Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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) is not the identity outside the principal interval

Example

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

arcsin⁡(sin⁡(3π/4))=π/4≠3π/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] (Principal inverse sine and inverse cosine).

[L2]

The supplementary identity is sin⁡(π−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). Since π/4 lies in the principal range of arcsine, [L1] gives arcsin⁡(sin⁡(3π/4))=π/4.

L1L2algebra
2.1

Since π/4≠3π/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 · two levels

13 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