| import Mathlib |
| import OAI.Geometry.SmoothYau.Geometry.PacketAxisShift |
| import OAI.Geometry.SmoothYau.Limits.CompactProfileUnitLipschitz |
| import OAI.Geometry.SmoothYau.SphereMetric.UnitCorrelationSymm |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology ENNReal RealInnerProductSpace |
| section NearFarVectors |
| variable {I : Type} [Fintype I] [DecidableEq I] |
|
|
| def maskComplex (s : Finset I) (v : I → ℂ) : I → ℂ := fun i => if i ∈ s then v i else 0 |
|
|
| omit [DecidableEq I] in |
| lemma complexValueVector_norm_le_sum (v : I → ℂ) : |
| ‖complexValueVector v‖ ≤ ∑ i, ‖v i‖ := by |
| have hs := Finset.sum_sq_le_sq_sum_of_nonneg (s := (Finset.univ : Finset I)) |
| (f := fun i => ‖v i‖) (fun _ _ => norm_nonneg _) |
| rw [←complexValueVector_norm_sq] at hs |
| nlinarith [norm_nonneg (complexValueVector v),Finset.sum_nonneg (s := Finset.univ) (f := fun i => ‖v i‖) |
| (fun _ _ => norm_nonneg _)] |
|
|
| omit [DecidableEq I] in |
| lemma complexValueVector_component_le (v : I → ℂ) (i : I) : |
| ‖v i‖ ≤ ‖complexValueVector v‖ := by |
| have hs := Finset.single_le_sum (f := fun j => ‖v j‖^2) (s := Finset.univ) |
| (fun _ _ => sq_nonneg _) (Finset.mem_univ i) |
| rw [←complexValueVector_norm_sq] at hs |
| nlinarith [norm_nonneg (v i),norm_nonneg (complexValueVector v)] |
|
|
| lemma maskComplex_far_error (s : Finset I) (v : I → ℂ) {τ : ℝ} (hτ : 0 ≤ τ) |
| (hfar : ∀ i ∉ s, ‖v i‖ ≤ τ) : |
| ‖complexValueVector (maskComplex s v)-complexValueVector v‖ ≤ (Fintype.card I : ℝ)*τ := by |
| rw [←complexValueVector_sub] |
| apply (complexValueVector_norm_le_sum _).trans |
| calc |
| _ ≤ ∑ _i : I, τ := Finset.sum_le_sum (fun i _ => by |
| by_cases hi : i ∈ s |
| · simp [maskComplex,hi,hτ] |
| · simpa only [maskComplex,ite_eq_right hi,zero_sub,norm_neg] using hfar i hi) |
| _ = _ := by simp |
|
|
| lemma maskComplex_norm_ge (s : Finset I) (v : I → ℂ) {A : ℝ} |
| (h : ∃ i ∈ s, A ≤ ‖v i‖) : A ≤ ‖complexValueVector (maskComplex s v)‖ := by |
| obtain ⟨i,hi,hA⟩ := h |
| exact hA.trans (by simpa only [maskComplex,ite_eq_left hi] using |
| complexValueVector_component_le (maskComplex s v) i) |
|
|
| lemma masked_endpoint_vector_error (s : Finset I) (v w : I → ℂ) (α : ℝ) (hα : 0 < α) |
| (θ : I → ℝ) {ε τ : ℝ} (hε : 0 ≤ ε) (hτ : 0 ≤ τ) |
| (hnear : ∀ i ∈ s, ‖(α : ℂ)*Complex.exp ((θ i : ℂ)*Complex.I)*v i-w i‖ ≤ |
| ε*‖(α : ℂ)*Complex.exp ((θ i : ℂ)*Complex.I)*v i‖) |
| (hfar : ∀ i ∉ s, ‖w i‖ ≤ τ) : |
| let z := maskComplex s v |
| let b := fun i => (α : ℂ)*rotatedWaveValues z θ i |
| ‖complexValueVector b-complexValueVector w‖ ≤ |
| ε*‖complexValueVector b‖+(Fintype.card I : ℝ)*τ := by |
| dsimp only |
| let b := fun i => (α : ℂ)*rotatedWaveValues (maskComplex s v) θ i |
| have hh : ‖complexValueVector b-complexValueVector (maskComplex s w)‖ ≤ |
| ε*‖complexValueVector b‖ := by |
| apply complexValueVector_relative_error _ _ hε |
| intro i |
| by_cases hi : i ∈ s |
| · simpa only [b,rotatedWaveValues,maskComplex,ite_eq_left hi,mul_assoc, |
| norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hα] using hnear i hi |
| · simp [b,rotatedWaveValues,maskComplex,hi] |
| have ht := norm_add_le (complexValueVector b-complexValueVector (maskComplex s w)) |
| (complexValueVector (maskComplex s w)-complexValueVector w) |
| rw [sub_add_sub_cancel] at ht |
| exact ht.trans (add_le_add hh (maskComplex_far_error s w hτ hfar)) |
|
|
| lemma masked_rotated_norm (s : Finset I) (v : I → ℂ) {α : ℝ} (hα : 0 < α) |
| (θ : I → ℝ) : |
| ‖complexValueVector (fun i => (α : ℂ)*rotatedWaveValues (maskComplex s v) θ i)‖ = |
| α*‖complexValueVector (maskComplex s v)‖ := by |
| rw [complexValueVector_real_smul,norm_smul,Real.norm_eq_abs,abs_of_pos hα, |
| complexValueVector_rotated_norm] |
|
|
| end NearFarVectors |
|
|
| theorem finite_near_far_sign_probability {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) : |
| ∃ p > 0, ∀ (I : Type) [Fintype I] [DecidableEq I] |
| (s : Finset I) (v : I → ℂ) (w : Fin 3 → I → ℂ) (α : Fin 3 → ℝ) |
| (θ : I → Fin 3 → ℝ) (m : ℝ) (m' : Fin 3 → ℝ) (A ε δ τ₀ : ℝ) (τ : Fin 3 → ℝ), |
| 0 < A → 0 ≤ ε → ε ≤ δ → 0 ≤ τ₀ → (∀ j, 0 ≤ τ j) → |
| (∃ i ∈ s, A ≤ ‖v i‖) → (∀ j, 0 < α j) → δ ≤ 1/16 → δ ≤ κ/96 → |
| (∀ i ∈ s, ∀ j, |θ i j| ≤ 1) → (∀ i ∈ s, κ ≤ ∑ j, (θ i j)^2) → |
| (∀ i ∉ s, ‖v i‖ ≤ τ₀) → (Fintype.card I : ℝ)*τ₀ ≤ δ*A → |
| (∀ j, (Fintype.card I : ℝ)*τ j ≤ (δ-ε)*α j*A) → |
| (∀ j i, i ∈ s → |
| ‖(α j : ℂ)*Complex.exp ((θ i j : ℂ)*Complex.I)*v i-w j i‖ ≤ |
| ε*‖(α j : ℂ)*Complex.exp ((θ i j : ℂ)*Complex.I)*v i‖) → |
| (∀ j i, i ∉ s → ‖w j i‖ ≤ τ j) → |
| |m/‖complexValueVector v‖| ≤ 1 → (∀ j, |m' j/‖complexValueVector (w j)‖| ≤ 1) → |
| ∃ j, ENNReal.ofReal p ≤ (Measure.pi (fun _ : I => stdGaussian ℂ)) |
| {γ | (m+complexGaussianRealValue v γ)*(m' j+complexGaussianRealValue (w j) γ) < 0} := by |
| obtain ⟨p,hp,hprob⟩ := common_near_wave_sign_probability hκ hκ1 |
| refine ⟨p,hp,?_⟩ |
| intro I _ _ s v w α θ m m' A ε δ τ₀ τ hA hε hεδ hτ₀ hτ hlarge hα hδ1 hδκ hθ heng hfar hbasebudget hendbudget hnear hfar' hm hm' |
| let z := maskComplex s v |
| have hAz : A ≤ ‖complexValueVector z‖ := maskComplex_norm_ge s v hlarge |
| have hz : complexValueVector z ≠ 0 := norm_pos_iff.mp (hA.trans_le hAz) |
| obtain ⟨i₀,hi₀,_hlarge⟩ := hlarge |
| let θ' := fun i j => if i ∈ s then θ i j else θ i₀ j |
| have hθ' (i j) : |θ' i j| ≤ 1 := by |
| dsimp [θ']; split_ifs with hi |
| · exact hθ i hi j |
| · exact hθ i₀ hi₀ j |
| have heng' (i) : κ ≤ ∑ j, (θ' i j)^2 := by |
| by_cases hi : i ∈ s |
| · simpa only [θ',ite_eq_left hi] using heng i hi |
| · simpa only [θ',ite_eq_right hi] using heng i₀ hi₀ |
| have hδ : 0 ≤ δ := hε.trans hεδ |
| have hbase : ‖complexValueVector z-complexValueVector v‖ ≤ δ*‖complexValueVector z‖ := |
| (maskComplex_far_error s v hτ₀ hfar).trans (hbasebudget.trans |
| (mul_le_mul_of_nonneg_left hAz hδ)) |
| have hend (j) : |
| ‖complexValueVector (fun i => (α j : ℂ)*rotatedWaveValues z (fun i => θ' i j) i) |
| -complexValueVector (w j)‖ ≤ |
| δ*‖complexValueVector (fun i => (α j : ℂ)*rotatedWaveValues z (fun i => θ' i j) i)‖ := by |
| have hh := masked_endpoint_vector_error s v (w j) (α j) (hα j) (fun i => θ' i j) |
| hε (hτ j) (fun i hi => by simpa only [θ',ite_eq_left hi] using hnear j i hi) (hfar' j) |
| dsimp only at hh |
| change _ ≤ _ at hh |
| refine hh.trans ?_ |
| have hnorm := masked_rotated_norm s v (hα j) (fun i => θ' i j) |
| change ‖complexValueVector (fun i => (α j : ℂ)*rotatedWaveValues z (fun i => θ' i j) i)‖ = |
| α j*‖complexValueVector z‖ at hnorm |
| rw [hnorm] |
| have hb := (hendbudget j).trans (mul_le_mul_of_nonneg_left hAz |
| (mul_nonneg (sub_nonneg.mpr hεδ) (hα j).le)) |
| nlinarith |
| exact hprob I z v w α θ' m m' δ hz hα hδ1 hδκ hθ' heng' hbase hend hm hm' |
| end |
| |
| |
| |
| section |
| open Set Filter |
| open scoped Topology ContDiff RealInnerProductSpace |
| |
| lemma finite_packet_variance_lower {I : Type} [Fintype I] |
| (φ : NormalWaveSpace → ℝ) {K : Set NormalWaveSpace} |
| (q : I → NormalWaveSpace) (U : I → NormalWaveSpace → ℂ) |
| (β : I → NormalWaveSpace →L[ℝ] ℝ) {n H M r : ℝ} |
| (hn : 1 ≤ n) (hH : 0 ≤ H) (hr : 0 ≤ r) (hnr : n⁻¹ ≤ r) (hMr : M*r ≤ 1/2) |
| (hq : ∀ i, q i ∈ K) |
| (hφ : ∀ q ∈ K, ∀ x, ‖x-q‖ ≤ 1 → ‖fderiv ℝ φ x‖ ≤ H ∧ |φ x-φ q| ≤ H*‖x-q‖) |
| (hu : ∀ i, U i (q i) = Complex.exp ((n : ℂ)*(φ (q i) : ℂ))) |
| (hend : ∀ i x, ‖x-q i‖ ≤ r → n*‖x-q i‖ ≤ 1 → |
| ‖U i x-(Real.exp (n*fderiv ℝ φ (q i) (x-q i)) : ℂ)* |
| Complex.exp (((n*β i (x-q i) : ℝ) : ℂ)*Complex.I)*U i (q i)‖ ≤ |
| (M*r)*‖(Real.exp (n*fderiv ℝ φ (q i) (x-q i)) : ℂ)* |
| Complex.exp (((n*β i (x-q i) : ℝ) : ℂ)*Complex.I)*U i (q i)‖) |
| (x : NormalWaveSpace) (hcov : ∃ i, n*‖x-q i‖ ≤ 1) : |
| ∃ i, n*‖x-q i‖ ≤ 1 ∧ |
| (Real.exp (-2*H)/2)*Real.exp (n*φ x) ≤ ‖U i x‖ := by |
| have hn0 : 0 < n := zero_lt_one.trans_le hn |
| obtain ⟨i,hi⟩ := hcov |
| have hdist : ‖x-q i‖ ≤ n⁻¹ := by |
| simpa only [mul_one] using (le_inv_mul_iff₀ hn0).mpr hi |
| have hdist1 : ‖x-q i‖ ≤ 1 := hdist.trans ((inv_le_one₀ hn0).mpr hn) |
| have hq0 : ‖q i-q i‖ ≤ 1 := by simp |
| have hφx := hφ (q i) (hq i) x hdist1 |
| have hφq := hφ (q i) (hq i) (q i) hq0 |
| exact ⟨i,hi,packet_value_lower_of_endpoint φ hn hH hr hMr hi hφq.1 hφx.2 |
| (U i) (β i) (hu i) (hend i x (hdist.trans hnr) hi)⟩ |
| |
| lemma finite_packet_standardized_mean {I : Type} [Fintype I] |
| (v : I → ℂ) {A m : ℝ} (hA : 0 < A) (hv : ∃ i, A ≤ ‖v i‖) (hm : |m| ≤ A) : |
| |m/‖complexValueVector v‖| ≤ 1 := by |
| classical |
| obtain ⟨i,hi⟩ := hv |
| have hnorm : A ≤ ‖complexValueVector v‖ := hi.trans (complexValueVector_component_le v i) |
| rw [abs_div,abs_of_pos (hA.trans_le hnorm)] |
| exact (div_le_one (hA.trans_le hnorm)).mpr (hm.trans hnorm) |
| end |
| |
| |
| |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology ENNReal RealInnerProductSpace |
| |
| theorem localized_packet_sign_probability {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) : |
| ∃ p > 0, ∀ (I : Type) [Fintype I] [DecidableEq I] |
| (φ : NormalWaveSpace → ℝ) (K : Set NormalWaveSpace) |
| (q : I → NormalWaveSpace) (β : I → NormalWaveSpace →L[ℝ] ℝ) |
| (U : I → NormalWaveSpace → ℂ) (b : NormalWaveSpace → ℝ) |
| (H Hs ε ℓ t T c M δ : ℝ) (x : NormalWaveSpace), |
| 0 ≤ H → 0 ≤ Hs → 0 < ε → ε ≤ 1 → 1 ≤ ℓ → 2 ≤ t → |
| 0 < T → 0 < c → 0 ≤ M → 0 < δ → δ ≤ 1/16 → δ ≤ κ/96 → |
| M*(2/t) ≤ δ/2 → (∀ i, q i ∈ K) → x ∈ K → |
| (∀ j, packetAxisShift x (t^4) ε ℓ j ∈ K) → |
| (∀ y ∈ K, ∃ i, t^4*‖y-q i‖ ≤ 1) → |
| (∀ q ∈ K, ∀ y, ‖y-q‖ ≤ 1 → ‖fderiv ℝ φ y‖ ≤ H ∧ |φ y-φ q| ≤ H*‖y-q‖) → |
| (∀ j, Real.exp (-Hs) ≤ Real.exp ((ε/ℓ)*fderiv ℝ φ x |
| (EuclideanSpace.basisFun (Fin 3) ℝ j))) → |
| (∀ i, U i (q i) = Complex.exp (((t^4 : ℝ) : ℂ)*(φ (q i) : ℂ))) → |
| (∀ i y, ‖U i y‖ ≤ T*Real.exp (t^4*φ y)*Real.exp (-c*t^4*‖y-q i‖^2)) → |
| (∀ i a y, ‖a-q i‖ ≤ 2/t → ‖y-q i‖ ≤ 2/t → t^4*‖y-a‖ ≤ 1 → |
| ‖U i y-(Real.exp (t^4*fderiv ℝ φ a (y-a)) : ℂ)* |
| Complex.exp (((t^4*β i (y-a) : ℝ) : ℂ)*Complex.I)*U i a‖ ≤ |
| (M*(2/t))*‖(Real.exp (t^4*fderiv ℝ φ a (y-a)) : ℂ)* |
| Complex.exp (((t^4*β i (y-a) : ℝ) : ℂ)*Complex.I)*U i a‖) → |
| (∀ i, ‖x-q i‖ ≤ t⁻¹ → |
| (∀ j, |ε/ℓ*β i (EuclideanSpace.basisFun (Fin 3) ℝ j)| ≤ 1) ∧ |
| κ ≤ ∑ j, (ε/ℓ*β i (EuclideanSpace.basisFun (Fin 3) ℝ j))^2) → |
| (Fintype.card I : ℝ)*T*Real.exp H*Real.exp (-c*t^2/4) ≤ |
| (δ/2)*Real.exp (-Hs)*(Real.exp (-2*H)/2) → |
| (∀ y ∈ K, |b y| ≤ (Real.exp (-2*H)/2)*Real.exp (t^4*φ y)) → |
| ∃ j, ENNReal.ofReal p ≤ (Measure.pi (fun _ : I => stdGaussian ℂ)) |
| {γ | (b x+complexGaussianRealValue (fun i => U i x) γ)* |
| (b (packetAxisShift x (t^4) ε ℓ j)+ |
| complexGaussianRealValue (fun i => U i (packetAxisShift x (t^4) ε ℓ j)) γ) < 0} := by |
| obtain ⟨p,hp,hprob⟩ := finite_near_far_sign_probability hκ hκ1 |
| refine ⟨p,hp,?_⟩ |
| intro I _ _ φ K q β U b H Hs ε ℓ t T c M δ x hH hHs hε hε1 hℓ ht hT hc hM hδ hδ1 hδκ hMr hq hx hy hcov hφ hαlo hcenter hloc hend hang hbudget hb |
| have ht1 : 1 ≤ t := by linarith |
| obtain ⟨ht0,hn1,hinv,hsqrt⟩ := quartic_inverse_bounds ht1 |
| have hn : 0 < t^4 := by positivity |
| have hr : 0 ≤ 2/t := by positivity |
| have hnr : (t^4)⁻¹ ≤ 2/t := hinv.trans (by rw [div_eq_mul_inv]; nlinarith [inv_pos.mpr ht0]) |
| have hMrhalf : M*(2/t) ≤ 1/2 := by linarith |
| let y : Fin 3 → NormalWaveSpace := packetAxisShift x (t^4) ε ℓ |
| let α : Fin 3 → ℝ := fun j => Real.exp ((ε/ℓ)*fderiv ℝ φ x (EuclideanSpace.basisFun (Fin 3) ℝ j)) |
| let θ : I → Fin 3 → ℝ := fun i j => ε/ℓ*β i (EuclideanSpace.basisFun (Fin 3) ℝ j) |
| let A := (Real.exp (-2*H)/2)*Real.exp (t^4*φ x) |
| let τ := T*Real.exp H*Real.exp (t^4*φ x)*Real.exp (-c*t^2/4) |
| let s : Finset I := Finset.univ.filter (fun i => ‖x-q i‖ ≤ t⁻¹) |
| have hA : 0 < A := by dsimp [A]; positivity |
| have hτ : 0 ≤ τ := by dsimp [τ]; positivity |
| have hstep (j) : t^4*‖y j-x‖ ≤ 1 := packetAxisShift_step hn hε.le hε1 hℓ x j |
| have hlow (a) (ha : a ∈ K) : ∃ i, t^4*‖a-q i‖ ≤ 1 ∧ |
| (Real.exp (-2*H)/2)*Real.exp (t^4*φ a) ≤ ‖U i a‖ := by |
| apply finite_packet_variance_lower φ q U β hn1 hH hr hnr hMrhalf hq hφ hcenter |
| (fun i a ha hstep => hend i (q i) a (by simpa using hr) ha hstep) a (hcov a ha) |
| obtain ⟨i₀,hi₀,hlo₀⟩ := hlow x hx |
| have hi₀dist : ‖x-q i₀‖ ≤ t⁻¹ := by |
| have hh : ‖x-q i₀‖ ≤ (t^4)⁻¹ := by |
| simpa only [mul_one] using (le_inv_mul_iff₀ hn).mpr hi₀ |
| exact hh.trans hinv |
| have hlarge : ∃ i ∈ s, A ≤ ‖U i x‖ := ⟨i₀,by simp [s,hi₀dist],hlo₀⟩ |
| have hnear (j i) (hi : i ∈ s) : |
| ‖(α j : ℂ)*Complex.exp ((θ i j : ℂ)*Complex.I)*U i x-U i (y j)‖ ≤ |
| (M*(2/t))*‖(α j : ℂ)*Complex.exp ((θ i j : ℂ)*Complex.I)*U i x‖ := by |
| have hix : ‖x-q i‖ ≤ t⁻¹ := (Finset.mem_filter.mp hi).2 |
| obtain ⟨hxq,hyq⟩ := quartic_near_endpoint ht1 x (y j) (q i) hix (hstep j) |
| have hh := hend i x (y j) hxq hyq (hstep j) |
| rw [packetAxisShift_covector hn.ne' (by linarith : ℓ ≠ 0) x j (fderiv ℝ φ x), |
| packetAxisShift_covector hn.ne' (by linarith : ℓ ≠ 0) x j (β i)] at hh |
| change ‖U i (y j)-(α j : ℂ)*Complex.exp ((θ i j : ℂ)*Complex.I)*U i x‖ ≤ _ at hh |
| rw [norm_sub_rev] |
| exact hh |
| have hfar_at (a : NormalWaveSpace) (ha : t^4*‖a-x‖ ≤ 1) |
| (hφa : Real.exp (t^4*φ a) ≤ Real.exp H*Real.exp (t^4*φ x)) |
| (i : I) (hi : i ∉ s) : ‖U i a‖ ≤ τ := by |
| have hix : t⁻¹ < ‖x-q i‖ := lt_of_not_ge (by simpa [s] using hi) |
| have hd := quartic_far_endpoint ht x a (q i) hix ha |
| have hexp : Real.exp (-c*t^4*‖a-q i‖^2) ≤ Real.exp (-c*t^2/4) := by |
| apply Real.exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_left hd hc.le] |
| apply (hloc i a).trans |
| dsimp [τ] |
| calc |
| _ ≤ T*(Real.exp H*Real.exp (t^4*φ x))*Real.exp (-c*t^2/4) := by gcongr |
| _ = _ := by ring |
| have hfar₀ (i) (hi : i ∉ s) : ‖U i x‖ ≤ τ := by |
| apply hfar_at x (by simp) ?_ i hi |
| exact le_mul_of_one_le_left (Real.exp_pos _).le (Real.one_le_exp_iff.mpr hH) |
| have hfar₁ (j i) (hi : i ∉ s) : ‖U i (y j)‖ ≤ τ := by |
| have hd : ‖y j-x‖ ≤ 1 := by nlinarith [hstep j,norm_nonneg (y j-x)] |
| exact hfar_at (y j) (hstep j) (profile_exp_close φ hn.le hH x (y j) (hstep j) |
| (hφ x hx (y j) hd).2) i hi |
| have hα (j) : 0 < α j := Real.exp_pos _ |
| have htail : (Fintype.card I : ℝ)*τ ≤ (δ/2)*Real.exp (-Hs)*A := by |
| have hh := mul_le_mul_of_nonneg_right hbudget (Real.exp_pos (t^4*φ x)).le |
| dsimp [τ,A] |
| calc |
| _ = ((Fintype.card I : ℝ)*T*Real.exp H*Real.exp (-c*t^2/4))*Real.exp (t^4*φ x) := by ring |
| _ ≤ ((δ/2)*Real.exp (-Hs)*(Real.exp (-2*H)/2))*Real.exp (t^4*φ x) := hh |
| _ = _ := by ring |
| have ha01 : Real.exp (-Hs) ≤ 1 := Real.exp_le_one_iff.mpr (by linarith) |
| have hbasebudget : (Fintype.card I : ℝ)*τ ≤ δ*A := by |
| apply htail.trans |
| have hh := mul_le_mul_of_nonneg_left ha01 (mul_nonneg (half_pos hδ).le hA.le) |
| nlinarith |
| have hendbudget (j) : (Fintype.card I : ℝ)*τ ≤ (δ-M*(2/t))*α j*A := by |
| apply htail.trans |
| have hal : Real.exp (-Hs) ≤ α j := hαlo j |
| have hh' : (δ/2)*Real.exp (-Hs) ≤ (δ-M*(2/t))*α j := by |
| have hn' : δ/2 ≤ δ-M*(2/t) := by linarith |
| exact mul_le_mul hn' hal (Real.exp_pos (-Hs)).le (by linarith) |
| exact mul_le_mul_of_nonneg_right hh' hA.le |
| have hstd : |b x/‖complexValueVector (fun i => U i x)‖| ≤ 1 := |
| finite_packet_standardized_mean _ hA ⟨i₀,hlo₀⟩ (hb x hx) |
| have hstd' (j) : |b (y j)/‖complexValueVector (fun i => U i (y j))‖| ≤ 1 := by |
| obtain ⟨i,_hi,hlo⟩ := hlow (y j) (hy j) |
| exact finite_packet_standardized_mean _ (by positivity) ⟨i,hlo⟩ (hb (y j) (hy j)) |
| exact hprob I s (fun i => U i x) (fun j i => U i (y j)) α θ (b x) (fun j => b (y j)) |
| A (M*(2/t)) δ τ (fun _ => τ) hA (mul_nonneg hM hr) (by linarith) hτ (fun _ => hτ) |
| hlarge hα hδ1 hδκ |
| (fun i hi => (hang i (Finset.mem_filter.mp hi).2).1) |
| (fun i hi => (hang i (Finset.mem_filter.mp hi).2).2) |
| hfar₀ hbasebudget hendbudget hnear hfar₁ hstd hstd' |
| |
| end YauCounterexamples |
| end |
| |
| end OAI |
| |
