| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory Filter | |
| open scoped Topology | |
| namespace ProductCells | |
| abbrev Plane := ℝ × ℝ | |
| def closedCell (A B : ℝ) : Set Plane := | |
| {p | |p.1| ≤ A ∧ |p.2| ≤ A ∧ |p.1 * p.2| ≤ B} | |
| def openCell (A B : ℝ) : Set Plane := | |
| {p | |p.1| < A ∧ |p.2| < A ∧ |p.1 * p.2| < B} | |
| def parameterCell (tau : ℝ) : Set Plane := closedCell 1 tau | |
| def interceptCell (tau gamma : ℝ) : Set Plane := openCell 3 ((1 + gamma) * tau) | |
| def logarithmicCell (m : ℝ) : Set Plane := closedCell m 1 | |
| def logarithmicIntercept (m gamma : ℝ) : Set Plane := openCell (3 * m) (1 + gamma) | |
| def dilation (m : ℝ) (p : Plane) : Plane := (m * p.1, m * p.2) | |
| def coordinateEquiv : Plane ≃L[ℝ] (Fin 2 → ℝ) := | |
| (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm | |
| def coordinateParameterCell (tau : ℝ) : Set (Fin 2 → ℝ) := | |
| coordinateEquiv '' parameterCell tau | |
| def coordinateInterceptCell (tau gamma : ℝ) : Set (Fin 2 → ℝ) := | |
| coordinateEquiv '' interceptCell tau gamma | |
| def areaRatio (tau gamma : ℝ) : ℝ := | |
| (1 + gamma) * (1 + Real.log (9 / ((1 + gamma) * tau))) / | |
| (1 + Real.log (1 / tau)) | |
| def CellAreaStatement : Prop := | |
| ∀ A B : ℝ, 0 < A → 0 < B → B ≤ A^2 → | |
| IsCompact (closedCell A B) ∧ IsOpen (openCell A B) ∧ | |
| Bornology.IsBounded (openCell A B) ∧ | |
| volume.real (closedCell A B) = 4*B*(1+Real.log (A^2/B)) ∧ | |
| volume.real (openCell A B) = 4*B*(1+Real.log (A^2/B)) ∧ | |
| volume (frontier (closedCell A B)) = 0 ∧ volume (frontier (openCell A B)) = 0 | |
| def CellDilationStatement : Prop := | |
| ∀ m gamma : ℝ, 1 < m → | |
| logarithmicCell m = dilation m '' parameterCell (1/m^2) ∧ | |
| logarithmicIntercept m gamma = dilation m '' interceptCell (1/m^2) gamma | |
| def CellRatioStatement : Prop := | |
| (∀ tau gamma : ℝ, 0 < tau → tau < 1 → 0 ≤ gamma → (1+gamma)*tau < 1 → | |
| volume.real (interceptCell tau gamma) / volume.real (parameterCell tau) = | |
| areaRatio tau gamma) ∧ | |
| (∀ gamma : ℝ, 0 ≤ gamma → | |
| Tendsto (fun tau : ℝ => areaRatio tau gamma) (𝓝[>] 0) (𝓝 (1+gamma))) | |
| def CellParameterStatement : Prop := | |
| (∀ gamma eta : ℝ, 0 ≤ gamma → gamma < eta → | |
| ∃ tau : ℝ, 0 < tau ∧ tau < 1 ∧ (1+gamma)*tau < 1 ∧ | |
| areaRatio tau gamma < 1+eta) ∧ | |
| (∀ tau eta : ℝ, 0 < tau → tau < 1 → areaRatio tau 0 < 1+eta → | |
| ∃ gamma : ℝ, 0 < gamma ∧ (1+gamma)*tau < 1 ∧ areaRatio tau gamma < 1+eta) ∧ | |
| (∀ eta : ℝ, 0 < eta → | |
| ∃ m : ℝ, 1 < m ∧ ∃ gamma : ℝ, 0 < gamma ∧ | |
| (1+gamma)*(1/m^2) < 1 ∧ areaRatio (1/m^2) gamma < 1+eta) | |
| def CellPerturbationStatement : Prop := | |
| ∀ z tau gamma eps : ℝ, ∀ s e : Plane, | |
| |z| ≤ 1 → s ∈ parameterCell tau → |e.1| ≤ eps → |e.2| ≤ eps → | |
| eps < 1/2 → eps < gamma*tau/5 → | |
| ((1+z)*s.1+e.1, (1-z)*s.2+e.2) ∈ interceptCell tau gamma | |
| def GeometryStatement : Prop := | |
| CellAreaStatement ∧ CellDilationStatement ∧ CellRatioStatement ∧ | |
| CellParameterStatement ∧ CellPerturbationStatement | |
| end ProductCells | |
| end | |
| end OAI | |