LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

Relaxed inequalities for squared Euclidean norms

Statement

For vectors $u, v \in R^{d}$ and $a > 0$ , $∥ u + v ∥_{2}^{2} \leq (1 + a) ∥ u ∥_{2}^{2} + (1 + a^{- 1}) ∥ v ∥_{2}^{2}, 2 ∣ ⟨ u, v ⟩ ∣ \leq a ∥ u ∥_{2}^{2} + a^{- 1} ∥ v ∥_{2}^{2} .$ For $m \geq 1$ vectors $u_{1}, \dots, u_{m} \in R^{d}$ , ${∥ \sum_{j = 1}^{m} u_{j} ∥}_{2}^{2} \leq m \sum_{j = 1}^{m} ∥ u_{j} ∥_{2}^{2} .$

Facts & Assumptions

Given: The standard Euclidean inner product and norm on $R^{d}$ , with $∥ x ∥_{2}^{2} = ⟨ x, x ⟩$ .

[A1]

The real inner product is symmetric and bilinear, and squared norms are nonnegative.

Proof

1.1givenA1algebra

Expanding the first inequality's right side minus its left side gives [given] $a ∥ u ∥_{2}^{2} + a^{- 1} ∥ v ∥_{2}^{2} - 2 ⟨ u, v ⟩ = ∥ \sqrt{a} u - a^{- 1 / 2} v ∥_{2}^{2} \geq 0.$ Replacing $v$ by $- v$ gives the same bound with $+ 2 ⟨ u, v ⟩$ ; together these two bounds yield both displayed two-vector inequalities.

1.2A1algebra∎

Expanding each squared difference and collecting diagonal and cross terms [given] gives $m \sum_{j = 1}^{m} ∥ u_{j} ∥_{2}^{2} - {∥ \sum_{j = 1}^{m} u_{j} ∥}_{2}^{2} = \sum_{1 \leq i < j \leq m} ∥ u_{i} - u_{j} ∥_{2}^{2} \geq 0.$ This proves the finite-sum inequality, including $m = 1$ , when the right difference is zero.

Used by

Dependency tree · 0 levels

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

Sources

Sai Praneeth Karimireddy et al., SCAFFOLD, Appendix B, Lemma 3 (standard reference, not scraped)