| import OAI.Probability.InvariantIsing.Spectral.ScaledSpectralLaw |
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| namespace OAI |
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| /-! Spectral support under a scalar of either sign. -/ |
| noncomputable section |
| open MeasureTheory Set |
| namespace InvariantIsing |
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| lemma signed_scaled_spectral_support (ν : ProbabilityMeasure ℝ) (a b c : ℝ) |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) : |
| (scaledSpectralLaw ν c : Measure ℝ).support ⊆ |
| Icc (min (c*a) (c*b)) (max (c*a) (c*b)) := by |
| apply Measure.support_subset_of_isClosed isClosed_Icc |
| change ∀ᵐ y ∂((ν : Measure ℝ).map (fun x => c*x)), |
| y∈Icc (min (c*a) (c*b)) (max (c*a) (c*b)) |
| apply (ae_map_iff (measurable_const.mul measurable_id).aemeasurable measurableSet_Icc).mpr |
| filter_upwards [(ν : Measure ℝ).support_mem_ae] with x hx |
| by_cases hc : 0 ≤ c |
| · exact ⟨(min_le_left _ _).trans (mul_le_mul_of_nonneg_left (hbound hx).1 hc), |
| (mul_le_mul_of_nonneg_left (hbound hx).2 hc).trans (le_max_right _ _)⟩ |
| · exact ⟨(min_le_right _ _).trans (mul_le_mul_of_nonpos_left (hbound hx).2 (le_of_not_ge hc)), |
| (mul_le_mul_of_nonpos_left (hbound hx).1 (le_of_not_ge hc)).trans (le_max_left _ _)⟩ |
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| lemma signed_scaled_spectral_endpoints (ν : ProbabilityMeasure ℝ) (a b c : ℝ) |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) : |
| min (c*a) (c*b)∈(scaledSpectralLaw ν c : Measure ℝ).support ∧ |
| max (c*a) (c*b)∈(scaledSpectralLaw ν c : Measure ℝ).support := by |
| by_cases h : c*a ≤ c*b |
| · simpa only [min_eq_left h,max_eq_right h] using |
| And.intro (scaled_spectral_support_mem ν c a ha) (scaled_spectral_support_mem ν c b hb) |
| · simpa only [min_eq_right (le_of_not_ge h),max_eq_left (le_of_not_ge h)] using |
| And.intro (scaled_spectral_support_mem ν c b hb) (scaled_spectral_support_mem ν c a ha) |
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| end InvariantIsing |
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| end |
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| end OAI |
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