Writing Lean Proofs

---
name: writing-lean-proofs
description: 'Use when asked to design, write, review, refactor, lint, or performance-diagnose Lean 4 proofs, libraries, or tactic extensions under Mathlib conventions. Not for non-Lean code or changes outside Lean source, library API, proof structure, and linter configuration.'
---

# Writing Lean proofs

## Contract

| Field | Bound contract |
|---|---|
| Trigger | The task is to design, write, review, refactor, lint, or performance-diagnose Lean 4 proofs, libraries, or tactic extensions under Mathlib conventions. |
| Authority | Reversible local writes to Lean source files, library API, proof structure, project linter configuration, and scoped mechanical Lean checks. Rollback via version control restore of changed files. |
| Side effect | Local writes to Lean source, library API, proof structure, project linter configuration, and scoped mechanical Lean checks. No remote mutation, credential change, paid action, or deployment. |
| Done | The requested Lean declarations have stable statements and structured proofs, compile under the project toolchain, and satisfy the selected axiom and linter policy. |

## Inputs

A Lean 4 project with a working `lakefile.lean` and toolchain (`lean-toolchain`) is required, along with target theorem statements, definitions to formalize, or proof obligations to discharge. Optional inputs include project-specific linter configuration, axiom policy (default: `[propext, Classical.choice, Quot.sound]`), `maxHeartbeats` budget, and Mathlib dependency.

## Procedure

1. **Design definitions and their API first.** Prefer total functions with junk values over subtypes or `Option` in signatures. Bundle morphisms with `FunLike`, subobjects with `SetLike`. Pick the canonical simp-normal form for every concept. Write `ext`, `@[simp]`, coercion, and injectivity lemmas in the same file immediately after the definition. Never use `unfold` or `show ... from rfl` downstream. Done when: every definition has its API lemmas co-located and the simp-normal form is chosen.
2. **Build a sorry skeleton.** State the target theorem and every lemma it needs with `:= sorry`. Verify the file compiles. Each `sorry` is an independent work unit. Inside a proof, lay out `have`/`suffices`/`calc` skeleton with `sorry` justifications and verify Lean accepts the structure before filling. Done when: the skeleton compiles and every `sorry` is an identified work unit.
3. **Fill goals one focused goal at a time.** Every subgoal gets a focusing dot `·` with an indented block. Open each block with a redundant `show` stating its goal; use `change` instead if `show` would alter the goal. Chained rewrites of (in)equalities become `calc` blocks with relations aligned vertically. Use `have` for forward stepping stones, `suffices` for backward reduction. Annotate goal state as a comment before non-obvious tactics; in headless workflows insert `trace_state` or deliberate `done`, run `lake env lean Path/To/File.lean`, and copy reported hypotheses and target. Strip probes after the proof works. Done when: every `sorry` is replaced with a structured proof and probes are stripped.
4. **Verify mechanically.** Run `lake build`: a green build is the floor, not the gate, because `sorry` exits 0. Gate unproved obligations by asking the kernel: `#print axioms myTheorem` for spot checks; for CI, collect axioms per declaration with `Lean.collectAxioms` and assert the whole expected footprint (`[propext, Classical.choice, Quot.sound]` unless deliberately widened) so stray `sorry` or new trust assumptions like `native_decide` fail loudly. Never grep for `sorry`: it matches comments and misses unproved helpers. Done when: `lake build` passes and the axiom footprint matches the declared policy.
5. **Apply the extraction ladder.** Before extracting, state the fragment type in a scratch `example`, run `exact?` and `apply?` on the bare goal, then try type-pattern and source search. Level 0: sub-argument repeats within one proof → local `have`. Level 1: statement is independently interesting or extraction sheds hypotheses → standalone lemma. Level 2: proof reads as long and unwieldy → split; if a fragment has a clean statement, it wanted to be a lemma. Done when: every extractable fragment is at the right level of the ladder.
6. **Run project linters.** Self-contained proof: `linter.auxLemma`, `linter.style.maxHeartbeats`, `linter.style.multiGoal`, `linter.style.setOption`, `linter.style.show`. Reusable library: also `linter.flexible`, `linter.style.missingEnd`, `linter.style.openClassical`, `unused*InType`. Treat `nativeDecide` as a trust-policy choice. Run Batteries' declaration-level `#lint` checks including `simpNF` separately. Verify every option against pinned Mathlib source with a known-trigger fixture. No warning gates anything unless warnings fail the build. Done when: linter output is clean under the selected profile.
7. **Write a custom linter for every project-specific convention.** A declaration-level `@[env_linter]` is one structure. It is the only mechanism that reliably catches missing attributes across declarations. Include vacuity anchors, prove-it-can-fail fixtures, and allowlists. Done when: the custom linter is written with vacuity anchors and failure fixtures.
8. **Diagnose performance.** Measure per-declaration cost with `#count_heartbeats` before adjusting `maxHeartbeats`. Every `maxHeartbeats` override is an unproven claim. Conditional simp lemma fires shallow but not deep → raise `maxDischargeDepth` (default 2). Re-derive every `simp only` list with `simp?` at its own site. Done when: performance is measured and every `maxHeartbeats` override is justified by measurement.

## Failure and recovery
On compilation failure, fix the source error and rebuild; do not widen scope. On sorry leakage, replace with structured proof or gate with `collectAxioms`/`#print axioms`; a build that exits 0 with sorries present is not done. On a linter violation, fix the code or suppress with explicit justification; no blanket `#nolint`. On a performance regression, measure with `#count_heartbeats`, restructure the definition or decompose the goal; do not raise `maxHeartbeats` without measurement. On a scope violation, stop and roll back to the last clean state; do not widen authority. On a non-convergent proof, report the stuck goal, the tactics tried, and the hypotheses; do not invent evidence or weaken the statement.

## Output
Lean source files with stable declarations, structured proofs, and no ungated `sorry`; axiom footprint matching the declared policy; linter output clean under the selected profile; for custom tactics, failure-surface tests and structured tracing.