Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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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.

Tangent-curve reachability is an equivalence relation

Statement

For an integrable distribution D, the relation ∼D is an equivalence relation on M.

Facts & Assumptions

Given: An integrable distribution D on M.

[A1]

The relation is defined by piecewise smooth curves tangent to D.

Proof

technique · direct
1.1

Reflexivity holds because the constant curve at any point is piecewise [given] smooth and has derivative 0∈D.

given
1.2

Symmetry holds because reversing a tangent piecewise smooth curve negates [given] its derivative but keeps it inside the same linear subspaces.

given
1.3

Transitivity holds because concatenating two tangent piecewise smooth curves [given] produces another piecewise smooth curve with the same tangency property.

givenconstruct
2.1

Therefore ∼D is an equivalence relation. [given] ∎

given

Depends on

Used by

Dependency tree · two levels

2 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