| import Mathlib |
|
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| 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 |
|
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| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| lemma det_updateRow_identity (j : ι) (v : ι → ℝ) : |
| (Matrix.updateRow (1 : Matrix ι ι ℝ) j v).det = v j := by |
| have he : (∑ i : ι, v i • (1 : Matrix ι ι ℝ) i) = v := by |
| ext l |
| simp [Matrix.one_apply] |
| rw [←he,Matrix.det_updateRow_sum] |
| simp [Matrix.one_apply] |
|
|
| theorem hasDerivAt_det_at_identity {A : ℝ → Matrix ι ι ℝ} |
| {B : Matrix ι ι ℝ} {t : ℝ} (hA : A t = 1) (hd : HasDerivAt A B t) : |
| HasDerivAt (fun s => (A s).det) B.trace t := by |
| let D : ContinuousMultilinearMap ℝ (fun _ : ι => ι → ℝ) ℝ := |
| ⟨Matrix.detRowAlternating.toMultilinearMap,continuous_id.matrix_det⟩ |
| have hD := (D.hasFDerivAt (A t)).comp_hasDerivAt t hd |
| change HasDerivAt (fun s => (A s).det) (D.linearDeriv (A t) B) t at hD |
| have hv : D.linearDeriv (fun i j => A t i j) (fun i j => B i j) = B.trace := by |
| rw [ContinuousMultilinearMap.linearDeriv_apply] |
| change (∑ i, (Matrix.updateRow (A t) i (B i)).det) = B.trace |
| rw [hA] |
| simp only [det_updateRow_identity,Matrix.trace,Matrix.diag] |
| exact hD.congr_deriv hv |
|
|
| def spatialRadialForm (g r t : ℝ) (α γ : ι → ℝ) : Matrix ι ι ℝ := |
| Matrix.of fun i j => (1-t*g)^2 * (if i=j then 1 else 0) - |
| (1-t*g)*t*r*(α i*γ j+γ i*α j) + t^2*r^2*γ i*γ j |
|
|
| omit [Fintype ι] in |
| lemma spatialRadialForm_zero (g r : ℝ) (α γ : ι → ℝ) : |
| spatialRadialForm g r 0 α γ = 1 := by |
| ext i j |
| simp [spatialRadialForm,Matrix.one_apply] |
|
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| lemma spatialRadialForm_hasDerivAt_zero (g r : ℝ) (α γ : ι → ℝ) : |
| HasDerivAt (fun t => spatialRadialForm g r t α γ) |
| (Matrix.of fun i j => -2*g*(if i=j then 1 else 0)-r*(α i*γ j+γ i*α j)) 0 := by |
| apply hasDerivAt_pi.2 |
| intro i |
| apply hasDerivAt_pi.2 |
| intro j |
| have hl := (hasDerivAt_const (0:ℝ) 1).sub ((hasDerivAt_id (0:ℝ)).mul_const g) |
| have he := (((hl.fun_pow 2).mul_const (if i=j then 1 else 0)).sub |
| (((hl.mul (hasDerivAt_id (0:ℝ))).mul_const r).mul_const (α i*γ j+γ i*α j))).add |
| ((((hasDerivAt_id (0:ℝ)).fun_pow 2).mul_const (r^2)).mul_const (γ i*γ j)) |
| convert he using 1 |
| · ext t |
| simp only [spatialRadialForm,Matrix.of_apply,Pi.sub_apply,Pi.add_apply,Pi.mul_apply,id_eq] |
| ring |
| · simp |
|
|
| theorem spatialRadialForm_sqrt_det_hasDerivAt_zero (g r : ℝ) (α γ : ι → ℝ) : |
| HasDerivAt (fun t => Real.sqrt (spatialRadialForm g r t α γ).det) |
| (-((Fintype.card ι : ℝ)*g+r*∑ i, α i*γ i)) 0 := by |
| have hd := hasDerivAt_det_at_identity (spatialRadialForm_zero g r α γ) |
| (spatialRadialForm_hasDerivAt_zero g r α γ) |
| have hs := hd.sqrt (by simp [spatialRadialForm_zero]) |
| convert hs using 1 |
| simp only [spatialRadialForm_zero,Matrix.det_one,Real.sqrt_one,Matrix.trace, |
| Matrix.diag,Matrix.of_apply,ite_true, mul_one, Finset.sum_sub_distrib, |
| Finset.sum_const, nsmul_eq_mul, Finset.card_univ] |
| rw [←Finset.mul_sum] |
| have hh : (∑ i, (α i*γ i+γ i*α i)) = 2*(∑ i, α i*γ i) := by |
| rw [Finset.sum_add_distrib] |
| simp only [mul_comm (γ _) (α _)] |
| ring |
| rw [hh] |
| ring |
|
|
| end CAT0Fillings |
|
|
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
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| end |
| end |
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| end OAI |
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