ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 one-step scalar Kalman update

Example

Take a one-step scalar model with $X_{0} \sim N (0, 1)$ , transition $X_{1} = X_{0} + q_{1}$ for $q_{1} \sim N (0, 1)$ , and observation $Y_{1} = X_{1} + r_{1}$ for $r_{1} \sim N (0, 1)$ , with the three Gaussian variables independent. At the observation $y_{1} = 2$ , the predicted law is $N (0, 2)$ , the innovation variance is $3$ , the gain is $2 / 3$ , and the selected filtering-kernel value is $N (4 / 3, 2 / 3)$ .

Facts & Assumptions

Given: AC and the stated independent-noise linear-Gaussian model.

[A1]

AC is assumed through the Gaussian model and conditional-law theorem. The Axiom of Choice.

[F1]

Under AC, the Kalman recursion with positive innovation covariance defines an everywhere filtering-kernel version of the exact conditional state law. The finite-horizon linear-Gaussian Kalman filter is exact.

[F2]

The model permits singular positive-semidefinite initial, process, and measurement covariances with mutually independent centered Gaussian noises. Linear-Gaussian state-space models with independent noises.

[F3]

Every positive-semidefinite covariance defines a multivariate-normal law under AC. Multivariate normal law, including singular covariance.

Verification

1.1A1F2F3givenalgebra

Set $m_{0 ∣ 0} = 0$ , $P_{0 ∣ 0} = 1$ , $F_{1} = H_{1} = 1$ , and $Q_{1} = R_{1} = 1$ . The covariances are positive semidefinite and the noises satisfy [F2]. The prediction is $m_{1 ∣ 0} = 0$ and $P_{1 ∣ 0} = 1 \cdot 1 \cdot 1 + 1 = 2$ . Thus $S_{1} = 1 \cdot 2 \cdot 1 + 1 = 3 > 0$ , and for $y_{1} = 2$ the innovation is $v_{1} = 2$ .

2.1A1F1step 1.1algebra∎

Solving the scalar innovation system gives $K_{1} = 2 / 3$ . Therefore $m_{1 ∣ 1} = 0 + (2 / 3) 2 = 4 / 3$ and $P_{1 ∣ 1} = 2 - (2 / 3) 3 (2 / 3) = 2 / 3$ . By [F1], the selected conditional-law version at $y_{1} = 2$ is exactly $N (4 / 3, 2 / 3)$ .

Depends on

Used by

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Sources

Simo Särkkä, Bayesian Filtering and Smoothing, Chapter 4, §§4.1–4.3 (standard reference, not scraped)
Kody Law, Andrew M. Stuart, and Konstantinos Zygalakis, Data Assimilation: A Mathematical Introduction, Chapter 4, §4.1 (standard reference, not scraped)