Nonpositive signed row sums bound the solution by its initial norm

Proved

BianchiIX.norm_le_initial_of_signedRowSum_nonpos

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

bianchi-ix-era-statisticsdynamical-systems

For a finite index type $\iota$ , a real matrix $A=(A_{ij})$ , and $i\in\iota$ , define its signed row sum by

\sigma_i(A)=A_{ii}+\sum_{j\ne i}|A_{ij}|.

Vectors in $\mathbb R^\iota$ carry the supremum norm. Assume $\iota$ is nonempty. Let $f$ be continuous on $[a,b]$ and satisfy $f'(t)=A(t)f(t)$ at every $t\in[a,b)$ . If $\sigma_i(A(t))\leq0$ for every such $t$ and every $i$ , then

\|f(t)\|_\infty\leq\|f(a)\|_\infty\quad(t\in[a,b]).

Preamble

import Definitions.Def_BianchiIX_Analysis_SignedRowGrowth
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.FunProp

noncomputable section
open BianchiIX
open Set Filter
open scoped Topology
variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι]

Formal statement

theorem BianchiIX.norm_le_initial_of_signedRowSum_nonpos {f : ℝ → (ι → ℝ)}
    {A : ℝ → Matrix ι ι ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    (hderiv : ∀ t ∈ Ico a b, HasDerivAt f ((A t).mulVec (f t)) t)
    (hA : ∀ t ∈ Ico a b, ∀ i, signedRowSum (A t) i ≤ 0) :
    ∀ t ∈ Icc a b, ‖f t‖ ≤ ‖f a‖ := by sorry

Source

https://github.com/l46430640-del/ERA-STATISTICS-FORMAL/blob/d74c241819f896ab8d050b21ace80c8e90eabd9e/BianchiIX/Analysis/SignedRowGrowth.lean#L84-L96

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