| import OAI.MeasureTheory.Falconer.Shells.HighShellStationaryScale | |
| namespace OAI | |
| namespace PlanarFalconer | |
| theorem source_block_dominates_slack {sig kap : ℝ} (hsig : 0 ≤ sig) | |
| (hpar : sig < kap/1000) (N : ℕ) : | |
| sig*(N:ℝ) ≤ (Nat.ceil (kap*N):ℝ) := by | |
| apply le_trans _ (Nat.le_ceil _) | |
| have : sig ≤ kap := by linarith | |
| exact mul_le_mul_of_nonneg_right this (Nat.cast_nonneg N) | |
| theorem high_shell_descriptor_scale {sig v : ℝ} (hsig : 0 ≤ sig) | |
| (N l a t : ℕ) (hl : 2*(l:ℝ) ≤ (t:ℝ)-a) | |
| (hgap : 0 ≤ v-a) (hv : (t:ℝ)-sig*N ≤ v) : | |
| (sourceAngularRadius N sig l)⁻¹ ≤ (2:ℝ)^((v-a)/2) := by | |
| rw [sourceAngularRadius_inv] | |
| apply max_le | |
| · exact Real.one_le_rpow (by norm_num) (by linarith) | |
| · rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2),← Real.rpow_natCast, | |
| ← Real.rpow_add (by norm_num : (0:ℝ) < 2)] | |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num : (1:ℝ) ≤ 2) | |
| nlinarith [mul_nonneg (Nat.cast_nonneg N) hsig] | |
| theorem high_shell_extracted_scale {ι : Type*} (S : Finset ι) | |
| (b : ι → PolicyDescriptor) (k : ι → ℕ) {sig v : ℝ} | |
| (hsig : 0 ≤ sig) (N a t j : ℕ) | |
| (hl : ∀ i ∈ S, 2*(descriptorLength (b i):ℝ) ≤ (t:ℝ)-a) | |
| (hgap : 0 ≤ v-a) (hv : (t:ℝ)-sig*N ≤ v) (hk : ∑ i ∈ S, k i ≤ 2*j) : | |
| (∏ i ∈ S, ((sourceAngularRadius N sig (descriptorLength (b i)))⁻¹)^(k i)) ≤ | |
| (2:ℝ)^((j:ℝ)*(v-a)) := by | |
| calc | |
| _ ≤ ∏ i ∈ S, ((2:ℝ)^((v-a)/2))^(k i) := by | |
| apply Finset.prod_le_prod₀ (fun index _ => | |
| pow_nonneg (inv_nonneg.mpr (sourceAngularRadius_pos N sig (descriptorLength (b index))).le) _) | |
| intro i hi | |
| exact pow_le_pow_left₀ (inv_nonneg.mpr (sourceAngularRadius_pos _ _ _).le) | |
| (high_shell_descriptor_scale hsig N _ a t (hl i hi) hgap hv) _ | |
| _ = ((2:ℝ)^((v-a)/2))^(∑ i ∈ S, k i) := Finset.prod_pow_eq_pow_sum _ _ _ | |
| _ ≤ ((2:ℝ)^((v-a)/2))^(2*j) := | |
| pow_le_pow_right₀ (Real.one_le_rpow (by norm_num) (by linarith)) hk | |
| _ = _ := by | |
| rw [← Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 2)] | |
| congr 1 | |
| push_cast | |
| ring | |
| theorem high_shell_inverse_power_split (a v r d : ℝ) (j : ℕ) : | |
| ((r*d)⁻¹)^j = ((2:ℝ)^(-(j:ℝ)*(v-a))) * | |
| ((r/(2:ℝ)^v)⁻¹)^j * ((d/(2:ℝ)^(-a))⁻¹)^j := by | |
| have hp : (2:ℝ)^(v-a) ≠ 0 := (Real.rpow_pos_of_pos (by norm_num) _).ne' | |
| have he : (2:ℝ)^(-(j:ℝ)*(v-a)) = (((2:ℝ)^(v-a))⁻¹)^j := by | |
| rw [← Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2), | |
| ← Real.rpow_mul_natCast (by norm_num : (0:ℝ) ≤ 2)] | |
| congr 1 | |
| ring | |
| rw [he,← mul_pow,← mul_pow] | |
| congr 1 | |
| rw [inv_div,inv_div,mul_inv_rev] | |
| have he : (2:ℝ)^v*(2:ℝ)^(-a) = (2:ℝ)^(v-a) := by | |
| rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2)]; rfl | |
| symm | |
| calc | |
| _ = r⁻¹*d⁻¹*(((2:ℝ)^(v-a))⁻¹*((2:ℝ)^v*(2:ℝ)^(-a))) := by ring | |
| _ = _ := by rw [he,inv_mul_cancel₀ hp]; ring | |
| theorem high_shell_weight_cancellation {r d s : ℝ} | |
| (hr : 0 < r) (hd : 0 < d) (hs : 0 < s) : | |
| (d/s)^(-(1/2:ℝ))*(r*d)^(1/2:ℝ) = (r*s)^(1/2:ℝ) := by | |
| rw [Real.div_rpow hd.le hs.le, Real.mul_rpow hr.le hd.le, | |
| Real.mul_rpow hr.le hs.le] | |
| have he : d^(-(1/2:ℝ))*d^(1/2:ℝ) = 1 := by | |
| rw [← Real.rpow_add hd]; norm_num | |
| have hs' : s^(-(1/2:ℝ)) = (s^(1/2:ℝ))⁻¹ := Real.rpow_neg hs.le _ | |
| rw [hs',div_inv_eq_mul] | |
| calc | |
| _ = r^(1/2:ℝ)*s^(1/2:ℝ)*(d^(-(1/2:ℝ))*d^(1/2:ℝ)) := by ring | |
| _ = _ := by rw [he,mul_one] | |
| theorem high_shell_energy_prefactor (a v : ℝ) : | |
| (2:ℝ)^(-a)*(2:ℝ)^v*(2:ℝ)^(v-a) = (2:ℝ)^(2*(v-a)) := by | |
| rw [← Real.rpow_add (by norm_num : (0:ℝ) < 2), | |
| ← Real.rpow_add (by norm_num : (0:ℝ) < 2)] | |
| congr 1 | |
| ring | |
| end PlanarFalconer | |
| end OAI | |