| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SharedBinSupportCollision |
|
|
| namespace OAI |
|
|
| /-! Canonical spherical measures and differential identities. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4FullFar |
| open MeasureTheory Set Metric Module |
| open scoped RealInnerProductSpace Topology ENNReal Pointwise |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
|
|
| noncomputable def sphereChartScaleND (u : V) : ℝ := Real.sqrt (1-‖u‖^2/4) |
| noncomputable def sphereChartND (u : V) : WithLp 2 (V × ℝ) := |
| WithLp.toLp 2 (sphereChartScaleND u • u, 1-‖u‖^2/2) |
|
|
| omit [InnerProductSpace ℝ V] in |
| lemma sphereChartScaleND_pos {u : V} (hu : ‖u‖ < 2) : |
| 0 < sphereChartScaleND u := by |
| apply Real.sqrt_pos.mpr |
| nlinarith [norm_nonneg u] |
|
|
| omit [InnerProductSpace ℝ V] in |
| lemma sphereChartScaleND_sq {u : V} (hu : ‖u‖ ≤ 2) : |
| sphereChartScaleND u ^ 2 = 1-‖u‖^2/4 := by |
| apply Real.sq_sqrt |
| nlinarith [norm_nonneg u] |
|
|
| lemma sphereChartND_norm {u : V} (hu : ‖u‖ ≤ 2) : |
| ‖sphereChartND u‖ = 1 := by |
| have hs : 0 ≤ sphereChartScaleND u := Real.sqrt_nonneg _ |
| have he : ‖sphereChartND u‖^2 = 1 := by |
| rw [sphereChartND, WithLp.prod_norm_sq_eq_of_L2] |
| change ‖sphereChartScaleND u • u‖^2 + ‖(1-‖u‖^2/2:ℝ)‖^2 = 1 |
| simp only [norm_smul, Real.norm_eq_abs, mul_pow, sq_abs] |
| rw [sphereChartScaleND_sq hu] |
| ring |
| nlinarith [norm_nonneg (sphereChartND u)] |
|
|
| lemma sphereChartND_mem_sphere {u : V} (hu : ‖u‖ ≤ 2) : |
| sphereChartND u ∈ sphere (0 : WithLp 2 (V × ℝ)) 1 := by |
| simpa only [mem_sphere, dist_zero_right] using sphereChartND_norm hu |
|
|
| lemma sphereChartND_phase (u : V) : |
| inner ℝ (sphereChartND u) (WithLp.toLp 2 (0, (1:ℝ))) = 1-‖u‖^2/2 := by |
| simp [sphereChartND,WithLp.prod_inner_apply] |
|
|
| lemma sphereChartND_north : sphereChartND (0:V) = WithLp.toLp 2 (0,(1:ℝ)) := by |
| simp [sphereChartND] |
|
|
| lemma sphereChartND_snd_gt_neg_one {u : V} (hu : ‖u‖ < 2) : |
| -1 < (WithLp.ofLp (sphereChartND u)).2 := by |
| change -1 < 1-‖u‖^2/2 |
| nlinarith [norm_nonneg u] |
|
|
| noncomputable def sphereChartInvND (w : WithLp 2 (V × ℝ)) : V := |
| (Real.sqrt ((1+(WithLp.ofLp w).2)/2))⁻¹ • (WithLp.ofLp w).1 |
|
|
| lemma sphereChartND_left_inv {u : V} (hu : ‖u‖ < 2) : |
| sphereChartInvND (sphereChartND u) = u := by |
| have hs : sphereChartScaleND u ≠ 0 := (sphereChartScaleND_pos hu).ne' |
| have he : (1+(1-‖u‖^2/2))/2 = 1-‖u‖^2/4 := by ring |
| simp only [sphereChartInvND, sphereChartND, WithLp.ofLp_toLp, he] |
| change (sphereChartScaleND u)⁻¹ • (sphereChartScaleND u • u) = u |
| rw [← mul_smul, inv_mul_cancel₀ hs, one_smul] |
|
|
| lemma sphereChartND_injOn : InjOn (sphereChartND : V → WithLp 2 (V × ℝ)) (ball 0 2) := by |
| intro u hu v hv he |
| have hu' : ‖u‖ < 2 := by simpa using hu |
| have hv' : ‖v‖ < 2 := by simpa using hv |
| have := congrArg sphereChartInvND he |
| simpa [sphereChartND_left_inv hu', sphereChartND_left_inv hv'] using this |
|
|
| lemma sphereChartInvND_norm_sq {w : WithLp 2 (V × ℝ)} (hw : ‖w‖ = 1) |
| (ht : -1 < (WithLp.ofLp w).2) : |
| ‖sphereChartInvND w‖^2 = 2*(1-(WithLp.ofLp w).2) := by |
| have hp : 0 < (1+(WithLp.ofLp w).2)/2 := by linarith |
| have hs : Real.sqrt ((1+(WithLp.ofLp w).2)/2)^2 = (1+(WithLp.ofLp w).2)/2 := |
| Real.sq_sqrt hp.le |
| have hn : ‖(WithLp.ofLp w).1‖^2 + ((WithLp.ofLp w).2)^2 = 1 := by |
| have he := WithLp.prod_norm_sq_eq_of_L2 w |
| rw [hw] at he |
| simpa [Real.norm_eq_abs, sq_abs] using he.symm |
| unfold sphereChartInvND |
| rw [norm_smul, Real.norm_eq_abs, mul_pow, sq_abs, inv_pow, hs] |
| field_simp [show 1+(WithLp.ofLp w).2 ≠ 0 by linarith] |
| rw [div_eq_iff (show 1+(WithLp.ofLp w).2 ≠ 0 by linarith)] |
| nlinarith [hn] |
|
|
| lemma sphereChartInvND_mem_ball {w : WithLp 2 (V × ℝ)} (hw : ‖w‖ = 1) |
| (ht : -1 < (WithLp.ofLp w).2) : sphereChartInvND w ∈ ball (0:V) 2 := by |
| have he := sphereChartInvND_norm_sq hw ht |
| rw [mem_ball, dist_zero_right] |
| nlinarith [norm_nonneg (sphereChartInvND w)] |
|
|
| lemma sphereChartND_right_inv {w : WithLp 2 (V × ℝ)} (hw : ‖w‖ = 1) |
| (ht : -1 < (WithLp.ofLp w).2) : sphereChartND (sphereChartInvND w) = w := by |
| have he := sphereChartInvND_norm_sq hw ht |
| have hp : 0 < (1+(WithLp.ofLp w).2)/2 := by linarith |
| have hs : Real.sqrt ((1+(WithLp.ofLp w).2)/2) ≠ 0 := (Real.sqrt_pos.mpr hp).ne' |
| have hscale : sphereChartScaleND (sphereChartInvND w) = |
| Real.sqrt ((1+(WithLp.ofLp w).2)/2) := by |
| unfold sphereChartScaleND |
| rw [he] |
| congr 1 |
| ring |
| apply WithLp.ofLp_injective 2 |
| apply Prod.ext |
| · change sphereChartScaleND (sphereChartInvND w) • sphereChartInvND w = (WithLp.ofLp w).1 |
| rw [hscale, sphereChartInvND, ← mul_smul, mul_inv_cancel₀ hs, one_smul] |
| · change 1-‖sphereChartInvND w‖^2/2 = (WithLp.ofLp w).2 |
| rw [he] |
| ring |
|
|
| lemma sphereChartScaleND_contDiffOn : |
| ContDiffOn ℝ ⊤ (sphereChartScaleND : V → ℝ) (ball 0 2) := by |
| apply ContDiffOn.sqrt |
| ((contDiffOn_const.sub ((contDiff_norm_sq ℝ).contDiffOn.div_const 4))) |
| intro u hu |
| have hu' : ‖u‖ < 2 := by simpa using hu |
| have : 0 < 1-‖u‖^2/4 := by nlinarith [norm_nonneg u] |
| exact this.ne' |
|
|
| lemma sphereChartND_contDiffOn : |
| ContDiffOn ℝ ⊤ (sphereChartND : V → WithLp 2 (V × ℝ)) (ball 0 2) := by |
| exact (WithLp.prodContinuousLinearEquiv 2 ℝ V ℝ).symm.contDiff.comp_contDiffOn |
| ((sphereChartScaleND_contDiffOn.smul contDiffOn_id).prodMk |
| (contDiffOn_const.sub ((contDiff_norm_sq ℝ).contDiffOn.div_const 2))) |
|
|
| lemma sphereChartScaleND_hasStrictFDerivAt {u : V} (hu : ‖u‖ < 2) : |
| HasStrictFDerivAt sphereChartScaleND |
| ((-1/(4*sphereChartScaleND u)) • innerSL ℝ u) u := by |
| have hp : 0 < 1-‖u‖^2/4 := by nlinarith [norm_nonneg u] |
| have hq : HasStrictFDerivAt (fun x : V => 1-‖x‖^2/4) |
| ((-(1/2:ℝ)) • innerSL ℝ u) u := by |
| convert! ((hasStrictFDerivAt_norm_sq u).mul_const (1/4:ℝ)).const_sub 1 using 1 |
| · funext x |
| ring |
| · apply ContinuousLinearMap.ext |
| intro v |
| simp only [smul_apply, neg_apply, smul_eq_mul] |
| ring |
| convert! hq.sqrt hp.ne' using 1 |
| apply ContinuousLinearMap.ext |
| intro v |
| simp only [sphereChartScaleND, smul_apply, smul_eq_mul] |
| ring |
|
|
| noncomputable def sphereConeND (p : V × ℝ) : V × ℝ := |
| (p.2 • (sphereChartScaleND p.1 • p.1), p.2 * (1-‖p.1‖^2/2)) |
|
|
| noncomputable def sphereConeDerivND (p : V × ℝ) : (V × ℝ) →L[ℝ] (V × ℝ) := |
| let u := p.1 |
| let r := p.2 |
| let s := sphereChartScaleND u |
| let F := ContinuousLinearMap.fst ℝ V ℝ |
| let T := ContinuousLinearMap.snd ℝ V ℝ |
| let H := (innerSL ℝ u).comp F |
| ((r*s) • F + (s • T - (r/(4*s)) • H).smulRight u).prod |
| ((1-‖u‖^2/2) • T - r • H) |
|
|
| lemma sphereConeND_hasStrictFDerivAt {p : V × ℝ} (hp : ‖p.1‖ < 2) : |
| HasStrictFDerivAt sphereConeND (sphereConeDerivND p) p := by |
| have hs := (sphereChartScaleND_hasStrictFDerivAt hp).comp p hasStrictFDerivAt_fst |
| have h₁ := hasStrictFDerivAt_snd.smul (hs.smul hasStrictFDerivAt_fst) |
| have hv : HasStrictFDerivAt (fun x : V => 1-‖x‖^2/2) (-(innerSL ℝ p.1)) p.1 := by |
| convert! ((hasStrictFDerivAt_norm_sq p.1).mul_const (1/2:ℝ)).const_sub 1 using 1 |
| · funext x |
| ring |
| · apply ContinuousLinearMap.ext |
| intro v |
| simp only [smul_apply, neg_apply, smul_eq_mul] |
| ring |
| have h₂ := (hasStrictFDerivAt_snd (𝕜 := ℝ)).mul |
| (hv.comp p hasStrictFDerivAt_fst) |
| convert! h₁.prodMk h₂ using 1 |
| apply ContinuousLinearMap.ext |
| intro z |
| apply Prod.ext |
| · simp [sphereConeDerivND, smul_add, sub_smul, smul_smul]; module |
| · simp [sphereConeDerivND]; ring |
|
|
| lemma sphereConeDerivND_apply (p z : V × ℝ) : |
| sphereConeDerivND p z = |
| ((p.2*sphereChartScaleND p.1) • z.1 + |
| (sphereChartScaleND p.1*z.2-p.2/(4*sphereChartScaleND p.1)*inner ℝ p.1 z.1) • p.1, |
| (1-‖p.1‖^2/2)*z.2-p.2*inner ℝ p.1 z.1) := by |
| simp [sphereConeDerivND] |
|
|
| end Attempt4FullFar |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
