Lean Proof

lean-proof

⚠️ MANDATORY (hot-tier): the gates in §Behavioural rules and the Persist step in §Workflow are enforced. There is no CI in this repo — the gates are enforced by the proving agent itself plus @lean-proof-review downstream. Skipping Persist = incomplete, regardless of artefact quality (FSIA-R-11-09).

Routing

• USE FOR: writing Lean 4 proofs one tactic at a time with done-driven diagnostics; resolving errors in syntax → type → unsolved-goals → linter order; planning sorry placeholders so the hardest case is attacked first; cleaning a working proof into its minimal form; recovering from motive is not type correct via generalise-then-instantiate.
• DO NOT USE FOR: reviewing or auditing existing proofs (use @lean-proof-review); project-level QA orchestration (use @lean-quality-engine); minimising a failing proof into a bug-report MWE (use @lean-mwe); searching Mathlib or exploring tactics in the abstract (use @lean-research); writing or revising the theorem statement itself (use @lean-specification); repairing the Lean toolchain or lake env (use @lean-setup).
• TRIGGERS: prove, tactic, sorry, unsolved goals, motive is not type correct.

Behavioural rules (G-*)

• G-1 (MUST): The skill MUST write exactly one tactic, then read diagnostics, before writing the next. [Trace: AC-01]
• G-2 (MUST): The skill MUST use done to surface unsolved goals whenever an active proof has expected next steps. [Trace: AC-02]
• G-3 (MUST): Errors MUST be addressed in the order syntax → type → unsolved goals → linter. A lower-priority diagnostic MUST NOT be touched while a higher-priority one is open in the same file. [Trace: AC-03]
• G-4 (MUST): When an "unsolved goals" error appears on a by or => line alongside a tactic error on a later line, the tactic error MUST be fixed first. [Trace: AC-04]
• G-5 (MUST NOT): The skill MUST NOT write further tactics after any unresolved error. [Trace: AC-05]
• G-6 (MUST): When working a target theorem, the skill MUST go directly to that theorem and MAY leave dependent helper lemmas as sorry. [Trace: AC-06]
• G-7 (SHOULD): Within a case split, the skill SHOULD sorry the easy branches and prove the hardest branch first. [Trace: AC-07]
• G-8 (MUST): After a proof closes, the skill MUST attempt cleanup (combine rewrites, test whether simp subsumes earlier steps) before declaring the proof done. [Trace: AC-08]
• G-9 (SHOULD): On motive is not type correct or analogous dependent-type rewrite failure, the skill SHOULD apply the generalise-then-instantiate pattern (suffices ∀ s, … + convert) rather than fighting the rewrite. [Trace: AC-09]
• G-10 (MUST NOT): The skill MUST NOT declare a proof complete while any sorry or error diagnostic remains in the closed term. [Trace: AC-10]
• G-11 (MUST): The skill MUST persist (commit + state-tracker tick) before handing off to @lean-proof-review. [Trace: AC-11]

Workflow

1. Discover [discover] — read the target theorem, locate its spec, list open sorrys and existing diagnostics. Confirm lake env lean resolves (else hand back to @lean-setup).
2. Plan [discover] — pick the target theorem (not a helper lemma); within it, pick the hardest case. STOP if confidence < 80 % on overall strategy and hand off to @lean-research.
3. Execute [execute] — write one tactic at a time, gated on done/diagnostics. Honour the error-priority order (G-3, G-4). Apply generalise-then-instantiate (G-9) if a rewrite hits the motive trap.
4. Verify [validate] — re-read diagnostics on the closed proof; cleanup pass (G-8); confirm no sorry or error remains (G-10). Max 3 cleanup attempts before escalating.
5. Persist [persist] (MANDATORY, FSIA-R-11-09) — commit the proof, update the state tracker, tick tasks.md, and (if a recurring pattern surfaced) emit a fleeting note via @lean-zettelkasten. Skipping Persist = incomplete.

Recovery & STOP

• Same error class fails ×3 attempts → STOP, escalate to human or hand off to @lean-mwe to isolate the failure.
• Scope drift (edit outside the target theorem, its helper lemmas, or its imports) → immediate STOP, re-anchor on the target.
• Confidence < 80 % on tactic strategy → STOP and hand off to @lean-research.
• Context degradation signals (≥2 from AGENTS.md — repeated tactic, ignored prior diagnostic, persona drift) → recommend a fresh session and re-anchor the theorem.

Handoffs

• Predecessors / successors: see FM handoffs (grammar from ADR-0080). Typical inbound: @lean-setup confirms toolchain; @lean-specification supplies the theorem; @lean-research supplies tactic strategy. Typical outbound: @lean-proof-review audits; @lean-mwe if stuck; @lean-zettelkasten records the pattern.
• Source spec: specs/lean/proof/requirements.md — every G-rule traces to an AC there.
• Related ADRs: ADR-0076 (skill-as-contract), ADR-0080 (handoff DAG), ADR-0079 (tier loading).

Common failure modes

AI agents commonly: write 3-5 tactics before reading diagnostics; chase a linter warning while an unsolved-goals error is open; fill helper-lemma sorrys before touching the target theorem; declare success while a sorry remains; fight motive is not type correct with more rw instead of generalising. Full registry: GUARDRAILS.md §Agent failure taxonomy.

See also

• This SKILL.md is the canonical v2 proof-writing contract for this package.
• ../../../templates/Template_ProofStrategy.md — proof methodology cheat sheet.
• ../../../references/lean4-proof-strategy.md — one-step-at-a-time, error priority, hardest case first.
• ../../../references/lean4-tactic-hierarchy.md — tactic priority table.