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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
end OAI
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