plurigrid Lean Proof Walk
Generate formal Lean 4 proof state chains using GF(3)-balanced random walks.
Triad Structure
| Agent | Trit | Role | Action |
|---|---|---|---|
| Generator | +1 | Create | Propose next proof state |
| Coordinator | 0 | Transport | Formalize transition, derive seed |
| Validator | -1 | Verify | Check soundness, GF(3) conservation |
Invariant: trit(G) + trit(C) + trit(V) = (+1) + 0 + (-1) = 0
State Chain Format
State N: Γ ⊢ G
where:
Γ = context (hypotheses: x : τ, h : P)
⊢ = turnstile (entailment)
G = goal (proposition to prove)
Example Chain
State 0: a : ℤ, b : ℤ, h : a + b = 0 ⊢ b = -a
State 1: a : ℤ, b : ℤ, h : a + b = 0 ⊢ a + b - a = 0 - a
State 2: a : ℤ, b : ℤ, h : a + b = 0 ⊢ b = -a
State 3: No Goals
Protocol
1. Initialize
seed := 0x42D (or user-provided)
state := State 0 with full context and goal
triad := spawn 3 parallel agents with trits {-1, 0, +1}
2. Walk Step (repeat until No Goals)
Generator (+1): propose tactic τ, predict State n+1
Coordinator (0): formalize Γₙ ⊢ Gₙ → Γₙ₊₁ ⊢ Gₙ₊₁
Validator (-1): verify transition sound, Σ trits = 0
Commit: seed_{n+1} = hash(seed_n ⊕ state_n)
3. Terminate
State m = "No Goals" → QED
Emit: formal statement, informal proof, detailed proof, state chain
Invocation
/lean-proof-walk "∀ a b : ℤ, a + b = b + a"
/lean-proof-walk --seed=1069 --theorem="commutativity of addition"
Output Structure
- Formal Statement (Lean 4 syntax)
- Informal Proof (1-2 sentences)
- Detailed Informal Proof (numbered steps)
- Chain of States (with interleaved explanations)
Tactics Vocabulary
| Tactic | State Transition |
|---|---|
intro x |
Γ ⊢ ∀x.P → Γ, x:τ ⊢ P |
apply h |
Γ, h:P→Q ⊢ Q → Γ ⊢ P |
exact h |
Γ, h:P ⊢ P → No Goals |
rfl |
Γ ⊢ a = a → No Goals |
simp |
Γ ⊢ P → Γ ⊢ P' (simplified) |
ring |
Γ ⊢ polynomial_eq → No Goals |
omega |
Γ ⊢ linear_arith → No Goals |
cases h |
Γ, h:P∨Q ⊢ R → Γ, h:P ⊢ R and Γ, h:Q ⊢ R |
induction n |
Γ ⊢ P(n) → base case + inductive step |
GF(3) Seed Derivation
γ = 0x9E3779B97F4A7C15 # golden ratio constant
def next_seed(seed, state_hash, trit):
return (seed ^ (state_hash * γ) ^ trit) & ((1 << 64) - 1)
Bundled Triad Skills
lean-proof-walk (0) ⊗ bdd-mathematical-verification (+1) ⊗ chromatic-walk (-1) = 0 ✓
Quick Reference
⟦State n⟧ = (Γₙ, Gₙ)
⟦S → S'⟧ = tactic application
⟦No Goals⟧ = proof complete
⟦Σ trits⟧ ≡ 0 (mod 3) always