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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 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 | 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
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