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r-irber-irbe/proof-skills/skills/lean-convergence-verification/SKILL.md

lean-convergence-verification

Skill3 starsChanged 24 days agoApache-2.0View on GitHub

USE FOR: discrete Lyapunov energy contraction, loop termination bounds, GPU power iteration limits. DO NOT USE FOR: continuous Lyapunov stability via differential equations (@lean-math-dynamical). TRIGGERS: lyapunov, convergence, termination bound, gpu bound.

The file

We split this skill into the rules it gives an agent

4 rules. About 0 are common, 1 is distinctive, and 3 are very specific to this project.

  • ≈0

    Common · estimate

    Many files like this have a rule of this kind.

  • ≈1

    Distinctive · estimate

    This author's own choices. Worth a look.

  • 3

    Very specific

    Tied to their project's names, files or tools. Probably skip.

  • Common
  • Distinctive
  • Very specific
  1. Very specific:

    Run lean_diagnostics ({"filePath": "..."}) to trigger the lean-analyzer static abstract interpretation engine (Zone DBM decreasing_by …

    line 42
  2. Distinctive:

    STOP if estimated belief in inductive convergence hypothesis is below 0.90 -- ask user via HITL channel.

    line 53
  3. Very specific:

    STOP if lean-analyzer reports NatSubTruncation or NatDivModByZero on the ceiling bound `(v0 + delta -

    line 54
  4. Very specific:

    STOP if omega fails on non-linear division -- check that Nat.div_add_mod has been explicitly introduced into the local context.

    line 59

≈ “Common” is an estimate: at least 1 in 5 of the 350,495 skill files we read have a rule of the same kind, not the same words. “Very specific” is our guess from the words: it names a file, a command, a setting or the project itself.

Keep what fits and drop the rest. Your version, ready for your agents.
Files
SKILL.md

Lean 4 Discrete Lyapunov Convergence & Loop Bound Verification

Guide to proving finite convergence and loop termination bounds for iterative algorithms, numerical solvers, and GPU kernels in Lean 4.

Routing

  • USE FOR: discrete Lyapunov energy contraction, loop termination bounds, GPU power iteration limits, Euclidean integer ceiling division proofs via Nat.div_add_mod and Nat.mod_lt, and finite convergence verification in Lean 4.
  • DO NOT USE FOR: continuous Lyapunov stability via differential equations (delegate to @lean-math-dynamical); simple for-loop syntax verification without energy functions (delegate to @lean-build).
  • TRIGGERS: lyapunov, convergence, termination bound, gpu bound, power iteration, ceiling division, div_add_mod, residue decay.

Workflow

  1. Define state space and non-negative integer Lyapunov energy function lyapunov : State -> Nat.
  2. Formulate the strict step decay condition forall t < n, lyapunov (traj (t + 1)) + delta <= lyapunov (traj t) with delta > 0.
  3. Run lean_diagnostics ({"filePath": "..."}) to trigger the lean-analyzer static abstract interpretation engine (Zone DBM decreasing_by synthesis, Octagon conservation ±x_i ± x_j <= c, Bourdoncle WTO loop widening, and trace partitioning) and verify that ceiling division (v0 + delta - 1) / delta and step-decay expressions are free of NatSubTruncation and NatDivModByZero defects (delta > 0 and v0 + delta >= 1). Run

Recovery & STOP

  • STOP if estimated belief in inductive convergence hypothesis is below 0.90 -- ask user via HITL channel.
  • STOP if lean-analyzer reports NatSubTruncation or NatDivModByZero on the ceiling bound `(v0 + delta -
    1. / delta-- add the missing0 < deltahypothesis before attemptingomega`.
  • STOP if omega fails on non-linear division -- check that Nat.div_add_mod has been explicitly introduced into the local context.

Handoffs

  • Predecessors: agent:gateway, skill:lean-math-discrete (discrete structures), skill:lean-proof (tactics).
  • Successors: skill:lean-proof (inductive step discharge), skill:lean-proof-review (lockdown audit).

Detailed reference

Full encyclopedic reference, decision trees, and lookup tables live in references/lean-convergence-verification-handbook.md .

See also

  • ../../references/lean-convergence-verification-handbook.md

    Complete lean-convergence-verification Handbook
  • ../lean-gateway/SKILL.md -- Top-level skill router

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