Formal knowledge
Quotient soundness
A trusted rule that identifies quotient representatives related by the chosen relation.
Definition
Quotient soundness
Quotient soundness states that related representatives become equal after they enter a quotient type.
Quotients construct values modulo an equivalence relation, such as integers from pairs of natural numbers. Lean treats the soundness rule as part of its trusted quotient machinery.
01
Example
Different fraction pairs can represent the same rational number after quotienting by cross-multiplication equality.
02
Important distinction
The relation must satisfy the required equivalence properties before it can support a well-formed quotient construction.
After reading this article, you should be able to
- Define Quotient soundness in the sense used on this site.
- Explain why quotient representatives can differ while denoting the same quotient value.