Prove2Me

Closing a scalar phase coupled to two transverse variations

Proved
BianchiIX.coupled_clock_variation_bounds

by trppppp · Nov 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

bianchi-ix-era-statisticsdynamical-systems

For C,A≥0C,A\geq0 define

SC,A,x,y(c)=∣c1∣∣x∣+∣c2∣∣y∣+CA(∣c1∣+∣c2∣),S_{C,A,x,y}(c)=|c_1||x|+|c_2||y|+CA(|c_1|+|c_2|), KC,l(c)=(C+1)(∣c0∣+l(∣c1∣+∣c2∣)).K_{C,l}(c)=(C+1)(|c_0|+l(|c_1|+|c_2|)).

Let s≤ts\leq t, C,A≥0C,A\geq0, and let q≥0q\geq0 and c:[s,t]→R3c:[s,t]\to\mathbb R^3 be continuous. Suppose l′=ql'=q, l(s)=0l(s)=0, ZZ is differentiable and Z′=q(c0Z+c1f0+c2f1)Z'=q(c_0Z+c_1f_0+c_2f_1). Assume

∣f0(r)−f0(s)∣+∣f1(r)−f1(s)∣≤C(A+∫sr∣Z∣q).|f_0(r)-f_0(s)|+|f_1(r)-f_1(s)| \leq C\left(A+\int_s^r|Z|q\right).

If ∫stSC,A,f0(s),f1(s)(c)q≤B\int_s^t S_{C,A,f_0(s),f_1(s)}(c)q\leq B and ∫stKC,l(c)q≤Λ<1\int_s^t K_{C,l}(c)q\leq\Lambda<1, set M=(∣Z(s)∣+B)/(1−Λ)M=(|Z(s)|+B)/(1-\Lambda). For every r∈[s,t]r\in[s,t],

∣Z(r)∣≤M,∣Z(r)−Z(s)∣≤B+ΛM,|Z(r)|\leq M,\quad |Z(r)-Z(s)|\leq B+\Lambda M, ∣f0(r)−f0(s)∣+∣f1(r)−f1(s)∣≤C(A+l(r)M).|f_0(r)-f_0(s)|+|f_1(r)-f_1(s)|\leq C(A+l(r)M).
Preamble
import Definitions.Def_BianchiIX_EntrancePhaseResponseData
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

noncomputable section
open BianchiIX
open Set MeasureTheory
Formal statement
theorem BianchiIX.coupled_clock_variation_bounds
    {Z Z' q l : ℝ → ℝ} {f : ℝ → Fin 2 → ℝ} {c : ℝ → Fin 3 → ℝ} {s t C A B Λ : ℝ}
    (hst : s ≤ t) (hC : 0 ≤ C) (hA : 0 ≤ A)
    (hZ : ∀ r ∈ Icc s t, HasDerivAt Z (Z' r) r)
    (hl : ∀ r ∈ Icc s t, HasDerivAt l (q r) r) (hls : l s = 0)
    (hqc : ContinuousOn q (Icc s t)) (hq : ∀ r ∈ Icc s t, 0 ≤ q r)
    (hcc : ContinuousOn c (Icc s t))
    (heq : ∀ r ∈ Icc s t, Z' r = q r * (c r 0 * Z r + c r 1 * f r 0 + c r 2 * f r 1))
    (htrans : ∀ r ∈ Icc s t, |f r 0 - f s 0| + |f r 1 - f s 1| ≤
      C * (A + ∫ z in s..r, |Z z| * q z))
    (hsource : (∫ r in s..t, coupledVariationSource C A (f s 0) (f s 1) (c r) * q r) ≤ B)
    (hbudget : (∫ r in s..t, coupledVariationFeedback C (l r) (c r) * q r) ≤ Λ)
    (hΛ : Λ < 1) :
    ∀ r ∈ Icc s t,
      |Z r| ≤ (|Z s| + B) / (1 - Λ) ∧
      |Z r - Z s| ≤ B + Λ * ((|Z s| + B) / (1 - Λ)) ∧
      |f r 0 - f s 0| + |f r 1 - f s 1| ≤ C * (A + l r * ((|Z s| + B) / (1 - Λ))) := by sorry
Source
https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/CoupledClockVariation.lean#L15-L103

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