB m) |
|
|
| theorem dimensionV_pos (theta B C : ℝ) (htheta : 0 < theta) |
| (hB : 0 < B) (hC : 1 ≤ C) (m : ℕ) : |
| 0 < dimensionV theta B C m := by |
| have hK : 0 < (dimensionK C m : ℝ) := by |
| exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one (dimensionK_one_le C hC m)) |
| exact mul_pos (mul_pos (mul_pos (by norm_num) hK) (pow_pos htheta m)) |
| (dimensionW_pos B hB m) |
|
|
| theorem dimension_volume (theta B C : ℝ) (htheta : 0 < theta) |
| (hB : 0 < B) (hC : 1 ≤ C) (m : ℕ) : |
| (dimensionK C m : ℝ) * (dimensionW B m / dimensionV theta B C m) * |
| theta ^ m = 1 / 2 := by |
| have hK : (dimensionK C m : ℝ) ≠ 0 := by |
| exact ne_of_gt (by exact_mod_cast |
| (lt_of_lt_of_le Nat.zero_lt_one (dimensionK_one_le C hC m))) |
| have ht : theta ^ m ≠ 0 := ne_of_gt (pow_pos htheta m) |
| have hw : dimensionW B m ≠ 0 := ne_of_gt (dimensionW_pos B hB m) |
| unfold dimensionV |
| field_simp |
|
|
| theorem dimensionK_div_W_le (B C : ℝ) (hB : 0 < B) (hC : 0 ≤ C) (m : ℕ) : |
| (dimensionK C m : ℝ) / dimensionW B m ≤ (C * B) ^ m := by |
| rw [dimensionW, div_inv_eq_mul, mul_pow] |
| exact mul_le_mul_of_nonneg_right (dimensionK_le C hC m) (pow_pos hB m).le |
|
|
| theorem dimensionV_geometric_lower (theta B C : ℝ) (htheta : 0 < theta) |
| (hB : 0 < B) (m : ℕ) (hlarge : 2 ≤ C ^ m) : |
| (C * theta / B) ^ m ≤ dimensionV theta B C m := by |
| have hfloor : C ^ m < (dimensionK C m : ℝ) + 1 := Nat.lt_floor_add_one _ |
| have hK : C ^ m ≤ 2 * (dimensionK C m : ℝ) := by linarith |
| rw [div_pow, mul_pow, div_eq_mul_inv] |
| unfold dimensionV dimensionW |
| exact mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hK (pow_pos htheta m).le) |
| (inv_pos.mpr (pow_pos hB m)).le |
|
|
| theorem dimension_volume_lt_one (theta C : ℝ) (htheta : 0 < theta) |
| (hC : 0 ≤ C) (hsmall : C * theta < 1) (m : ℕ) (hm : 1 ≤ m) : |
| (dimensionK C m : ℝ) * theta ^ m < 1 := by |
| calc |
| (dimensionK C m : ℝ) * theta ^ m ≤ C ^ m * theta ^ m := |
| mul_le_mul_of_nonneg_right (dimensionK_le C hC m) (pow_pos htheta m).le |
| _ = (C * theta) ^ m := (mul_pow _ _ _).symm |
| _ < 1 := pow_lt_one₀ (mul_nonneg hC htheta.le) hsmall (by omega) |
|
|
| theorem dimension_error_le (theta B C a b : ℝ) (htheta : 0 < theta) |
| (hB : 0 < B) (hC : 0 ≤ C) (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (m : ℕ) (hlarge : 2 ≤ C ^ m) : |
| (a * (m : ℝ) + b) / dimensionV theta B C m + |
| 100 * (dimensionK C m : ℝ) / dimensionW B m ≤ |
| (a * (m : ℝ) + b) / (C * theta / B) ^ m + 100 * (C * B) ^ m := by |
| have hr : 0 < (C * theta / B) ^ m := by |
| have hCp : 0 < C := by |
| by_contra hn |
| have hz : C = 0 := le_antisymm (le_of_not_gt hn) hC |
| subst C |
| rcases m with _ | m <;> norm_num at hlarge |
| positivity |
| apply add_le_add |
| · exact div_le_div_of_nonneg_left (by positivity) hr |
| (dimensionV_geometric_lower theta B C htheta hB m hlarge) |
| · rw [mul_div_assoc] |
| exact mul_le_mul_of_nonneg_left (dimensionK_div_W_le B C hB hC m) (by norm_num) |
|
|
| theorem dimension_collision_identity (theta A B C eta : ℝ) |
| (htheta : 0 < theta) (hA : 0 < A) (hB : 0 < B) (hC : 1 ≤ C) (m : ℕ) : |
| eta ^ 2 * (dimensionK C m : ℝ) * theta ^ m / |
| (((m : ℝ) + 1) * dimensionV theta B C m * A ^ m) = |
| eta ^ 2 * (B / A) ^ m / (2 * ((m : ℝ) + 1)) := by |
| have hK : (dimensionK C m : ℝ) ≠ 0 := by |
| exact ne_of_gt (by exact_mod_cast |
| (lt_of_lt_of_le Nat.zero_lt_one (dimensionK_one_le C hC m))) |
| have ht : theta ^ m ≠ 0 := ne_of_gt (pow_pos htheta m) |
| have hA' : A ^ m ≠ 0 := ne_of_gt (pow_pos hA m) |
| have hB' : B ^ m ≠ 0 := ne_of_gt (pow_pos hB m) |
| have hm : (m : ℝ) + 1 ≠ 0 := by positivity |
| unfold dimensionV dimensionW |
| rw [div_pow] |
| field_simp |
|
|
| theorem tendsto_dimension_error_majorant (theta B C a b : ℝ) |
| (hB : 0 < B) (hC : 0 < C) (hsmall : C * B < 1) |
| (hlarge : 1 < C * theta / B) : |
| Tendsto (fun m : ℕ => (a * (m : ℝ) + b) / (C * theta / B) ^ m + |
| 100 * (C * B) ^ m) atTop (𝓝 0) := by |
| have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 hlarge |
| have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 hlarge |
| have hlin : Tendsto (fun m : ℕ => (a * (m : ℝ) + b) / |
| (C * theta / B) ^ m) atTop (𝓝 0) := by |
| convert (h₁.const_mul a).add (h₀.const_mul b) using 1 |
| · ext m |
| simp only [pow_one, pow_zero] |
| ring |
| · simp |
| have hgeom := tendsto_pow_atTop_nhds_zero_of_lt_one (mul_pos hC hB).le hsmall |
| simpa only [mul_zero, add_zero] using hlin.add (hgeom.const_mul 100) |
|
|
| theorem tendsto_dimension_collision (A B eta : ℝ) |
| (hratio : 1 < B / A) (heta : 0 < eta) : |
| Tendsto (fun m : ℕ => eta ^ 2 * (B / A) ^ m / |
| (2 * ((m : ℝ) + 1))) atTop atTop := by |
| have h := (tendsto_pow_div_nat_add_one hratio).const_mul_atTop |
| (show 0 < eta ^ 2 / 2 by positivity) |
| convert h using 1 |
| ext m |
| rw [mul_div_mul_comm] |
|
|
| theorem exists_dimension_margin (theta A B C eta a b epsilon target : ℝ) |
| (htheta : 0 < theta) (hB : 0 < B) (hC : 1 < C) |
| (hsmall : C * B < 1) (hlarge : 1 < C * theta / B) |
| (hratio : 1 < B / A) (heta : 0 < eta) |
| (ha : 0 ≤ a) (hb : 0 ≤ b) (hepsilon : 0 < epsilon) : |
| ∃ m : ℕ, 1 ≤ m ∧ |
| (a * (m : ℝ) + b) / dimensionV theta B C m + |
| 100 * (dimensionK C m : ℝ) / dimensionW B m < epsilon ∧ |
| target < eta ^ 2 * (B / A) ^ m / (2 * ((m : ℝ) + 1)) := by |
| have herrors := (tendsto_dimension_error_majorant theta B C a b hB |
| (zero_lt_one.trans hC) hsmall hlarge).eventually (gt_mem_nhds hepsilon) |
| have hcollision := (tendsto_dimension_collision A B eta hratio heta).eventually |
| (eventually_gt_atTop target) |
| have hfloor := (tendsto_pow_atTop_atTop_of_one_lt hC).eventually |
| (eventually_ge_atTop (2 : ℝ)) |
| have hgood : ∀ᶠ m : ℕ in atTop, 1 ≤ m ∧ |
| (a * (m : ℝ) + b) / dimensionV theta B C m + |
| 100 * (dimensionK C m : ℝ) / dimensionW B m < epsilon ∧ |
| target < eta ^ 2 * (B / A) ^ m / (2 * ((m : ℝ) + 1)) := by |
| filter_upwards [herrors, hcollision, hfloor, eventually_ge_atTop 1] |
| with m herror hcollision hfloor hm |
| exact ⟨hm, (dimension_error_le theta B C a b htheta hB |
| (zero_lt_one.trans hC).le ha hb m hfloor).trans_lt herror, hcollision⟩ |
| exact hgood.exists |
|
|
| end PiExponent |
|
|
| end |
|
|
| end OAI |
|
|
