LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Sine chord and linear bounds

Statement

For $0 \leq u \leq 1 / 2$ and $v \geq 0$ , $\sin (π u) \geq 2 u, \sin (v) \leq v .$

Facts & Assumptions

Given: $u \in [0, 1 / 2]$ , $v \geq 0$ , and $π$ as defined by Pi as twice the smallest positive zero of cosine.

[F1]

$π > 0$ , $\sin 0 = 0$ , and $\sin (π / 2) = 1$ (Pi as twice the smallest positive zero of cosine, The derivatives of sine and cosine are cosine and minus sine, Quarter-turn values and shifts by pi/2 and pi).

[F2]

$\sin^{'} = \cos$ , and $\cos$ is nonincreasing on $[0, π / 2]$ (The derivatives of sine and cosine are cosine and minus sine, Signs, monotonicity intervals, and ranges of sine and cosine).

[F3]

Sums and scalar multiples of differentiable real functions have the corresponding sum and scalar-multiple derivatives (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F4]

The identity function has derivative $1$ , as its difference quotient is $1$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[F5]

The mean value theorem applies to a function continuous on a closed interval and differentiable on its interior (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

[F6]

Sine is $1$ -Lipschitz on $R$ : $∣ \sin a - \sin b ∣ \leq ∣ a - b ∣$ (Sine and cosine are $1$ -Lipschitz on $R$ ).

Proof

1.1givenF1F3F4construct

Define $g (t) = \sin t - 2 t / π$ on $[0, π / 2]$ . By [F1], [F3], and [F4], $g (0) = g (π / 2) = 0$ and $g^{'} (t) = \cos t - 2 / π$ on the interior.

2.1F2step 1.1

By [F2], $g^{'}$ is nonincreasing: if $s < t$ , then $\cos s \geq \cos t$ , so $g^{'} (s) \geq g^{'} (t)$ .

3.1F1F5step 2.1choosealgebra

Fix $t \in (0, π / 2)$ . Apply [F5] on $[0, t]$ and $[t, π / 2]$ to obtain $a \in (0, t)$ and $b \in (t, π / 2)$ with $g (t) = t g^{'} (a)$ and $- g (t) = (π / 2 - t) g^{'} (b)$ . Since $a < b$ , step 2.1 gives $g^{'} (a) \geq g^{'} (b)$ . Dividing by the positive numbers $t$ and $π / 2 - t$ , then multiplying by their product, gives $g (t) π / 2 \geq 0$ ; as $π > 0$ , $g (t) \geq 0$ .

4.1F1step 3.1algebra

If $0 < u < 1 / 2$ , take $t = π u \in (0, π / 2)$ in step 3.1 to get $\sin (π u) \geq 2 u$ . At $u = 0$ and $u = 1 / 2$ , the same inequality is equality by [F1].

5.1F1F6step 4.1algebra∎

For $v \geq 0$ , [F6] and $\sin 0 = 0$ give $\sin v = \sin v - \sin 0 \leq ∣ \sin v - \sin 0 ∣ \leq v$ . Thus both stated inequalities hold.

Depends on

Used by

Dependency tree · two levels

32 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §4.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, version 6, §4.6 (standard reference, not scraped)
Lin Lin, Lecture Notes on Quantum Algorithms for Scientific Computation, §§16.1–16.3 (standard reference, not scraped)