openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationInverse.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationEndpoint | |
| namespace OAI | |
| /-! Ordinary continuation derivatives pulled back through the actual | |
| optimizing bias. The weighted second derivative has zero boundary limits. -/ | |
| noncomputable section | |
| open Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| def magneticContinuationPullback (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ := | |
| J.value (magneticBias h s) | |
| def magneticContinuationPullbackFirst (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ := | |
| J.first (magneticBias h s) / fieldBiasCurvature h (magneticBias h s) | |
| def magneticContinuationPullbackSecond (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ := | |
| J.second (magneticBias h s) / (fieldBiasCurvature h (magneticBias h s)) ^ 2 - | |
| J.first (magneticBias h s) * fieldBiasThird h (magneticBias h s) / | |
| (fieldBiasCurvature h (magneticBias h s)) ^ 3 | |
| lemma magneticContinuationPullback_hasDerivAt (h : FieldStep) (J : MagneticContinuationJet) | |
| {s : ℝ} (hs : |s| < 1) : | |
| HasDerivAt (magneticContinuationPullback h J) | |
| (magneticContinuationPullbackFirst h J s) s := by | |
| convert (J.dValue (magneticBias h s)).comp s (hasDerivAt_magneticBias h hs) using 1 | |
| · rfl | |
| · simp only [magneticContinuationPullbackFirst, div_eq_mul_inv] | |
| lemma magneticContinuationPullbackFirst_hasDerivAt (h : FieldStep) (J : MagneticContinuationJet) | |
| {s : ℝ} (hs : |s| < 1) : | |
| HasDerivAt (magneticContinuationPullbackFirst h J) | |
| (magneticContinuationPullbackSecond h J s) s := by | |
| have hZ : HasDerivAt (magneticBias h) (1 / fieldBiasCurvature h (magneticBias h s)) s := by | |
| simpa only [one_div] using hasDerivAt_magneticBias h hs | |
| exact inverse_mean_pullback_quotient (fieldBiasCurvature_pos h (magneticBias h s)).ne' | |
| hZ rfl rfl (J.dFirst (magneticBias h s)) (hasDerivAt_fieldBiasCurvature h (magneticBias h s)) | |
| lemma magneticContinuationPullback_second_deriv (h : FieldStep) (J : MagneticContinuationJet) | |
| {s : ℝ} (hs : |s| < 1) : | |
| deriv (deriv (magneticContinuationPullback h J)) s = | |
| magneticContinuationPullbackSecond h J s := by | |
| have hh := magneticContinuationPullbackFirst_hasDerivAt h J hs | |
| apply HasDerivAt.deriv | |
| apply hh.congr_of_eventuallyEq | |
| filter_upwards [Ioo_mem_nhds (abs_lt.mp hs).1 (abs_lt.mp hs).2] with t ht | |
| exact (magneticContinuationPullback_hasDerivAt h J (abs_lt.mpr ht)).deriv | |
| lemma magneticContinuation_weighted_second (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : | |
| (magneticCurvature h s) ^ 2 * magneticContinuationPullbackSecond h J s = | |
| J.second (magneticBias h s) - J.first (magneticBias h s) * | |
| (fieldBiasThird h (magneticBias h s) / fieldBiasCurvature h (magneticBias h s)) := by | |
| unfold magneticCurvature magneticContinuationPullbackSecond | |
| field_simp [(fieldBiasCurvature_pos h (magneticBias h s)).ne'] | |
| lemma magneticFieldLevel_weighted_second_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) : | |
| Tendsto (fun s => (magneticCurvature h s) ^ 2 * | |
| deriv (deriv (fun t => magneticFieldLevel h t i)) s) | |
| (𝓝[<] (1 : ℝ)) (𝓝 0) := by | |
| obtain ⟨J, hJ⟩ := magneticLevelAtBias_has_bounded_jet h i | |
| have hvalue : magneticContinuationPullback h J = fun t => magneticFieldLevel h t i := by | |
| funext t | |
| change J.value (magneticBias h t) = magneticLevelAtBias h (magneticBias h t) i | |
| exact congrFun hJ (magneticBias h t) | |
| apply (magneticContinuation_weighted_endpoint h i J hJ).congr' | |
| have hN : ∀ᶠ s : ℝ in 𝓝[<] (1 : ℝ), -1 < s := | |
| (show ∀ᶠ s : ℝ in 𝓝 (1 : ℝ), -1 < s from Ioi_mem_nhds (by norm_num)).filter_mono | |
| nhdsWithin_le_nhds | |
| filter_upwards [hN, self_mem_nhdsWithin] with s hs hs1 | |
| rw [← hvalue, magneticContinuationPullback_second_deriv h J (abs_lt.mpr ⟨hs, hs1⟩)] | |
| exact (magneticContinuation_weighted_second h J s).symm | |
| lemma magneticFieldLevel_weighted_second_neg_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) : | |
| Tendsto (fun s => (magneticCurvature h s) ^ 2 * | |
| deriv (deriv (fun t => magneticFieldLevel h t i)) s) | |
| (𝓝[>] (-1 : ℝ)) (𝓝 0) := by | |
| obtain ⟨J, hJ⟩ := magneticLevelAtBias_has_bounded_jet h i | |
| have hvalue : magneticContinuationPullback h J = fun t => magneticFieldLevel h t i := by | |
| funext t | |
| change J.value (magneticBias h t) = magneticLevelAtBias h (magneticBias h t) i | |
| exact congrFun hJ (magneticBias h t) | |
| apply (magneticContinuation_weighted_neg_endpoint h i J hJ).congr' | |
| have hN : ∀ᶠ s : ℝ in 𝓝[>] (-1 : ℝ), s < 1 := | |
| (show ∀ᶠ s : ℝ in 𝓝 (-1 : ℝ), s < 1 from Iio_mem_nhds (by norm_num)).filter_mono | |
| nhdsWithin_le_nhds | |
| filter_upwards [hN, self_mem_nhdsWithin] with s hs hs1 | |
| rw [← hvalue, magneticContinuationPullback_second_deriv h J (abs_lt.mpr ⟨hs1, hs⟩)] | |
| exact (magneticContinuation_weighted_second h J s).symm | |
| end InvariantIsing | |
| end | |
| end OAI | |