ExampleConstruction: AI-generatedVerification: 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 Gaussian coefficient law with mass outside the uniform-ellipticity domain

Example

Assume AC. Let parameter and data spaces be $X = Y = R$ with their Borel sigma-algebras, let the admissible truth set be $U = R$ , and let $F : U \to Y$ be the identity map. Regard $θ \in R$ as the constant coefficient field $a_{θ} (x) = θ$ on $[0, 1]$ . Let $D = [1, 2] \subseteq U$ be the uniformly elliptic class with lower bound $1$ and upper bound $2$ , meaning $1 \leq a_{θ} (x) \leq 2$ for every $x \in [0, 1]$ ; the restricted map $G = F ∣_{D}$ is measurable. Give $X$ the standard normal law $μ = N (0, 1)$ . Since $μ ((- \infty, 1)) > 0$ , $μ (D) < 1$ ; consequently $μ$ is a Gaussian probability on the parameter space but is not a prior on this forward-map domain.

Example references

a prior must assign probability one to its declared forward-map domain

the admissible truth set, parameter and data spaces, and forward-map domain

the standard normal law has positive density on the real line

the Axiom of Choice required by the library's standard-normal interface

measurability is defined by measurable inverse images

the Borel sigma-algebra contains open and closed sets

a probability measure has total mass one

measures are monotone

Dashti and Stuart, §1.3 and §3.4, full text

Giné and Nickl, §1.1.2, full book

Facts & Assumptions

Given: AC, the standard normal probability $μ = N (0, 1)$ on $X = R$ , the closed domain $D = [1, 2]$ , and the identity map $G : D \to R$ .

[A1]

AC is required by the library's standard normal law definition; the measure comparison below does not use choice. The Axiom of Choice

[F1]

The standard normal law has Borel density $ϕ (t) = (2 π)^{- 1 / 2} e^{- t^{2} / 2}$ , positive for every $t \in R$ . Standard normal and normal laws

[F2]

A prior on the forward domain $D$ requires a Borel probability on $X$ with $μ (D) = 1$ . A Borel prior concentrated on a forward-map domain

[F3]

A measurable map is one whose measurable-set preimages are measurable; for the identity map on $D$ , $G^{- 1} (E) = E \cap D$ . A measurable function between measurable spaces

[F4]

The Borel sigma-algebra contains open sets and their complements, so the closed interval $D$ and the half-line $(- \infty, 1)$ are Borel. The Borel sigma-algebra of a topological space

[F5]

Measures are monotone under measurable set inclusion. Measures are monotone

[F6]

A probability measure has total mass one and is countably additive. Probability measures and probability spaces

Verification

Given: The parameter space, ellipticity domain, standard normal law, and identity forward map in the Example.

1.1F3F4given

The interval $D = [1, 2]$ is closed and hence Borel by [F4]. For every Borel $E \subseteq R$ , $G^{- 1} (E) = E \cap D$ is in the trace Borel sigma-algebra on $D$ by [F3], so $G$ is an admissible measurable forward map.

1.2A1F1givenalgebra

The density in [F1] is continuous and strictly positive on $[0, \frac{1}{2}]$ , so it has a positive minimum there and $μ ((- \infty, 1)) \geq \int_{0}^{1 / 2} ϕ (t) d t > 0$ .

2.1F2F5F6step 1.2algebra∎

Since $D \subseteq (- \infty, 1)^{c}$ , [F5] gives $μ (D) \leq μ ((- \infty, 1)^{c}) = 1 - μ ((- \infty, 1)) < 1$ by [F6] and step 1.2. This violates the full-measure condition in [F2], so $μ$ is not a prior on the declared domain.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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.

Sources

M. Dashti and A. M. Stuart, The Bayesian Approach to Inverse Problems (arXiv:1302.6989) (standard reference, not scraped)
E. Giné and R. Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models (Cambridge University Press, 2016) (standard reference, not scraped)