| import OAI.Probability.InvariantIsing.Magnetic.MagneticHeightSupport |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticBoundaryContinuity |
|
|
| namespace OAI |
|
|
| /-! The constrained supporting inequality on the full nonnegative height |
| cone, including repeated levels and zero variances. -/ |
|
|
| noncomputable section |
| open Set Filter |
| open scoped BigOperators Topology |
|
|
| namespace InvariantIsing |
|
|
| theorem magneticHeight_support (h : FieldStep) {m : ℝ} (hm : |m| < 1) |
| (r : Fin (h.depth + 1) → ℝ) (hr0 : ∀ i, 0 ≤ r i) (hrmono : Monotone r) : |
| constrainedFieldValue (fieldWithHeights h r hr0 hrmono) m ≤ constrainedFieldValue h m + |
| ∑ i, (-(h.cut i.succ - h.cut i.castSucc) / 2 * magneticFieldLevel h m i) * |
| (r i - h.height i) := by |
| let k := fieldWithHeights h r hr0 hrmono |
| have hleft := continuous_fieldRegularizedConstrainedValue k hm |
| have hright : Continuous (fun t => constrainedFieldValue (fieldRegularized h t) m + |
| ∑ i : Fin (h.depth + 1), (-(h.cut i.succ - h.cut i.castSucc) / 2 * |
| magneticFieldLevel (fieldRegularized h t) m i) * (r i - h.height i)) := by |
| have hval := continuous_fieldRegularizedConstrainedValue h hm |
| have hB := continuous_fieldRegularizedMagneticLevel h hm |
| fun_prop |
| have hineq : ∀ t ∈ Ioc (0 : ℝ) 1, |
| constrainedFieldValue (fieldRegularized k t) m ≤ |
| constrainedFieldValue (fieldRegularized h t) m + |
| ∑ i : Fin (h.depth + 1), (-(h.cut i.succ - h.cut i.castSucc) / 2 * |
| magneticFieldLevel (fieldRegularized h t) m i) * (r i - h.height i) := by |
| intro t ht |
| have hh : (fieldRegularized h t).height ∈ fieldStrictHeightCone h.depth := |
| fieldRegularized_strict h ht |
| have hk : (fieldRegularized k t).height ∈ fieldStrictHeightCone h.depth := |
| fieldRegularized_strict k ht |
| have heh : fieldStepOfStrictHeights h (fieldRegularized h t).height hh = |
| fieldRegularized h t := by |
| unfold fieldStepOfStrictHeights fieldRegularized fieldWithHeights |
| rfl |
| have hek : fieldStepOfStrictHeights h (fieldRegularized k t).height hk = |
| fieldRegularized k t := by |
| unfold fieldStepOfStrictHeights fieldRegularized fieldWithHeights k |
| rfl |
| have hi := magneticHeight_support_strict h hm _ _ hh hk |
| have hevh := congrArg (fun q : FieldStep => constrainedFieldValue q m) heh |
| have hevk := congrArg (fun q : FieldStep => constrainedFieldValue q m) hek |
| rw [hevh, hevk] at hi |
| have hd (i : Fin (h.depth + 1)) : (fieldRegularized k t).height i - |
| (fieldRegularized h t).height i = r i - h.height i := by |
| change (r i + fieldHeightEpsilon t * (i.val + 1 : ℕ)) - |
| (h.height i + fieldHeightEpsilon t * (i.val + 1 : ℕ)) = _ |
| ring |
| have heBsum : |
| (∑ i : Fin (h.depth + 1), (-(h.cut i.succ - h.cut i.castSucc) / 2 * |
| magneticFieldLevel (fieldStepOfStrictHeights h (fieldRegularized h t).height hh) m i) * |
| ((fieldRegularized k t).height i - (fieldRegularized h t).height i)) = |
| ∑ i : Fin (h.depth + 1), (-(h.cut i.succ - h.cut i.castSucc) / 2 * |
| magneticFieldLevel (fieldRegularized h t) m i) * |
| (r i - h.height i) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [magneticFieldLevel_congr heh m i i rfl, hd] |
| exact hi.trans_eq (congrArg |
| (fun a : ℝ => constrainedFieldValue (fieldRegularized h t) m + a) heBsum) |
| have hevent : ∀ᶠ t in 𝓝[>] (0 : ℝ), |
| constrainedFieldValue (fieldRegularized k t) m ≤ |
| constrainedFieldValue (fieldRegularized h t) m + |
| ∑ i : Fin (h.depth + 1), (-(h.cut i.succ - h.cut i.castSucc) / 2 * |
| magneticFieldLevel (fieldRegularized h t) m i) * (r i - h.height i) := by |
| filter_upwards [self_mem_nhdsWithin, |
| mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds (show (0 : ℝ) < 1 by norm_num))] with t ht ht1 |
| exact hineq t ⟨ht, ht1.le⟩ |
| have hh := le_of_tendsto_of_tendsto |
| (hleft.continuousAt.tendsto.mono_left nhdsWithin_le_nhds) |
| (hright.continuousAt.tendsto.mono_left nhdsWithin_le_nhds) hevent |
| have hz (i : Fin (h.depth + 1)) := |
| magneticFieldLevel_congr (fieldRegularized_zero h) m i i rfl |
| simp_rw [hz] at hh |
| simpa only [fieldRegularized_zero, k] using hh |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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