---
title: "Functions, mappings, and bound variables"
description: "Read an input-output contract and distinguish free inputs from names bound inside its rule."
canonical_url: "https://fanout.sh/ml-math/decoder/functions-mappings-and-bound-variables"
md_url: "https://fanout.sh/ml-math/decoder/functions-mappings-and-bound-variables.md"
last_updated: "2026-08-09"
access: "public"
---

# Functions, mappings, and bound variables

Read an input-output contract and distinguish free inputs from names bound inside its rule.

## Public overview

Which names belong to the function contract, and which are local to an operator?

Assign a provisional mathematical type to each important object.

Check whether the operation is valid for the stated types and shapes.

Use a type or shape mismatch to locate a reading error.

---
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