ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Length and distance on the circle

Example

On the unit circle with induced metric, $d (e^{i a}, e^{i b}) = \min_{k \in Z} ∣ b - a + 2 π k ∣$ . Antipodes have two distinct minimizing semicircles.

Facts & Assumptions

Given: Real angles $a, b$ , and the circle parametrization $e^{i t} = (\cos t, \sin t)$ .

[F1]

Riemannian speed and length: The Riemannian speed on a $C^{1}$ piece is $∣ \dot{γ} (t) ∣_{g} = \sqrt{g_{γ (t)} (\dot{γ} (t), \dot{γ} (t))}$ . Its length is $L_{g} (γ) = \sum_{j} \int_{t_{j - 1}}^{t_{j}} ∣ \dot{γ} (t) ∣_{g} d t$ . The curve convention is def-piecewise-c-one-curve-on-a-manifold and the norm is def-pointwise-norm-and-angle-from-a-riemannian-metric. Each integrand is continuous on its closed piece with the one-sided endpoint derivative, hence Riemann integrable and nonnegative. Values chosen at the finitely many corners do not change its integral. For a singleton interval the empty sum is zero; a constant curve also has zero length. Partition independence is established next.

[F2]

Riemannian distance on a connected manifold: On a connected Riemannian manifold define $d_{g} (p, q) = \inf {L_{g} (γ) : γ is piecewise C^{1} from p to q}$ . Lengths are those of def-riemannian-speed-and-length. For each pair $p, q$ , lem-any-two-points-in-a-connected-smooth-manifold-can-be-joined-by-a-piecewise-c-one-curve supplies a curve, so the set of lengths is nonempty, contains a finite real number and is bounded below by zero. Applying the least-upper-bound property cor-cauchy-reals-lub-complete to the negatives gives a finite nonnegative infimum. On the empty connected manifold this defines the empty distance function; there are no pairs to evaluate. No minimizing curve is part of this definition.

[F3]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: Let $a < b$ . Suppose $G : [a, b] \to R$ is continuous on $[a, b]$ and differentiable on $(a, b)$ . If $f : [a, b] \to R$ is Riemann integrable and $f (x) = G^{'} (x) (a < x < b),$ then $\int_{a}^{b} f = G (b) - G (a) .$ No derivative of $G$ at either endpoint is assumed, and the two endpoint values assigned to the integrable extension $f$ do not enter the conclusion.

[F4]

Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ : Every real $x$ has a unique integer $m = ⌊ x ⌋$ with $m \leq x < m + 1$ .

Verification

technique · direct

1.1given

For any piecewise $C^{1}$ circle path, the inverse images of smooth angle arcs form an open cover of its compact parameter interval. A finite subcover has a positive Lebesgue number; subdividing more finely than it, and at the original differentiability breakpoints, puts each piece in one angle arc. Start the first angle at $a$ , and add a multiple of $2 π$ to each successive local angle to match the preceding endpoint. This yields a continuous piecewise $C^{1}$ lift $θ$ starting at $a$ , ending at $b + 2 π k$ for some integer $k$ .

2.1F1F3step 1.1

Differentiation of $(\cos θ, \sin θ)$ gives squared speed $θ^{' 2} (\sin^{2} θ + \cos^{2} θ) = θ^{' 2}$ . Consequently $L = \int ∣ θ^{'} ∣ \geq ∣ \int θ^{'} ∣ = ∣ b - a + 2 π k ∣$ , where Newton–Leibniz is applied on each closed smooth piece and the endpoint increments telescope.

3.1F2F4step 2.1

Put $x = (a - b) / (2 π)$ and $m = ⌊ x + 1 / 2 ⌋$ . By [F4], $- 1 / 2 \leq x - m < 1 / 2$ . Set $k_{0} = m$ and $δ = b - a + 2 π k_{0} = 2 π (m - x) \in (- π, π]$ . For any integer $k$ , the two neighboring integers $m - 1, m + 1$ are at least as far from $x$ as $m$ , and every more distant integer is farther still; thus $∣ b - a + 2 π k ∣ \geq ∣ δ ∣$ . The path $t \mapsto e^{i (a + t δ)}$ on $[0, 1]$ has constant speed $∣ δ ∣$ and attains that lower bound, proving the distance formula.

4.1step 3.1∎

For $b - a = π$ , the representatives $δ = π$ and $δ = - π$ both minimize. The paths $e^{i (a + π t)}$ and $e^{i (a - π t)}$ have length $π$ and disjoint interior semicircle images. For equal endpoints $δ = 0$ , the same construction is a constant path of length zero.

Source locator

Lee, pp. 331 and 337–338, induced metric and distance; the finite angle lift and minimization over integers are proved above.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)