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.

Two paths with the same endpoints in a convex subset of $R^{n}$ are path homotopic relative to their endpoints

Example

Let $C \subseteq R^{n}$ be convex, with $n \geq 1$ , and let $α, β : I \to C$ be paths with the same initial and terminal points. Then

$H (s, t) = (1 - t) α (s) + t β (s)$

is a path homotopy from $α$ to $β$ relative to the endpoints.

Facts & Assumptions

Given: A convex $C \subseteq R^{n}$ and paths $α, β : I \to C$ with $α (0) = β (0)$ and $α (1) = β (1)$ .

[L1]

The straight-line formula defines a continuous map $H : I \times I \to C$ (For continuous maps into a convex subset of $R^{n}$ , the straight-line formula defines a continuous homotopy).

[A1]

A path homotopy relative endpoints is a homotopy $H : I \times I \to C$ with the path parameter on the first coordinate and with $H (0, t)$ and $H (1, t)$ fixed (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

Verification

technique · direct

1.1

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

1.2

One has $H (s, 0) = α (s)$ and $H (s, 1) = β (s)$ . At the endpoints, $H (0, t) = (1 - t) α (0) + t β (0) = α (0)$ and $H (1, t) = (1 - t) α (1) + t β (1) = α (1)$ .

algebra

2.1

Thus $H$ satisfies all clauses of [A1], so it is a path homotopy from $α$ to $β$ relative to the endpoints.

step 1.1step 1.2A1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)
A. Hatcher, Algebraic Topology, Section 0 (standard reference, not scraped)

Two paths with the same endpoints in a convex subset of RnRn are path homotopic relative to their endpoints