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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 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 | import OAI.Probability.InvariantIsing.Core.FiniteVariationalBounds
namespace OAI
/-! Continuity under changes of the positive masses of a fixed finite spectral
alphabet. The inverse is compared using its defining resolvent equation. -/
noncomputable section
open Filter Set
open scoped BigOperators Topology
namespace InvariantIsing
variable {ι : Type*} [Fintype ι]
theorem finiteResolvent_tendsto_weights (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hρs : Tendsto ρs atTop (𝓝 ρ)) (b : ℝ) :
Tendsto (fun k => finiteResolvent (ρs k) eig b) atTop (𝓝 (finiteResolvent ρ eig b)) := by
exact tendsto_finsetSum Finset.univ (fun a _ =>
((tendsto_pi_nhds.mp hρs) a).div_const (b - eig a))
theorem finiteInverse_tendsto_weights (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ))
(xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 < x) :
Tendsto (fun k => finiteInverse (ρs k) eig (hspos k) (hssum k) (xs k))
atTop (𝓝 (finiteInverse ρ eig hpos hsum x)) := by
have hs := finiteInverse_spec ρ eig hpos hsum hx
have hxp : ∀ᶠ k in atTop, 0 < xs k := hxs.eventually (Ioi_mem_nhds hx)
apply tendsto_order.2
constructor
· intro l hl
by_cases he : ∀ a, eig a < l
· have hR : x < finiteResolvent ρ eig l := by
rw [← hs.2]
exact finiteResolvent_strictAnti (fun a => (hpos a).le) hsum he hl
have ht := hxs.eventually_lt (finiteResolvent_tendsto_weights ρs ρ eig hρs l) hR
filter_upwards [hxp, ht] with k hk hlt
have hsk := finiteInverse_spec (ρs k) eig (hspos k) (hssum k) hk
by_contra hnot
rcases (le_of_not_gt hnot).eq_or_lt with h | h
· rw [← h, hsk.2] at hlt
exact (lt_irrefl _ hlt)
· have hc := finiteResolvent_strictAnti (fun a => (hspos k a).le)
(hssum k) hsk.1 h
rw [hsk.2] at hc
exact (not_lt_of_ge hc.le hlt)
· push Not at he
obtain ⟨a, ha⟩ := he
filter_upwards [hxp] with k hk
exact ha.trans_lt ((finiteInverse_spec (ρs k) eig (hspos k) (hssum k) hk).1 a)
· intro u hu
have he : ∀ a, eig a < u := fun a => (hs.1 a).trans hu
have hR : finiteResolvent ρ eig u < x := by
rw [← hs.2]
exact finiteResolvent_strictAnti (fun a => (hpos a).le) hsum hs.1 hu
have ht := (finiteResolvent_tendsto_weights ρs ρ eig hρs u).eventually_lt hxs hR
filter_upwards [hxp, ht] with k hk hlt
have hsk := finiteInverse_spec (ρs k) eig (hspos k) (hssum k) hk
by_contra hnot
rcases (le_of_not_gt hnot).eq_or_lt with h | h
· rw [h, hsk.2] at hlt
exact (lt_irrefl _ hlt)
· have hc := finiteResolvent_strictAnti (fun a => (hspos k a).le) (hssum k) he h
rw [hsk.2] at hc
exact (not_lt_of_ge hc.le hlt)
theorem finiteR_tendsto_weights_pos (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ))
(xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 < x) :
Tendsto (fun k => finiteR (ρs k) eig (hspos k) (hssum k) (xs k))
atTop (𝓝 (finiteR ρ eig hpos hsum x)) := by
have hb := finiteInverse_tendsto_weights ρs ρ eig hspos hssum hpos hsum hρs xs hxs hx
have hi : Tendsto (fun k => 1 / xs k) atTop (𝓝 (1 / x)) :=
tendsto_const_nhds.div hxs hx.ne'
have he : (fun k => finiteR (ρs k) eig (hspos k) (hssum k) (xs k)) =ᶠ[atTop]
(fun k => finiteInverse (ρs k) eig (hspos k) (hssum k) (xs k) - 1 / xs k) := by
filter_upwards [hxs.eventually (Ioi_mem_nhds hx)] with k hk
simp only [finiteR, ite_eq_left hk]
have hl : Tendsto
(fun k => finiteInverse (ρs k) eig (hspos k) (hssum k) (xs k) - 1 / xs k)
atTop (𝓝 (finiteR ρ eig hpos hsum x)) := by
simpa only [finiteR, ite_eq_left hx] using hb.sub hi
exact hl.congr' he.symm
theorem finiteR_tendsto_weights_zero (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ))
(xs : ℕ → ℝ) (hxs : Tendsto xs atTop (𝓝 0)) :
Tendsto (fun k => finiteR (ρs k) eig (hspos k) (hssum k) (xs k))
atTop (𝓝 (finiteR ρ eig hpos hsum 0)) := by
let K := 1 + ∑ a, |eig a|
have hK : 0 < K := by
have := Finset.sum_nonneg (s := Finset.univ) (fun a _ => abs_nonneg (eig a))
dsimp only [K]
linarith
have heig a : |eig a| ≤ K := by
have := Finset.single_le_sum (f := fun a => |eig a|)
(fun a _ => abs_nonneg (eig a)) (Finset.mem_univ a)
dsimp only [K]
linarith
have hsmall : ∀ᶠ k in atTop, xs k < 1 / (4 * K) :=
hxs.eventually (Iio_mem_nhds (by positivity))
have herr : ∀ᶠ k in atTop,
‖finiteR (ρs k) eig (hspos k) (hssum k) (xs k) - ∑ a, ρs k a * eig a‖ ≤
8 * K ^ 2 * |xs k| := by
filter_upwards [hsmall] with k hk
rw [Real.norm_eq_abs]
by_cases hxk : 0 < xs k
· rw [abs_of_nonneg (sub_nonneg.mpr (mean_le_finiteR (ρs k) eig
(hspos k) (hssum k) hxk)), abs_of_pos hxk]
apply finiteR_sub_mean_le (ρs k) eig (hspos k) (hssum k) hK.le heig hxk
have := (lt_div_iff₀ (show 0 < 4 * K by positivity)).mp hk
nlinarith
· simp only [finiteR, ite_eq_right hxk, sub_self, abs_zero]
positivity
have hb : Tendsto (fun k => 8 * K ^ 2 * |xs k|) atTop (𝓝 0) := by
simpa using hxs.abs.const_mul (8 * K ^ 2)
have hz := squeeze_zero_norm' herr hb
have hm : Tendsto (fun k => ∑ a, ρs k a * eig a) atTop (𝓝 (∑ a, ρ a * eig a)) :=
tendsto_finsetSum Finset.univ (fun a _ => ((tendsto_pi_nhds.mp hρs) a).mul_const _)
simpa only [sub_add_cancel, zero_add, finiteR, lt_self_iff_false, ite_false] using hz.add hm
theorem finiteR_tendsto_weights_nonneg (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ))
(xs : ℕ → ℝ) {x : ℝ} (hxs : Tendsto xs atTop (𝓝 x)) (hx : 0 ≤ x) :
Tendsto (fun k => finiteR (ρs k) eig (hspos k) (hssum k) (xs k))
atTop (𝓝 (finiteR ρ eig hpos hsum x)) := by
rcases hx.eq_or_lt with hx | hx
· subst x
exact finiteR_tendsto_weights_zero ρs ρ eig hspos hssum hpos hsum hρs xs hxs
· exact finiteR_tendsto_weights_pos ρs ρ eig hspos hssum hpos hsum hρs xs hxs hx
/-- Uniformity on the whole interval includes the transform's value at zero. -/
theorem finiteR_uniform_weights (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ)) :
∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, ∀ x ∈ Icc (0 : ℝ) 1,
|finiteR (ρs k) eig (hspos k) (hssum k) x - finiteR ρ eig hpos hsum x| < ε := by
classical
intro ε hε
by_contra hbad
have hbad' : ∃ᶠ k in atTop, ∃ x ∈ Icc (0 : ℝ) 1,
ε ≤ |finiteR (ρs k) eig (hspos k) (hssum k) x - finiteR ρ eig hpos hsum x| := by
simpa only [Filter.not_eventually, not_forall, not_imp, not_lt, exists_prop] using hbad
obtain ⟨φ, hφ, hφbad⟩ := extraction_of_frequently_atTop hbad'
choose xs hxs herror using hφbad
obtain ⟨x, hx, ψ, hψ, hlim⟩ := isCompact_Icc.tendsto_subseq hxs
have hweights := hρs.comp (hφ.comp hψ).tendsto_atTop
have hR := finiteR_tendsto_weights_nonneg
(fun k => ρs (φ (ψ k))) ρ eig (fun k => hspos (φ (ψ k)))
(fun k => hssum (φ (ψ k))) hpos hsum hweights (fun k => xs (ψ k)) hlim hx.1
have hfixed := (continuous_finiteR ρ eig hpos hsum).continuousAt.tendsto.comp hlim
have hdiff := (hR.sub hfixed).abs
have hzero : Tendsto (fun k =>
|finiteR (ρs (φ (ψ k))) eig (hspos (φ (ψ k))) (hssum (φ (ψ k))) (xs (ψ k)) -
finiteR ρ eig hpos hsum (xs (ψ k))|) atTop (𝓝 0) := by
simpa only [Function.comp_apply, sub_self, abs_zero] using hdiff
have hsmall : ∀ᶠ k in atTop,
|finiteR (ρs (φ (ψ k))) eig (hspos (φ (ψ k))) (hssum (φ (ψ k))) (xs (ψ k)) -
finiteR ρ eig hpos hsum (xs (ψ k))| < ε := by
exact hzero.eventually (Iio_mem_nhds hε)
obtain ⟨k, hk⟩ := hsmall.exists
exact (not_le_of_gt hk) (herror (ψ k))
theorem finiteVariational_tendsto_weights (ρs : ℕ → ι → ℝ) (ρ eig : ι → ℝ)
(hspos : ∀ k a, 0 < ρs k a) (hssum : ∀ k, ∑ a, ρs k a = 1)
(hpos : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a = 1)
(hρs : Tendsto ρs atTop (𝓝 ρ)) :
Tendsto (fun k => (variationalFunctional (finiteR (ρs k) eig (hspos k) (hssum k))).toReal)
atTop (𝓝 (variationalFunctional (finiteR ρ eig hpos hsum)).toReal) := by
have hf := finiteVariational_ne_top_bot ρ eig hpos hsum
apply Metric.tendsto_nhds.mpr
intro ε hε
filter_upwards [finiteR_uniform_weights ρs ρ eig hspos hssum hpos hsum hρs ε hε] with k hk
have hfk := finiteVariational_ne_top_bot (ρs k) eig (hspos k) (hssum k)
have hleft := variationalFunctional_le_add (finiteR (ρs k) eig (hspos k) (hssum k))
(finiteR ρ eig hpos hsum) (continuous_finiteR (ρs k) eig (hspos k) (hssum k))
(continuous_finiteR ρ eig hpos hsum) ε (fun x hx => (hk x hx).le)
have hright := variationalFunctional_le_add (finiteR ρ eig hpos hsum)
(finiteR (ρs k) eig (hspos k) (hssum k)) (continuous_finiteR ρ eig hpos hsum)
(continuous_finiteR (ρs k) eig (hspos k) (hssum k)) ε
(fun x hx => by simpa only [abs_sub_comm] using (hk x hx).le)
rw [← EReal.coe_toReal hfk.1 hfk.2, ← EReal.coe_toReal hf.1 hf.2,
← EReal.coe_add, EReal.coe_le_coe_iff] at hleft hright
rw [Real.dist_eq, abs_lt]
constructor <;> linarith
end InvariantIsing
end
end OAI
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