| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Assembly |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Inner |
| open MeasureTheory ProbabilityTheory BinaryFourier RowNoise GaussianHermite Subbox |
|
|
| structure InnerParameters (J : ℝ) where |
| s : ℕ |
| m : ℕ |
| k : ℕ |
| n : ℕ |
| H : ℕ |
| A₀ : ℕ |
| γ : ℚ |
| p : ℚ |
| θ : ℚ |
| hm : 2 ≤ m |
| hn : k+2 ≤ n |
| hk : 1 ≤ k |
| hs : ∀ t : ℕ, s+2 ≤ t → (t:ℝ)*(smoothingRho (sourceSigma J))^(2*t) ≤ 1/16 |
| hroom : sourceR J*m < (m-s).choose 2 |
| hH : 0 < H |
| hA : 0 < A₀ |
| hγ : 0 < (γ:ℝ) |
| hp : 0 < (p:ℝ) |
| hp1 : (p:ℝ) < 1 |
| hθ : 0 < (θ:ℝ) |
| hθ1 : (θ:ℝ) < 1 |
| hloc : (sourceSigma J)⁻¹*Real.sqrt (((s+sourceR J*m:ℕ):ℝ)*((k:ℝ)/((n:ℝ)-1))) < |
| classTarget (sourceT J) m/10 |
| henergy : (sourceSigma J)⁻¹^2/(H:ℝ) < classTarget (sourceT J) m/10 |
| hclip : 2*((sourceSigma J)⁻¹^2)/(A₀:ℝ) < classTarget (sourceT J) m/10 |
| hcollision : 2*(A₀:ℝ)*((m:ℝ)/((k:ℝ)+1))^(((2^m:ℕ):ℝ)⁻¹) < classTarget (sourceT J) m/10 |
| hfourth : Real.sqrt ((H:ℝ)*((γ:ℝ)+(H:ℝ)^2/(A₀:ℝ))) < classTarget (sourceT J) m/10 |
| hatom : Real.sqrt ((sourceSigma J)⁻¹^2)*Real.sqrt |
| (((4:ℝ)^Fintype.card (Row m n)*((Fintype.card (Point m n)+s).choose s:ℝ)* |
| Real.sqrt ((θ:ℝ)^2+(m:ℝ)*(p:ℝ)*(n^m:ℕ)/(sourceSigma J)^2))/(γ:ℝ)) < |
| classTarget (sourceT J) m/10 |
|
|
| lemma exists_innerParameters {J : ℝ} (hJ : 1 ≤ J) : Nonempty (InnerParameters J) := by |
| have hσ := sourceSigma_pos hJ |
| have hT : (0:ℝ) < sourceT J := by exact_mod_cast sourceT_pos hJ |
| obtain ⟨s,hs⟩ := exists_cutoff (smoothingRho_lt_one hσ.ne') (by norm_num : (0:ℝ) < 1/16) |
| obtain ⟨m,hm,_,hroom⟩ := exists_dimension (sourceR J) s |
| let e := classTarget (sourceT J) m/10 |
| have he : 0 < e := div_pos (classTarget_pos hT m) (by norm_num) |
| obtain ⟨H,A₀,γ,hH,hA,hγ,_,henergy,hclip,hfourth⟩ := |
| exists_reciprocal_processing (E := (sourceSigma J)⁻¹^2) he |
| have hAr : (0:ℝ) < A₀ := by exact_mod_cast hA |
| obtain ⟨k,hk,hcollision⟩ := exists_box_scale m hAr he |
| obtain ⟨n,hn,hloc⟩ := exists_ambient_scale s (sourceR J*m) k hσ he |
| let C := (4:ℝ)^Fintype.card (Row m n)*((Fintype.card (Point m n)+s).choose s:ℝ) |
| have hchoose : 0 < (Fintype.card (Point m n)+s).choose s := Nat.choose_pos (by omega) |
| have hC : 0 < C := by dsimp [C]; positivity |
| let L := (m:ℝ)*(n^m:ℕ)/(sourceSigma J)^2 |
| have hnpos : 0 < n := by omega |
| have hmpos : 0 < m := by omega |
| have hL : 0 < L := by dsimp [L]; positivity |
| obtain ⟨p,θ,hp,hp1,hθ,hθ1,hQ⟩ := exists_small_rational_atoms hL |
| (show 0 < (e^2*(sourceSigma J)^2*(γ:ℝ)/C)^2 by positivity) |
| have hQ' : (θ:ℝ)^2+(m:ℝ)*(p:ℝ)*(n^m:ℕ)/(sourceSigma J)^2 < |
| (e^2*(sourceSigma J)^2*(γ:ℝ)/C)^2 := by |
| convert hQ using 1 |
| dsimp [L] |
| ring |
| have hatom := nested_atom_lt hσ he hγ hC hQ' |
| exact ⟨⟨s,m,k,n,H,A₀,γ,p,θ,hm,hn,hk,hs,hroom,hH,hA,hγ,hp,hp1,hθ,hθ1, |
| hloc,henergy,hclip,hcollision,hfourth,hatom⟩⟩ |
|
|
| theorem InnerParameters.decode {J : ℝ} (hJ : 1 ≤ J) (P : InnerParameters J) |
| {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)] (w : Ξ → ℝ) (hw : ∀ ξ, 0 ≤ w ξ) (hw1 : ∑ ξ, w ξ=1) |
| (f : ∀ ξ, FoldedProof (A ξ)) |
| (h1 : (∑ ξ, w ξ*firstError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J*(10*sourceSigma J)) |
| (h2 : (∑ ξ, w ξ*secondError (m := P.m) (n := P.n) (f ξ) |
| ((sourceEta J)^2/(P.m:ℝ)) (sourceSigma J) (sourceEta J)) ≤ J*(10*(sourceEta J+sourceEta J))) |
| (h3 : (∑ ξ, w ξ*thirdError (m := P.m) (n := P.n) (f ξ) (sourceSigma J) (sourceEta J)) ≤ |
| J/(Fintype.card (Code P.m P.n):ℝ)) : |
| (P.p:ℝ) ≤ ∑ ξ, w ξ*atomProbability (m := P.m) (n := P.n) (f ξ) |
| (sourceSigma J) (sourceEta J) (P.θ:ℝ) := by |
| apply inner_atom_of_errors V A w hw hw1 f hJ P.hm (by have := P.hn; omega) |
| (by have := P.hn; omega) P.hs P.hroom (by exact_mod_cast P.hH) P.hγ |
| (by exact_mod_cast P.hA) P.hθ.le P.hloc P.henergy P.hclip P.hcollision P.hfourth P.hatom |
| h1 h2 h3 |
| end MinUncut.Inner |
|
|
| end |
| end |
|
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| end OAI |
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