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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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