Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Probability.SKRatio.Matrices.GaussianCoordinateMemLp
import OAI.Probability.SKRatio.Matrices.IntegralAbsSet

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
namespace SKRatioGaussian
open Matrix MeasureTheory ProbabilityTheory Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal
variable {ι : Type*} [Fintype ι] [DecidableEq ι]

omit [DecidableEq ι] in
lemma abs_mul_entry_bound (M K : Matrix ι ι ℝ) (C : ℝ) (hK : ∀ a b, |K a b| ≤ C) (i : ι) :
    |(M*K) i i| ≤ C*∑ b, |M i b| := by
  rw [Matrix.mul_apply]
  apply (Finset.abs_sum_le_sum_abs _ _).trans
  rw [Finset.mul_sum]
  apply Finset.sum_le_sum
  intro b _
  rw [abs_mul,mul_comm C]
  exact mul_le_mul_of_nonneg_left (hK b i) (abs_nonneg _)

omit [DecidableEq ι] in

theorem goe_diagonal_inverse_loop {r : ℝ} (hr : 0 ≤ r) (a c : ι → ℝ) (i : ι)
    {K : (MatrixCoordinates ι → ℝ) → Matrix ι ι ℝ} {C : ℝ}
    {s : Set (MatrixCoordinates ι → ℝ)} (hK : Continuous K)
    (hC : ∀ g a b, |K g a b| ≤ C) (hs : MeasurableSet s)
    (he : ∀ g ∈ s, K g i i = 1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) :
    let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1)
    |(∫ g, K g i i ∂μ)-
      (1+a i*((∫ g, (goeMatrix r g*K g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)))| ≤
      (C+1+|a i*c i| *C)*μ.real sᶜ+
        (|a i| *C)*(Fintype.card ι:ℝ)*sqrt (2*r)*sqrt (μ.real sᶜ) := by
  intro μ
  have hC0 : 0 ≤ C := (abs_nonneg _).trans (hC 0 i i)
  have hKi (a b : ι) : Integrable (fun g => K g a b) μ :=
    Integrable.mono' (integrable_const C) (hK.matrix_elem a b).aestronglyMeasurable
      (ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g a b))
  have hW (b : ι) : Integrable (fun g => goeMatrix r g i b) μ :=
    (goeEntry_memLp r i b).integrable (by norm_num)
  have hWK (b : ι) : Integrable (fun g => goeMatrix r g i b*K g b i) μ := by
    simpa only [mul_comm] using (hW b).bdd_mul (hK.matrix_elem b i).aestronglyMeasurable
      (ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g b i))
  have hWI : Integrable (fun g => (goeMatrix r g*K g) i i) μ :=
    integrable_finsetSum _ (fun b _ => hWK b)
  let A := C+1+|a i*c i| *C
  let B := |a i| *C
  let H := fun g => A+B*∑ b, |goeMatrix r g i b|
  have hHA : Integrable (fun g => ∑ b, |goeMatrix r g i b|) μ :=
    integrable_finsetSum _ (fun b _ => (hW b).norm)
  have hHI : Integrable H μ := (integrable_const A).add (hHA.const_mul B)
  have hGI : Integrable (fun g => 1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) μ :=
    (integrable_const 1).add ((hWI.sub ((hKi i i).const_mul _)).const_mul _)
  have hb (g : MatrixCoordinates ι → ℝ) :
      ‖K g i i-(1+a i*((goeMatrix r g*K g) i i-c i*K g i i))‖ ≤ H g := by
    rw [Real.norm_eq_abs]
    have hid : K g i i-(1+a i*((goeMatrix r g*K g) i i-c i*K g i i)) =
        (K g i i-1)+(a i*c i)*K g i i+(-a i)*(goeMatrix r g*K g) i i := by ring
    rw [hid]
    apply (abs_add_le _ _).trans
    apply (add_le_add (abs_add_le _ _) le_rfl).trans
    have hk1 : |K g i i-1| ≤ C+1 := (abs_sub _ _).trans (by simpa using add_le_add_right (hC g i i) 1)
    have hk2 : |(a i*c i)*K g i i| ≤ |a i*c i| *C := by
      rw [abs_mul]; exact mul_le_mul_of_nonneg_left (hC g i i) (abs_nonneg _)
    have hk3 : |(-a i)*(goeMatrix r g*K g) i i| ≤ |a i| *(C*∑ b, |goeMatrix r g i b|) := by
      rw [abs_mul,abs_neg]
      exact mul_le_mul_of_nonneg_left (abs_mul_entry_bound _ _ C (hC g) i) (abs_nonneg _)
    exact (add_le_add (add_le_add hk1 hk2) hk3).trans_eq (by dsimp [H,A,B]; ring)
  have hraw := integral_exceptional_equal_bound hs.compl (hKi i i) hGI hHI
    (fun g hg => he g (by simpa using hg)) (fun g _ => hb g)
  have hbad (b : ι) : (∫ g in sᶜ, |goeMatrix r g i b| ∂μ) ≤
      sqrt (2*r)*sqrt (μ.real sᶜ) := by
    apply (integral_abs_on_set_le (goeEntry_memLp r i b) hs.compl).trans
    exact mul_le_mul_of_nonneg_right (sqrt_le_sqrt (goeEntry_integral_sq_le hr i b)) (sqrt_nonneg _)
  have hHr : (∫ g in sᶜ, H g ∂μ) ≤ A*μ.real sᶜ+B*(Fintype.card ι:ℝ)*sqrt (2*r)*sqrt (μ.real sᶜ) := by
    rw [show H = fun g => A+B*∑ b, |goeMatrix r g i b| from rfl,
      integral_add (integrable_const A) ((hHA.const_mul B).mono_measure Measure.restrict_le_self),
      integral_const,integral_const_mul]
    have hWa (b : ι) : Integrable (fun g => |goeMatrix r g i b|) (μ.restrict sᶜ) := by
      simpa only [Real.norm_eq_abs] using (hW b).norm.mono_measure Measure.restrict_le_self
    rw [integral_finsetSum _ (fun b _ => hWa b)]
    simp only [measureReal_restrict_apply_univ,smul_eq_mul]
    calc
      _ ≤ μ.real sᶜ*A+B*(∑ _b : ι, sqrt (2*r)*sqrt (μ.real sᶜ)) :=
        add_le_add le_rfl (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun b _ => hbad b))
          (mul_nonneg (abs_nonneg _) hC0))
      _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
  have heI : (∫ g, 1+a i*((goeMatrix r g*K g) i i-c i*K g i i) ∂μ) =
      1+a i*((∫ g, (goeMatrix r g*K g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)) := by
    have ht := integral_add (integrable_const (1:ℝ))
      ((hWI.sub ((hKi i i).const_mul (c i))).const_mul (a i))
    change (∫ g, 1+a i*((goeMatrix r g*K g) i i-c i*K g i i) ∂μ) = _ at ht
    rw [ht,integral_const_mul]
    simp only [Pi.sub_apply]
    rw [integral_sub hWI ((hKi i i).const_mul (c i)),integral_const_mul]
    simp
  rw [heI] at hraw
  exact hraw.trans hHr
end SKRatioGaussian
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI

	import Mathlib
	import OAI.Probability.SKRatio.Matrices.GaussianCoordinateMemLp
	import OAI.Probability.SKRatio.Matrices.IntegralAbsSet

	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
	namespace SKRatioGaussian
	open Matrix MeasureTheory ProbabilityTheory Real
	open scoped BigOperators Matrix.Norms.Frobenius NNReal ENNReal
	variable {ι : Type*} [Fintype ι] [DecidableEq ι]

	omit [DecidableEq ι] in
	lemma abs_mul_entry_bound (M K : Matrix ι ι ℝ) (C : ℝ) (hK : ∀ a b, \|K a b\| ≤ C) (i : ι) :
	\|(MK) i i\| ≤ C∑ b, \|M i b\| := by
	rw [Matrix.mul_apply]
	apply (Finset.abs_sum_le_sum_abs _ _).trans
	rw [Finset.mul_sum]
	apply Finset.sum_le_sum
	intro b _
	rw [abs_mul,mul_comm C]
	exact mul_le_mul_of_nonneg_left (hK b i) (abs_nonneg _)

	omit [DecidableEq ι] in

	theorem goe_diagonal_inverse_loop {r : ℝ} (hr : 0 ≤ r) (a c : ι → ℝ) (i : ι)
	{K : (MatrixCoordinates ι → ℝ) → Matrix ι ι ℝ} {C : ℝ}
	{s : Set (MatrixCoordinates ι → ℝ)} (hK : Continuous K)
	(hC : ∀ g a b, \|K g a b\| ≤ C) (hs : MeasurableSet s)
	(he : ∀ g ∈ s, K g i i = 1+a i((goeMatrix r gK g) i i-c i*K g i i)) :
	let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1)
	\|(∫ g, K g i i ∂μ)-
	(1+a i((∫ g, (goeMatrix r gK g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)))\| ≤
	(C+1+\|a ic i\| C)*μ.real sᶜ+
	(\|a i\| C)(Fintype.card ι:ℝ)sqrt (2r)*sqrt (μ.real sᶜ) := by
	intro μ
	have hC0 : 0 ≤ C := (abs_nonneg _).trans (hC 0 i i)
	have hKi (a b : ι) : Integrable (fun g => K g a b) μ :=
	Integrable.mono' (integrable_const C) (hK.matrix_elem a b).aestronglyMeasurable
	(ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g a b))
	have hW (b : ι) : Integrable (fun g => goeMatrix r g i b) μ :=
	(goeEntry_memLp r i b).integrable (by norm_num)
	have hWK (b : ι) : Integrable (fun g => goeMatrix r g i b*K g b i) μ := by
	simpa only [mul_comm] using (hW b).bdd_mul (hK.matrix_elem b i).aestronglyMeasurable
	(ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hC g b i))
	have hWI : Integrable (fun g => (goeMatrix r g*K g) i i) μ :=
	integrable_finsetSum _ (fun b _ => hWK b)
	let A := C+1+\|a ic i\| C
	let B := \|a i\| *C
	let H := fun g => A+B*∑ b, \|goeMatrix r g i b\|
	have hHA : Integrable (fun g => ∑ b, \|goeMatrix r g i b\|) μ :=
	integrable_finsetSum _ (fun b _ => (hW b).norm)
	have hHI : Integrable H μ := (integrable_const A).add (hHA.const_mul B)
	have hGI : Integrable (fun g => 1+a i((goeMatrix r gK g) i i-c i*K g i i)) μ :=
	(integrable_const 1).add ((hWI.sub ((hKi i i).const_mul _)).const_mul _)
	have hb (g : MatrixCoordinates ι → ℝ) :
	‖K g i i-(1+a i((goeMatrix r gK g) i i-c i*K g i i))‖ ≤ H g := by
	rw [Real.norm_eq_abs]
	have hid : K g i i-(1+a i((goeMatrix r gK g) i i-c i*K g i i)) =
	(K g i i-1)+(a ic i)K g i i+(-a i)(goeMatrix r gK g) i i := by ring
	rw [hid]
	apply (abs_add_le _ _).trans
	apply (add_le_add (abs_add_le _ _) le_rfl).trans
	have hk1 : \|K g i i-1\| ≤ C+1 := (abs_sub _ _).trans (by simpa using add_le_add_right (hC g i i) 1)
	have hk2 : \|(a ic i)K g i i\| ≤ \|a ic i\| C := by
	rw [abs_mul]; exact mul_le_mul_of_nonneg_left (hC g i i) (abs_nonneg _)
	have hk3 : \|(-a i)(goeMatrix r gK g) i i\| ≤ \|a i\| (C∑ b, \|goeMatrix r g i b\|) := by
	rw [abs_mul,abs_neg]
	exact mul_le_mul_of_nonneg_left (abs_mul_entry_bound _ _ C (hC g) i) (abs_nonneg _)
	exact (add_le_add (add_le_add hk1 hk2) hk3).trans_eq (by dsimp [H,A,B]; ring)
	have hraw := integral_exceptional_equal_bound hs.compl (hKi i i) hGI hHI
	(fun g hg => he g (by simpa using hg)) (fun g _ => hb g)
	have hbad (b : ι) : (∫ g in sᶜ, \|goeMatrix r g i b\| ∂μ) ≤
	sqrt (2r)sqrt (μ.real sᶜ) := by
	apply (integral_abs_on_set_le (goeEntry_memLp r i b) hs.compl).trans
	exact mul_le_mul_of_nonneg_right (sqrt_le_sqrt (goeEntry_integral_sq_le hr i b)) (sqrt_nonneg _)
	have hHr : (∫ g in sᶜ, H g ∂μ) ≤ Aμ.real sᶜ+B(Fintype.card ι:ℝ)sqrt (2r)*sqrt (μ.real sᶜ) := by
	rw [show H = fun g => A+B*∑ b, \|goeMatrix r g i b\| from rfl,
	integral_add (integrable_const A) ((hHA.const_mul B).mono_measure Measure.restrict_le_self),
	integral_const,integral_const_mul]
	have hWa (b : ι) : Integrable (fun g => \|goeMatrix r g i b\|) (μ.restrict sᶜ) := by
	simpa only [Real.norm_eq_abs] using (hW b).norm.mono_measure Measure.restrict_le_self
	rw [integral_finsetSum _ (fun b _ => hWa b)]
	simp only [measureReal_restrict_apply_univ,smul_eq_mul]
	calc
	_ ≤ μ.real sᶜA+B(∑ _b : ι, sqrt (2r)sqrt (μ.real sᶜ)) :=
	add_le_add le_rfl (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun b _ => hbad b))
	(mul_nonneg (abs_nonneg _) hC0))
	_ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
	have heI : (∫ g, 1+a i((goeMatrix r gK g) i i-c i*K g i i) ∂μ) =
	1+a i((∫ g, (goeMatrix r gK g) i i ∂μ)-c i*(∫ g, K g i i ∂μ)) := by
	have ht := integral_add (integrable_const (1:ℝ))
	((hWI.sub ((hKi i i).const_mul (c i))).const_mul (a i))
	change (∫ g, 1+a i((goeMatrix r gK g) i i-c i*K g i i) ∂μ) = _ at ht
	rw [ht,integral_const_mul]
	simp only [Pi.sub_apply]
	rw [integral_sub hWI ((hKi i i).const_mul (c i)),integral_const_mul]
	simp
	rw [heI] at hraw
	exact hraw.trans hHr

	end SKRatioGaussian

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

	end OAI