ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03

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 fundamental group of the unit interval is trivial at every basepoint

Example

For every $x_{0} \in I = [0, 1]$ , the fundamental group $π_{1} (I, x_{0})$ has one element. A loop $α$ contracts relative to its endpoints by

$H (s, t) = (1 - t) α (s) + t x_{0} .$

Facts & Assumptions

Given: A basepoint $x_{0} \in I$ and a based loop $α : I \to I$ .

[L1]

The unit interval is convex: if $x, y, t \in [0, 1]$ , then $0 \leq (1 - t) x + t y \leq 1$ , so the convex combination remains in $[0, 1]$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, A convex subset of $R^{m}$ contains every line segment between two of its points, algebra).

[L2]

Every nonempty convex Euclidean subset is simply connected, with the displayed straight-line endpoint homotopy (Every nonempty convex subset of $R^{n}$ is simply connected).

[L3]

Elements of $π_{1} (I, x_{0})$ are endpoint-homotopy classes of based loops (Based loops and the fundamental group).

Verification

technique · direct

1.1

By [L1], the interval satisfies the hypotheses of [L2], so the displayed formula is an endpoint-fixed homotopy from $α$ to the constant loop at $x_{0}$ .

L1L2

2.1

Thus every class in $π_{1} (I, x_{0})$ equals the constant-loop class, and the group has exactly one element.

step 1.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

A. Hatcher, Algebraic Topology, Chapter 1, Example 1.4 (standard reference, not scraped)