Aristotle Lean

Trit: -1 (MINUS) Domain: Formal Verification / Theorem Proving Provider: Harmonic (harmonic.fun)

Overview

Aristotle is an IMO Gold Medal level Lean4 theorem prover that fills sorry holes in proofs, auto-generates counterexamples for false statements, and integrates with Mathlib and lake dependencies.

API Configuration

Endpoint: aristotle.harmonic.fun
Auth: Auth0-based (requires signup/login at harmonic.fun)

Capabilities

1. Sorry Hole Filling: Completes incomplete Lean4 proofs
2. Dual Input: Accepts English descriptions or Lean4 code
3. Counterexample Generation: Auto-generates counterexamples for false statements
4. Project Integration: Works with project theorems, lake dependencies, Mathlib
5. PROVIDED SOLUTION Tag: Use comment tag to mark solution regions

Benchmarks

Usage Pattern

-- English prompt in comment
-- "Prove that the sum of two even numbers is even"

theorem sum_even (a b : ℕ) (ha : Even a) (hb : Even b) : Even (a + b) := by
  sorry  -- Aristotle fills this

-- PROVIDED SOLUTION: explicit solution marker
theorem my_theorem : P → Q := by
  -- PROVIDED SOLUTION
  sorry

Integration with GF(3)

This skill participates in triadic composition:

• Trit -1 (MINUS): Verification/validation/analysis
• Conservation: Σ trits ≡ 0 (mod 3) across skill triplets

Related Skills

• lean4-metaprogramming (trit +1)
• mathlib-tactics (trit 0)
• proof-assistant (trit -1)
• formal-verification (trit -1)

Skill Name: aristotle-lean Type: Formal Verification / Theorem Proving Trit: -1 (MINUS) GF(3): Conserved in triplet composition

Non-Backtracking Geodesic Qualification

Condition: μ(n) ≠ 0 (Möbius squarefree)

This skill is qualified for non-backtracking geodesic traversal:

1. Prime Path: No state revisited in skill invocation chain
2. Möbius Filter: Composite paths (backtracking) cancel via μ-inversion
3. GF(3) Conservation: Trit sum ≡ 0 (mod 3) across skill triplets
4. Spectral Gap: Ramanujan bound λ₂ ≤ 2√(k-1) for k-regular expansion

Geodesic Invariant:
  ∀ path P: backtrack(P) = ∅ ⟹ μ(|P|) ≠ 0
  
Möbius Inversion:
  f(n) = Σ_{d|n} g(d) ⟹ g(n) = Σ_{d|n} μ(n/d) f(d)

SDF Interleaving

This skill connects to Software Design for Flexibility (Hanson & Sussman, 2021):

Primary Chapter: 4. Pattern Matching

Concepts: unification, match, segment variables, pattern

GF(3) Balanced Triad

aristotle-lean (−) + SDF.Ch4 (+) + [balancer] (○) = 0

Skill Trit: -1 (MINUS - verification)

Connection Pattern

Pattern matching extracts structure. This skill recognizes and transforms patterns.