Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

A consistent underdetermined system has an affine two-parameter solution set

Example

Over R\mathbb R, the system x+2y−z=3,2x+4y−2z=6x+2y-z=3,\qquad 2x+4y-2z=6 has the affine two-parameter solution set (x,y,z)=(3,0,0)+s(−2,1,0)+t(1,0,1),s,t∈R.(x,y,z)=(3,0,0)+s(-2,1,0)+t(1,0,1),\qquad s,t\in\mathbb R.

Facts & Assumptions

Given: The displayed real system.

[L1]

RREF parametrises solutions by the nonpivot variables (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

A nonempty solution set is a particular solution plus the nullspace (If Ax=bAx=b has one solution xpx_p, then its full solution set is the affine subspace xp+N(A)x_p+N(A)).

[L3]

The real numbers form a field (The reals form a field).

Verification

technique · direct
1.1

Subtracting twice the first row from the second leaves one nonzero row x+2y−z=3x+2y-z=3. Put y=sy=s and z=tz=t; then the pivot variable is x=3−2s+tx=3-2s+t.

L1L3algebra
2.1

This is the displayed affine form. The base point maps to 33, while each direction maps to 00, so direct multiplication verifies that (3,0,0)(3,0,0) is particular and the two direction vectors lie in the nullspace, as [L2] requires.

step 1.1L2algebra∎

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: 45 results over 11 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