Datasets:

roseking
/

openAI-math-dataset

import OAI.Analysis.DirectCrouzeix.ExponentialSupport

namespace OAI

noncomputable section

open scoped Matrix Matrix.Norms.L2Operator Kronecker

noncomputable section

open MeasureTheory Set Filter Metric

open scoped Topology Interval ENNReal NNReal ComplexConjugate

noncomputable section

open Filter Metric Set

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set Filter Metric

open scoped Topology ComplexConjugate

noncomputable section

open Set

open scoped ComplexConjugate Matrix

namespace DirectCrouzeix.Geometry

theorem exists_exponential_outer {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
    (hv : Convex ℝ K) {ε : ℝ} (hε : 0 < ε) :
    ∃ (D : Finset ℂ) (v : D → ℂ) (b : D → ℝ),
      (∀ z ∈ K, expLevel v b z < 1) ∧
      IsCompact {z | expLevel v b z ≤ 1} ∧
      (∀ z, expLevel v b z ≤ 1 → ∃ w ∈ K, ‖z-w‖ < ε) := by
  classical
  obtain ⟨M,hM0,hM⟩ := hK.isBounded.exists_pos_norm_le
  let L := 2*(M+1)
  let C := 1+L+M
  have hC : 0 < C := by dsimp [C,L]; linarith
  let δ := min (1/2) (ε/(2*C))
  have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (by positivity))
  have hδ1 : δ < 1 := lt_of_le_of_lt (min_le_left _ _) (by norm_num)
  have hδε : δ*C < ε := by
    have hh : δ ≤ ε/(2*C) := min_le_right _ _
    have hc := (le_div_iff₀ (by positivity : 0 < 2*C)).mp hh
    nlinarith
  obtain ⟨D,hD,h1,hm1,hI,hmI,hnet⟩ := exists_axis_net hδ
  let N : ℝ := Fintype.card D
  have hN : 0 ≤ N := by dsimp [N]; positivity
  let t := (N+1)/δ
  have ht : 0 < t := div_pos (by linarith) hδ
  have htδ : t*δ = N+1 := div_mul_cancel₀ _ (ne_of_gt hδ)
  let v : D → ℂ := fun ν => t • (ν:ℂ)
  let b : D → ℝ := fun ν => -t*(convexSupport K ν+δ)
  have he (z : ℂ) (ν : D) : inner ℝ (v ν) z+b ν = t*(inner ℝ (ν:ℂ) z-convexSupport K ν-δ) := by
    dsimp only [v,b]
    rw [real_inner_smul_left]
    ring
  have hsumlt : N*Real.exp (-t*δ) < 1 := by
    rw [neg_mul,htδ,Real.exp_neg,mul_inv_lt_iff₀ (Real.exp_pos _),one_mul]
    linarith [Real.add_one_le_exp (N+1)]
  have hinside (z : ℂ) (hz : z ∈ K) : expLevel v b z < 1 := by
    calc
      expLevel v b z ≤ ∑ _ν : D, Real.exp (-t*δ) := by
        apply Finset.sum_le_sum
        intro ν hν
        rw [he]
        apply Real.exp_le_exp.mpr
        have hh := mul_le_mul_of_nonneg_left (le_convexSupport hK hz (ν:ℂ)) (le_of_lt ht)
        nlinarith
      _ = N*Real.exp (-t*δ) := by simp [N]
      _ < 1 := hsumlt
  have hsupport (z : ℂ) (hz : expLevel v b z ≤ 1) (ν : ℂ) (hν : ν ∈ D) :
      inner ℝ ν z ≤ convexSupport K ν+δ := by
    have hh : Real.exp (inner ℝ (v ⟨ν,hν⟩) z+b ⟨ν,hν⟩) ≤ expLevel v b z :=
      Finset.single_le_sum (f := fun i : D => Real.exp (inner ℝ (v i) z+b i))
        (fun i _ => Real.exp_nonneg _) (Finset.mem_univ (⟨ν,hν⟩ : D))
    have hex := (Real.exp_le_one_iff).mp (hh.trans hz)
    rw [he] at hex
    nlinarith
  have hbound (z : ℂ) (hz : expLevel v b z ≤ 1) : ‖z‖ ≤ L := by
    have hh := axis_support_bound hne hM h1 hm1 hI hmI (hsupport z hz)
    dsimp [L]
    linarith
  refine ⟨D,v,b,hinside,?_,?_⟩
  · apply (isCompact_closedBall (0:ℂ) L).of_isClosed_subset
    · exact isClosed_le (expLevel_contDiff v b).continuous continuous_const
    · intro z hz
      exact mem_closedBall_zero_iff.mpr (hbound z hz)
  · intro z hz
    obtain ⟨w,hw,hzw⟩ := direction_net_distance hK hne hv hδ hM hnet (hbound z hz) (hsupport z hz)
    exact ⟨w,hw,lt_of_le_of_lt hzw hδε⟩

end DirectCrouzeix.Geometry

namespace DirectCrouzeix

@[simp] theorem matrixPolynomialValue_coefficientPolynomial {m d : ℕ}
    (B : Fin (d+1) → Matrix (Fin m) (Fin m) ℂ) (z : ℂ) :
    matrixPolynomialValue z (coefficientPolynomial B) = polynomialValue B z := by
  simp only [coefficientPolynomial,map_sum,map_mul,map_pow,matrixPolynomialValue_C,
    matrixPolynomialValue_X,smul_pow,one_pow,Matrix.mul_smul,Matrix.mul_one,polynomialValue]

@[simp] theorem matrixPolynomialValue_smul {m : ℕ} (F : MatrixPolynomial m) (s z : ℂ) :
    matrixPolynomialValue z (s • F) = s • matrixPolynomialValue z F := by
  rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
    map_mul,matrixPolynomialValue_C,Matrix.smul_mul,Matrix.one_mul]

@[simp] theorem tensorPolynomial_smul {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)
    (F : MatrixPolynomial m) (s : ℂ) :
    tensorPolynomial A (s • F) = s • tensorPolynomial A F := by
  rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
    map_mul,tensorPolynomial_C,Matrix.kronecker_smul,Matrix.one_kronecker_one,
    Matrix.smul_mul,Matrix.one_mul]

end DirectCrouzeix

namespace DirectCrouzeix

open Set Filter Metric

end DirectCrouzeix

end

end

end

end

end

end

end

end

end

end OAI

	import OAI.Analysis.DirectCrouzeix.ExponentialSupport

	namespace OAI

	noncomputable section

	open scoped Matrix Matrix.Norms.L2Operator Kronecker

	noncomputable section

	open MeasureTheory Set Filter Metric

	open scoped Topology Interval ENNReal NNReal ComplexConjugate

	noncomputable section

	open Filter Metric Set

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set Filter Metric

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set Filter Metric

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set Filter Metric

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set Filter Metric

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set Filter Metric

	open scoped Topology ComplexConjugate

	noncomputable section

	open Set

	open scoped ComplexConjugate Matrix

	namespace DirectCrouzeix.Geometry

	theorem exists_exponential_outer {K : Set ℂ} (hK : IsCompact K) (hne : K.Nonempty)
	(hv : Convex ℝ K) {ε : ℝ} (hε : 0 < ε) :
	∃ (D : Finset ℂ) (v : D → ℂ) (b : D → ℝ),
	(∀ z ∈ K, expLevel v b z < 1) ∧
	IsCompact {z \| expLevel v b z ≤ 1} ∧
	(∀ z, expLevel v b z ≤ 1 → ∃ w ∈ K, ‖z-w‖ < ε) := by
	classical
	obtain ⟨M,hM0,hM⟩ := hK.isBounded.exists_pos_norm_le
	let L := 2*(M+1)
	let C := 1+L+M
	have hC : 0 < C := by dsimp [C,L]; linarith
	let δ := min (1/2) (ε/(2*C))
	have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (by positivity))
	have hδ1 : δ < 1 := lt_of_le_of_lt (min_le_left _ _) (by norm_num)
	have hδε : δ*C < ε := by
	have hh : δ ≤ ε/(2*C) := min_le_right _ _
	have hc := (le_div_iff₀ (by positivity : 0 < 2*C)).mp hh
	nlinarith
	obtain ⟨D,hD,h1,hm1,hI,hmI,hnet⟩ := exists_axis_net hδ
	let N : ℝ := Fintype.card D
	have hN : 0 ≤ N := by dsimp [N]; positivity
	let t := (N+1)/δ
	have ht : 0 < t := div_pos (by linarith) hδ
	have htδ : t*δ = N+1 := div_mul_cancel₀ _ (ne_of_gt hδ)
	let v : D → ℂ := fun ν => t • (ν:ℂ)
	let b : D → ℝ := fun ν => -t*(convexSupport K ν+δ)
	have he (z : ℂ) (ν : D) : inner ℝ (v ν) z+b ν = t*(inner ℝ (ν:ℂ) z-convexSupport K ν-δ) := by
	dsimp only [v,b]
	rw [real_inner_smul_left]
	ring
	have hsumlt : NReal.exp (-tδ) < 1 := by
	rw [neg_mul,htδ,Real.exp_neg,mul_inv_lt_iff₀ (Real.exp_pos _),one_mul]
	linarith [Real.add_one_le_exp (N+1)]
	have hinside (z : ℂ) (hz : z ∈ K) : expLevel v b z < 1 := by
	calc
	expLevel v b z ≤ ∑ _ν : D, Real.exp (-t*δ) := by
	apply Finset.sum_le_sum
	intro ν hν
	rw [he]
	apply Real.exp_le_exp.mpr
	have hh := mul_le_mul_of_nonneg_left (le_convexSupport hK hz (ν:ℂ)) (le_of_lt ht)
	nlinarith
	_ = NReal.exp (-tδ) := by simp [N]
	_ < 1 := hsumlt
	have hsupport (z : ℂ) (hz : expLevel v b z ≤ 1) (ν : ℂ) (hν : ν ∈ D) :
	inner ℝ ν z ≤ convexSupport K ν+δ := by
	have hh : Real.exp (inner ℝ (v ⟨ν,hν⟩) z+b ⟨ν,hν⟩) ≤ expLevel v b z :=
	Finset.single_le_sum (f := fun i : D => Real.exp (inner ℝ (v i) z+b i))
	(fun i _ => Real.exp_nonneg _) (Finset.mem_univ (⟨ν,hν⟩ : D))
	have hex := (Real.exp_le_one_iff).mp (hh.trans hz)
	rw [he] at hex
	nlinarith
	have hbound (z : ℂ) (hz : expLevel v b z ≤ 1) : ‖z‖ ≤ L := by
	have hh := axis_support_bound hne hM h1 hm1 hI hmI (hsupport z hz)
	dsimp [L]
	linarith
	refine ⟨D,v,b,hinside,?_,?_⟩
	· apply (isCompact_closedBall (0:ℂ) L).of_isClosed_subset
	· exact isClosed_le (expLevel_contDiff v b).continuous continuous_const
	· intro z hz
	exact mem_closedBall_zero_iff.mpr (hbound z hz)
	· intro z hz
	obtain ⟨w,hw,hzw⟩ := direction_net_distance hK hne hv hδ hM hnet (hbound z hz) (hsupport z hz)
	exact ⟨w,hw,lt_of_le_of_lt hzw hδε⟩

	end DirectCrouzeix.Geometry

	namespace DirectCrouzeix

	@[simp] theorem matrixPolynomialValue_coefficientPolynomial {m d : ℕ}
	(B : Fin (d+1) → Matrix (Fin m) (Fin m) ℂ) (z : ℂ) :
	matrixPolynomialValue z (coefficientPolynomial B) = polynomialValue B z := by
	simp only [coefficientPolynomial,map_sum,map_mul,map_pow,matrixPolynomialValue_C,
	matrixPolynomialValue_X,smul_pow,one_pow,Matrix.mul_smul,Matrix.mul_one,polynomialValue]

	@[simp] theorem matrixPolynomialValue_smul {m : ℕ} (F : MatrixPolynomial m) (s z : ℂ) :
	matrixPolynomialValue z (s • F) = s • matrixPolynomialValue z F := by
	rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
	map_mul,matrixPolynomialValue_C,Matrix.smul_mul,Matrix.one_mul]

	@[simp] theorem tensorPolynomial_smul {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)
	(F : MatrixPolynomial m) (s : ℂ) :
	tensorPolynomial A (s • F) = s • tensorPolynomial A F := by
	rw [← smul_one_smul (Matrix (Fin m) (Fin m) ℂ) s F,Polynomial.smul_eq_C_mul,
	map_mul,tensorPolynomial_C,Matrix.kronecker_smul,Matrix.one_kronecker_one,
	Matrix.smul_mul,Matrix.one_mul]

	end DirectCrouzeix

	namespace DirectCrouzeix

	open Set Filter Metric

	end DirectCrouzeix

	end

	end

	end

	end

	end

	end

	end

	end

	end

	end OAI