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feat(Regularized): prove to_SupRegularized and of_Subadditive #1022

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feat(Regularized): prove to_SupRegularized and of_Subadditive #1022

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18 changes: 12 additions & 6 deletions QuantumInfo/Regularized.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -4,8 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Meiburg
-/
import QuantumInfo.ForMathlib.Superadditive
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Order.MonotoneConvergence
import Mathlib.Algebra.Order.Ring.Star
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.Subadditive

/-! Definition of "Regularized quantities" as are common in information theory,
from one-shot versions, and good properties coming from Fekete's lemma.
Expand Down Expand Up @@ -120,7 +121,9 @@ variable {fn : ℕ → ℝ} {_lb _ub : ℝ} {hl : ∀ n, _lb ≤ fn n} {hu : ∀

theorem InfRegularized.to_SupRegularized : InfRegularized fn hl hu = -SupRegularized (-fn ·)
(lb := -_ub) (ub := -_lb) (neg_le_neg_iff.mpr <| hu ·) (neg_le_neg_iff.mpr <| hl ·) := by
sorry
have liminf_neg : Filter.liminf fn Filter.atTop = -(Filter.limsup (-fn) Filter.atTop) := by
simp [Filter.limsup_eq, Filter.liminf_eq, Real.sInf_def]
exact Real.ext_cauchy (congrArg Real.cauchy liminf_neg)
Comment on lines +124 to +126
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@jstoobysmith jstoobysmith Apr 6, 2026

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Suggested change
have liminf_neg : Filter.liminf fn Filter.atTop = -(Filter.limsup (-fn) Filter.atTop) := by
simp [Filter.limsup_eq, Filter.liminf_eq, Real.sInf_def]
exact Real.ext_cauchy (congrArg Real.cauchy liminf_neg)
refine Real.ext_cauchy (congrArg Real.cauchy ?_)
simp [Filter.limsup_eq, Filter.liminf_eq, Real.sInf_def]

Haven't checked, and will leave it to the QuantumInfo folk for a proper review. But this might be shorter.


theorem SupRegularized.to_InfRegularized : SupRegularized fn hl hu = -InfRegularized (-fn ·)
(lb := -_ub) (ub := -_lb) (neg_le_neg_iff.mpr <| hu ·) (neg_le_neg_iff.mpr <| hl ·) := by
Expand Down Expand Up @@ -159,6 +162,9 @@ theorem InfRegularized.of_Subadditive (hf : Subadditive (fun n ↦ fn n * n))
convert Or.inr (hl (n+1))
field_simp
)
apply tendsto_nhds_unique h₁
have := InfRegularized.anti_tendsto (fn := fn) (hl := hl) (hu := hu) (sorry)
sorry
have h₂ : Filter.Tendsto fn .atTop (nhds hf.lim) := by
refine h₁.congr' ?_
filter_upwards [Filter.eventually_ne_atTop 0] with n hn
have : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hn
field_simp
exact h₂.liminf_eq.symm
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