| import Mathlib | |
| namespace OAI | |
| namespace CylinderCovering.RadialInducedField | |
| def Statement : Prop := | |
| ∀ (F : Type) [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F], | |
| ∀ (a b : F) (α β : ℝ) (w : ℝ → F) (lam : ℝ → ℝ), | |
| Module.finrank ℝ F = 2 → LinearIndependent ℝ ![a, b] → | |
| ContDiffOn ℝ 1 w (Set.Icc α β) → | |
| (∀ t ∈ Set.Ioo α β, HasDerivAt w (lam t • (a + t • b)) t) → | |
| ∃ e : (ℝ × ℝ) ≃L[ℝ] F, | |
| (∀ u v : ℝ, e (u, v) = u • a + v • b) ∧ | |
| (∀ ρ t : ℝ, 0 < ρ → | |
| w ((e.symm (ρ • (a + t • b))).2 / (e.symm (ρ • (a + t • b))).1) = w t) ∧ | |
| ContDiffOn ℝ 1 (fun z : F => w ((e.symm z).2 / (e.symm z).1)) | |
| {z : F | 0 < (e.symm z).1 ∧ (e.symm z).2 / (e.symm z).1 ∈ Set.Ioo α β} ∧ | |
| ∀ z : F, 0 < (e.symm z).1 → | |
| (e.symm z).2 / (e.symm z).1 ∈ Set.Ioo α β → | |
| (fderiv ℝ (fun x : F => w ((e.symm x).2 / (e.symm x).1)) z = | |
| (fderiv ℝ (fun x : F => (e.symm x).2 / (e.symm x).1) z).smulRight | |
| (lam ((e.symm z).2 / (e.symm z).1) • | |
| (a + ((e.symm z).2 / (e.symm z).1) • b))) ∧ | |
| (fderiv ℝ (fun x : F => (e.symm x).2 / (e.symm x).1) z) | |
| (a + ((e.symm z).2 / (e.symm z).1) • b) = 0 ∧ | |
| (fderiv ℝ (fun x : F => w ((e.symm x).2 / (e.symm x).1)) z).comp | |
| (fderiv ℝ (fun x : F => w ((e.symm x).2 / (e.symm x).1)) z) = 0 | |
| end CylinderCovering.RadialInducedField | |
| end OAI | |