| import OAI.Combinatorics.EditApproximation.Geometry.RepresentativeBands |
| import OAI.Combinatorics.EditApproximation.Geometry.ActionVectors |
| import OAI.Combinatorics.EditApproximation.Refinement.ComputedGroupNumerical |
|
|
| namespace OAI |
|
|
| /-! The dimension charged by entropy is the number of actual child coordinates. -/ |
|
|
| namespace EditApproximation |
|
|
| open Finset |
|
|
| theorem wideChildStateList_length_le {nx ny M b P : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (hP : 0 < P) (i : Fin M) : |
| (wideChildStateList parent r b P i).length ≤ (40 * P + 2) ^ 2 := by |
| have h := gridStateBox_length_le ny (seedGridSpacing b P) (10 * b) |
| (r.lo + ((sourceChild parent i).lo - parent.lo)) |
| (r.lo + ((sourceChild parent i).hi - parent.lo)) (by unfold seedGridSpacing; omega) |
| exact h.trans (Nat.pow_le_pow_left (by |
| simpa only [show 2 * (10 * b) = 20 * b by omega] using seedGridSpacing_radius_count b P hP) 2) |
| |
| theorem bandCoordinates_card_le {nx ny M b P : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (hP : 0 < P) (band : List (Fin M → TargetInterval ny)) |
| (hband : ∀ action ∈ band, action ∈ wideActionList M parent r b P) : |
| (bandCoordinates band.toFinset).card ≤ M * (40 * P + 2) ^ 2 := by |
| classical |
| let coordinateSet := univ.biUnion fun i : Fin M => |
| (wideChildStateList parent r b P i).toFinset.image fun s => (i, s) |
| have hsub : bandCoordinates band.toFinset ⊆ coordinateSet := by |
| intro coordinate hcoordinate |
| obtain ⟨action, ha, hcoord⟩ := mem_biUnion.mp hcoordinate |
| obtain ⟨i, _, rfl⟩ := mem_image.mp hcoord |
| apply mem_biUnion.mpr |
| refine ⟨i, mem_univ _, mem_image.mpr ⟨action i, ?_, rfl⟩⟩ |
| exact List.mem_toFinset.mpr (finiteAssignments_sound M _ action |
| (hband action (List.mem_toFinset.mp ha)) i) |
| have hsum : coordinateSet.card ≤ ∑ i : Fin M, (40 * P + 2) ^ 2 := by |
| apply (card_biUnion_le).trans |
| apply sum_le_sum |
| intro i _ |
| exact (card_image_le).trans ((List.toFinset_card_le _).trans |
| (wideChildStateList_length_le parent r hP i)) |
| exact (card_le_card hsub).trans (by simpa only [sum_const, card_univ, Fintype.card_fin, |
| Nat.nsmul_eq_mul] using hsum) |
|
|
| theorem computed_band_dimension_log (N : ℕ) (hN : 0 < N) |
| (hell : 42 ≤ inputSmallLog N) {nx ny b : ℕ} (parent : TargetInterval nx) |
| (r : TargetInterval ny) (band : List (Fin (integerParameters N).M → TargetInterval ny)) |
| (hband : ∀ action ∈ band, action ∈ wideActionList (integerParameters N).M parent r b (integerParameters N).P) : |
| Real.log (bandCoordinates band.toFinset).card ≤ |
| 4 * (coarseExponent (inputSmallLog N) + 21 : ℕ) * (inputSmallLog N : ℝ) := by |
| let ell := inputSmallLog N |
| let p := coarseExponent ell |
| let H := inputHeight N |
| let P := (integerParameters N).P |
| let M := (integerParameters N).M |
| let d := (bandCoordinates band.toFinset).card |
| have hellR : (42 : ℝ) ≤ ell := by exact_mod_cast hell |
| have hell0 : (0 : ℝ) ≤ ell := Nat.cast_nonneg _ |
| have hPpos : 0 < P := pow_pos (Nat.two_pow_pos (heightExponent N)) _ |
| have hPone : (1 : ℝ) ≤ P := by exact_mod_cast hPpos |
| have hM : M ≤ H := pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (Nat.div_le_self _ _) |
| have hMexp : (M : ℝ) ≤ Real.exp ell := |
| (show (M : ℝ) ≤ H by exact_mod_cast hM).trans (computed_height_exp_upper N hN) |
| have hPexp : (P : ℝ) ≤ Real.exp (((p : ℝ) + 20) * ell) := by |
| change ((H ^ (p + 20) : ℕ) : ℝ) ≤ _ |
| rw [Nat.cast_pow] |
| have h := pow_le_pow_left₀ (Nat.cast_nonneg H) (computed_height_exp_upper N hN) (p + 20) |
| rw [← Real.exp_nat_mul] at h |
| simpa only [Nat.cast_add, Nat.cast_ofNat] using h |
| have h42 : (42 : ℝ) ≤ Real.exp ell := by linarith only [Real.add_one_le_exp (ell : ℝ), hellR] |
| have hbox : (40 * P + 2 : ℝ) ≤ Real.exp (((p : ℝ) + 21) * ell) := by |
| calc |
| (40 * P + 2 : ℝ) ≤ 42 * P := by linarith only [hPone] |
| _ ≤ Real.exp ell * Real.exp (((p : ℝ) + 20) * ell) := |
| mul_le_mul h42 hPexp (Nat.cast_nonneg P) (Real.exp_pos _).le |
| _ = Real.exp (((p : ℝ) + 21) * ell) := by rw [← Real.exp_add]; congr 1; ring |
| have hcount : (d : ℝ) ≤ M * (40 * P + 2 : ℝ) ^ 2 := by |
| exact_mod_cast bandCoordinates_card_le parent r hPpos band hband |
| have hexp : (d : ℝ) ≤ Real.exp (4 * ((p : ℝ) + 21) * ell) := by |
| apply hcount.trans |
| have h := mul_le_mul hMexp (pow_le_pow_left₀ (by positivity) hbox 2) (by positivity) |
| (Real.exp_pos _).le |
| apply h.trans |
| rw [← Real.exp_nat_mul, ← Real.exp_add] |
| norm_num only [Nat.cast_ofNat] |
| apply Real.exp_le_exp.mpr |
| have hp : (0 : ℝ) ≤ p := Nat.cast_nonneg _ |
| nlinarith only [mul_nonneg hp hell0, hell0] |
| by_cases hd : d = 0 |
| · change Real.log (d : ℝ) ≤ _ |
| rw [hd, Nat.cast_zero, Real.log_zero] |
| exact mul_nonneg (mul_nonneg (by norm_num) (Nat.cast_nonneg _)) (Nat.cast_nonneg _) |
| · have hdR : (0 : ℝ) < d := by exact_mod_cast (Nat.pos_of_ne_zero hd) |
| have hlog := (Real.log_le_iff_le_exp hdR).2 hexp |
| simpa only [p, ell, d, Nat.cast_mul, Nat.cast_add, Nat.cast_ofNat] using hlog |
|
|
| end EditApproximation |
|
|
| end OAI |
|
|
