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.

A zero-likelihood observation has no posterior update

Example

Let $S = {s_{0}, s_{1}}$ , $A = {a}$ , $Ω = {x, y}$ , reward alphabet ${0}$ , horizon $H = 1$ , and discount $γ = 1$ . From each state, the joint next-state/reward law assigns probability one to that same state with reward zero. Put $b_{0} (s_{0}) = 1$ , $b_{0} (s_{1}) = 0$ , and let $O (x ∣ s_{0}) = 1$ , $O (y ∣ s_{0}) = 0$ , $O (x ∣ s_{1}) = 0$ , and $O (y ∣ s_{1}) = 1$ . Then the observation distribution has $q (y ∣ b_{0}, a) = 0$ and $q (x ∣ b_{0}, a) = 1$ , so $τ (b_{0}, a, y)$ is undefined and the belief transition uses only the positive-mass $x$ branch.

Facts & Assumptions

Given: The finite POMDP model in the example, the prior $b_{0} = δ_{s_{0}}$ , and the action $a$ .

[F1]

The finite POMDP uses probability transition and observation laws; the reward is not part of the observed history. Finite partially observable Markov decision process

[F2]

The predicted state distribution, observation normalizer, positive-domain Bayes update, and omission of zero-normalizer observations are defined by the finite belief filter. Belief states and the finite Bayes filter

[F3]

Observation probabilities sum to one, and the induced belief transition is a probability kernel using only positive-probability branches. The finite POMDP Bayes filter is a probability kernel

[F4]

Finite sums over a finite index set are well defined. The sum $\sum_{i \in S} a_{i}$ over a finite index set, and its product form

Verification

1.1givenF1F2

The hidden-state transition is the identity, so the predicted state distribution after $a$ is $\overset{ˉ}{b} (s_{0}) = 1$ and $\overset{ˉ}{b} (s_{1}) = 0$ .

2.1step 1.1F2F4algebra

The normalizers are $q (y ∣ b_{0}, a) = O (y ∣ s_{0}) \overset{ˉ}{b} (s_{0}) + O (y ∣ s_{1}) \overset{ˉ}{b} (s_{1}) = 0$ and $q (x ∣ b_{0}, a) = 1$ .

3.1step 2.1F2F3∎

Since $q (y ∣ b_{0}, a) = 0$ , the definition makes $τ (b_{0}, a, y)$ undefined and omits the $y$ term from $K$ . The only positive observation branch is $x$ with mass one by step 2.1, so $K$ assigns total mass one to $τ (b_{0}, a, x)$ and no mass to an impossible $y$ update, as required by [F3].

Depends on

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