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.

The formula $H (x, t) = (1 - t) f (x) + t g (x)$ gives an explicit homotopy between maps into $R^{n}$

Example

Let $n \geq 1$ , let $X$ be a topological space, and let $f, g : X \to R^{n}$ be continuous. Since $R^{n}$ is convex, the straight-line formula

$H (x, t) = (1 - t) f (x) + t g (x)$

deforms $f$ to $g$ .

Facts & Assumptions

Given: Continuous maps $f, g : X \to R^{n}$ with $n \geq 1$ .

[L1]

The straight-line formula is continuous for maps into a convex subspace of $R^{n}$ (For continuous maps into a convex subset of $R^{n}$ , the straight-line formula defines a continuous homotopy).

[L2]

Any two continuous maps into a nonempty convex subset of $R^{n}$ are homotopic by that formula (Any two continuous maps into a nonempty convex subset of $R^{n}$ are homotopic by straight lines).

Verification

technique · direct

1.1

For $u, v \in R^{n}$ and $t \in I$ , the vector $(1 - t) u + t v$ lies in $R^{n}$ , so $R^{n}$ is convex.

algebra

1.2

The map $H$ is continuous by [L1].

2.1

Substitution gives $H (x, 0) = f (x)$ and $H (x, 1) = g (x)$ , so [L2] identifies $H$ as a homotopy from $f$ to $g$ .

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

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)

The formula H(x,t)=(1−t)f(x)+tg(x)H(x,t)=(1−t)f(x)+tg(x) gives an explicit homotopy between maps into RnRn