| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Geometry.Determinant |
| import OAI.Geometry.CAT0Fillings.Radial.SpatialForm |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace CAT0Fillings |
| open Matrix |
| open scoped BigOperators |
|
|
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| omit [Fintype ι] in |
| lemma spatialRadialForm_rankOne_decomposition (g r t : ℝ) (α γ : ι → ℝ) : |
| spatialRadialForm g r t α γ = |
| (1-t*g)^2 • (1-vecMulVec α α) + |
| vecMulVec ((1-t*g) • α-(t*r) • γ) ((1-t*g) • α-(t*r) • γ) := by |
| ext i j |
| simp only [spatialRadialForm,Matrix.of_apply,Matrix.add_apply,Matrix.sub_apply, |
| Matrix.smul_apply,Matrix.one_apply,Matrix.vecMulVec_apply,Pi.sub_apply,Pi.smul_apply, |
| smul_eq_mul] |
| ring |
|
|
| omit [DecidableEq ι] in |
| lemma dotProduct_vecMulVec_self (u v : ι → ℝ) : |
| v ⬝ᵥ (vecMulVec u u).mulVec v = (u ⬝ᵥ v)^2 := by |
| rw [Matrix.vecMulVec_mulVec,dotProduct_smul,op_smul_eq_smul,smul_eq_mul, |
| dotProduct_comm v u,pow_two] |
|
|
| theorem spatialRadialForm_posSemidef (g r t : ℝ) (α γ : ι → ℝ) |
| (hα : ∀ v : ι → ℝ, (α ⬝ᵥ v)^2 ≤ v ⬝ᵥ v) : |
| (spatialRadialForm g r t α γ).PosSemidef := by |
| have hQ : (1-vecMulVec α α : Matrix ι ι ℝ).PosSemidef := by |
| apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg |
| · apply Matrix.IsHermitian.sub Matrix.isHermitian_one |
| simpa only [star_trivial] using (Matrix.posSemidef_vecMulVec_self_star α).isHermitian |
| · intro v |
| simp only [star_trivial] |
| rw [Matrix.sub_mulVec,Matrix.one_mulVec,dotProduct_sub,dotProduct_vecMulVec_self] |
| exact sub_nonneg.mpr (hα v) |
| rw [spatialRadialForm_rankOne_decomposition] |
| apply Matrix.PosSemidef.add (hQ.smul (sq_nonneg _)) |
| simpa only [star_trivial] using Matrix.posSemidef_vecMulVec_self_star |
| ((1-t*g) • α-(t*r) • γ) |
|
|
| lemma spatialRadialForm_quadratic (g r t : ℝ) (α γ v : ι → ℝ) : |
| v ⬝ᵥ (spatialRadialForm g r t α γ).mulVec v = |
| ((1-t*g)*(α ⬝ᵥ v)-t*r*(γ ⬝ᵥ v))^2+ |
| (1-t*g)^2*(v ⬝ᵥ v-(α ⬝ᵥ v)^2) := by |
| rw [spatialRadialForm_rankOne_decomposition,Matrix.add_mulVec,dotProduct_add, |
| dotProduct_vecMulVec_self,Matrix.smul_mulVec,dotProduct_smul,Matrix.sub_mulVec, |
| Matrix.one_mulVec,dotProduct_sub,dotProduct_vecMulVec_self, |
| sub_dotProduct,smul_dotProduct,smul_dotProduct] |
| simp only [smul_eq_mul] |
| ring |
|
|
| theorem abs_det_le_spatialRadialJacobian {n : ℕ} |
| (L : Matrix (Fin n) (Fin n) ℝ) (g r t : ℝ) (α γ : Fin n → ℝ) |
| (hα : ∀ v : Fin n → ℝ, (α ⬝ᵥ v)^2 ≤ v ⬝ᵥ v) |
| (hL : ∀ i v, (L i ⬝ᵥ v)^2 ≤ |
| ((1-t*g)*(α ⬝ᵥ v)-t*r*(γ ⬝ᵥ v))^2+ |
| (1-t*g)^2*(v ⬝ᵥ v-(α ⬝ᵥ v)^2)) : |
| |L.det| ≤ Real.sqrt (spatialRadialForm g r t α γ).det := by |
| apply abs_det_le_sqrt_det_of_posSemidef L _ (spatialRadialForm_posSemidef g r t α γ hα) |
| intro i v |
| have hn : 0 ≤ v ⬝ᵥ (spatialRadialForm g r t α γ).mulVec v := by |
| rw [spatialRadialForm_quadratic] |
| exact (sq_nonneg _).trans (hL i v) |
| apply (Real.le_sqrt (abs_nonneg _) hn).2 |
| simpa only [sq_abs,spatialRadialForm_quadratic] using hL i v |
|
|
| end CAT0Fillings |
|
|
| end |
| end |
|
|
| end OAI |
|
|
