Index Multiplication

n * m = n + n + … + n (m times): multiplication earned by structural recursion on addition.

Index Multiplication

Summary

n * m = n + n + … + n (m times): multiplication earned by structural recursion on addition.

Statement

%
\label{def:idx-mul}
For $\underline{n}, \underline{m} \in \tau\text{-Idx}$,
\textbf{index multiplication} is defined by
structural recursion on $m$:
\begin{align*}
    \underline{n} \times \underline{0} &:= \underline{0}, \\
    \underline{n} \times \rho(\underline{m})
    &:= (\underline{n} \times \underline{m}) + \underline{n}.
\end{align*}
That is, $\underline{n} \times \underline{m}
= \underbrace{\underline{n} + \cdots + \underline{n}}_{m \text{ times}}$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 25
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex lines 159-172

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Denotation.Arithmetic
• Name: Tau.Denotation.idx_mul

Dependencies

• Canonical: I.D10, I.D06

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.