| import Mathlib |
| import OAI.Geometry.BallPacking.Toric.CubicRadialFamilyExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped BigOperators ContDiff Manifold Topology |
| open Set Function Filter Manifold |
| open DiagonalQuadrics DiagonalQuadrics.Explicit Hamiltonian FiniteMoment |
| section |
| variable {ι : Type*} [Fintype ι] |
|
|
| theorem radialProbability_nonneg {k : ι → ℕ × ℕ} (h0 : ∃ i,k i=(0,0)) |
| (D m : ι → ℕ) (p : Radial.Plane) (x : BaseCurve) (i : ι) : |
| 0≤radialProbability k D m p x i := |
| Radial.selected_nonneg (fun j => coefficientNorm_pos (D j) (m j) x) h0 p i |
|
|
| theorem radialProbability_sum {k : ι → ℕ × ℕ} (h0 : ∃ i,k i=(0,0)) |
| (D m : ι → ℕ) (p : Radial.Plane) (x : BaseCurve) : |
| ∑ i,radialProbability k D m p x i=1 := |
| Radial.selected_sum (fun j => coefficientNorm_pos (D j) (m j) x) h0 p |
|
|
| theorem radialProbability_mean {A B : ℕ} (hAB : A≤B) |
| (D m : TrapezoidWeight A B → ℕ) {p : Radial.Plane} |
| (hp : p∈Radial.lowerTrapezoid A B) (x : BaseCurve) : |
| ∑ i,radialProbability Radial.latticeIndex D m p x i • trapezoidWeight A B i= |
| planeVector p.1 p.2 := |
| Radial.trapezoid_selected_mean hAB (fun j => coefficientNorm_pos (D j) (m j) x) hp |
|
|
| end |
| section |
|
|
| variable {ι : Type*} [Fintype ι] [Nonempty ι] |
|
|
| theorem reducedEndRegularPrimitive_norm_le (w : ι → MomentPlane) (D m : ι → ℕ) |
| (c : ℝ) (p : MomentPlane) (ε : EndIndex) (y : Plane) {M : ℝ} |
| (hM : ∀ i,‖weightedRegularEndPrimitive (D i) (m i) ε c y‖≤M) : |
| ‖reducedEndRegularPrimitive w D m c p ε y‖≤M := by |
| apply (norm_sum_le _ _).trans |
| have hq (i : ι) : 0<reducedEndProbability w D m p ε y i := |
| reducedProbability_pos w D m p _ _ |
| simp only [norm_smul,Real.norm_eq_abs,abs_of_pos (hq _)] |
| exact probability_weighted_le |
| (fun i => (reducedProbability_pos w D m p (infinityDiffeomorph (parameters 0) ε y) i).le) |
| (reducedProbability_sum w D m p _) hM |
|
|
| omit [Nonempty ι] in |
| theorem exists_regularCoefficient_log_bound (D m : ι → ℕ) {R : ℝ} |
| (hR : {s : ℂ | Complex.normSq s≤R}⊆endRegularRegion) : |
| ∃ M : ℝ,0≤M ∧ ∀ (i : ι) (ε : EndIndex) y,y∈radialAnnulus 0 R → |
| |Real.log (regularCoefficientNorm (D i) (m i) ε y)|≤M := by |
| have hc : ContinuousOn (fun y : Plane => fun j : ι × EndIndex => |
| Real.log (regularCoefficientNorm (D j.1) (m j.1) j.2 y)) (radialAnnulus 0 R) := by |
| apply continuousOn_pi.mpr |
| intro j y hy |
| have hRy : Complex.equivRealProdCLM.symm y∈endRegularRegion := by |
| apply hR |
| change Complex.normSq (Complex.equivRealProdCLM.symm y)≤R |
| rw [radiusSq_complex] |
| exact hy.2 |
| exact ((regularCoefficientNorm_contDiffAt (D j.1) (m j.1) j.2 hRy).continuousAt.log |
| (regularCoefficientNorm_pos _ _ _ _).ne').continuousWithinAt |
| obtain ⟨M,hM⟩ := (radialAnnulus_compact 0 R).exists_bound_of_continuousOn hc |
| refine ⟨max 0 M,le_max_left _ _,?_⟩ |
| intro i ε y hy |
| exact (show |Real.log (regularCoefficientNorm (D i) (m i) ε y)|≤ |
| ‖fun j : ι × EndIndex => Real.log (regularCoefficientNorm (D j.1) (m j.1) j.2 y)‖ from |
| (norm_le_pi_norm (fun j : ι × EndIndex => Real.log (regularCoefficientNorm (D j.1) (m j.1) j.2 y)) |
| (i,ε))).trans ((hM y hy).trans (le_max_right _ _)) |
|
|
| omit [Nonempty ι] in |
| theorem exists_regularPrimitive_bound (D m : ι → ℕ) (c : ℝ) {R : ℝ} |
| (hR : {s : ℂ | Complex.normSq s≤R}⊆endRegularRegion) : |
| ∃ M : ℝ,0≤M ∧ ∀ (i : ι) (ε : EndIndex) y,y∈radialAnnulus 0 R → |
| ‖weightedRegularEndPrimitive (D i) (m i) ε c y‖≤M := by |
| have hc : ContinuousOn (fun y : Plane => fun j : ι × EndIndex => |
| weightedRegularEndPrimitive (D j.1) (m j.1) j.2 c y) (radialAnnulus 0 R) := by |
| apply continuousOn_pi.mpr |
| intro j y hy |
| have hRy : Complex.equivRealProdCLM.symm y∈endRegularRegion := by |
| apply hR |
| change Complex.normSq (Complex.equivRealProdCLM.symm y)≤R |
| rw [radiusSq_complex] |
| exact hy.2 |
| exact (weightedRegularEndPrimitive_contDiffAt _ _ _ _ hRy).continuousAt.continuousWithinAt |
| obtain ⟨M,hM⟩ := (radialAnnulus_compact 0 R).exists_bound_of_continuousOn hc |
| refine ⟨max 0 M,le_max_left _ _,?_⟩ |
| intro i ε y hy |
| exact (norm_le_pi_norm (fun j : ι × EndIndex => |
| weightedRegularEndPrimitive (D j.1) (m j.1) j.2 c y) (i,ε)).trans |
| ((hM y hy).trans (le_max_right _ _)) |
|
|
| end |
| section |
|
|
| variable {ι : Type*} [Fintype ι] |
|
|
| theorem radialEndRegularPrimitive_norm_le {k : ι → ℕ × ℕ} (h0 : ∃ i,k i=(0,0)) |
| (D m : ι → ℕ) (c : ℝ) (p : Radial.Plane) (ε : EndIndex) (y : Plane) {M : ℝ} |
| (hM : ∀ i,‖weightedRegularEndPrimitive (D i) (m i) ε c y‖≤M) : |
| ‖radialEndRegularPrimitive k D m c p ε y‖≤M := by |
| apply (norm_sum_le _ _).trans |
| have hq (i : ι) : 0≤radialEndProbability k D m p ε y i := |
| radialProbability_nonneg h0 D m p _ _ |
| simp only [norm_smul,Real.norm_eq_abs,abs_of_nonneg (hq _)] |
| exact probability_weighted_le |
| (fun i => radialProbability_nonneg h0 D m p (infinityDiffeomorph (parameters 0) ε y) i) |
| (radialProbability_sum h0 D m p _) hM |
|
|
| end |
|
|
| def limitingEndOrder (D S : ℝ) (p : MomentPlane) (ε : EndIndex) : ℝ := |
| if ε 0 then (if ε 1 then D-3*(p 0+p 1) else D-3*(p 0+p 1)-markedHeight S p) |
| else (if ε 1 then D-3*(p 0+p 1)-markedHeight S p else 0) |
|
|
| theorem probability_affine_degree {ι : Type*} [Fintype ι] {w : ι → MomentPlane} |
| {q : ι → ℝ} {p : MomentPlane} (hs : ∑ i,q i=1) (he : ∑ i,q i • w i=p) (D C : ℝ) : |
| (∑ i,q i*(D-C*((w i) 0+(w i) 1)))=D-C*(p 0+p 1) := by |
| have hcoord (j : Fin 2) : ∑ i,q i*(w i) j=p j := by |
| simpa using congrArg (fun z : MomentPlane => z j) he |
| simp only [mul_sub,mul_add,Finset.sum_sub_distrib,Finset.sum_add_distrib] |
| simp_rw [mul_left_comm (q _) C] |
| rw [←Finset.mul_sum,←Finset.mul_sum,←Finset.sum_mul,hs,one_mul,hcoord 0,hcoord 1] |
|
|
| theorem reducedEndProbability_regularized {A B D S : ℕ} (hB : 3*B<D) (hS : 3*S<D) |
| (p : MomentPlane) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| reducedEndProbability (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) p ε y= |
| selected (trapezoidWeight A B) (fun k => |
| regularCoefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) ε y/ |
| (radiusSq y)^(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε)) p := by |
| unfold reducedEndProbability reducedProbability |
| have he : (fun k : TrapezoidWeight A B => coefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) |
| (infinityDiffeomorph (parameters 0) ε y))=(fun k => regularCoefficientNorm (cubicDegree D k) |
| (cubicMarkedOrder S k) ε y/(radiusSq y)^(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε)) := by |
| funext k |
| exact coefficientNorm_eq_regular_div (cubicMarked_lt_degree hB hS k).le |
| (cubicMarked_twice_complement hB hS k) ε hy hR |
| rw [he] |
|
|
| theorem reducedEndResidue_error {A B D S : ℕ} (hA : 0<A) (hS0 : 0<S) |
| (hAB : A≤B) (hSB : S≤B) (hB : 3*B<D) (hS : 3*S<D) |
| {p : MomentPlane} (hp : p∈openTrapezoid A B) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) |
| (ht : 0<radiusSq y) (ht1 : radiusSq y<1) {M : ℝ} (hM : 0≤M) |
| (hlog : ∀ k : TrapezoidWeight A B,|Real.log (regularCoefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) ε y)|≤M) |
| (c : ℝ) : |
| |reducedEndResidue (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) c p ε y- |
| c*limitingEndOrder D S p ε|≤ |
| |c| * ((2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y))) := by |
| have hs := trapezoidWeight_surrounds hAB hp |
| have hb : 0≤(2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y)) := |
| div_nonneg (by positivity) (neg_pos.mpr (Real.log_neg ht ht1)).le |
| have he : reducedEndResidue (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) c p ε y= |
| c*(∑ k,reducedEndProbability (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) p ε y k* |
| (weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ)) := by |
| simp only [reducedEndResidue,weightedEndResidue,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [he,←mul_sub,abs_mul] |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg c) |
| have hsum := reducedProbability_sum (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) p |
| (infinityDiffeomorph (parameters 0) ε y) |
| have hmean := reducedProbability_mean (cubicDegree D) (cubicMarkedOrder S) hs |
| (infinityDiffeomorph (parameters 0) ε y) |
| have hmarked (hn : ∀ k,(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ)= |
| (D:ℝ)-3*((trapezoidWeight A B k) 0+(trapezoidWeight A B k) 1)-markedHeight S (trapezoidWeight A B k)) : |
| |(∑ k,reducedEndProbability (trapezoidWeight A B) (cubicDegree D) (cubicMarkedOrder S) p ε y k* |
| (weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ))- |
| ((D:ℝ)-3*(p 0+p 1)-markedHeight S p)|≤ |
| (2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y)) := by |
| rw [reducedEndProbability_regularized hB hS p ε hy hR] |
| exact selected_marked_error hA hS0 hAB hSB |
| (fun k => regularCoefficientNorm_pos _ _ _ _) hlog ht ht1 hn hp |
| cases h0 : ε 0 <;> cases h1 : ε 1 |
| · simp only [limitingEndOrder,h0,h1,Bool.false_eq_true,↓reduceIte,weightedEndOrder,Nat.cast_zero,mul_zero,Finset.sum_const_zero,sub_self,abs_zero] |
| exact hb |
| · simpa only [limitingEndOrder,h0,h1,Bool.false_eq_true,↓reduceIte] using |
| hmarked (fun k => by simpa only [h0,h1,Bool.false_eq_true,↓reduceIte] using cubicEndOrder_cast hB hS ε k) |
| · simpa only [limitingEndOrder,h0,h1,Bool.false_eq_true,↓reduceIte] using |
| hmarked (fun k => by simpa only [h0,h1,Bool.false_eq_true,↓reduceIte] using cubicEndOrder_cast hB hS ε k) |
| · have hdeg (k : TrapezoidWeight A B) := cubicEndOrder_cast hB hS ε k |
| simp only [h0,h1,↓reduceIte] at hdeg |
| simp only [hdeg,limitingEndOrder,h0,h1,↓reduceIte] |
| rw [probability_affine_degree (by exact hsum) (by exact hmean),sub_self,abs_zero] |
| exact hb |
|
|
| end PackingSufficiencySupport.CubicModel |
|
|
| namespace PackingSufficiencySupport.FiniteMoment.Radial |
| open scoped Topology |
| open Set Filter |
| open scoped BigOperators ContDiff |
| open Function |
|
|
| theorem lowerTrapezoid_interior_sequence {A B : ℝ} {p : Plane} |
| (hp : p∈lowerTrapezoid A B) : |
| ∃ q : ℕ → Plane,(∀ n,planeVector (q n).1 (q n).2∈openTrapezoid A B) ∧ |
| Tendsto q atTop (𝓝 p) := by |
| let e : ℝ := min (A-p.1) (B-(p.1+p.2))/4 |
| have he : 0<e := div_pos (lt_min (sub_pos.mpr hp.2.1) (sub_pos.mpr hp.2.2.2)) (by norm_num) |
| have he0 : e≤(A-p.1)/4 := div_le_div_of_nonneg_right (min_le_left _ _) (by norm_num) |
| have he1 : e≤(B-(p.1+p.2))/4 := div_le_div_of_nonneg_right (min_le_right _ _) (by norm_num) |
| let d : ℕ → ℝ := fun n => e*(1/((n:ℝ)+1)) |
| have hd (n : ℕ) : 0<d n ∧ d n≤e := by |
| have hn : (1:ℝ)≤(n:ℝ)+1 := by have hh : (0:ℝ)≤n := Nat.cast_nonneg n; linarith |
| refine ⟨mul_pos he (one_div_pos.mpr (by positivity)),?_⟩ |
| simpa [d] using mul_le_mul_of_nonneg_left (one_div_le_one_div_of_le (by norm_num) hn) he.le |
| refine ⟨fun n => (p.1+d n,p.2+d n),?_,?_⟩ |
| · intro n |
| change 0<p.1+d n ∧ p.1+d n<A ∧ 0<p.2+d n ∧ (p.1+d n)+(p.2+d n)<B |
| have hh := hd n |
| exact ⟨by linarith [hp.1],by linarith,by linarith [hp.2.2.1],by linarith⟩ |
| · have ht : Tendsto d atTop (𝓝 0) := by |
| simpa [d] using tendsto_const_nhds.mul (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) |
| simpa only [add_zero,Prod.eta] using |
| (tendsto_const_nhds.add ht).prodMk_nhds (tendsto_const_nhds.add ht) |
|
|
| theorem trapezoid_selected_tendsto {A B : ℕ} (hA : 0<A) (hAB : A≤B) |
| {a : TrapezoidWeight A B → ℝ} (ha : ∀ i,0<a i) {p : Plane} |
| (hp : p∈lowerTrapezoid A B) {q : ℕ → Plane} |
| (hq : ∀ n,q n∈lowerTrapezoid A B) (ht : Tendsto q atTop (𝓝 p)) |
| (i : TrapezoidWeight A B) : |
| Tendsto (fun n => selected latticeIndex a (q n) i) atTop (𝓝 (selected latticeIndex a p i)) := by |
| have hf := (trapezoid_selected_contDiffWithinAt hA hAB ha hp i).continuousWithinAt |
| have hm : Tendsto (fun n => (a,q n)) atTop |
| (𝓝[{z : (TrapezoidWeight A B → ℝ) × Plane | (∀ j,0<z.1 j) ∧ z.2∈lowerTrapezoid A B}] (a,p)) := |
| tendsto_nhdsWithin_iff.mpr ⟨tendsto_const_nhds.prodMk_nhds ht,Eventually.of_forall (fun n => ⟨ha,hq n⟩)⟩ |
| exact hf.tendsto.comp hm |
|
|
| theorem selected_marked_error_lower {A B S : ℕ} (hA : 0<A) (hS : 0<S) |
| (hAB : A≤B) (hSB : S≤B) {a : TrapezoidWeight A B → ℝ} {n : TrapezoidWeight A B → ℕ} |
| {M t D C : ℝ} (ha : ∀ k,0<a k) (hloga : ∀ k,|Real.log (a k)|≤M) |
| (ht : 0<t) (ht1 : t<1) |
| (hn : ∀ k,(n k:ℝ)=D-C*((trapezoidWeight A B k) 0+(trapezoidWeight A B k) 1)- |
| markedHeight S (trapezoidWeight A B k)) {p : Plane} (hp : p∈lowerTrapezoid A B) : |
| |(∑ k,selected latticeIndex (fun j => a j/t^(n j)) p k*(n k:ℝ))- |
| (D-C*(p.1+p.2)-markedHeight S (planeVector p.1 p.2))|≤ |
| (2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log t) := by |
| let b : TrapezoidWeight A B → ℝ := fun j => a j/t^(n j) |
| have hb (j : TrapezoidWeight A B) : 0<b j := div_pos (ha j) (pow_pos ht _) |
| obtain ⟨q,hq,hqlim⟩ := lowerTrapezoid_interior_sequence hp |
| have hqlo (j : ℕ) : q j∈lowerTrapezoid A B := |
| ⟨(hq j).1.le,(hq j).2.1,(hq j).2.2.1.le,(hq j).2.2.2⟩ |
| have hs (j : ℕ) : |
| |(∑ k,selected latticeIndex b (q j) k*(n k:ℝ))- |
| (D-C*((q j).1+(q j).2)-markedHeight S (planeVector (q j).1 (q j).2))|≤ |
| (2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log t) := by |
| have hw : Surrounds (weight latticeIndex) (planeVector (q j).1 (q j).2) := |
| trapezoidWeight_surrounds hAB (hq j) |
| simp_rw [selected_eq_exponential hb (lattice_zero A B) (lattice_unit_fst hA hAB) |
| (lattice_unit_snd hA hAB) hw] |
| exact selected_marked_error hA hS hAB hSB ha hloga ht ht1 hn (hq j) |
| have hsum : Tendsto (fun j => ∑ k,selected latticeIndex b (q j) k*(n k:ℝ)) atTop |
| (𝓝 (∑ k,selected latticeIndex b p k*(n k:ℝ))) := |
| tendsto_finsetSum _ (fun i _ => (trapezoid_selected_tendsto hA hAB hb hp hqlo hqlim i).mul_const _) |
| have hh : Continuous (fun z : Plane => D-C*(z.1+z.2)-markedHeight S (planeVector z.1 z.2)) := by |
| simp only [markedHeight,planeVector] |
| fun_prop |
| exact le_of_tendsto ((hsum.sub (hh.continuousAt.tendsto.comp hqlim)).abs) (Eventually.of_forall hs) |
|
|
| end PackingSufficiencySupport.FiniteMoment.Radial |
|
|
| namespace PackingSufficiencySupport.CubicModel |
| open scoped BigOperators ContDiff Manifold Topology |
| open Set Function Filter Manifold |
| open DiagonalQuadrics DiagonalQuadrics.Explicit Hamiltonian FiniteMoment |
|
|
| theorem radialEndProbability_regularized {A B D S : ℕ} (hB : 3*B<D) (hS : 3*S<D) |
| (p : Radial.Plane) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| radialEndProbability (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) p ε y= |
| Radial.selected (Radial.latticeIndex (A := A) (B := B)) (fun k => |
| regularCoefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) ε y/ |
| (radiusSq y)^(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε)) p := by |
| unfold radialEndProbability radialProbability |
| have he : (fun k : TrapezoidWeight A B => coefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) |
| (infinityDiffeomorph (parameters 0) ε y))=(fun k => regularCoefficientNorm (cubicDegree D k) |
| (cubicMarkedOrder S k) ε y/(radiusSq y)^(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε)) := by |
| funext k |
| exact coefficientNorm_eq_regular_div (cubicMarked_lt_degree hB hS k).le |
| (cubicMarked_twice_complement hB hS k) ε hy hR |
| rw [he] |
|
|
| theorem radialEndResidue_error {A B D S : ℕ} (hA : 0<A) (hS0 : 0<S) |
| (hAB : A≤B) (hSB : S≤B) (hB : 3*B<D) (hS : 3*S<D) |
| {p : Radial.Plane} (hp : p∈Radial.lowerTrapezoid A B) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) |
| (ht : 0<radiusSq y) (ht1 : radiusSq y<1) {M : ℝ} (hM : 0≤M) |
| (hlog : ∀ k : TrapezoidWeight A B,|Real.log (regularCoefficientNorm (cubicDegree D k) (cubicMarkedOrder S k) ε y)|≤M) |
| (c : ℝ) : |
| |radialEndResidue (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p ε y- |
| c*limitingEndOrder D S (planeVector p.1 p.2) ε|≤ |
| |c| * ((2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y))) := by |
| have hb : 0≤(2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y)) := |
| div_nonneg (by positivity) (neg_pos.mpr (Real.log_neg ht ht1)).le |
| have he : radialEndResidue (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) c p ε y= |
| c*(∑ k,radialEndProbability (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) p ε y k* |
| (weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ)) := by |
| simp only [radialEndResidue,weightedEndResidue,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| rw [he,←mul_sub,abs_mul] |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg c) |
| have hsum := radialProbability_sum (Radial.lattice_zero A B) (cubicDegree D) (cubicMarkedOrder S) p |
| (infinityDiffeomorph (parameters 0) ε y) |
| have hmean := radialProbability_mean hAB (cubicDegree D) (cubicMarkedOrder S) hp |
| (infinityDiffeomorph (parameters 0) ε y) |
| have hmarked (hn : ∀ k,(weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ)= |
| (D:ℝ)-3*((trapezoidWeight A B k) 0+(trapezoidWeight A B k) 1)-markedHeight S (trapezoidWeight A B k)) : |
| |(∑ k,radialEndProbability (Radial.latticeIndex (A := A) (B := B)) (cubicDegree D) (cubicMarkedOrder S) p ε y k* |
| (weightedEndOrder (cubicDegree D k) (cubicMarkedOrder S k) ε:ℝ))- |
| ((D:ℝ)-3*(p.1+p.2)-markedHeight S (planeVector p.1 p.2))|≤ |
| (2*M+Fintype.card (TrapezoidWeight A B))/(-Real.log (radiusSq y)) := by |
| rw [radialEndProbability_regularized hB hS p ε hy hR] |
| exact Radial.selected_marked_error_lower hA hS0 hAB hSB |
| (fun k => regularCoefficientNorm_pos _ _ _ _) hlog ht ht1 hn hp |
| cases h0 : ε 0 <;> cases h1 : ε 1 |
| · simp only [limitingEndOrder,h0,h1,Bool.false_eq_true,↓reduceIte,weightedEndOrder,Nat.cast_zero,mul_zero,Finset.sum_const_zero,sub_self,abs_zero] |
| exact hb |
| · simpa [limitingEndOrder,h0,h1,planeVector] using |
| hmarked (fun k => by simpa only [h0,h1,Bool.false_eq_true,↓reduceIte] using cubicEndOrder_cast hB hS ε k) |
| · simpa [limitingEndOrder,h0,h1,planeVector] using |
| hmarked (fun k => by simpa only [h0,h1,Bool.false_eq_true,↓reduceIte] using cubicEndOrder_cast hB hS ε k) |
| · have hdeg (k : TrapezoidWeight A B) := cubicEndOrder_cast hB hS ε k |
| simp only [h0,h1,↓reduceIte] at hdeg |
| simp only [hdeg,limitingEndOrder,h0,h1,↓reduceIte] |
| rw [probability_affine_degree (by exact hsum) (by exact hmean),sub_self,abs_zero] |
| exact hb |
|
|
| variable {ι : Type*} [Fintype ι] [Nonempty ι] |
|
|
| theorem reducedEndProbability_contDiffAt {w : ι → MomentPlane} (D m : ι → ℕ) {p : MomentPlane} |
| (hp : Surrounds w p) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) (i : ι) : |
| ContDiffAt ℝ ∞ (fun z => reducedEndProbability w D m p ε z i) y := |
| ((reducedProbability_smooth D m hp i).contMDiffAt.comp y |
| ((infinityDiffeomorph (parameters 0) ε).contMDiffOn.contMDiffAt |
| ((infinityDiffeomorph (parameters 0) ε).open_source.mem_nhds hy))).contDiffAt |
|
|
| theorem reducedEndRegularPrimitive_contDiffAt {w : ι → MomentPlane} (D m : ι → ℕ) |
| (c : ℝ) {p : MomentPlane} (hp : Surrounds w p) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) |
| (hR : Complex.equivRealProdCLM.symm y∈endRegularRegion) : |
| ContDiffAt ℝ ∞ (reducedEndRegularPrimitive w D m c p ε) y := |
| ContDiffAt.sum (fun i _ => (reducedEndProbability_contDiffAt D m hp ε hy i).smul |
| (weightedRegularEndPrimitive_contDiffAt (D i) (m i) ε c hR)) |
|
|
| theorem reducedEndResidue_contDiffAt {w : ι → MomentPlane} (D m : ι → ℕ) |
| (c : ℝ) {p : MomentPlane} (hp : Surrounds w p) (ε : EndIndex) {y : Plane} |
| (hy : y∈(infinityDiffeomorph (parameters 0) ε).source) : |
| ContDiffAt ℝ ∞ (reducedEndResidue w D m c p ε) y := |
| ContDiffAt.sum (fun i _ => (reducedEndProbability_contDiffAt D m hp ε hy i).mul contDiffAt_const) |
|
|
| end PackingSufficiencySupport.CubicModel |
| end |
|
|
| end OAI |
|
|
