lemmaly — Algorithm-First Proof

The model already knows Big-O, hash tables, divide-and-conquer, dynamic programming, sorting, graph algorithms, and amortized analysis. It just does not apply them spontaneously. lemmaly fixes the behavior, not the knowledge.

This skill is the gateway for an algorithm-discipline suite of four skills (lemmaly, mathguard, invariant-guard, complexity-cuts). It enforces the hard rules that every other guard in the suite assumes.

Violating the letter of these rules is violating the spirit of the skill. "Just this once" is how O(n²) ships to production.

When to Use This Skill

Use lemmaly when:

• Writing, editing, or reviewing code that involves loops, collections, lookups, searches, joins, recursion, graphs, queries, or any computation over more than a handful of items.
• About to write a for inside a for, .find / .includes / .indexOf inside a loop, await inside for / map / forEach over independent items, or one query per item in a collection.
• Auditing a codebase / PR for known anti-patterns (await-in-loop, .includes inside .filter, string-concat in loop, SELECT *, N+1, etc.).
• Reviewing AI-generated code that "looks idiomatic" but might hide O(n²) or N+1.

When in doubt, start at lemmaly — it is the gateway and will tell you when to escalate to its three sibling skills.

Routing flow

Are you writing new code?
├── yes → lemmaly (state complexity, structure, family BEFORE coding)
│         ├── classical algorithm at its lower bound AND n is large? → mathguard
│         └── subtle correctness trap (invariant, base case, off-by-one)? → invariant-guard
└── no, refactoring existing slow / OOM / timed-out code → complexity-cuts
          └── still slow after classical fixes? → mathguard

One-line mental model

• lemmaly = think first (prevention).
• complexity-cuts = clean up bad Big-O (correction).
• invariant-guard = prove it's correct (verification).
• mathguard = beat the classical floor (acceleration).

The Iron Law

NO NON-TRIVIAL CODE WITHOUT STATED COMPLEXITY, DATA STRUCTURE, AND ALGORITHM FAMILY

Before you write a loop, a recursion, a query, or any computation over more than a handful of items, three things must appear in your message — in this order:

1. time = O(?), space = O(?), with the dominant input dimension named.
2. The data structure you will use, with a one-phrase reason.
3. The algorithm family (one of: linear scan, two-pointer, sliding window, binary search, sort+sweep, hash join, BFS/DFS, topo sort, Dijkstra/A*, union-find, DP, greedy, recursion+memo, prefix sum, segment tree, monoid reduction).

If you cannot state all three, you do not understand the problem yet. Ask, or read more code. Do not write code.

Non-negotiable rules

1. State complexity before writing any non-trivial code. In one line:

   • time = O(?), space = O(?)
   • Dominant input dimension: n = what, with realistic magnitude (e.g. n ~ 10^6 rows)
   • If you cannot state these, you do not yet understand the problem. Ask, or read more code.

2. Name the data structure with a one-phrase reason. Every collection-shaped value gets a deliberate choice from Array / List / Set / HashMap / TreeMap / Heap / Deque / Trie / Graph / BitSet / Counter / LinkedList — with the reason: "Set for O(1) membership inside the loop", "Heap for top-K in O(n log k)", "Counter to fold the nested loop into a single pass". Default to hashed structures (Set, Map) for lookup inside loops. Default to streaming/iterator over materialized list when n is large.

3. Identify the algorithm family before writing. Name one of: linear scan, divide and conquer, two-pointer, sliding window, binary search,