| import OAI.Analysis.PlaneSpectrum.ConditionalSpectrum |
| import OAI.MeasureTheory.CompactFactors.RelativeCenters |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped ComplexConjugate |
| open scoped BigOperators |
| open scoped BigOperators BoundedContinuousFunction |
| open Filter Topology |
| open MeasureTheory Filter |
| open scoped ENNReal ComplexConjugate |
| open MeasureTheory ProbabilityTheory |
| open scoped ComplexConjugate |
| open scoped BigOperators |
| open scoped BigOperators |
| open Polynomial |
| open MeasureTheory |
| open scoped BigOperators |
| open MeasureTheory |
| open scoped BigOperators |
| open MeasureTheory |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
| open MeasureTheory |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators |
| open MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory |
| open scoped ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| namespace EuclideanFiveColor.UniformEnumeration |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators ENNReal |
| variable {G Y : Type*} [MeasurableSpace G] [MeasurableSingletonClass G] |
| [MeasurableSpace Y] [Nonempty G] |
|
|
| omit [Nonempty G] in |
| theorem list_of_uniform_atoms (κ : Measure G) |
| (S : Finset G) (a : ℝ≥0∞) (hmass : ∀ d ∈ S, κ {d} = a) |
| (hsupp : ∀ᵐ d ∂κ, d ∈ S) (hc : (S.card : ℝ≥0∞)*a=1) : |
| ∃ g : Fin S.card → G, κ = (S.card : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g i) := by |
| classical |
| let e := S.equivFin |
| let g : Fin S.card → G := fun i => (e.symm i).val |
| have heq := Measure.ae_mem_finset_iff.mp hsupp |
| have ha : a = (S.card : ℝ≥0∞)⁻¹ := by |
| have hn : (S.card : ℝ≥0∞) ≠ 0 := by intro h |
| calc |
| a = 1*a := by simp |
| _ = ((S.card : ℝ≥0∞)⁻¹ * S.card)*a := by rw [ENNReal.inv_mul_cancel hn (by simp)] |
| _ = (S.card : ℝ≥0∞)⁻¹ := by rw [mul_assoc, hc, mul_one] |
| refine ⟨g, ?_⟩ |
| calc |
| κ = ∑ d ∈ S, κ {d} • Measure.dirac d := heq |
| _ = ∑ d ∈ S, (S.card : ℝ≥0∞)⁻¹ • Measure.dirac d := |
| Finset.sum_congr rfl (fun d hd => by rw [hmass d hd, ha]) |
| _ = (S.card : ℝ≥0∞)⁻¹ • ∑ d ∈ S, Measure.dirac d := Finset.smul_sum.symm |
| _ = (S.card : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g i) := by |
| congr 1 |
| rw [← Finset.sum_coe_sort S (fun d => Measure.dirac d)] |
| exact (Equiv.sum_comp e.symm (fun d : S => Measure.dirac d.val)).symm |
|
|
| theorem constant_uniform_lists (ν : Measure Y) [IsProbabilityMeasure ν] |
| (κ : Kernel Y G) [IsMarkovKernel κ] (a : ℝ≥0∞) (ha : 0 < a) |
| (hf : ∀ᵐ y ∂ν, ∃ S : Finset G, S.Nonempty ∧ |
| (∀ d ∈ S, κ y {d} = a) ∧ (∀ᵐ d ∂κ y, d ∈ S) ∧ (S.card : ℝ≥0∞)*a=1) : |
| ∃ m : ℕ, 0 < m ∧ ∃ g : Y → Fin m → G, |
| ∀ᵐ y ∂ν, κ y = (m : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g y i) := by |
| classical |
| obtain ⟨y₀, S₀, hS₀, _, _, hc₀⟩ := hf.exists |
| let m := S₀.card |
| have hm : 0 < m := Finset.card_pos.mpr hS₀ |
| have haT : a ≠ ⊤ := by |
| intro h |
| rw [h, ENNReal.mul_top (by exact_mod_cast hm.ne')] at hc₀ |
| exact ENNReal.top_ne_one hc₀ |
| have hlist : ∀ᵐ y ∂ν, ∃ g : Fin m → G, |
| κ y = (m : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g i) := by |
| filter_upwards [hf] with y hy |
| obtain ⟨S, _hS, hSa, hSupp, hc⟩ := hy |
| have hcard : S.card = m := by |
| have hh : (S.card : ℝ≥0∞) = (m : ℝ≥0∞) := |
| (ENNReal.mul_right_inj ha.ne' haT).mp (by simpa only [mul_comm] using hc.trans hc₀.symm) |
| exact_mod_cast hh |
| obtain ⟨g, hg⟩ := list_of_uniform_atoms (κ y) S a hSa hSupp hc |
| have hh : (∑ j : Fin S.card, Measure.dirac (g j)) = |
| ∑ i : Fin m, Measure.dirac (g (Fin.cast hcard.symm i)) := |
| (Equiv.sum_comp (finCongr hcard.symm) (fun j => Measure.dirac (g j))).symm |
| refine ⟨fun i => g (Fin.cast hcard.symm i), ?_⟩ |
| calc |
| κ y = (S.card : ℝ≥0∞)⁻¹ • ∑ i : Fin m, Measure.dirac (g (Fin.cast hcard.symm i)) := hg.trans (congrArg ((S.card : ℝ≥0∞)⁻¹ • ·) hh) |
| _ = _ := by congr 2 |
| let g : Y → Fin m → G := fun y => if h : ∃ g : Fin m → G, |
| κ y = (m : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g i) then h.choose else fun _ => Classical.choice inferInstance |
| refine ⟨m, hm, g, ?_⟩ |
| filter_upwards [hlist] with y hy |
| simpa only [g, dite_eq_left hy] using hy.choose_spec |
|
|
| end EuclideanFiveColor.UniformEnumeration |
|
|
| open MeasureTheory ProbabilityTheory |
| namespace EuclideanFiveColor.DifferenceInvariant |
| variable {G Y : Type*} [CommGroup G] [MeasurableSpace G] |
| [MeasurableMul₂ G] [MeasurableInv G] [MeasurableSpace Y] |
|
|
| noncomputable def differenceLaw (ν : Measure Y) (κ : Kernel Y G) [IsMarkovKernel κ] : Measure G := |
| ((κ ×ₖ κ) ∘ₘ ν).map (fun p : G × G => p.1 / p.2) |
|
|
| lemma invariant (ν : Measure Y) [IsProbabilityMeasure ν] |
| (κ : Kernel Y G) [IsMarkovKernel κ] |
| (R : Y → Y) (hR : MeasurePreserving R ν ν) |
| (T : G →* G) (hT : Measurable T) |
| (he : ∀ᵐ y ∂ν, κ (R y) = (κ y).map T) : |
| MeasurePreserving T (differenceLaw ν κ) (differenceLaw ν κ) := by |
| have hprod : (((κ ×ₖ κ) ∘ₘ ν).map (Prod.map T T)) = ((κ ×ₖ κ) ∘ₘ ν) := by |
| ext s hs |
| rw [Measure.map_apply (hT.prodMap hT) hs, |
| Measure.bind_apply ((hT.prodMap hT) hs) (κ ×ₖ κ).aemeasurable, |
| Measure.bind_apply hs (κ ×ₖ κ).aemeasurable] |
| calc |
| (∫⁻ y, (κ ×ₖ κ) y ((Prod.map T T) ⁻¹' s) ∂ν) = |
| ∫⁻ y, (κ ×ₖ κ) (R y) s ∂ν := by |
| apply lintegral_congr_ae |
| filter_upwards [he] with y hy |
| rw [Kernel.prod_apply, Kernel.prod_apply, hy, |
| Measure.map_prod_map _ _ hT hT, Measure.map_apply (hT.prodMap hT) hs] |
| _ = ∫⁻ y, (κ ×ₖ κ) y s ∂ν := hR.lintegral_comp (Kernel.measurable_coe _ hs) |
| have hdiv : Measurable (fun p : G × G => p.1 / p.2) := measurable_fst.div measurable_snd |
| refine ⟨hT, ?_⟩ |
| rw [differenceLaw, Measure.map_map hT hdiv] |
| calc |
| (((κ ×ₖ κ) ∘ₘ ν).map (T ∘ (fun p : G × G => p.1 / p.2))) = |
| ((κ ×ₖ κ) ∘ₘ ν).map ((fun p : G × G => p.1 / p.2) ∘ Prod.map T T) := by |
| congr 1 |
| funext p |
| exact map_div T p.1 p.2 |
| _ = (((κ ×ₖ κ) ∘ₘ ν).map (Prod.map T T)).map (fun p : G × G => p.1 / p.2) := |
| (Measure.map_map hdiv (hT.prodMap hT)).symm |
| _ = _ := by rw [hprod] |
| |
| end EuclideanFiveColor.DifferenceInvariant |
| |
| /-! The scalar finite bounded-module approximation in (source-4), and its |
| finite-vector consequence. All functions are measurable bounded representatives. -/ |
| namespace EuclideanFiveColor.CompactApproximation |
| open MeasureTheory |
| open scoped BigOperators |
| variable {K Y W : Type*} [SMul K Y] [MeasurableSpace Y] [MeasurableSpace W] |
| |
| def Scalar (ν : Measure Y) (q : Y → W) : Prop := |
| ∀ H : Y → ℂ, Measurable H → (∃ C : ℝ, ∀ y, ‖H y‖ ≤ C) → |
| ∀ ε : ℝ, 0 < ε → ∃ (r : ℕ) (B C : ℝ) (Hf : Fin r → Y → ℂ), |
| 0 ≤ B ∧ 0 ≤ C ∧ (∀ i, Measurable (Hf i)) ∧ (∀ y, ∑ i, ‖Hf i y‖ ≤ C) ∧ |
| ∀ k : K, ∃ a : W → Fin r → ℂ, Measurable a ∧ (∀ w, ‖a w‖ ≤ B) ∧ |
| Integrable (fun y => dist (H (k • y)) (∑ i, a (q y) i * Hf i y)^2) ν ∧ |
| (∫ y, dist (H (k • y)) (∑ i, a (q y) i * Hf i y)^2 ∂ν) < ε^2 |
| |
| def Vector (ν : Measure Y) (q : Y → W) : Prop := |
| ∀ (m : ℕ) (H : Y → Fin m → ℂ), Measurable H → |
| (∃ C : ℝ, ∀ y, ‖H y‖ ≤ C) → ∀ ε : ℝ, 0 < ε → |
| ∃ (r : ℕ) (B C : ℝ) (Hf : Fin r → Y → Fin m → ℂ), |
| 0 ≤ B ∧ 0 ≤ C ∧ (∀ i, Measurable (Hf i)) ∧ (∀ y, ∑ i, ‖Hf i y‖ ≤ C) ∧ |
| ∀ k : K, ∃ a : W → Fin r → ℂ, Measurable a ∧ (∀ w, ‖a w‖ ≤ B) ∧ |
| Integrable (fun y => dist (H (k • y)) (∑ i, a (q y) i • Hf i y)^2) ν ∧ |
| (∫ y, dist (H (k • y)) (∑ i, a (q y) i • Hf i y)^2 ∂ν) < ε^2 |
| |
| lemma pi_dist_sq_le_sum {I : Type*} [Fintype I] (a b : I → ℂ) : |
| dist a b ^ 2 ≤ ∑ i, dist (a i) (b i)^2 := by |
| have hs : 0 ≤ ∑ i, dist (a i) (b i)^2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have hn : dist a b ≤ Real.sqrt (∑ i, dist (a i) (b i)^2) := by |
| rw [dist_eq_norm, pi_norm_le_iff_of_nonneg (Real.sqrt_nonneg _)] |
| intro i |
| change ‖a i - b i‖ ≤ _ |
| rw [← dist_eq_norm] |
| apply (Real.le_sqrt (dist_nonneg) hs).mpr |
| exact Finset.single_le_sum (fun j _ => sq_nonneg (dist (a j) (b j))) (Finset.mem_univ i) |
| nlinarith [Real.sq_sqrt hs, Real.sqrt_nonneg (∑ i, dist (a i) (b i)^2), dist_nonneg (x := a) (y := b)] |
| |
| lemma vector_of_scalar (ν : Measure Y) [IsProbabilityMeasure ν] |
| (q : Y → W) (hq : Measurable q) [MeasurableConstSMul K Y] |
| (hs : Scalar (K := K) ν q) : Vector (K := K) ν q := by |
| classical |
| intro m H hH hHb ε hε |
| let δ : ℝ := ε / (m+1) |
| have hδ : 0 < δ := div_pos hε (by positivity) |
| have hcoord (j : Fin m) : ∃ (r : ℕ) (B C : ℝ) (Hf : Fin r → Y → ℂ), |
| 0 ≤ B ∧ 0 ≤ C ∧ (∀ i, Measurable (Hf i)) ∧ (∀ y, ∑ i, ‖Hf i y‖ ≤ C) ∧ |
| ∀ k : K, ∃ a : W → Fin r → ℂ, Measurable a ∧ (∀ w, ‖a w‖ ≤ B) ∧ |
| Integrable (fun y => dist (H (k • y) j) (∑ i, a (q y) i * Hf i y)^2) ν ∧ |
| (∫ y, dist (H (k • y) j) (∑ i, a (q y) i * Hf i y)^2 ∂ν) < δ^2 := by |
| apply hs (fun y => H y j) ((measurable_pi_iff.mp hH) j) _ δ hδ |
| obtain ⟨C, hC⟩ := hHb |
| exact ⟨C, fun y => (norm_le_pi_norm (H y) j).trans (hC y)⟩ |
| choose r B C Hf hB hC hHm hHb hA using hcoord |
| let I := (j : Fin m) × Fin (r j) |
| let e : Fin (Fintype.card I) ≃ I := (Fintype.equivFin I).symm |
| let V : I → Y → Fin m → ℂ := fun s y => Pi.single s.1 (Hf s.1 s.2 y) |
| have hV (s : I) : Measurable (V s) := by |
| apply Measurable.of_eval |
| intro j |
| by_cases h : s.1 = j |
| · subst j |
| simpa [V] using hHm s.1 s.2 |
| · simp only [V, Pi.single_apply, Ne.symm h, ite_false] |
| exact measurable_const |
| have hVnorm (s : I) (y : Y) : ‖V s y‖ ≤ ‖Hf s.1 s.2 y‖ := by |
| rw [pi_norm_le_iff_of_nonneg (norm_nonneg _)] |
| intro j |
| by_cases h : s.1 = j |
| · subst j; simp [V] |
| · simp [V, h] |
| refine ⟨Fintype.card I, ∑ j, B j, ∑ j, C j, fun i => V (e i), |
| Finset.sum_nonneg (fun j _ => hB j), Finset.sum_nonneg (fun j _ => hC j), |
| fun i => hV (e i), ?_, ?_⟩ |
| · intro y |
| change (∑ i, (fun s : I => ‖V s y‖) (e i)) ≤ _ |
| rw [Equiv.sum_comp e (fun s : I => ‖V s y‖)] |
| calc |
| ∑ s : I, ‖V s y‖ ≤ ∑ s : I, ‖Hf s.1 s.2 y‖ := Finset.sum_le_sum (fun s _ => hVnorm s y) |
| _ = ∑ j, ∑ i, ‖Hf j i y‖ := Fintype.sum_sigma _ |
| _ ≤ ∑ j, C j := Finset.sum_le_sum (fun j _ => hHb j y) |
| · intro k |
| choose a ham hab hai hae using fun j => hA j k |
| let a' : W → I → ℂ := fun w s => a s.1 w s.2 |
| have hap (w : W) : ‖fun i => a' w (e i)‖ ≤ ∑ j, B j := by |
| rw [pi_norm_le_iff_of_nonneg (Finset.sum_nonneg (fun j _ => hB j))] |
| intro i |
| exact ((norm_le_pi_norm (a (e i).1 w) (e i).2).trans (hab (e i).1 w)).trans |
| (Finset.single_le_sum (fun j _ => hB j) (Finset.mem_univ _)) |
| have ham' : Measurable (fun w i => a' w (e i)) := by |
| apply Measurable.of_eval |
| intro i |
| exact (measurable_pi_iff.mp (ham (e i).1)) (e i).2 |
| let A : Y → Fin m → ℂ := fun y => ∑ s : I, a' (q y) s • V s y |
| have hAeq (y : Y) (j : Fin m) : A y j = ∑ i, a j (q y) i * Hf j i y := by |
| change (∑ s : I, a' (q y) s • V s y) j = _ |
| simp only [Finset.sum_apply, Pi.smul_apply] |
| rw [Fintype.sum_sigma] |
| simp [a', V, Pi.single_apply] |
| have hAm : Measurable A := by |
| apply Measurable.of_eval |
| intro j |
| simp_rw [hAeq] |
| exact Finset.measurable_sum _ (fun i _ => ((measurable_pi_iff.mp (ham j)) i |>.comp hq).mul (hHm j i)) |
| have hem : AEStronglyMeasurable (fun y => dist (H (k • y)) (A y)^2) ν := |
| ((hH.comp (measurable_const_smul k)).dist hAm |>.pow_const 2).aestronglyMeasurable |
| have hsint : Integrable (fun y => ∑ j, dist (H (k • y) j) (A y j)^2) ν := by |
| apply integrable_finsetSum |
| intro j _ |
| simpa only [hAeq] using hai j |
| have hbound (y : Y) : dist (H (k • y)) (A y)^2 ≤ ∑ j, dist (H (k • y) j) (A y j)^2 := |
| pi_dist_sq_le_sum _ _ |
| have hint : Integrable (fun y => dist (H (k • y)) (A y)^2) ν := |
| hsint.mono' hem (Filter.Eventually.of_forall (fun y => by simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (dist (H (k • y)) (A y)))] using hbound y)) |
| have hbudget : (∫ y, dist (H (k • y)) (A y)^2 ∂ν) < ε^2 := by |
| calc |
| _ ≤ ∫ y, ∑ j, dist (H (k • y) j) (A y j)^2 ∂ν := integral_mono hint hsint hbound |
| _ = ∑ j, ∫ y, dist (H (k • y) j) (A y j)^2 ∂ν := integral_finsetSum _ (fun j _ => by simpa only [hAeq] using hai j) |
| _ ≤ ∑ _j : Fin m, δ^2 := Finset.sum_le_sum (fun j _ => by simpa only [hAeq] using (hae j).le) |
| _ = (m : ℝ) * δ^2 := by simp |
| _ < ε^2 := by |
| dsimp [δ] |
| have hm : (0 : ℝ) < m+1 := by positivity |
| rw [div_pow, ← mul_div_assoc, div_lt_iff₀ (sq_pos_of_pos hm)] |
| nlinarith [sq_nonneg (m : ℝ), sq_pos_of_pos hε] |
| have hAe (y : Y) : (∑ i, a' (q y) (e i) • V (e i) y) = A y := |
| Equiv.sum_comp e (fun s => a' (q y) s • V s y) |
| refine ⟨fun w i => a' w (e i), ham', hap, ?_, ?_⟩ |
| · simpa only [hAe] using hint |
| · simpa only [hAe] using hbudget |
| |
| end EuclideanFiveColor.CompactApproximation |
| |
| namespace EuclideanFiveColor.KernelSupport |
| open MeasureTheory |
| variable {K X Y : Type*} [Group K] [MeasurableSpace X] [MeasurableSpace Y] |
| [MulAction K X] [MulAction K Y] [MeasurableConstSMul K X] [MeasurableConstSMul K Y] |
| lemma factor_smul_preserving (μ : Measure X) [SMulInvariantMeasure K X μ] |
| (p : X → Y) (hp : Measurable p) |
| (hpe : ∀ k : K, ∀ᵐ x ∂μ, p (k • x) = k • p x) (k : K) : |
| MeasurePreserving (fun y : Y => k • y) (μ.map p) (μ.map p) := by |
| refine ⟨measurable_const_smul k, ?_⟩ |
| rw [Measure.map_map (measurable_const_smul k) hp] |
| calc |
| μ.map ((fun y : Y => k • y) ∘ p) = μ.map (p ∘ (fun x : X => k • x)) := |
| Measure.map_congr ((hpe k).mono fun _ hx => hx.symm) |
| _ = (μ.map (fun x : X => k • x)).map p := |
| (Measure.map_map hp (measurable_const_smul k)).symm |
| _ = μ.map p := by rw [measurePreserving_smul k μ |>.map_eq] |
| end EuclideanFiveColor.KernelSupport |
| |
| namespace PlaneFiveColor.Spectral |
| open MeasureTheory ProbabilityTheory EuclideanFiveColor |
| open scoped BigOperators ENNReal |
| variable {X Y W : Type*} [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace W] |
| [MulAction K X] [MulAction K Y] |
| [MeasurableConstSMul K X] [MeasurableConstSMul K Y] |
| |
| lemma relative_difference_invariant (μ : Measure X) [IsProbabilityMeasure μ] |
| [SMulInvariantMeasure K X μ] (b : X → D) (p : X → Y) |
| (hb : Measurable b) (hp : Measurable p) |
| (hbe : ∀ k : K, ∀ᵐ x ∂μ, b (k • x) = k • b x) |
| (hpe : ∀ k : K, ∀ᵐ x ∂μ, p (k • x) = k • p x) (k : K) : |
| MeasurePreserving (fun d : D => k • d) |
| (DifferenceFourier.differenceLaw (μ.map p) (condDistrib b p μ)) |
| (DifferenceFourier.differenceLaw (μ.map p) (condDistrib b p μ)) := by |
| have he := KernelSupport.condDistrib_equivariant μ b p hb hp |
| (MeasurableEquiv.smul k) (MeasurableEquiv.smul k) (fun d : D => k • d) |
| (measurable_const_smul k) (measurePreserving_smul k μ) (hpe k) (hbe k) |
| exact DifferenceInvariant.invariant (μ.map p) (condDistrib b p μ) |
| (fun y : Y => k • y) (KernelSupport.factor_smul_preserving μ p hp hpe k) |
| (scalarHom (kCoeff k)) (measurable_const_smul k) he |
| |
| lemma finite_encoding_approximation_of_compact |
| [StandardBorelSpace X] [Nonempty X] [StandardBorelSpace Y] [Nonempty Y] |
| [StandardBorelSpace W] [Nonempty W] |
| (μ : Measure X) [IsProbabilityMeasure μ] [SMulInvariantMeasure K X μ] |
| (p : X → Y) (q : Y → W) (b : X → D) |
| (hp : Measurable p) (hq : Measurable q) (hb : Measurable b) |
| (hpe : ∀ k : K, ∀ᵐ x ∂μ, p (k • x) = k • p x) |
| (hbe : ∀ k : K, ∀ᵐ x ∂μ, b (k • x) = k • b x) |
| (m : ℕ) (hm : 0 < m) (g : Y → Fin m → D) |
| (hg : ∀ᵐ y ∂μ.map p, condDistrib b p μ y = |
| (m : ℝ≥0∞)⁻¹ • ∑ i, Measure.dirac (g y i)) |
| (hc : CompactApproximation.Scalar (K := K) (μ.map p) q) : |
| FiniteEncodingApproximation μ p q b m := by |
| classical |
| let H₀ : Y → Fin m → ℂ := PhaseEncoding.kernelEncoding (condDistrib b p μ) |
| (fun d => (eval 1 d : ℂ)) m |
| have hH₀ : Measurable H₀ := |
| PhaseEncoding.measurable_kernelEncoding _ _ (measurable_eval_coe 1) m |
| obtain ⟨A, hA⟩ := PhaseEncoding.exists_encoding_bound m |
| have hHB : ∀ᵐ y ∂μ.map p, ‖H₀ y‖ ≤ max A 0 := by |
| filter_upwards [hg] with y hy |
| have he := PhaseEncoding.kernelEncoding_eq_encoding (condDistrib b p μ) (eval 1) m hm y (g y) hy |
| change ‖PhaseEncoding.kernelEncoding (condDistrib b p μ) (fun d => (eval 1 d : ℂ)) m y‖ ≤ max A 0 |
| rw [he] |
| exact (hA (fun i => eval 1 (g y i))).trans (le_max_left _ _) |
| let H : Y → Fin m → ℂ := fun y => if ‖H₀ y‖ ≤ max A 0 then H₀ y else 0 |
| have hH : Measurable H := hH₀.piecewise (measurableSet_le hH₀.norm measurable_const) measurable_const |
| have hHbound : ∀ y, ‖H y‖ ≤ max A 0 := by |
| intro y |
| by_cases hy : ‖H₀ y‖ ≤ max A 0 |
| · simpa only [H, ite_eq_left hy] using hy |
| · simp only [H, ite_eq_right hy, norm_zero] |
| exact le_max_right _ _ |
| have hHeq : H =ᵐ[μ.map p] H₀ := hHB.mono fun y hy => ite_eq_left hy |
| have hv := CompactApproximation.vector_of_scalar (μ.map p) q hq hc |
| intro γ hγ |
| obtain ⟨r, B, C, Hf, hB, hC, _hHm, hHf, ha⟩ := hv m H hH ⟨max A 0, hHbound⟩ γ hγ |
| refine ⟨r, B, C, Hf, hB, hC, hHf, ?_⟩ |
| intro k |
| obtain ⟨a, _ham, hab, hai, hae⟩ := ha k |
| have hker := KernelSupport.condDistrib_equivariant μ b p hb hp |
| (MeasurableEquiv.smul k) (MeasurableEquiv.smul k) (fun d : D => k • d) |
| (measurable_const_smul k) (measurePreserving_smul k μ) (hpe k) (hbe k) |
| have hcov := PhaseEncoding.kernelEncoding_covariance (μ.map p) (condDistrib b p μ) |
| (fun y : Y => k • y) (fun d : D => k • d) (measurable_const_smul k) |
| (fun d => (eval 1 d : ℂ)) (measurable_eval_coe 1) m hker |
| have heq : ∀ᵐ y ∂μ.map p, H (k • y) = |
| PhaseEncoding.kernelEncoding (condDistrib b p μ) (fun d => (eval (kCoeff k) d : ℂ)) m y := by |
| filter_upwards [(KernelSupport.factor_smul_preserving μ p hp hpe k).quasiMeasurePreserving.ae hHeq, |
| hcov] with y hy hc' |
| exact hy.trans (by simpa only [Function.comp_def, eval_smul_one] using hc') |
| have he : (fun y => dist (H (k • y)) (∑ i, a (q y) i • Hf i y)^2) =ᵐ[μ.map p] |
| (fun y => dist (PhaseEncoding.kernelEncoding (condDistrib b p μ) |
| (fun d => (eval (kCoeff k) d : ℂ)) m y) (∑ i, a (q y) i • Hf i y)^2) := by |
| filter_upwards [heq] with y hy |
| rw [hy] |
| have hai' := hai.congr he |
| refine ⟨a, hab, hai'.comp_measurable hp, ?_⟩ |
| rw [← integral_map hp.aemeasurable hai'.aestronglyMeasurable, |
| ← integral_congr_ae he] |
| exact hae |
|
|
| /-- The complete source successor implication. The predecessor assertion is |
| universal in the equivariant wild map, as required by the transfinite induction. -/ |
| theorem relative_singularity_successor |
| [StandardBorelSpace X] [Nonempty X] [StandardBorelSpace Y] [Nonempty Y] |
| [StandardBorelSpace W] [Nonempty W] |
| [MulAction K W] [MeasurableConstSMul K W] |
| (μ : Measure X) [IsProbabilityMeasure μ] [ErgodicSMul K X μ] |
| (d : X → D) (p : X → Y) (q : Y → W) |
| (hd : Measurable d) (hp : Measurable p) (hq : Measurable q) |
| (hde : ∀ k : K, ∀ᵐ x ∂μ, d (k • x) = k • d x) |
| (hpe : ∀ k : K, ∀ᵐ x ∂μ, p (k • x) = k • p x) |
| (hqe : ∀ k : K, ∀ᵐ y ∂μ.map p, q (k • y) = k • q y) |
| (hwild : IsWild (μ.map d)) |
| (hc : CompactApproximation.Scalar (K := K) (μ.map p) q) |
| (hprev : ∀ b : X → D, Measurable b → |
| (∀ k : K, ∀ᵐ x ∂μ, b (k • x) = k • b x) → IsWild (μ.map b) → |
| DifferenceFourier.differenceLaw (μ.map (q ∘ p)) (condDistrib b (q ∘ p) μ) C = 0) : |
| DifferenceFourier.differenceLaw (μ.map p) (condDistrib d p μ) C = 0 := by |
| by_contra hn |
| have hpos : 0 < RelativeCenters.differenceLaw (μ.map p) (condDistrib d p μ) C := |
| pos_iff_ne_zero.mpr hn |
| obtain ⟨b, hb, hbe, hbw, a, ha, hf⟩ := exact_finite_center_reduction μ d p hd hp hde hpe hwild hpos |
| obtain ⟨m, hm, g, hg⟩ := UniformEnumeration.constant_uniform_lists (μ.map p) |
| (condDistrib b p μ) a ha (hf.mono fun y ⟨S, hS, hSa, hSupp, hcS, _⟩ => |
| ⟨S, hS, hSa, hSupp, hcS⟩) |
| have hcomp : ∀ k : K, ∀ᵐ x ∂μ, (q ∘ p) (k • x) = k • (q ∘ p) x := by |
| intro k |
| filter_upwards [hpe k, ae_of_ae_map hp.aemeasurable (hqe k)] with x hx hy |
| simp only [Function.comp_def, hx, hy] |
| have hpwild : IsWild (DifferenceFourier.differenceLaw (μ.map (q ∘ p)) (condDistrib b (q ∘ p) μ)) := |
| ⟨relative_difference_invariant μ b (q ∘ p) hb (hq.comp hp) hbe hcomp, |
| hprev b hb hbe hbw⟩ |
| exact finite_uniform_successor_contradiction μ p q b hp hq hb m hm g hg hpwild |
| (finite_encoding_approximation_of_compact μ p q b hp hq hb hpe hbe m hm g hg hc) |
|
|
| end PlaneFiveColor.Spectral |
|
|
| namespace EuclideanFiveColor.CompactApproximation |
| open MeasureTheory |
| open scoped BigOperators |
| variable {K Y Z W : Type*} [SMul K Y] [SMul K Z] |
| [MeasurableSpace Y] [MeasurableSpace Z] [MeasurableSpace W] |
|
|
| def At (ν : Measure Y) (q : Y → W) (m : ℕ) (H : Y → Fin m → ℂ) (ε : ℝ) : Prop := |
| ∃ (r : ℕ) (B C : ℝ) (Hf : Fin r → Y → Fin m → ℂ), |
| 0 ≤ B ∧ 0 ≤ C ∧ (∀ i, Measurable (Hf i)) ∧ (∀ y, ∑ i, ‖Hf i y‖ ≤ C) ∧ |
| ∀ k : K, ∃ a : W → Fin r → ℂ, Measurable a ∧ (∀ w, ‖a w‖ ≤ B) ∧ |
| Integrable (fun y => dist (H (k • y)) (∑ i, a (q y) i • Hf i y)^2) ν ∧ |
| (∫ y, dist (H (k • y)) (∑ i, a (q y) i • Hf i y)^2 ∂ν) < ε^2 |
|
|
| lemma lift_at |
| (ν : Measure Y) [IsProbabilityMeasure ν] |
| (p : Y → Z) (q : Z → W) (hp : Measurable p) (hq : Measurable q) |
| [MeasurableConstSMul K Y] [MeasurableConstSMul K Z] |
| (hpres : ∀ k : K, MeasurePreserving (fun y : Y => k • y) ν ν) |
| (hpe : ∀ k : K, ∀ᵐ y ∂ν, p (k • y) = k • p y) |
| (m : ℕ) (H : Y → Fin m → ℂ) (H₀ : Z → Fin m → ℂ) |
| (hH : Measurable H) (hH₀ : Measurable H₀) |
| (hB₀ : ∃ C : ℝ, ∀ z, ‖H₀ z‖ ≤ C) |
| (ε : ℝ) (hε : 0 < ε) |
| (hi : Integrable (fun y => dist (H y) (H₀ (p y))^2) ν) |
| (he : (∫ y, dist (H y) (H₀ (p y))^2 ∂ν) < (ε/2)^2) |
| (hc : Vector (K := K) (ν.map p) q) : At (K := K) ν (q ∘ p) m H ε := by |
| classical |
| obtain ⟨r, B, C, Hf, hB, hC, hHm, hHf, ha⟩ := hc m H₀ hH₀ hB₀ (ε/2) (by positivity) |
| refine ⟨r, B, C, fun i => Hf i ∘ p, hB, hC, fun i => (hHm i).comp hp, |
| fun y => hHf (p y), ?_⟩ |
| intro k |
| obtain ⟨a, ham, hab, hai, hae⟩ := ha k |
| let F : Y → Fin m → ℂ := fun y => ∑ i, a (q (p y)) i • Hf i (p y) |
| have hFm : Measurable F := by |
| exact Finset.measurable_sum _ (fun i _ => |
| (((measurable_pi_iff.mp ham) i).comp (hq.comp hp)).smul ((hHm i).comp hp)) |
| let e₁ : Y → ℝ := fun y => dist (H (k • y)) (H₀ (k • p y))^2 |
| let e₂ : Y → ℝ := fun y => dist (H₀ (k • p y)) (F y)^2 |
| have heq : (fun y => dist (H (k • y)) (H₀ (p (k • y)))^2) =ᵐ[ν] e₁ := |
| (hpe k).mono fun y hy => by simp only [e₁, hy] |
| have hi₁ : Integrable e₁ ν := ((hpres k).integrable_comp hi.aestronglyMeasurable |>.mpr hi).congr heq |
| have he₁ : (∫ y, e₁ y ∂ν) < (ε/2)^2 := by |
| have hm : AEStronglyMeasurable (fun y => dist (H y) (H₀ (p y))^2) |
| (ν.map (fun y : Y => k • y)) := by |
| rw [(hpres k).map_eq] |
| exact hi.aestronglyMeasurable |
| rw [← integral_congr_ae heq, ← integral_map (hpres k).measurable.aemeasurable hm, |
| (hpres k).map_eq] |
| exact he |
| have hi₂ : Integrable e₂ ν := hai.comp_measurable hp |
| have he₂ : (∫ y, e₂ y ∂ν) < (ε/2)^2 := by |
| change (∫ y, dist (H₀ (k • p y)) (∑ i, a (q (p y)) i • Hf i (p y))^2 ∂ν) < _ |
| rw [← integral_map hp.aemeasurable hai.aestronglyMeasurable] |
| exact hae |
| have hmajor (y : Y) : dist (H (k • y)) (F y)^2 ≤ 2*(e₁ y + e₂ y) := by |
| have ht := dist_triangle (H (k • y)) (H₀ (k • p y)) (F y) |
| have hx := dist_nonneg (x := H (k • y)) (y := H₀ (k • p y)) |
| have hy := dist_nonneg (x := H₀ (k • p y)) (y := F y) |
| have hz := dist_nonneg (x := H (k • y)) (y := F y) |
| dsimp [e₁, e₂] |
| nlinarith [sq_nonneg (dist (H (k • y)) (H₀ (k • p y)) - dist (H₀ (k • p y)) (F y))] |
| have himaj : Integrable (fun y => 2*(e₁ y + e₂ y)) ν := (hi₁.add hi₂).const_mul 2 |
| have hint : Integrable (fun y => dist (H (k • y)) (F y)^2) ν := |
| himaj.mono' (((hH.comp (measurable_const_smul k)).dist hFm).pow_const 2).aestronglyMeasurable |
| (Filter.Eventually.of_forall fun y => by simpa only [Real.norm_eq_abs, abs_sq] using hmajor y) |
| refine ⟨a, ham, hab, hint, ?_⟩ |
| calc |
| (∫ y, dist (H (k • y)) (F y)^2 ∂ν) ≤ ∫ y, 2*(e₁ y + e₂ y) ∂ν := |
| integral_mono hint himaj hmajor |
| _ = 2*((∫ y, e₁ y ∂ν) + ∫ y, e₂ y ∂ν) := by |
| rw [integral_const_mul, integral_add hi₁ hi₂] |
| _ < ε^2 := by nlinarith |
| |
| end EuclideanFiveColor.CompactApproximation |
| |
| end |
| |
| end OAI |
| |
