Datasets:

roseking
/

openAI-math-dataset

import Mathlib

namespace OAI

section
section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open MeasureTheory.Measure
open scoped ENNReal NNReal MeasureTheory Topology
open MeasureTheory
noncomputable section
noncomputable section
open MeasureTheory Set NormedSpace
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
noncomputable section
open Set Real
open scoped Topology
noncomputable section
open Matrix Set Filter
open scoped Topology Matrix.Norms.Frobenius
noncomputable section
open Matrix NormedSpace ContinuousLinearMap
open scoped Matrix.Norms.Frobenius
noncomputable section
open Matrix
noncomputable section
open MeasureTheory ProbabilityTheory Real Set
open scoped ENNReal NNReal
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory
open Real
noncomputable section
open scoped BigOperators Topology
open Filter Real
namespace SKRatioGaussian
section MatrixScale
variable {ι : Type*} [Fintype ι]

lemma diagonal_sum_le {a : ι → ℝ} {A : ℝ} (hA : ∀ i, a i ≤ A) :
    (∑ i, a i) ≤ (Fintype.card ι : ℝ) * A := by
  simpa only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] using
    Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hA i)

lemma diagonal_sq_sum_le {a : ι → ℝ} {A : ℝ}
    (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) :
    (∑ i, (a i)^2) ≤ (Fintype.card ι : ℝ) * A^2 :=
  diagonal_sum_le (fun i => by nlinarith [ha i, hA i])

lemma scaled_mean_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
    (hj : 0 ≤ j) (hA : ∀ i, a i ≤ A) :
    (j / (Fintype.card ι : ℝ)) * (∑ i, a i) ≤ j*A := by
  have hn : 0 < (Fintype.card ι : ℝ) := by exact_mod_cast Fintype.card_pos
  have h := mul_le_mul_of_nonneg_left (diagonal_sum_le hA) (div_nonneg hj hn.le)
  calc
    _ ≤ _ := h
    _ = _ := by field_simp

lemma scaled_variance_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
    (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) :
    2*(j / (Fintype.card ι : ℝ))^2 * (∑ i, (a i)^2) ≤
      2*j^2*A^2 / (Fintype.card ι : ℝ) := by
  have hn : 0 < (Fintype.card ι : ℝ) := by exact_mod_cast Fintype.card_pos
  have h := mul_le_mul_of_nonneg_left (diagonal_sq_sum_le ha hA)
    (show 0 ≤ 2*(j / (Fintype.card ι : ℝ))^2 by positivity)
  calc
    _ ≤ _ := h
    _ = _ := by field_simp

lemma scaled_sqrt_mean_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
    (hj : 0 ≤ j) (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) (hA0 : 0 ≤ A) :
    sqrt ((j / (Fintype.card ι : ℝ)) * (∑ i, a i)) * sqrt A ≤ sqrt j * A := by
  have hn : 0 ≤ (Fintype.card ι : ℝ) := by positivity
  have hq : 0 ≤ (j / (Fintype.card ι : ℝ)) * ∑ i, a i :=
    mul_nonneg (div_nonneg hj hn) (Finset.sum_nonneg (fun i _ => ha i))
  rw [← sqrt_mul hq]
  calc
    _ ≤ sqrt (j * A^2) := sqrt_le_sqrt (by
      have h := mul_le_mul_of_nonneg_right (scaled_mean_bound hj hA) hA0
      nlinarith only [h])
    _ = sqrt j * A := by rw [sqrt_mul hj, sqrt_sq hA0]

lemma subcritical_endpoint_margin {b : ℝ} (hb0 : 0 ≤ b) (hb1 : b ≤ 1) :
    b^2 ≤ 1 - (1-b)^2/2 := by
  nlinarith [mul_nonneg (sub_nonneg.mpr hb1) (show 0 ≤ 3*b+1 by positivity)]

lemma scaled_qA_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
    (hj : 0 ≤ j) (hA : ∀ i, a i ≤ A) (hA0 : 0 ≤ A)
    (hsub : sqrt j * A ≤ 1) :
    ((j / (Fintype.card ι : ℝ)) * (∑ i, a i))*A ≤
      1-(1-sqrt j*A)^2/2 := by
  have h := mul_le_mul_of_nonneg_right (scaled_mean_bound hj hA) hA0
  have hm := subcritical_endpoint_margin (show 0 ≤ sqrt j*A by positivity) hsub
  have he : (sqrt j*A)^2 = j*A^2 := by rw [mul_pow, sq_sqrt hj]
  rw [he] at hm
  nlinarith only [h, hm]

end MatrixScale

lemma diagonal_error_tendsto (j A : ℝ) :
    Tendsto (fun n : ℕ => 2*sqrt A * sqrt (sqrt (2*j^2*A^2/(n:ℝ)))) atTop (𝓝 0) := by
  have hd : Tendsto (fun n : ℕ => 2*j^2*A^2/(n:ℝ)) atTop (𝓝 0) :=
    tendsto_const_div_atTop_nhds_zero_nat (2*j^2*A^2)
  simpa only [sqrt_zero, mul_zero, Function.comp_def] using
    (tendsto_const_nhds.mul (Real.continuous_sqrt.continuousAt.tendsto.comp
      (Real.continuous_sqrt.continuousAt.tendsto.comp hd)))

end SKRatioGaussian

end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI

	import Mathlib

	namespace OAI

	section
	section
	noncomputable section
	open MeasureTheory ProbabilityTheory InformationTheory Real Set
	open scoped NNReal ENNReal
	open Filter
	open scoped Topology
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius NNReal
	noncomputable section
	open MeasureTheory ProbabilityTheory Real Set Filter
	open MeasureTheory.Measure
	open scoped ENNReal NNReal MeasureTheory Topology
	open MeasureTheory
	noncomputable section
	noncomputable section
	open MeasureTheory Set NormedSpace
	open scoped Topology
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius
	noncomputable section
	open Set Real
	open scoped Topology
	noncomputable section
	open Matrix Set Filter
	open scoped Topology Matrix.Norms.Frobenius
	noncomputable section
	open Matrix NormedSpace ContinuousLinearMap
	open scoped Matrix.Norms.Frobenius
	noncomputable section
	open Matrix
	noncomputable section
	open MeasureTheory ProbabilityTheory Real Set
	open scoped ENNReal NNReal
	noncomputable section
	open MeasureTheory ProbabilityTheory InformationTheory Real Set
	open scoped NNReal ENNReal
	noncomputable section
	open scoped BigOperators
	open MeasureTheory ProbabilityTheory
	open Real
	noncomputable section
	open scoped BigOperators Topology
	open Filter Real
	namespace SKRatioGaussian
	section MatrixScale
	variable {ι : Type*} [Fintype ι]

	lemma diagonal_sum_le {a : ι → ℝ} {A : ℝ} (hA : ∀ i, a i ≤ A) :
	(∑ i, a i) ≤ (Fintype.card ι : ℝ) * A := by
	simpa only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul] using
	Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) => hA i)

	lemma diagonal_sq_sum_le {a : ι → ℝ} {A : ℝ}
	(ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) :
	(∑ i, (a i)^2) ≤ (Fintype.card ι : ℝ) * A^2 :=
	diagonal_sum_le (fun i => by nlinarith [ha i, hA i])

	lemma scaled_mean_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
	(hj : 0 ≤ j) (hA : ∀ i, a i ≤ A) :
	(j / (Fintype.card ι : ℝ)) * (∑ i, a i) ≤ j*A := by
	have hn : 0 < (Fintype.card ι : ℝ) := by exact_mod_cast Fintype.card_pos
	have h := mul_le_mul_of_nonneg_left (diagonal_sum_le hA) (div_nonneg hj hn.le)
	calc
	_ ≤ _ := h
	_ = _ := by field_simp

	lemma scaled_variance_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
	(ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) :
	2(j / (Fintype.card ι : ℝ))^2 (∑ i, (a i)^2) ≤
	2j^2A^2 / (Fintype.card ι : ℝ) := by
	have hn : 0 < (Fintype.card ι : ℝ) := by exact_mod_cast Fintype.card_pos
	have h := mul_le_mul_of_nonneg_left (diagonal_sq_sum_le ha hA)
	(show 0 ≤ 2*(j / (Fintype.card ι : ℝ))^2 by positivity)
	calc
	_ ≤ _ := h
	_ = _ := by field_simp

	lemma scaled_sqrt_mean_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
	(hj : 0 ≤ j) (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A) (hA0 : 0 ≤ A) :
	sqrt ((j / (Fintype.card ι : ℝ)) * (∑ i, a i)) * sqrt A ≤ sqrt j * A := by
	have hn : 0 ≤ (Fintype.card ι : ℝ) := by positivity
	have hq : 0 ≤ (j / (Fintype.card ι : ℝ)) * ∑ i, a i :=
	mul_nonneg (div_nonneg hj hn) (Finset.sum_nonneg (fun i _ => ha i))
	rw [← sqrt_mul hq]
	calc
	_ ≤ sqrt (j * A^2) := sqrt_le_sqrt (by
	have h := mul_le_mul_of_nonneg_right (scaled_mean_bound hj hA) hA0
	nlinarith only [h])
	_ = sqrt j * A := by rw [sqrt_mul hj, sqrt_sq hA0]

	lemma subcritical_endpoint_margin {b : ℝ} (hb0 : 0 ≤ b) (hb1 : b ≤ 1) :
	b^2 ≤ 1 - (1-b)^2/2 := by
	nlinarith [mul_nonneg (sub_nonneg.mpr hb1) (show 0 ≤ 3*b+1 by positivity)]

	lemma scaled_qA_bound [Nonempty ι] {j A : ℝ} {a : ι → ℝ}
	(hj : 0 ≤ j) (hA : ∀ i, a i ≤ A) (hA0 : 0 ≤ A)
	(hsub : sqrt j * A ≤ 1) :
	((j / (Fintype.card ι : ℝ)) * (∑ i, a i))*A ≤
	1-(1-sqrt j*A)^2/2 := by
	have h := mul_le_mul_of_nonneg_right (scaled_mean_bound hj hA) hA0
	have hm := subcritical_endpoint_margin (show 0 ≤ sqrt j*A by positivity) hsub
	have he : (sqrt jA)^2 = jA^2 := by rw [mul_pow, sq_sqrt hj]
	rw [he] at hm
	nlinarith only [h, hm]

	end MatrixScale

	lemma diagonal_error_tendsto (j A : ℝ) :
	Tendsto (fun n : ℕ => 2sqrt A sqrt (sqrt (2j^2A^2/(n:ℝ)))) atTop (𝓝 0) := by
	have hd : Tendsto (fun n : ℕ => 2j^2A^2/(n:ℝ)) atTop (𝓝 0) :=
	tendsto_const_div_atTop_nhds_zero_nat (2j^2A^2)
	simpa only [sqrt_zero, mul_zero, Function.comp_def] using
	(tendsto_const_nhds.mul (Real.continuous_sqrt.continuousAt.tendsto.comp
	(Real.continuous_sqrt.continuousAt.tendsto.comp hd)))

	end SKRatioGaussian

	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end

	end OAI