Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02
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.

u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions

Statement refuted

The tangent addition formula cannot be asserted for all real u,v without domain restrictions. The conventions and prerequisite facts used below are recorded in Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains, Tangent, cotangent, secant, and cosecant on their exact natural domains, Quarter-turn values and shifts by pi/2 and pi.

Facts & Assumptions

Given: u=v=π/4.

Counterexample

1.1

Both tan⁡u and tan⁡v equal 1, so the formal right side has denominator 1−1=0.

given
2.1

Meanwhile u+v=π/2, where tangent itself is undefined.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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