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 two-state POMDP update changes a three-to-one prior to nine-to-one

Example

Let $S = {s_{0}, s_{1}}$ , $A = {a}$ , $Ω = {x, y}$ , reward alphabet ${0}$ , and horizon $H = 1$ with $γ = 1$ . For each $i \in {0, 1}$ , the joint next-state/reward law assigns probability one to $(s_{i}, 0)$ from $(s_{i}, a)$ , so the hidden state remains fixed and the reward is zero. Let $b_{0} (s_{0}) = 3 / 4$ , $b_{0} (s_{1}) = 1 / 4$ , and set $O (x ∣ s_{0}) = 3 / 4$ , $O (x ∣ s_{1}) = 1 / 4$ , $O (y ∣ s_{0}) = 1 / 4$ , and $O (y ∣ s_{1}) = 3 / 4$ . The probability of observing $x$ is $5 / 8$ , and after $x$ the belief is $b_{1} (s_{0}) = 9 / 10$ , $b_{1} (s_{1}) = 1 / 10$ .

Facts & Assumptions

Given: The finite POMDP model in the example and the observation $x$ .

[F1]

The state transition is the marginal of the specified finite joint reward-transition law, and each $O (\cdot ∣ s^{'})$ is a probability distribution. Finite partially observable Markov decision process

[F2]

The predicted belief is $\overset{ˉ}{b} (s^{'}) = \sum_{s} T (s^{'} ∣ s, a) b (s)$ , the observation probability is $q (o ∣ b, a) = \sum_{s^{'}} O (o ∣ s^{'}) \overset{ˉ}{b} (s^{'})$ , and the positive-probability update is $τ (b, a, o) (s^{'}) = O (o ∣ s^{'}) \overset{ˉ}{b} (s^{'}) / q (o ∣ b, a)$ . Belief states and the finite Bayes filter

[F3]

The predicted law and observation probabilities normalize, and $τ$ is a probability vector when $q > 0$ . The finite POMDP Bayes filter is a probability kernel

[F4]

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

Verification

1.1givenF1F2

The transition marginal is the identity on $S$ , so the predicted law after action $a$ is ${\overset{ˉ}{b}}_{1} (s_{0}) = 3 / 4$ and ${\overset{ˉ}{b}}_{1} (s_{1}) = 1 / 4$ .

2.1step 1.1F2F4algebra

The observation normalizer is $q (x ∣ b_{0}, a) = (3 / 4) (3 / 4) + (1 / 4) (1 / 4) = 9 / 16 + 1 / 16 = 5 / 8 > 0$ .

3.1step 1.1step 2.1F2F3algebra∎

The update gives $b_{1} (s_{0}) = (3 / 4) (3 / 4) / (5 / 8) = (9 / 16) / (10 / 16) = 9 / 10$ and $b_{1} (s_{1}) = (1 / 4) (1 / 4) / (5 / 8) = (1 / 16) / (10 / 16) = 1 / 10$ . Both coordinates are nonnegative and sum to one, in agreement with [F3].

Depends on

Used by

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