ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31verified 2026-08-08 (gpt-5.6-terra-codex-subscription)

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.

Every nonempty interval and every $\mathbb{R}^n$ with $n\ge1$ contracts to any chosen point

Example

Every nonempty interval $J\subseteq\mathbb R$ is contractible. If $c\in J$ , the contraction is

$H(x,t)=(1-t)x+tc.$

Likewise, for every $n\ge1$ , every chosen $c\in\mathbb R^n$ gives a contraction of $\mathbb R^n$ by the same formula.

Facts & Assumptions

Given: A nonempty interval $J\subseteq\mathbb R$ , a point $c\in J$ , and a natural $n\ge1$ .

[A1]

Intervals are order-convex: if $x,c\in J$ and $x\le z\le c$ , or $c\le z\le x$ , then $z\in J$ (Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length).

[L1]

Every nonempty convex subset of $\mathbb R^m$ with $m\ge1$ is contractible to any chosen point by straight lines (Every nonempty convex subset of $\mathbb{R}^n$ is contractible).

Verification

technique · direct

1.1

For $x,c\in J$ and $t\in I$ , $(1-t)x+tc$ lies between $x$ and $c$ , so it lies in $J$ by [A1]. Thus $J$ , viewed as a subset of $\mathbb R^1$ , is convex.

A1algebra

1.2

The whole space $\mathbb R^n$ is convex, since it is closed under vector addition and scalar multiplication.

algebra

2.1

Apply [L1] to step 1.1 and the point $c$ to obtain the stated contraction of $J$ , and apply it to step 1.2 and any chosen point of $\mathbb R^n$ to obtain the Euclidean contraction.

step 1.1step 1.2L1∎

Depends on

Used by

A one-point space and ℝ are homotopy equivalent but not homeomorphic Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Algebraic Topology lecture notes (UC Riverside) (standard reference, not scraped)

Every nonempty interval and every Rn\mathbb{R}^nRn with n≥1n\ge1n≥1 contracts to any chosen point