| import Mathlib |
| import OAI.Computability.MinUncut.Analysis.LowGradientEnergyLowerWeighted |
| import OAI.Computability.MinUncut.Analysis.FreshCorrelation |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory GaussianHermite RowNoise |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| def sourceSigma (J : ℝ) : ℝ := 1/(2560*J) |
| def sourceEta (J : ℝ) : ℝ := (sourceSigma J)^2/(2560*J) |
| def sourceR (J : ℝ) : ℕ := ⌈(sourceEta J)⁻¹^2⌉₊ |
| def sourceT (J : ℝ) : ℕ := ⌈32*J/(sourceEta J*(sourceSigma J)^4)⌉₊ |
|
|
| lemma sourceSigma_pos {J : ℝ} (hJ : 1 ≤ J) : 0 < sourceSigma J := by |
| unfold sourceSigma |
| positivity |
| lemma sourceEta_pos {J : ℝ} (hJ : 1 ≤ J) : 0 < sourceEta J := by |
| unfold sourceEta |
| have hs := sourceSigma_pos hJ |
| positivity |
| lemma sourceSigma_le_one {J : ℝ} (hJ : 1 ≤ J) : sourceSigma J ≤ 1 := by |
| unfold sourceSigma |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < 2560*J)).mpr |
| linarith |
| lemma sourceEta_le_sigma {J : ℝ} (hJ : 1 ≤ J) : sourceEta J ≤ sourceSigma J := by |
| have hs := sourceSigma_pos hJ |
| have hs1 := sourceSigma_le_one hJ |
| have hJpos : 0 < J := by linarith |
| unfold sourceEta |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < 2560*J)).mpr |
| nlinarith |
| lemma sourceR_bound {J : ℝ} (hJ : 1 ≤ J) : 1 ≤ (sourceR J:ℝ)*(sourceEta J)^2 := by |
| have hη := sourceEta_pos hJ |
| have hh := Nat.le_ceil ((sourceEta J)⁻¹^2) |
| have hw := mul_le_mul_of_nonneg_right hh (sq_nonneg (sourceEta J)) |
| have he : (sourceEta J)⁻¹^2*(sourceEta J)^2=1 := by field_simp |
| simpa only [he,sourceR] using hw |
| lemma sourceT_pos {J : ℝ} (hJ : 1 ≤ J) : 0 < sourceT J := by |
| apply Nat.ceil_pos.mpr |
| have hη := sourceEta_pos hJ |
| have hσ := sourceSigma_pos hJ |
| positivity |
| lemma sourceT_bound {J : ℝ} (hJ : 1 ≤ J) : |
| 32*J ≤ (sourceT J:ℝ)*sourceEta J*(sourceSigma J)^4 := by |
| have hη := sourceEta_pos hJ |
| have hσ := sourceSigma_pos hJ |
| have hh := Nat.le_ceil (32*J/(sourceEta J*(sourceSigma J)^4)) |
| exact (div_le_iff₀ (by positivity)).mp hh |>.trans_eq (by unfold sourceT; ring) |
|
|
| lemma source_row_rate {J : ℝ} (hJ : 1 ≤ J) {m : ℕ} (hm : 0 < m) : |
| 0 ≤ (sourceEta J)^2/(m:ℝ) ∧ (sourceEta J)^2/(m:ℝ) ≤ 1 ∧ |
| ∀ k : ℕ, sourceR J*m < k → 1 ≤ (k:ℝ)*((sourceEta J)^2/(m:ℝ)) := by |
| have hm' : (0:ℝ) < m := by exact_mod_cast hm |
| have hm1 : (1:ℝ) ≤ m := by exact_mod_cast hm |
| have hetapos := sourceEta_pos hJ |
| have heta1 : sourceEta J ≤ 1 := (sourceEta_le_sigma hJ).trans (sourceSigma_le_one hJ) |
| refine ⟨by positivity,?_,?_⟩ |
| · apply (div_le_iff₀ hm').mpr |
| nlinarith |
| · intro k hk |
| have hk' : (sourceR J:ℝ)*(m:ℝ) ≤ k := by exact_mod_cast hk.le |
| have hh := mul_le_mul_of_nonneg_right hk' (show 0 ≤ (sourceEta J)^2/(m:ℝ) by positivity) |
| have he : (sourceR J:ℝ)*(m:ℝ)*((sourceEta J)^2/(m:ℝ)) = |
| (sourceR J:ℝ)*(sourceEta J)^2 := by field_simp |
| rw [he] at hh |
| exact (sourceR_bound hJ).trans hh |
| end MinUncut.Inner |
|
|
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory GaussianHermite RowNoise |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {Ξ : Type*} [Fintype Ξ] (V A : Ξ → Type*) |
| [∀ ξ, AddCommGroup (V ξ)] [∀ ξ, Module F₂ (V ξ)] [∀ ξ, AddTorsor (V ξ) (A ξ)] |
| [∀ ξ, Fintype (A ξ)] {m n : ℕ} |
|
|
| theorem source_lowGradient_energy (w : Ξ → ℝ) (hw : ∀ ξ, 0 ≤ w ξ) |
| (hw1 : ∑ ξ, w ξ=1) (f : ∀ ξ, FoldedProof (A ξ)) {J : ℝ} (hJ : 1 ≤ J) |
| (hm : 2 ≤ m) (hn : 2 ≤ n) (s : ℕ) |
| (hs : ∀ k : ℕ, s+2 ≤ k → (k:ℝ)*(smoothingRho (sourceSigma J))^(2*k) ≤ 1/16) |
| (h1 : (∑ ξ, w ξ*firstError (m := m) (n := n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J*(10*sourceSigma J)) |
| (h2 : (∑ ξ, w ξ*secondError (m := m) (n := n) (f ξ) |
| ((sourceEta J)^2/(m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ J*(10*(sourceEta J+sourceEta J))) : |
| 1/2 ≤ (∑ ξ, w ξ*averagedEnergy |
| (lowGradient (m := m) (n := n) (sourceR J*m) s (f ξ) (sourceSigma J) (sourceEta J))) ∧ |
| (∑ ξ, w ξ*averagedEnergy |
| (lowGradient (m := m) (n := n) (sourceR J*m) s (f ξ) (sourceSigma J) (sourceEta J))) ≤ |
| (sourceSigma J)⁻¹^2 := by |
| have hm0 : 0 < m := by omega |
| have hn0 : 0 < n := by omega |
| have hσ := sourceSigma_pos hJ |
| have hη := sourceEta_pos hJ |
| obtain ⟨ha,ha1,hD⟩ := source_row_rate hJ hm0 |
| have hL := lowGradient_energy_lower_weighted (m := m) (n := n) V A w hw hw1 f hn0 hσ.ne' hη.ne' ha ha1 |
| (by norm_num : (0:ℝ) ≤ 1/16) (sourceR J*m) s hD hs |
| have hJ0 : J≠0 := ne_of_gt (by linarith) |
| have hc1 : 4*(J*(10*sourceSigma J))=1/64 := by |
| unfold sourceSigma |
| field_simp |
| ring |
| have hc2 : 8*(sourceSigma J)⁻¹^2*(J*(10*(sourceEta J+sourceEta J)))=1/16 := by |
| unfold sourceEta sourceSigma |
| field_simp |
| ring |
| have hf1 := mul_le_mul_of_nonneg_left h1 (by norm_num : (0:ℝ) ≤ 4) |
| have hf2 := mul_le_mul_of_nonneg_left h2 (show 0 ≤ 8*(sourceSigma J)⁻¹^2 by positivity) |
| rw [hc1] at hf1 |
| rw [hc2] at hf2 |
| constructor |
| · linarith |
| · calc |
| _ ≤ ∑ ξ, w ξ*(sourceSigma J)⁻¹^2 := Finset.sum_le_sum |
| (fun ξ _ => mul_le_mul_of_nonneg_left |
| (averagedEnergy_lowGradient_le (m := m) (sourceR J*m) s (f ξ) hn0 hσ.ne' (sourceEta J)) (hw ξ)) |
| _ = _ := by rw [← Finset.sum_mul,hw1,one_mul] |
|
|
| theorem source_lowGradient_correlation (w : Ξ → ℝ) (hw : ∀ ξ, 0 ≤ w ξ) |
| (hw1 : ∑ ξ, w ξ=1) (f : ∀ ξ, FoldedProof (A ξ)) {J : ℝ} (hJ : 1 ≤ J) |
| (hm : 2 ≤ m) (hn : 2 ≤ n) (s : ℕ) |
| (hs : ∀ k : ℕ, s+2 ≤ k → (k:ℝ)*(smoothingRho (sourceSigma J))^(2*k) ≤ 1/16) |
| (h1 : (∑ ξ, w ξ*firstError (m := m) (n := n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J*(10*sourceSigma J)) |
| (h2 : (∑ ξ, w ξ*secondError (m := m) (n := n) (f ξ) |
| ((sourceEta J)^2/(m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ J*(10*(sourceEta J+sourceEta J))) |
| (h3 : (∑ ξ, w ξ*thirdError (m := m) (n := n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J/(Fintype.card (Code m n):ℝ)) : |
| 1/(4*(sourceT J:ℝ)) ≤ ∑ ξ, w ξ*averagedCorrelation |
| (lowGradient (m := m) (n := n) (sourceR J*m) s (f ξ) (sourceSigma J) (sourceEta J)) := by |
| have hn0 : 0 < n := by omega |
| have hσ := sourceSigma_pos hJ |
| have hη := sourceEta_pos hJ |
| have hT : (0:ℝ) < sourceT J := by exact_mod_cast sourceT_pos hJ |
| obtain ⟨hE,hE'⟩ := source_lowGradient_energy V A w hw hw1 f hJ hm hn s hs h1 h2 |
| have hU := lowGradient_correlation_truncated_weighted (m := m) (n := n) V A (sourceR J*m) s w hw f hn0 hσ.ne' hη hT |
| (show 0 < (sourceSigma J)^2/8 by positivity) |
| have he := mul_le_mul_of_nonneg_left hE' |
| (show 0 ≤ (sourceSigma J)^2/8 by positivity) |
| have hea : ((sourceSigma J)^2/8)*(sourceSigma J)⁻¹^2=1/8 := by field_simp |
| rw [hea] at he |
| have hc : (0:ℝ) < Fintype.card (Code m n) := by positivity |
| have hz := mul_le_mul_of_nonneg_left h3 |
| (show 0 ≤ ((sourceSigma J)⁻¹^2/(4*((sourceSigma J)^2/8)*(sourceT J:ℝ)))* |
| ((Fintype.card (Code m n):ℝ)/sourceEta J) by positivity) |
| have hzb : ((sourceSigma J)⁻¹^2/(4*((sourceSigma J)^2/8)*(sourceT J:ℝ)))* |
| ((Fintype.card (Code m n):ℝ)/sourceEta J)*(J/(Fintype.card (Code m n):ℝ)) = |
| 2*J/(sourceEta J*(sourceSigma J)^4*(sourceT J:ℝ)) := by |
| field_simp |
| ring |
| rw [hzb] at hz |
| have hzt : 2*J/(sourceEta J*(sourceSigma J)^4*(sourceT J:ℝ)) ≤ 1/16 := by |
| apply (div_le_iff₀ (by positivity)).mpr |
| nlinarith [sourceT_bound hJ] |
| apply (div_le_iff₀ (by positivity : (0:ℝ) < 4*(sourceT J:ℝ))).mpr |
| nlinarith |
| end MinUncut.Inner |
|
|
| end |
| end |
|
|
| end OAI |
|
|
