Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-03
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.

arcsin⁡(sin⁡x)\arcsin(\sin x) is not the identity outside the principal interval

Example

The identity arcsin⁡(sin⁡x)=x\arcsin(\sin x)=x is not valid for every x∈Rx\in\mathbb R. For example,

arcsin⁡(sin⁡(3π/4))=π/4≠3π/4.\arcsin(\sin(3\pi/4))=\pi/4\ne3\pi/4.

Facts & Assumptions

Given: No hypotheses beyond those quantified in the statement.

[L1]

Principal arcsine is the inverse of sine with values restricted to [−π/2,π/2][-\pi/2,\pi/2] (Principal inverse sine and inverse cosine).

[L2]

The supplementary identity is sin⁡(π−x)=sin⁡x\sin(\pi-x)=\sin x (Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions).

Proof

technique · direct
1.1

By [L2], sin⁡(3π/4)=sin⁡(π/4)\sin(3\pi/4)=\sin(\pi/4). Since π/4\pi/4 lies in the principal range of arcsine, [L1] gives arcsin⁡(sin⁡(3π/4))=π/4\arcsin(\sin(3\pi/4))=\pi/4.

L1L2algebra
2.1

Since π/4≠3π/4\pi/4\ne3\pi/4, step 1.1 is the claimed counterexample.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 52 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources