LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A conjugate bound for residual and Bregman errors

Statement

Let $ψ : [0, \infty) \to [0, \infty)$ be continuous, nondecreasing, concave, and satisfy $ψ (0) = 0$ . Extend $g = - ψ$ to $R$ by $g (s) = + \infty$ for $s < 0$ , and define $H (q) : = g^{*} (- q) = \sup_{s \geq 0} (ψ (s) - q s)$ for $q > 0$ , assuming the displayed suprema below are finite. For $c_{1}, c_{2}, C_{3}, α > 0$ and $d, r, e \geq 0$ , if

$c_{1} r + c_{2} α e \leq d + α ψ (C_{3} r),$

then

$e \leq \frac{1}{c_{2}} (\frac{d}{α} + H ⁣ (\frac{c_{1}}{C_{3} α})), r \leq \frac{2}{c_{1}} (d + α H ⁣ (\frac{c_{1}}{2 C_{3} α})) .$

Statement references

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Lemma 2 and Appendix A.1

Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20

Facts & Assumptions

Given: $ψ$ , $H$ , $c_{1}, c_{2}, C_{3}, α, d, r, e$ satisfy the hypotheses and inequality in the Statement.

[A1]

The two values $H (c_{1} / (C_{3} α))$ and $H (c_{1} / (2 C_{3} α))$ are finite.

Proof

1.1givenA1algebra

Put $t = C_{3} r$ . Rearranging the given inequality and dividing by $c_{2} α$ gives $e \leq \frac{d}{c_{2} α} + \frac{1}{c_{2}} (ψ (t) - \frac{c_{1}}{C_{3} α} t)$ . The parenthesized expression is bounded above by the defining supremum $H (c_{1} / (C_{3} α))$ , proving the first bound.

2.1givenA1step 1.1algebra∎

Subtracting $c_{1} r / 2$ from the given inequality and using $c_{2} α e \geq 0$ gives $\frac{c_{1} r}{2} \leq d + α (ψ (t) - \frac{c_{1}}{2 C_{3} α} t) \leq d + α H ⁣ (\frac{c_{1}}{2 C_{3} α})$ . Multiplication by $2 / c_{1}$ proves the second bound.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography, Lemma 2 (standard reference, not scraped)
Frederic Weidling, Variational Source Conditions and Conditional Stability Estimates for Inverse Problems in PDEs, Theorem 2.20 (standard reference, not scraped)