Alphabeta MathHow 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 system has no solution because its augmented column contains a pivot
Example
The real system is inconsistent.
Facts & Assumptions
Given: The displayed system and its augmented matrix.
A pivot in the augmented column of the RREF is equivalent to inconsistency (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).
Consistency is equivalent to equality of coefficient and augmented ranks (The system is consistent if and only if ).
Verification
Subtracting twice row from row gives the row , which normalizes to a pivot in the augmented column and represents .
The coefficient matrix has rank because rows and are independent, while the augmented matrix has the additional pivot and rank . Thus [L2] independently gives inconsistency.
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: 18 results over 9 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
- D. Margalit and J. Rabinoff, Interactive Linear Algebra, §1.3 (standard reference, not scraped)