DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Countable-state l1 distance for mass functions

Definition

Let $S$ be at most countable. For nonnegative functions $a, b : S \to [0, \infty)$ with finite total masses, define

$∥ a - b ∥_{ℓ^{1} (S)} : = \sum_{x \in S} ∣ a (x) - b (x) ∣ .$

For a nonnegative family, $\sum_{x \in S}$ means the supremum of its finite subsums; this agrees with the ordinary series under any enumeration of $S$ . The distance is finite because $∣ a (x) - b (x) ∣ \leq a (x) + b (x)$ termwise, so $∥ a - b ∥_{ℓ^{1} (S)} \leq \sum_{x} a (x) + \sum_{x} b (x)$ . In particular it is defined for probability and subprobability mass functions.

Depends on

Finite, countably infinite, countable, uncountable

Used by

Killed finite-state projection and cemetery lift Definition
Finite-state killed projection has an exact lost-mass certificate Theorem

Dependency tree · two levels

6 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

Brian Munsky and Mustafa Khammash, The Finite State Projection Algorithm for the Solution of the Chemical Master Equation (standard reference, not scraped)