| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphJetMul |
|
|
| namespace OAI |
|
|
| /-! Graph energies and pair estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace CurrentFalconer |
| open Set Metric |
| open scoped RealInnerProductSpace |
| variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
|
|
| noncomputable def graphPerpND (v x : V) : V := x-inner ℝ x v • v |
|
|
| lemma graphPerpND_sub (v x y : V) : |
| graphPerpND v (x-y) = graphPerpND v x-graphPerpND v y := by |
| simp only [graphPerpND,inner_sub_left,sub_smul]; abel |
|
|
| lemma graphPerpND_add (v x y : V) : |
| graphPerpND v (x+y) = graphPerpND v x+graphPerpND v y := by |
| simp only [graphPerpND,inner_add_left,add_smul]; abel |
|
|
| lemma graphPerpND_norm_le (v x : V) (hv : ‖v‖=1) : ‖graphPerpND v x‖ ≤ ‖x‖ := by |
| have he : ‖graphPerpND v x‖^2=‖x‖^2-(inner ℝ x v)^2 := by |
| rw [graphPerpND,norm_sub_sq_real,norm_smul,inner_smul_right,hv,Real.norm_eq_abs] |
| rw [mul_one,sq_abs] |
| ring |
| nlinarith [norm_nonneg (graphPerpND v x),norm_nonneg x,sq_nonneg (inner ℝ x v)] |
|
|
| lemma graphPerpND_change_direction (v w x : V) (hv : ‖v‖=1) (hw : ‖w‖=1) : |
| ‖graphPerpND w x-graphPerpND v x‖ ≤ 2*‖x‖*dist v w := by |
| have he : graphPerpND w x-graphPerpND v x = |
| inner ℝ x v • (v-w)+inner ℝ x (v-w) • w := by |
| simp only [graphPerpND,inner_sub_right,smul_sub,sub_smul]; abel |
| rw [he] |
| calc |
| _ ≤ ‖inner ℝ x v • (v-w)‖+‖inner ℝ x (v-w) • w‖ := norm_add_le _ _ |
| _ = |inner ℝ x v| *‖v-w‖+|inner ℝ x (v-w)| := by |
| simp [norm_smul,Real.norm_eq_abs,hw] |
| _ ≤ (‖x‖*‖v‖)*‖v-w‖+‖x‖*‖v-w‖ := by |
| gcongr <;> exact abs_real_inner_le_norm _ _ |
| _ = _ := by rw [hv,dist_eq_norm]; ring |
|
|
| theorem graph_witness_tubeND (v w a x y : V) (hv : ‖v‖=1) (hw : ‖w‖=1) |
| {τ δ D : ℝ} (_hδ : 0 ≤ δ) (hxy : dist x y ≤ D) (hvw : dist v w ≤ δ) |
| (hx : ‖graphPerpND v (x-a)‖ ≤ τ) (hy : ‖graphPerpND v (y-a)‖ ≤ τ) : |
| ‖graphPerpND w (x-y)‖ ≤ 2*τ+2*D*δ := by |
| have hD : 0 ≤ D := dist_nonneg.trans hxy |
| have hp : ‖graphPerpND v (x-y)‖ ≤ 2*τ := by |
| have he : x-y=(x-a)-(y-a) := by abel |
| rw [he,graphPerpND_sub] |
| exact (norm_sub_le _ _).trans (by linarith) |
| calc |
| _ ≤ ‖graphPerpND w (x-y)-graphPerpND v (x-y)‖+‖graphPerpND v (x-y)‖ := |
| norm_le_norm_sub_add _ _ |
| _ ≤ 2*‖x-y‖*dist v w+2*τ := add_le_add (graphPerpND_change_direction v w (x-y) hv hw) hp |
| _ ≤ 2*D*δ+2*τ := by gcongr; simpa [dist_eq_norm] using hxy |
| _ = _ := by ring |
|
|
| theorem graph_scalar_perp_ballND (v a x : V) (hv : ‖v‖=1) {t τ : ℝ} |
| (hp : ‖graphPerpND v (x-a)‖ ≤ τ) |
| (hs : |t+inner ℝ (x-a) v| ≤ τ) : |
| dist x (a-t • v) ≤ 2*τ := by |
| have he : x-(a-t • v)=graphPerpND v (x-a)+(t+inner ℝ (x-a) v) • v := by |
| simp only [graphPerpND,add_smul]; abel |
| rw [dist_eq_norm,he] |
| calc |
| _ ≤ ‖graphPerpND v (x-a)‖+‖(t+inner ℝ (x-a) v) • v‖ := norm_add_le _ _ |
| _ ≤ τ+τ := by simpa [norm_smul,Real.norm_eq_abs,hv] using add_le_add hp hs |
| _ = _ := by ring |
|
|
| lemma graph_phase_changeND (v w d Δ : V) (hw : ‖w‖=1) : |
| |inner ℝ w Δ-inner ℝ v d| ≤ dist w v*‖d‖+‖Δ-d‖ := by |
| have he : inner ℝ w Δ-inner ℝ v d=inner ℝ (w-v) d+inner ℝ w (Δ-d) := by |
| simp only [inner_sub_left,inner_sub_right]; ring |
| rw [he] |
| calc |
| _ ≤ |inner ℝ (w-v) d|+|inner ℝ w (Δ-d)| := abs_add_le _ _ |
| _ ≤ ‖w-v‖*‖d‖+‖w‖*‖Δ-d‖ := add_le_add (abs_real_inner_le_norm _ _) (abs_real_inner_le_norm _ _) |
| _ = _ := by rw [hw,one_mul,dist_eq_norm] |
|
|
| lemma graph_perp_center_changeND (v d Δ : V) (hv : ‖v‖=1) : |
| ‖graphPerpND v d‖ ≤ ‖graphPerpND v Δ‖+‖Δ-d‖ := by |
| calc |
| _ = ‖graphPerpND v Δ+graphPerpND v (d-Δ)‖ := by rw [← graphPerpND_add]; congr 2; abel |
| _ ≤ ‖graphPerpND v Δ‖+‖graphPerpND v (d-Δ)‖ := norm_add_le _ _ |
| _ ≤ ‖graphPerpND v Δ‖+‖d-Δ‖ := add_le_add le_rfl (graphPerpND_norm_le v (d-Δ) hv) |
| _ = _ := by rw [norm_sub_rev] |
|
|
| def graphAffineLineND (v c : V) : Set V := {z | ∃ t : ℝ, z=c+t • v} |
|
|
| lemma graphAffineLineND_nonempty (v c : V) : (graphAffineLineND v c).Nonempty := by |
| exact ⟨c,0,by simp⟩ |
|
|
| lemma graphPerpND_sub_smul (v x : V) (hv : ‖v‖=1) (t : ℝ) : |
| graphPerpND v (x-t • v)=graphPerpND v x := by |
| simp only [graphPerpND,inner_sub_left,inner_smul_left,real_inner_self_eq_norm_sq,hv,one_pow,mul_one,sub_smul,starRingEnd_apply,star_trivial] |
| abel |
|
|
| lemma infDist_graphAffineLineND (v c x : V) (hv : ‖v‖=1) : |
| infDist x (graphAffineLineND v c)=‖graphPerpND v (x-c)‖ := by |
| apply le_antisymm |
| · have hy : c+inner ℝ (x-c) v • v ∈ graphAffineLineND v c := |
| ⟨inner ℝ (x-c) v,rfl⟩ |
| calc |
| _ ≤ dist x (c+inner ℝ (x-c) v • v) := infDist_le_dist_of_mem hy |
| _ = _ := by rw [dist_eq_norm,graphPerpND]; congr 1; abel |
| · apply (le_infDist (graphAffineLineND_nonempty v c)).mpr |
| rintro y ⟨t,rfl⟩ |
| have he : x-(c+t • v)=(x-c)-t • v := by abel |
| rw [dist_eq_norm,he] |
| rw [← graphPerpND_sub_smul v (x-c) hv t] |
| exact graphPerpND_norm_le v ((x-c)-t • v) hv |
|
|
| end CurrentFalconer |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
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|
