ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

No observed failures with positive versus zero exposure

Example

Assume Countable Choice.

Compare two right-censored exponential samples with no observed failures. In each scenario, assume the two event times are iid $E x p (λ)$ . In the first, two subjects are followed until fixed censoring time $5$ and both are censored there. In the second, both subjects are censored immediately at time $0$ . The deterministic censoring times make the subject pairs iid in each sample. Here $λ > 0$ is the common event-rate parameter in both samples.

Facts & Assumptions

Given: Right-censored records use $U_{i} = \min (T_{i}, C_{i})$ and $δ_{i} = 1 {T_{i} \leq C_{i}}$ ; in each scenario the two event times are iid $E x p (λ)$ and the censoring time is fixed and independent of the event times.

[F1]

For $d = \sum_{i} δ_{i}$ and $E = \sum_{i} U_{i}$ , the event likelihood factor is $λ^{d} e^{- λ E}$ . If $d = 0, E > 0$ , its supremum occurs as $λ ↓ 0$ and is not attained for $λ > 0$ ; if $d = 0, E = 0$ , it is constant. The boundary $λ = 0$ is not a proper exponential law on finite event times. Right-censored exponential likelihood and its boundary maximum

[F2]

An exponential event time has $P (T > 0) = 1$ . Exponential and Weibull survival specifications define probability laws

[A1]

The two censoring designs form the family $G_{c} = δ_{c}$ for $c \in {0, 5}$ , dominated by the fixed finite measure $ν = δ_{0} + δ_{5}$ with specified density $g_{c} (u) = 1_{{c}} (u)$ ; in particular $g_{c} (c) = 1$ . In scenario one $C_{i} = 5$ , and in scenario two $C_{i} = 0$ . Event-time observation, censoring, and truncation schemes, The measure with density $f$ relative to $μ$

[A2]

Countable Choice is inherited from the exponential-law supplier [F2] and the likelihood supplier [F1]. The Axiom of Countable Choice ( ${A C}_{ω}$ )

Verification

technique · direct

1.1A1A2F1givenalgebra

In scenario one the records are $(U_{i}, δ_{i}) = (5, 0)$ for both subjects, so $d = 0$ , $E = 10$ , and $C_{5} = g_{5} (5)^{2} = 1$ . By [F1], the event likelihood factor is $e^{- 10 λ}$ , whose supremum $1$ is approached only as $λ ↓ 0$ .

1.2A1A2F1F2givenalgebra

In scenario two, [F2] gives $T_{i} > 0$ almost surely, while $C_{i} = 0$ . Thus $U_{i} = 0$ and $δ_{i} = 0$ almost surely for both subjects, so $d = 0$ , $E = 0$ , and $C_{0} = g_{0} (0)^{2} = 1$ . By [F1], the event likelihood factor is $1$ for every $λ > 0$ .

1.3A2F1given

The value $λ = 0$ is only a formal likelihood boundary for the finite-time exponential model; including it as an actual law requires an extended event-time model that admits $T = \infty$ .

2.1F1step 1.1step 1.2algebra

These samples have the same failure count but different likelihood maximizers: positive exposure gives only a boundary supremum, whereas zero exposure gives every positive rate as a maximizer. Thus the no-failure boundary claim requires $E > 0$ , in agreement with the full case distinction in [F1].

3.1step 1.1step 1.2step 1.3∎

Earlier steps 1.1, 1.2, and 1.3 establish the two boundary cases and the required positive exposure condition.

Depends on

Used by

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Dependency tree · two levels

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Sources

NIST/SEMATECH Engineering Statistics Handbook, Chapter 8, §8.4.1.2, PDF pp. 159–161 (standard reference, not scraped)
Lu Tian, STAT331 Unit 2: Likelihood Construction, Inference for Parametric Survival Distributions, §2.2, PDF pp. 5–6 (standard reference, not scraped)