ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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 $R^{n}$ with $n \geq 1$ contracts to any chosen point

Example

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

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

Likewise, for every $n \geq 1$ , every chosen $c \in R^{n}$ gives a contraction of $R^{n}$ by the same formula.

Facts & Assumptions

Given: A nonempty interval $J \subseteq R$ , a point $c \in J$ , and a natural $n \geq 1$ .

[A1]

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

[L1]

Every nonempty convex subset of $R^{m}$ with $m \geq 1$ is contractible to any chosen point by straight lines (Every nonempty convex subset of $R^{n}$ is contractible).

Verification

technique · direct

1.1

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

A1algebra

1.2

The whole space $R^{n}$ 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 $R^{n}$ 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 · two levels

8 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

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

Every nonempty interval and every RnRn with n≥1n≥1 contracts to any chosen point