LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Minimum, maximum, and bounded shifts of stopping times

Statement

If $σ, τ$ are stopping times, then $σ \land τ$ and $σ \lor τ$ are stopping times. If $c \in N_{0}$ , then $τ + c$ is a stopping time for $(F_{n})$ , with $\infty + c = \infty$ . More generally, if $ρ$ is stopping for $G_{n} = F_{n + c}$ , then $ρ + c$ is stopping for $(F_{n})$ . The earlier shift $(τ - c)^{+}$ is stopping for $(F_{n + c})$ but need not be stopping for $(F_{n})$ . Finally, for every deterministic $N \in N_{0}$ , $τ \land N$ is a bounded stopping time.

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Discrete stopping time supplies the finite-horizon test.

[F2]

Equivalent event tests for a discrete stopping time supplies the complementary tests.

Proof

1.1F1

The identities ${σ \land τ \leq n} = {σ \leq n} \cup {τ \leq n}, {σ \lor τ \leq n} = {σ \leq n} \cap {τ \leq n}$ prove the first two claims by F1.

1.2F1

If $n < c$ , ${τ + c \leq n} = \emptyset$ ; if $n \geq c$ , it is ${τ \leq n - c} \in F_{n - c} \subseteq F_{n}$ . If $ρ$ is stopping for $G$ , the same event is in $G_{n - c} = F_{n}$ .

1.3F1

For the earlier shift, ${(τ - c)^{+} \leq n} = {τ \leq n + c} \in F_{n + c},$ so it is stopping for the shifted filtration. The right side need not lie in $F_{n}$ , which is why no unshifted assertion is made.

2.1F1F2∎

A deterministic $N$ is a stopping time, so step 1.1 makes $τ \land N$ a stopping time, and it is bounded by $N$ . All empty and infinite-value cases follow from the displayed identities.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Durrett, Probability: Theory and Examples, 5th ed., §4.4 (standard reference, not scraped)