ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-16

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.

Assembling a keyhole contour from two radial segments and two circular arcs

Example

Let $0 < r < R$ . A keyhole contour about the positive real axis is the concatenation of the upper radial segment $r \to R$ , the outer circle once counterclockwise, the lower radial segment $R \to r$ , and the inner circle clockwise. For every continuous integrand on the trace, its integral is the signed sum of the four piece integrals.

Facts & Assumptions

Given: Radii $0 < r < R$ and four oriented pieces with matching endpoints.

[L1]

Concatenation and reversal of rectifiable complex contours are defined in Rectifiable complex contours, reversal, concatenation, closedness, and orientation.

[L2]

Complex line integrals add under concatenation and change sign under reversal (Complex line integrals change sign under reversal and add under concatenation).

[L3]

The absolute integral of $1$ is contour length (The absolute line integral of the constant function 1 is the length of the path).

Verification

technique · direct

1.1L1construct

On $[0, 1]$ , parametrize the pieces by $r + (R - r) t$ , $R e^{2 π i t}$ , $R - (R - r) t$ , and $r e^{2 π i (1 - t)}$ , respectively. Their endpoints match in this order, so [L1] defines a closed concatenation.

2.1step 1.1L2

Repeated application of [L2] gives the total integral as the sum of the four oriented integrals, with the reversed radial and inner-circle orientations carrying their signs.

3.1step 1.1L3∎

By [L3], the piece lengths are $R - r$ , $2 π R$ , $R - r$ , and $2 π r$ . This verifies rectifiability and bookkeeping without evaluating the integral by Cauchy's theorem or choosing a logarithm branch.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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