feat(InformationTheory/Coding): add Kraft inequality for prefix-free codes

[elazarg]

This PR defines prefix-free codes and, over a finite, nonempty alphabet, proves Kraft's inequality for finite and arbitrary prefix-free codes. It is a direct continuation of #34108, which introduced uniquely decodable codes and the Kraft–McMillan inequality.

The results live in the InformationTheory namespace under InformationTheory/Coding.

Main definitions

• InformationTheory.PrefixFree: a set of codewords is prefix-free if two codewords in the set cannot be distinct when one is a prefix of the other.

Main results

• InformationTheory.PrefixFree.uniquelyDecodable: a prefix-free code not containing the empty word is uniquely decodable.
• InformationTheory.PrefixFree.uniquelyDecodable_of_nontrivial: every nontrivial prefix-free code is uniquely decodable.
• InformationTheory.kraft_inequality: the Kraft sum of a finite prefix-free code is at most one.
• InformationTheory.summable_kraft_sum: the Kraft series of an arbitrary prefix-free code is summable.
• InformationTheory.kraft_inequality_infinite: the Kraft sum of an arbitrary prefix-free code is at most one.

---

This is another part of https://github.com/elazarg/kraft. Comments about next items are welcome.

[github-actions]

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[github-actions]

PR summary fd95666751

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.InformationTheory.Coding.PrefixFree (new file)	195
Mathlib.InformationTheory.Coding.Kraft (new file)	1603

---

Declarations diff (regex)

+ PrefixFree
+ PrefixFree.eq_singleton_empty_of_empty_mem
+ PrefixFree.mono
+ PrefixFree.uniquelyDecodable
+ PrefixFree.uniquelyDecodable_of_nontrivial
+ kraft_inequality
+ kraft_inequality_infinite
+ sum_kraft_le_one
+ summable_kraft_sum

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit fd95666).

• +8 new declarations
• −0 removed declarations

+InformationTheory.PrefixFree
+InformationTheory.PrefixFree.eq_singleton_empty_of_empty_mem
+InformationTheory.PrefixFree.mono
+InformationTheory.PrefixFree.uniquelyDecodable
+InformationTheory.PrefixFree.uniquelyDecodable_of_nontrivial
+InformationTheory.kraft_inequality
+InformationTheory.kraft_inequality_infinite
+InformationTheory.summable_kraft_sum

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type (weak)
5051	2	exposed public sections

Current commit fd95666751
Reference commit 639923353f

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[wwylele]

Mostly looks good to me