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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 | import OAI.MeasureTheory.Falconer.Estimates.RestorationSeries
namespace OAI
/-! # Converting the original-pair deletion bound to its summable majorant -/
namespace Falconer
open scoped ENNReal
theorem dyadic_leaf_error (N : ℕ) :
((2:ℝ≥0∞)^(2*N))⁻¹ = ENNReal.ofReal (((2:ℝ)^N)^(-2:ℝ)) := by
have hfin : ((2:ℝ≥0∞)^(2*N))⁻¹ ≠ ∞ := by simp
rw [← ENNReal.ofReal_toReal hfin]
congr 1
simp only [ENNReal.toReal_inv,ENNReal.toReal_pow,ENNReal.toReal_ofNat]
rw [Real.rpow_neg (by positivity),Real.rpow_two]
congr 1
rw [← pow_mul,Nat.mul_comm N 2]
theorem class_discard_numeric (N k : ℕ) (lam : ℝ) :
((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam)) ≤
ENNReal.ofReal (((2:ℝ)^N)^(-2:ℝ) + ((N:ℝ)+1)^k*((2:ℝ)^N)^(-lam)) := by
have hp : ((N^k:ℕ):ℝ≥0∞) ≤ ENNReal.ofReal (((N:ℝ)+1)^k) := by
rw [← ENNReal.ofReal_natCast]
apply ENNReal.ofReal_le_ofReal
simpa only [Nat.cast_pow] using pow_le_pow_left₀ (Nat.cast_nonneg N)
(show (N:ℝ) ≤ N+1 by linarith) k
rw [dyadic_leaf_error]
apply (add_le_add le_rfl (mul_le_mul' hp le_rfl)).trans_eq
rw [← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_add (by positivity) (by positivity)]
theorem restoration_numeric_bound (N k₁ k₂ : ℕ) {Cw Ccap c lam σ : ℝ}
(hw : 0 ≤ Cw) (hcap : 0 ≤ Ccap) {D X₁ X₂ V : ℝ≥0∞}
(hD : D ≤ ENNReal.ofReal (Ccap*((2:ℝ)^N)^(-c)))
(h₁ : X₁ ≤ ((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k₁:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam)))
(h₂ : X₂ ≤ ((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k₂:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam)))
(hV : V ≤ ENNReal.ofReal Cw * (D + (X₁ + X₂ +
(4*(N+1)^2:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-σ))))) :
V ≤ ENNReal.ofReal (restorationError Cw Ccap c lam σ k₁ k₂ N) := by
apply hV.trans
have ht : (4*(N+1)^2:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-σ)) =
ENNReal.ofReal (4*((N:ℝ)+1)^2*((2:ℝ)^N)^(-σ)) := by
rw [← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (by positivity)]
simp only [Nat.cast_mul,Nat.cast_ofNat,Nat.cast_pow,Nat.cast_add,Nat.cast_one]
rw [ht,restorationError,ENNReal.ofReal_mul hw]
apply mul_le_mul' le_rfl
apply (add_le_add hD (add_le_add (add_le_add (h₁.trans (class_discard_numeric N k₁ lam))
(h₂.trans (class_discard_numeric N k₂ lam))) le_rfl)).trans_eq
rw [← ENNReal.ofReal_add (by positivity) (by positivity),
← ENNReal.ofReal_add (by positivity) (by positivity),
← ENNReal.ofReal_add (by positivity) (by positivity)]
congr 1
ring
end Falconer
end OAI
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