openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Profiles /ProfileDetectionBridge.lean
| import OAI.Combinatorics.BalancedRyser.Profiles.AnalyticDetection | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileCoefficients | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileDetection | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileIdentifiability | |
| namespace OAI | |
| open MeasureTheory Filter Set | |
| open scoped Topology | |
| namespace Balanced.ProfileIdentifiability | |
| theorem power_series_ne_zero_ae {ι : Type*} (a w : ι → ℝ) | |
| (ha : Summable (fun i => |a i|)) (hw0 : ∀ i, 0 ≤ w i) | |
| (hw : Function.Injective w) (hfinite : ∀ t : ℝ, {i | w i ≤ t}.Finite) | |
| (hne : ∃ i, a i ≠ 0) : | |
| ∀ᵐ z : ℝ ∂volume.restrict (Ioo 0 1), (∑' i, a i * z ^ w i) ≠ 0 := by | |
| have hn := exists_laplaceSeries_ne_zero a w ha hw hfinite hne | |
| have hz := Balanced.power_series_zero_null_of_laplace_nonzero a w ha hw0 hn | |
| simpa only [ae_iff, not_not] using hz | |
| end Balanced.ProfileIdentifiability | |
| namespace Balanced.ProfileDetectionBridge | |
| noncomputable def planeTest (P : ℕ × ℕ → ℝ) (z : ℝ) : ℝ := | |
| ∑' a : ℕ × ℕ, P a * z ^ ((a.1 : ℝ) + Real.sqrt 2 * a.2) | |
| noncomputable def dTest (x : ℕ × ℕ → ℝ) (z : ℝ) : ℝ := | |
| ∑' a : ℕ × ℕ, x a * z ^ ((a.1 : ℝ) - 1 + Real.sqrt 2 * a.2) | |
| noncomputable def cTest (x : ℕ × ℕ → ℝ) (z : ℝ) : ℝ := | |
| ∑' a : ℕ × ℕ, x a * z ^ ((a.1 : ℝ) + Real.sqrt 2 * ((a.2 : ℝ) - 1)) | |
| theorem d_visible_ae_ne (x P : ℕ × ℕ → ℝ) | |
| (hx : Summable x) (hP : Summable P) (hzero : ∀ j, x (0, j) = 0) | |
| (hvisible : ¬ ∀ i j, x (i, j) = if i = 0 then 0 else P (i - 1, j)) : | |
| ∀ᵐ z : ℝ ∂volume.restrict (Ioo 0 1), dTest x z ≠ planeTest P z := by | |
| have hc := ProfileCoefficients.summable_abs_shift_sub x P hx hP | |
| (fun a : ℕ × ℕ => (a.1 + 1, a.2)) ProfileCoefficients.shift_fst_injective | |
| have hne : ∃ a : ℕ × ℕ, x (a.1 + 1, a.2) - P a ≠ 0 := by | |
| by_contra h | |
| push Not at h | |
| exact hvisible ((ProfileCoefficients.d_invisibility_iff x P).mpr ⟨hzero, h⟩) | |
| have hdet := ProfileIdentifiability.power_series_ne_zero_ae | |
| (fun a : ℕ × ℕ => x (a.1 + 1, a.2) - P a) ProfileDetection.profileExponent | |
| hc ProfileDetection.profileExponent_nonneg ProfileDetection.profileExponent_injective | |
| ProfileDetection.profileExponent_finite_sublevel hne | |
| filter_upwards [hdet, ae_restrict_mem measurableSet_Ioo] with z hz hzone | |
| apply sub_ne_zero.mp | |
| change (∑' a : ℕ × ℕ, x a * z ^ ((a.1 : ℝ) - 1 + Real.sqrt 2 * a.2)) - | |
| (∑' a : ℕ × ℕ, P a * z ^ ((a.1 : ℝ) + Real.sqrt 2 * a.2)) ≠ 0 | |
| rw [ProfileCoefficients.original_d_transform_sub x P hx hP hzero | |
| (Real.sqrt_nonneg 2) hzone.1.le hzone.2.le] | |
| exact hz | |
| theorem c_visible_ae_ne (x P : ℕ × ℕ → ℝ) | |
| (hx : Summable x) (hP : Summable P) (hzero : ∀ i, x (i, 0) = 0) | |
| (hvisible : ¬ ∀ i j, x (i, j) = if j = 0 then 0 else P (i, j - 1)) : | |
| ∀ᵐ z : ℝ ∂volume.restrict (Ioo 0 1), cTest x z ≠ planeTest P z := by | |
| have hc := ProfileCoefficients.summable_abs_shift_sub x P hx hP | |
| (fun a : ℕ × ℕ => (a.1, a.2 + 1)) ProfileCoefficients.shift_snd_injective | |
| have hne : ∃ a : ℕ × ℕ, x (a.1, a.2 + 1) - P a ≠ 0 := by | |
| by_contra h | |
| push Not at h | |
| exact hvisible ((ProfileCoefficients.c_invisibility_iff x P).mpr ⟨hzero, h⟩) | |
| have hdet := ProfileIdentifiability.power_series_ne_zero_ae | |
| (fun a : ℕ × ℕ => x (a.1, a.2 + 1) - P a) ProfileDetection.profileExponent | |
| hc ProfileDetection.profileExponent_nonneg ProfileDetection.profileExponent_injective | |
| ProfileDetection.profileExponent_finite_sublevel hne | |
| filter_upwards [hdet, ae_restrict_mem measurableSet_Ioo] with z hz hzone | |
| apply sub_ne_zero.mp | |
| change (∑' a : ℕ × ℕ, x a * z ^ ((a.1 : ℝ) + Real.sqrt 2 * ((a.2 : ℝ) - 1))) - | |
| (∑' a : ℕ × ℕ, P a * z ^ ((a.1 : ℝ) + Real.sqrt 2 * a.2)) ≠ 0 | |
| rw [ProfileCoefficients.original_c_transform_sub x P hx hP hzero | |
| (Real.sqrt_nonneg 2) hzone.1.le hzone.2.le] | |
| exact hz | |
| theorem d_invisible_eq (x P : ℕ × ℕ → ℝ) | |
| (hinvisible : ∀ i j, x (i, j) = if i = 0 then 0 else P (i - 1, j)) | |
| (z : ℝ) : dTest x z = planeTest P z := by | |
| obtain ⟨hzero, hc⟩ := (ProfileCoefficients.d_invisibility_iff x P).mp hinvisible | |
| unfold dTest planeTest | |
| rw [ProfileCoefficients.d_transform_reindex x hzero] | |
| apply tsum_congr | |
| intro a | |
| rw [sub_eq_zero.mp (hc a)] | |
| theorem c_invisible_eq (x P : ℕ × ℕ → ℝ) | |
| (hinvisible : ∀ i j, x (i, j) = if j = 0 then 0 else P (i, j - 1)) | |
| (z : ℝ) : cTest x z = planeTest P z := by | |
| obtain ⟨hzero, hc⟩ := (ProfileCoefficients.c_invisibility_iff x P).mp hinvisible | |
| unfold cTest planeTest | |
| rw [ProfileCoefficients.c_transform_reindex x hzero] | |
| apply tsum_congr | |
| intro a | |
| rw [sub_eq_zero.mp (hc a)] | |
| end Balanced.ProfileDetectionBridge | |
| end OAI | |