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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | 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
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