LemmaStatement: AI-generatedProof: AI-generated

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.

Rational inverse-sine brackets on the principal branch

Statement

Let $N \geq 0$ be an integer and let $z, a, b$ be rational with $0 \leq z \leq 1$ and $0 \leq a \leq b \leq π / 2$ . For $x \in {a, b}$ put $q_{N} (x) = x^{2} / ((2 N + 4) (2 N + 5))$ and assume $q_{N} (x) < 1$ . Let $S_{N}, R_{N}$ be the finite sums and tails in the sine-enclosure lemma. If $S_{N} (a) + R_{N} (a) \leq z$ and $S_{N} (b) - R_{N} (b) \geq z$ , then $a \leq \arcsin z \leq b$ . A rational search may certify its final endpoints using these two inequalities under the stated enclosure conditions; it need not assign a value when a trial enclosure straddles $z$ .

Facts & Assumptions

Given: The integer $N$ , rational inputs, enclosure conditions and endpoint inequalities in the Statement.

[F1]

Principal inverse sine and inverse cosine defines $\arcsin$ as the inverse of strictly increasing sine on $[- π / 2, π / 2]$ .

[F2]

Rational sine enclosures from a finite factorial sum gives $S_{N} (x) - R_{N} (x) \leq \sin x \leq S_{N} (x) + R_{N} (x)$ for rational $x$ , an integer $N \geq 0$ and $q_{N} (x) < 1$ .

Proof

Given: The integer $N$ , rational inputs, enclosure conditions and endpoint inequalities in the Statement.

Goal: Prove the inverse-sine bracketing rule.

1.1givenF1F2

The integer $N \geq 0$ , rational endpoints $a, b$ and inequalities $q_{N} (a) < 1$ , $q_{N} (b) < 1$ meet the hypotheses of [F2] at both endpoints. Its enclosures and the endpoint tests give $\sin a \leq S_{N} (a) + R_{N} (a) \leq z \leq S_{N} (b) - R_{N} (b) \leq \sin b$ . By [F1], $\sin (\arcsin z) = z$ , and all three arguments $a, \arcsin z, b$ lie in $[- π / 2, π / 2]$ .

2.1step 1.1F1∎

Applying strict monotonicity to the two comparisons yields $a \leq \arcsin z \leq b$ . In particular, a search whose final endpoints pass these exact rational tests yields a certificate regardless of how its trials were selected; an unclassified trial is not evidence.

Depends on

Used by · 2

Dependency tree · 2 levels

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