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7f37411 | 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 | import OAI.MeasureTheory.Falconer.Shells.HighShellCoefficientBounds
namespace OAI
namespace PlanarFalconer
theorem high_shell_radial_power_bound {v r : ℝ} (hr : (2:ℝ)^(v-1) ≤ r) (j : ℕ) :
0 ≤ ((r/(2:ℝ)^v)⁻¹)^j ∧ ((r/(2:ℝ)^v)⁻¹)^j ≤ (2:ℝ)^j := by
have hrpos : 0 < r := (Real.rpow_pos_of_pos (by norm_num) _).trans_le hr
have hd : (1/2:ℝ) ≤ r/(2:ℝ)^v := by
apply (le_div_iff₀ (by positivity)).mpr
have he : (1/2:ℝ)*(2:ℝ)^v = (2:ℝ)^(v-1) := by
rw [Real.rpow_sub (by norm_num : (0:ℝ) < 2),Real.rpow_one]
ring
rw [he]
exact hr
have hi : (r/(2:ℝ)^v)⁻¹ ≤ 2 := by
simpa only [one_div, inv_inv] using inv_anti₀ (by norm_num : (0:ℝ) < 1/2) hd
exact ⟨pow_nonneg (by positivity) _,pow_le_pow_left₀ (by positivity) hi j⟩
theorem uniform_minimal_inverseLeibniz_subpower {eps : ℝ} (heps : 0 < eps) (j : ℕ) :
∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (A : ℕ → ℝ) (q a t : ℕ), 0 < q →
((inverseLeibnizTerms j (pairMaskIndices (policyDescriptorList A N q a t))).map
(fun b => ‖b.1‖)).sum * (2:ℝ)^j ≤ ((2:ℝ)^N)^eps := by
let C := (inverseStationaryExpr j).cost * (2:ℝ)^(2*j) * (2:ℝ)^j
obtain ⟨N₁,h₁⟩ := fixed_constant_absorbed (half_pos heps) C
obtain ⟨N₂,h₂⟩ := fixed_polynomial_subpower (6*j) (half_pos heps)
refine ⟨max N₁ N₂,?_⟩
intro N hN A q a t hq
have hpoly : (((N:ℝ)+1)^3)^(2*j) = ((N:ℝ)+1)^(6*j) := by
rw [← pow_mul]
congr 1
omega
have hbound := mul_le_mul_of_nonneg_right (minimal_inverseLeibniz_cost A (N := N) (a := a) (t := t) hq j)
(by positivity : 0 ≤ (2:ℝ)^j)
apply hbound.trans
calc
_ = C*((N:ℝ)+1)^(6*j) := by rw [mul_pow,hpoly]; dsimp [C]; ring
_ ≤ C*(4*(N:ℝ)+1)^(6*j) := mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (by positivity) (by have := Nat.cast_nonneg (α := ℝ) N; linarith) _)
(by dsimp [C]; exact mul_nonneg (mul_nonneg (inverseStationaryExpr j).cost_nonneg (by positivity)) (by positivity))
_ ≤ ((2:ℝ)^N)^(eps/2)*((2:ℝ)^N)^(eps/2) :=
mul_le_mul (h₁ N ((le_max_left _ _).trans hN)) (h₂ N ((le_max_right _ _).trans hN))
(by positivity) (by positivity)
_ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring
end PlanarFalconer
end OAI
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