ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Positive coordinate weights define an inner product and change lengths and projections

Example

For positive real weights $a_{0}, \dots, a_{n - 1}$ ,

$⟨ x, y ⟩_{a} = \sum_{k < n} a_{k} x_{k} \overline{y_{k}}$

is an inner product on $F^{n}$ , where $F = R$ or $C$ . With weights $(1, 4)$ on $R^{2}$ , the vector $(1, 1)$ has length $\sqrt{5}$ , and the projection of $(1, 0)$ onto $span (1, 1)$ is $(1 / 5, 1 / 5)$ rather than the standard projection $(1 / 2, 1 / 2)$ .

Facts & Assumptions

Given: Positive weights $a_{k}$ and the displayed pairing.

[L1]

A real or complex inner product is linear in the first argument, conjugate symmetric, positive on the diagonal, and definite (Real and complex inner product spaces, with the inner product linear in the first argument).

[L2]

The induced norm is $\sqrt{⟨ x, x ⟩}$ (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product).

[L3]

For an orthonormal basis $(e_{i})_{i < r}$ of a subspace $W$ of a finite-dimensional inner product space, $P_{W} v = \sum_{i < r} ⟨ v, e_{i} ⟩ e_{i}$ (Orthogonal projection is linear, and an orthonormal basis $(e_{i})$ of $W$ gives $P_{W} v = \sum_{i} ⟨ v, e_{i} ⟩ e_{i}$ ).

[L4]

The empty finite sum has additive value zero (The product $g_{0} g_{1} \dots g_{n - 1}$ of a finite list in a monoid, by recursion, with the empty product ( $n = 0$ ) equal to the identity).

Verification

technique · direct

1.1L1L4algebra

Finite-sum algebra gives linearity, conjugate symmetry, and $⟨ x, x ⟩_{a} = \sum_{k} a_{k} ∣ x_{k} ∣^{2} \geq 0$ . Because every $a_{k} > 0$ , this sum vanishes exactly when every coordinate of $x$ vanishes, so [L1] holds. For $n = 0$ , [L4] leaves only the zero vector.

2.1L2L3algebra∎

For weights $(1, 4)$ , [L2] gives $∥ (1, 1) ∥_{a}^{2} = 1 + 4 = 5$ . Put $u = (1, 1)$ , $v = (1, 0)$ and $W = span (u)$ . Then $e = ∥ u ∥_{a}^{- 1} u$ is an orthonormal basis of $W$ , so [L3] gives $P_{W} v = ⟨ v, e ⟩_{a} e = ⟨ v, u ⟩_{a} ⟨ u, u ⟩_{a}^{- 1} u$ . Here $⟨ v, u ⟩_{a} = 1$ and $⟨ u, u ⟩_{a} = 5$ , so $P_{W} v = (1 / 5) u$ . With both weights $1$ the same computation uses $⟨ v, u ⟩ = 1$ and $⟨ u, u ⟩ = 2$ and gives $(1 / 2) u$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.