openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /MeasureInverseContinuity.lean
| import OAI.Probability.InvariantIsing.Spectral.MeasureResolvent | |
| namespace OAI | |
| /-! Order tests and continuity for the inverse across the capped branch. -/ | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology Classical | |
| namespace InvariantIsing | |
| theorem measureInverse_le_iff (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| {e x b : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) (hb : e < b) : | |
| measureInverse μ e x ≤ b ↔ measureResolvent μ b ≤ x := by | |
| by_cases hex : ∃ r, e < r ∧ measureResolvent μ r = x | |
| · rw [measureInverse, dite_eq_left hex] | |
| have hs := hex.choose_spec | |
| constructor | |
| · intro hrb | |
| rcases hrb.eq_or_lt with h | h | |
| · rw [← h, hs.2] | |
| · have ht := measureResolvent_strictAnti μ hμ hs.1 h | |
| rw [hs.2] at ht | |
| exact ht.le | |
| · intro hR | |
| by_contra hnot | |
| have ht := measureResolvent_strictAnti μ hμ hb (lt_of_not_ge hnot) | |
| rw [hs.2] at ht | |
| exact (not_lt_of_ge hR) ht | |
| · rw [measureInverse, dite_eq_right hex] | |
| constructor | |
| · intro _ | |
| by_contra hnot | |
| exact hex (exists_measureResolvent_solution μ hμ hb hx (lt_of_not_ge hnot).le) | |
| · intro _ | |
| exact hb.le | |
| theorem measureInverse_lt_iff (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| {e x b : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) (hb : e < b) : | |
| measureInverse μ e x < b ↔ measureResolvent μ b < x := by | |
| by_cases hex : ∃ r, e < r ∧ measureResolvent μ r = x | |
| · rw [measureInverse, dite_eq_left hex] | |
| have hs := hex.choose_spec | |
| constructor | |
| · intro hrb | |
| have ht := measureResolvent_strictAnti μ hμ hs.1 hrb | |
| rwa [hs.2] at ht | |
| · intro hR | |
| by_contra hnot | |
| rcases (le_of_not_gt hnot).eq_or_lt with h | h | |
| · rw [h, hs.2] at hR | |
| exact lt_irrefl _ hR | |
| · have ht := measureResolvent_strictAnti μ hμ hb h | |
| rw [hs.2] at ht | |
| exact (not_lt_of_ge ht.le) hR | |
| · rw [measureInverse, dite_eq_right hex] | |
| have hv : measureResolvent μ b < x := by | |
| by_contra hnot | |
| exact hex (exists_measureResolvent_solution μ hμ hb hx (le_of_not_gt hnot)) | |
| exact ⟨fun _ => hv, fun _ => hb⟩ | |
| theorem continuousAt_measureInverse (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| {e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) : | |
| ContinuousAt (measureInverse μ e) x := by | |
| have hbound := (measureInverse_bounds μ hμ hx).1 | |
| have hxp : ∀ᶠ y in 𝓝 x, 0 < y := Ioi_mem_nhds hx | |
| apply tendsto_order.2 | |
| constructor | |
| · intro l hl | |
| by_cases hle : l < e | |
| · filter_upwards [hxp] with y hy | |
| exact hle.trans_le (measureInverse_bounds μ hμ hy).1 | |
| · let c := (l + measureInverse μ e x) / 2 | |
| have hec : e < c := by dsimp only [c]; linarith [le_of_not_gt hle] | |
| have hlc : l < c := by dsimp only [c]; linarith | |
| have hci : c < measureInverse μ e x := by dsimp only [c]; linarith | |
| have hR : x < measureResolvent μ c := by | |
| by_contra hnot | |
| exact (not_le_of_gt hci) | |
| ((measureInverse_le_iff μ hμ hx hec).mpr (le_of_not_gt hnot)) | |
| filter_upwards [hxp, Iio_mem_nhds hR] with y hy hRy | |
| have hnot : ¬ measureInverse μ e y ≤ c := by | |
| intro hle | |
| exact (not_le_of_gt hRy) ((measureInverse_le_iff μ hμ hy hec).mp hle) | |
| exact hlc.trans (lt_of_not_ge hnot) | |
| · intro u hu | |
| have heu : e < u := hbound.trans_lt hu | |
| have hR := (measureInverse_lt_iff μ hμ hx heu).mp hu | |
| filter_upwards [hxp, Ioi_mem_nhds hR] with y hy hRy | |
| exact (measureInverse_lt_iff μ hμ hy heu).mpr hRy | |
| theorem continuousAt_measureR_pos (μ : Measure ℝ) [IsProbabilityMeasure μ] | |
| {e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ≤ e) (hx : 0 < x) : | |
| ContinuousAt (measureR μ e) x := by | |
| have hc : ContinuousAt (fun y => measureInverse μ e y - 1 / y) x := | |
| (continuousAt_measureInverse μ hμ hx).sub | |
| (continuousAt_const.div continuousAt_id hx.ne') | |
| apply hc.congr_of_eventuallyEq | |
| filter_upwards [Ioi_mem_nhds hx] with y hy | |
| simp only [measureR, ite_eq_left (show 0 < y from hy)] | |
| end InvariantIsing | |
| end | |
| end OAI | |