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e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | 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
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