القائمة
Quantitative insights on formalization speed by item type and proof strategy. Use when estimating difficulty from textbook statements, planning batches, or predicting project timelines.
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| name | formalization-velocity |
| description | Quantitative insights on formalization speed by item type and proof strategy. Use when estimating difficulty from textbook statements, planning batches, or predicting project timelines. |
| allowed-tools | Read, Bash, Glob, Grep |
Data from the first complete FormalFrontier formalization: Sutherland Number Theory Lecture 1 (27 items, 46 PRs, ~5.3 hours total with 2 parallel agents).
Measured from Sutherland Lecture 1. Times include scaffolding + proof filling per item.
| Item Type | Count | Avg Lines | Avg Time | Mathlib Wrapper % | Notes |
|---|---|---|---|---|---|
| Definition | 9 | 25 | 5 min | 100% | Always Mathlib wrappers or abbrev |
| Proposition | 5 | 28 | 10 min | 80% | Usually inferInstance or 1-line API |
| Corollary | 3 | 30 | 8 min | 100% | Always follow from parent theorem |
| Theorem | 3 | 63 | 30 min | 67% | Named theorems often in Mathlib; product formula was outlier |
| Lemma | 1 | 85 | 45 min | 0% | Required original calc chain proof |
| Example | 4 | 69 | 35 min | 25% | Counterexamples never in Mathlib |
Key insight: Definitions and corollaries are always fast. Lemmas and examples (especially counterexamples) are the slowest.
| Strategy | Items | Avg Lines | Avg Time | When to Expect |
|---|---|---|---|---|
inferInstance | 6 | 23 | 3 min | Typeclass properties (IC, UFD, field) |
| Direct API call | 8 | 28 | 5 min | Named theorems, standard results |
| Instance transfer | 2 | 30 | 8 min | Properties via equivalence/isomorphism |
| Short tactic (2-10 lines) | 5 | 35 | 15 min | Setup + exact, localization results |
| Case-split construction | 1 | 39 | 20 min | Building new structures field-by-field |
| Algebraic identity + order | 1 | 63 | 25 min | Char p results, finite field proofs |
| Calc chain / helper | 2 | 99 | 45 min | Inequality chains, density arguments |
| Contradiction / counterexample | 2 | 102 | 60 min | "Not X" proofs, specific ring examples |
For introductory algebraic number theory (undergraduate/early graduate level):
| Content Type | Expected Mathlib Coverage |
|---|---|
| Standard definitions (DVR, PID, UFD, integral closure) | 95%+ |
| Named theorems (Ostrowski, Dedekind, Minkowski) | 80%+ |
| "Follows from" propositions | 70%+ |
| Worked examples (specific rings like Z_p, k[[t]]) | 50% |
| Counterexamples (Z[sqrt(5)] not IC) | 0% |
| Product/summation formulas | 30% |
Rule of thumb: coverage% * 0.7 + 30 = expected % of items provable in <10 min
For this project: 74% * 0.7 + 30 = 82%. Actual: 22/27 = 81%.
Based on observed throughput with auto-merge PRs:
| Batch Type | Items per PR | Agent Time | Rationale |
|---|---|---|---|
| Scaffolding (Stage 3.1) | 5-8 | 30 min | Parallel-safe; no proof work |
| Easy proofs (inferInstance, API) | 3-5 | 20 min | All trivial; group by topic |
| Medium proofs | 2-3 | 30 min | Related items share imports |
| Hard proofs | 1 | 45-90 min | Need full agent attention |
Total throughput: 2 agents can do ~12-15 items/hour on easy, ~4-6/hour on medium, ~1-2/hour on hard.
Given a textbook section with N formalizable items and C% Mathlib coverage:
Phase 1 (source prep): 1 + 0.15 * pages hours
Phase 2 (dep mapping): 1 + 0.05 * N hours
Phase 3 (formalization):
Easy items: N * C% * 0.15 hours
Medium items: N * (1-C%) * 0.6 * 0.4 hours
Hard items: N * (1-C%) * 0.4 * 1.2 hours
Total with 2 agents: (Phase 1 + Phase 2 + Phase 3) / 1.6
For Sutherland Lecture 1 (9 pages, 27 items, 74% coverage):
Actual: ~5.3h (faster due to high parallelism in Phase 3 and aggressive batching).
Useful for estimating file sizes and review effort:
| Difficulty | Lean Lines (with imports) | Proof Lines Only |
|---|---|---|
| Trivial (inferInstance) | 20-25 | 1 |
| Easy (API call) | 25-35 | 1-5 |
| Medium (short tactic) | 30-50 | 5-15 |
| Hard (calc chain) | 60-100 | 30-70 |
| Very Hard (counterexample) | 80-140 | 40-100 |
Total project: ~1,200 lines of Lean across 27 files.
Use when deciding whether a formalized proof should delegate to Mathlib, when refactoring standalone proofs into Mathlib wrappers, or when reviewing delegation refactors for correctness.
Use when running Stage 3.4 (Proof Polishing) to bring sorry-free proofs to Mathlib quality. Use when assessing whether a proof needs cleanup, when working through proof-polish GitHub issues, or when deciding which tactic to prefer for a given goal.
Use when a Lean proof has failed 2-3 attempts and needs to be sent to Aristotle for automated proving, or when checking the status of a previously submitted Aristotle job.
Use when filling sorry placeholders in Lean 4 proof files, when choosing between proof approaches for a FormalFrontier item, or when a proof attempt fails and you need an alternative strategy.
Use when searching for Mathlib declarations to fill a proof, when imports fail, when you need to find the right Mathlib module for a concept, or when exact? or apply? don't find what you need.
Transferable lessons from the first complete FormalFrontier book formalization (Sutherland Number Theory Lecture 1). Use when planning or executing future book formalizations.