openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticContinuationEndpoint.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticScalarContinuation | |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticBiasEndpoint | |
| namespace OAI | |
| /-! Endpoint decay for the literal finite-field conditional square levels. | |
| The inverse-coordinate weighted curvature has zero boundary values, proved | |
| from bounded ordinary derivatives rather than an asymptotic expansion. -/ | |
| noncomputable section | |
| open Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma fieldBiasMean_tendsto_one (h : FieldStep) : | |
| Tendsto (fieldBiasMean h) atTop (𝓝 (1 : ℝ)) := by | |
| apply tendsto_order.2 | |
| constructor | |
| · intro l hl | |
| let s := (max l (-1) + 1) / 2 | |
| have hm : max l (-1 : ℝ) < 1 := max_lt hl (by norm_num) | |
| have hs : |s| < 1 := by | |
| apply abs_lt.mpr | |
| dsimp only [s] | |
| constructor <;> linarith [le_max_right l (-1 : ℝ)] | |
| have hls : l < s := by dsimp only [s]; linarith [le_max_left l (-1 : ℝ)] | |
| filter_upwards [eventually_gt_atTop (magneticBias h s)] with b hb | |
| have hh := strictMono_fieldBiasMean h hb | |
| rw [fieldBiasMean_magneticBias h hs] at hh | |
| exact hls.trans hh | |
| · intro u hu | |
| exact Eventually.of_forall fun b => (fieldBiasMean_strict_bounds h b).2.trans hu | |
| lemma fieldBiasMean_tendsto_neg_one (h : FieldStep) : | |
| Tendsto (fieldBiasMean h) atBot (𝓝 (-1 : ℝ)) := by | |
| apply tendsto_order.2 | |
| constructor | |
| · intro l hl | |
| exact Eventually.of_forall fun b => hl.trans (fieldBiasMean_strict_bounds h b).1 | |
| · intro u hu | |
| let s := (min u 1 - 1) / 2 | |
| have hm : (-1 : ℝ) < min u 1 := lt_min hu (by norm_num) | |
| have hs : |s| < 1 := by | |
| apply abs_lt.mpr | |
| dsimp only [s] | |
| constructor <;> linarith [min_le_right u (1 : ℝ)] | |
| have hsu : s < u := by dsimp only [s]; linarith [min_le_left u (1 : ℝ)] | |
| filter_upwards [eventually_lt_atBot (magneticBias h s)] with b hb | |
| have hh := strictMono_fieldBiasMean h hb | |
| rw [fieldBiasMean_magneticBias h hs] at hh | |
| exact hh.trans hsu | |
| lemma magneticLevelAtBias_tendsto_one (h : FieldStep) (i : Fin (h.depth + 1)) : | |
| Tendsto (fun b => magneticLevelAtBias h b i) atTop (𝓝 (1 : ℝ)) := by | |
| have hs := (fieldBiasMean_tendsto_one h).pow 2 | |
| norm_num only [one_pow] at hs | |
| exact hs.squeeze tendsto_const_nhds (fun b => fieldBiasMean_sq_le_magneticLevel h b i) | |
| (fun b => (magneticLevelAtBias_mem_unit h b i).2) | |
| lemma magneticLevelAtBias_tendsto_neg_one (h : FieldStep) (i : Fin (h.depth + 1)) : | |
| Tendsto (fun b => magneticLevelAtBias h b i) atBot (𝓝 (1 : ℝ)) := by | |
| have hs := (fieldBiasMean_tendsto_neg_one h).pow 2 | |
| norm_num only [neg_sq, one_pow] at hs | |
| exact hs.squeeze tendsto_const_nhds (fun b => fieldBiasMean_sq_le_magneticLevel h b i) | |
| (fun b => (magneticLevelAtBias_mem_unit h b i).2) | |
| lemma magnetic_derivative_tendsto_zero_atBot {F G D : ℝ → ℝ} {c L : ℝ} | |
| (hL : 0 ≤ L) (hF : ∀ x, HasDerivAt F (G x) x) | |
| (hG : ∀ x, HasDerivAt G (D x) x) (hD : ∀ x, |D x| ≤ L) | |
| (hlimit : Tendsto F atBot (𝓝 c)) : Tendsto G atBot (𝓝 0) := by | |
| have hdF (x : ℝ) : HasDerivAt (fun y => F (-y)) (-G (-x)) x := by | |
| convert (hF (-x)).comp x (hasDerivAt_neg x) using 1 <;> first | rfl | simp | |
| have hdG (x : ℝ) : HasDerivAt (fun y => -G (-y)) (D (-x)) x := by | |
| convert ((hG (-x)).comp x (hasDerivAt_neg x)).neg using 1 <;> first | rfl | simp | |
| have hh := magnetic_derivative_tendsto_zero hL hdF hdG (fun x => hD (-x)) | |
| (hlimit.comp tendsto_neg_atTop_atBot) | |
| simpa only [Function.comp_def, neg_neg, neg_zero] using hh.neg.comp tendsto_neg_atBot_atTop | |
| lemma MagneticContinuationJet.first_tendsto_zero (J : MagneticContinuationJet) | |
| {c : ℝ} (hJ : Tendsto J.value atTop (𝓝 c)) : Tendsto J.first atTop (𝓝 0) := by | |
| obtain ⟨L, hL, bL⟩ := J.bSecond | |
| exact magnetic_derivative_tendsto_zero hL J.dValue J.dFirst bL hJ | |
| lemma MagneticContinuationJet.second_tendsto_zero (J : MagneticContinuationJet) | |
| {c : ℝ} (hJ : Tendsto J.value atTop (𝓝 c)) : Tendsto J.second atTop (𝓝 0) := by | |
| obtain ⟨L, hL, bL⟩ := J.bThird | |
| exact magnetic_derivative_tendsto_zero hL J.dFirst J.dSecond bL (J.first_tendsto_zero hJ) | |
| lemma MagneticContinuationJet.first_tendsto_zero_atBot (J : MagneticContinuationJet) | |
| {c : ℝ} (hJ : Tendsto J.value atBot (𝓝 c)) : Tendsto J.first atBot (𝓝 0) := by | |
| obtain ⟨L, hL, bL⟩ := J.bSecond | |
| exact magnetic_derivative_tendsto_zero_atBot hL J.dValue J.dFirst bL hJ | |
| lemma MagneticContinuationJet.second_tendsto_zero_atBot (J : MagneticContinuationJet) | |
| {c : ℝ} (hJ : Tendsto J.value atBot (𝓝 c)) : Tendsto J.second atBot (𝓝 0) := by | |
| obtain ⟨L, hL, bL⟩ := J.bThird | |
| exact magnetic_derivative_tendsto_zero_atBot hL J.dFirst J.dSecond bL | |
| (J.first_tendsto_zero_atBot hJ) | |
| lemma magneticContinuation_weighted_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) | |
| (J : MagneticContinuationJet) (hJ : J.value = fun b => magneticLevelAtBias h b i) : | |
| Tendsto (fun s => J.second (magneticBias h s) - J.first (magneticBias h s) * | |
| (fieldBiasThird h (magneticBias h s) / fieldBiasCurvature h (magneticBias h s))) | |
| (𝓝[<] (1 : ℝ)) (𝓝 0) := by | |
| obtain ⟨L, hL, bL⟩ := J.bSecond | |
| obtain ⟨K, hK, bK⟩ := J.bThird | |
| have hv : Tendsto J.value atTop (𝓝 (1 : ℝ)) := by | |
| rw [hJ] | |
| exact magneticLevelAtBias_tendsto_one h i | |
| exact (magnetic_weighted_continuation_limit h hL hK J.dValue J.dFirst J.dSecond bL bK hv).comp | |
| (magneticBias_tendsto_one h) | |
| lemma magneticContinuation_weighted_neg_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) | |
| (J : MagneticContinuationJet) (hJ : J.value = fun b => magneticLevelAtBias h b i) : | |
| Tendsto (fun s => J.second (magneticBias h s) - J.first (magneticBias h s) * | |
| (fieldBiasThird h (magneticBias h s) / fieldBiasCurvature h (magneticBias h s))) | |
| (𝓝[>] (-1 : ℝ)) (𝓝 0) := by | |
| have hv : Tendsto J.value atBot (𝓝 (1 : ℝ)) := by | |
| rw [hJ] | |
| exact magneticLevelAtBias_tendsto_neg_one h i | |
| have hg := J.first_tendsto_zero_atBot hv | |
| have hd := J.second_tendsto_zero_atBot hv | |
| have hp : Tendsto (fun b => J.first b * (fieldBiasThird h b / fieldBiasCurvature h b)) | |
| atBot (𝓝 0) := by | |
| apply squeeze_zero_norm (fun b => ?_) | |
| (show Tendsto (fun b => |J.first b| * magneticThirdRatioCap (scalarFieldIncrements h)) | |
| atBot (𝓝 0) by | |
| simpa only [Real.norm_eq_abs, abs_zero, zero_mul] using | |
| hg.norm.mul_const (magneticThirdRatioCap (scalarFieldIncrements h))) | |
| rw [Real.norm_eq_abs, abs_mul] | |
| exact mul_le_mul_of_nonneg_left (fieldBiasThird_quotient_bound h b) (abs_nonneg _) | |
| have hh := hd.sub hp | |
| simp only [sub_zero] at hh | |
| exact hh.comp (magneticBias_tendsto_neg_one h) | |
| end InvariantIsing | |
| end | |
| end OAI | |