ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

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A $3 \times 3$ system row reduces to a unique solution

Example

Over $R$ , the system $x + y + z = 6, 2 x - y + z = 3, x + 2 y - z = 2$ has the unique solution $(x, y, z) = (1, 2, 3)$ .

Facts & Assumptions

Given: The displayed real linear system.

[L1]

An RREF with a pivot in every variable column gives one solution and no free variables (Reduced row echelon form detects consistency and parametrises every solution by the nonpivot variables).

[L2]

Row operations on the augmented matrix preserve the solution set (Elementary row operations on an augmented matrix preserve exactly the solutions of its linear system).

Verification

technique · direct

1.1

Its augmented matrix reduces as follows: subtract twice row $0$ from row $1$ , subtract row $0$ from row $2$ , swap rows $1, 2$ , add three times row $1$ to row $2$ , scale row $2$ by $- 1 / 7$ , clear above its pivot, and then clear above the pivot in column $1$ ; the result is $[I_{3} ∣ (1, 2, 3)^{T}]$ .

L1L2algebra

2.1

The RREF gives the unique solution. Direct substitution yields $1 + 2 + 3 = 6$ , $2 - 2 + 3 = 3$ , and $1 + 4 - 3 = 2$ .

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

D. Margalit and J. Rabinoff, Interactive Linear Algebra, §§1.2–1.3 (standard reference, not scraped)

A 3×33×3 system row reduces to a unique solution