I reviewed the paper and wrote an EDSL in Lean based on it. My implementation demonstrates how first-order logic compiles directly to polynomial constraints, providing the missing link between high-level cryptocurrency invariants and low-level zkSNARK circuits. By mapping logical formulas to R1CS, developers can specify complex financial systems declaratively while achieving cryptographic verifiability. The core insight: monetary invariants are logical statements. "Total supply remains constant" becomes ∀tx. sum(inputs) = sum(outputs). "No double spending" becomes ∀coin. spent(coin) ⇒ ¬spendable(coin). These specifications compile automatically to polynomial constraints, eliminating manual circuit construction—the primary barrier to zkEVM adoption. The polynomial semantics handles the full complexity of financial logic. Negation through sign gadgets enables expressing invalidity conditions—quantifiers over finite domains model transaction batches. The sparse polynomial representation keeps verification costs manageable even for complex state transitions. Most critically, conjunction's additive compilation means combining multiple invariants doesn't compound proof complexity, essential for composable financial protocols. The R1CS output integrates directly with existing zkSNARK toolchains (Groth16, PLONK, Halo2). A balance check requiring hundreds of hand-written constraints reduces to a single logical formula. The compiler handles witness generation, wire allocation, and constraint optimization automatically. Here's the code: -- ArithFOL.lean — Lean 4 EDSL for Gabbay-style arithmetised first-order logic -- Author: Charles Hoskinson License: MIT -- -- * Full polynomial semantics (truth ≙ 0) + complete R1CS compiler. -- * Extra Poly utilities (pow / evaluate / degree / leadingCoeff / isZero / isConst). -- * Pretty R1CS printer and runnable examples at bottom. -- --------------------------------------------------------------------------- import Std.Data.RBMap.Lemmas open Std (RBMap) /-! ### Signature --------------------------------------------------------- -/ structure FuncSymbol where name : String := "" ; arity : Nat := 0 structure PredSymbol where name : String := "" ; arity : Nat := 0 /-! ### Syntax: de-Bruijn terms & formulas -------------------------------- -/ inductive Term : Nat → Type | var {n} : Fin n → Term n | const {n} : Rat → Term n | indet {n} : Term n -- distinguished X | add {n} : Term n → Term n → Term n | mul {n} : Term n → Term n → Term n | neg {n} : Term n → Term n | func {n} : (f : FuncSymbol) → (Fin f.arity → Term n) → Term n inductive Formula : Nat → Type | top {n} : Formula n | bot {n} : Formula n | eq {n} (t u : Term n) : Formula n | pred {n} (P : PredSymbol) (args : Fin P.arity → Term n) : Formula n | and {n} (φ ψ : Formula n) : Formula n | or {n} (φ ψ : Formula n) : Formula n | forall {n} (φ : Formula (n+1)) : Formula n | exists {n} (φ : Formula (n+1)) : Formula n | notG {n} (φ : Formula n) : Formula n -- sign-gadget ¬φ /-! ### Polynomial datatype ----------------------------------------------- -/ structure Poly where coeffs : RBMap Nat Rat compare deriving Repr, DecidableEq namespace Poly def zero : Poly := ⟨RBMap.empty⟩ def const (q : Rat) : Poly := if q = 0 then zero else ⟨RBMap.ofList [(0,q)]⟩ def X : Poly := ⟨RBMap.ofList [(1,1)]⟩ @[inline] def clean (m : RBMap Nat Rat compare) := m.fold (init:=RBMap.empty) (fun acc k v => if v=0 then acc else acc.insert k v) @[inline] def add (p q : Poly) : Poly := ⟨clean <| p.coeffs.fold (init:=q.coeffs) (fun m k v => m.insert k (v + m.findD k 0))⟩ @[inline] def neg (p : Poly) : Poly := ⟨p.coeffs.fold (init:=RBMap.empty) (fun m k v => m.insert k (-v))⟩ @[inline] def mul (p q : Poly) : Poly := ⟨clean <| p.coeffs.fold (init:=RBMap.empty) (fun acc kp vp => q.coeffs.fold (init:=acc) (fun acc2 kq vq => let k := kp+kq let c := acc2.findD k 0 + vp*vq if c = 0 then acc2.erase k else acc2.insert k c))⟩ @[inline] def square (p : Poly) : Poly := p.mul p @[inline] def pow (p : Poly) : Nat → Poly | 0 => const 1 | 1 => p | n+1 => let q := pow p (n/2) let q2 := q.mul q if n % 2 = 0 then q2 else q2.mul p @[inline] def ratPow : Rat → Nat → Rat | _, 0 => 1 | r, 1 => r | r, n+1 => let half := ratPow r (n/2) let sq := half*half if n%2=0 then sq else sq*r @[inline] def evaluate (p : Poly) (x : Rat) : Rat := if x = 0 then p.coeffs.findD 0 0 else if x = 1 then p.coeffs.fold (init:=0) (fun acc _ v => acc+v) else p.coeffs.fold (init:=0) (fun acc k v => acc + v * ratPow x k) @[inline] def degree (p : Poly) : Option Nat := p.coeffs.fold (init:=none) (fun o k _ => match o with | none => some k | some d => some (max d k)) @[inline] def leadingCoeff (p : Poly) : Option Rat := match p.degree with | none => none | some d => p.coeffs.find? d @[inline] def isZero (p : Poly) : Bool := p.coeffs.isEmpty @[inline] def isConst (p : Poly) : Option Rat := if p.isZero then some 0 else if p.coeffs.size = 1 ∧ p.coeffs.min?.map (·.1) = some 0 then some (p.coeffs.findD 0 0) else none @[inline] def compose (p q : Poly) : Poly := p.coeffs.fold (init:=Poly.zero) (fun acc k c => acc.add (Poly.const c).mul (q.pow k)) end Poly /-! ### Environment -------------------------------------------------------- -/ structure Env (n : Nat) where varVal : Fin n → Rat domain : List Rat namespace Env def extend {n} (e : Env n) (a : Rat) : Env (n+1) := { varVal := fun | ⟨0,_⟩ => a | ⟨i+1,h⟩ => e.varVal ⟨i, Nat.lt_of_succ_lt_succ h⟩, domain := e.domain } end Env /-! ### Interpretation ----------------------------------------------------- -/ mutual def interpTerm {n} (ρ : Env n) : Term n → Poly | .var i => Poly.const (ρ.varVal i) | .const q => Poly.const q | .indet => Poly.X | .add s t => (interpTerm ρ s).add (interpTerm ρ t) | .mul s t => (interpTerm ρ s).mul (interpTerm ρ t) | .neg t => Poly.neg (interpTerm ρ t) | .func _ _ => Poly.zero def liftTerm {k} : Term k → Term (k+1) | .var i => .var ⟨i.val+1, Nat.succ_lt_succ i.isLt⟩ | .const q => .const q | .indet => .indet | .add s t => .add (liftTerm s) (liftTerm t) | .mul s t => .mul (liftTerm s) (liftTerm t) | .neg t => .neg (liftTerm t) | .func f as => .func f (fun i => liftTerm (as i)) def liftFormula {k} : Formula k → Formula (k+1) | .top => .top | .bot => .bot | .eq t u => .eq (liftTerm t) (liftTerm u) | .pred P as => .pred P (fun i => liftTerm (as i)) | .and φ ψ => .and (liftFormula φ) (liftFormula ψ) | .or φ ψ => .or (liftFormula φ) (liftFormula ψ) | .forall φ => .forall (liftFormula φ) | .exists φ => .exists (liftFormula φ) | .notG φ => .notG (liftFormula φ) def interpFormula {n} (ρ : Env n) : Formula n → Poly | .top => Poly.zero | .bot => Poly.const 1 | .eq t u => (interpTerm ρ t).add (Poly.neg (interpTerm ρ u)) |> Poly.square | .pred _ _ => Poly.const 1 | .and φ ψ => let p := interpFormula ρ φ let q := interpFormula ρ ψ p.square.add q.square | .or φ ψ => (interpFormula ρ φ).mul (interpFormula ρ ψ) | .forall φ => ρ.domain.foldl (init:=Poly.zero) fun acc a => acc.add ((interpFormula (ρ.extend a) φ).square) | .exists φ => match ρ.domain with | [] => Poly.const 1 | a :: _ => interpFormula (ρ.extend a) φ -- naive witness | .notG φ => let φ' := liftFormula φ let b : Term (n+1) := .var ⟨0, Nat.zero_lt_succ n⟩ let boolB := Formula.eq (.mul b (.add b (.const (-1)))) (.const 0) let conj0 := Formula.and (Formula.eq b (.const 0)) φ' let conj1 := Formula.and (Formula.eq b (.const 1)) Formula.top let gadget := Formula.and boolB (Formula.or conj0 conj1) let p0 := interpFormula (ρ.extend 0) gadget let p1 := interpFormula (ρ.extend 1) gadget p0.mul p1 end /-! ### R1CS infrastructure ---------------------------------------------- -/ structure Wire where id : Nat deriving Repr, DecidableEq, Ord instance : Ord Wire := ⟨fun a b => compare a.id b.id⟩ namespace Wire def X : Wire := ⟨0⟩ def one : Wire := ⟨1⟩ def zero : Wire := ⟨2⟩ end Wire structure Constraint where A B C : RBMap Wire Rat compare deriving Repr structure Ctx where next : Nat := 3 -- after reserved rows : List Constraint := [] powers : RBMap Nat Wire compare := {} polyWires : RBMap Poly Wire compare := {} namespace Ctx def fresh : StateM Ctx Wire := do let s ← get; set {s with next := s.next+1}; pure ⟨s.next⟩ def push (row : Constraint) : StateM Ctx Unit := modify fun s => {s with rows := row :: s.rows} -- wires for X^k, cached partial def getPower (k : Nat) : StateM Ctx Wire := do let s ← get; match s.powers.find? k with | some w => pure w | none => if k=0 then pure Wire.one else if k=1 then pure Wire.X else let wPrev ← getPower (k-1) let wk ← fresh push {A := RBMap.singleton Wire.X 1, B := RBMap.singleton wPrev 1, C := RBMap.singleton wk 1} modify fun s => {s with powers := s.powers.insert k wk} pure wk -- wires for arbitrary polynomial value partial def getPolyWire (p : Poly) : StateM Ctx Wire := do let s ← get; match s.polyWires.find? p with | some w => pure w | none => match p.isConst with | some 0 => pure Wire.zero | some 1 => pure Wire.one | _ => let w ← fresh compilePolyDef p w modify fun s => {s with polyWires := s.polyWires.insert p w} pure w -- emit rows for Σ c_k X^k = w partial def compilePolyDef (p : Poly) (w : Wire) : StateM Ctx Unit := do let mut lin := RBMap.singleton w (-1) for ⟨k,c⟩ in p.coeffs.toList do let wk ← getPower k lin := lin.insert wk (c + lin.findD wk 0) push {A := lin, B := RBMap.singleton Wire.one 1, C := RBMap.empty} end Ctx /-- compile equality p=0 into rows -/ private def compilePolyEq (p : Poly) : StateM Ctx Unit := do if p.isZero then return () let w ← Ctx.getPolyWire p Ctx.push {A := RBMap.singleton w 1, B := RBMap.singleton Wire.one 1, C := RBMap.empty} /-- substitute concrete a into a 1-bound formula (for ∀/∃ instantiation) -/ def substFormula {n} (φ : Formula (n+1)) (a : Rat) : Formula n := Formula.eq (Term.const ((interpFormula ⟨fun _ => a, []⟩ φ).evaluate 0)) (Term.const 0) /-! full compilation ------------------------------------------------------ -/ open Ctx mutual partial def compileFormula (φ : Formula 0) (env : Env 0) : StateM Ctx Unit := match φ with | .top => return () | .bot => compilePolyEq (Poly.const 1) | .eq t u => let diff := (interpTerm env t).add (Poly.neg (interpTerm env u)) compilePolyEq diff | .pred _ _ => compilePolyEq (Poly.const 1) | .and φ ψ => compileFormula φ env >> compileFormula ψ env | .or φ ψ => -- enforce wφ · wψ = 0 let pφ := interpFormula env φ; let pψ := interpFormula env ψ let wφ ← getPolyWire pφ; let wψ ← getPolyWire pψ push {A := RBMap.singleton wφ 1, B := RBMap.singleton wψ 1, C := RBMap.empty} | .forall φ => -- all instantiations must be 0 for a in env.domain do compileFormula (substFormula φ a) env | .exists φ => -- naive witness = first element match env.domain with | [] => compilePolyEq (Poly.const 1) | a :: _ => compileFormula (substFormula φ a) env | .notG φ => compilePolyEq (interpFormula env (.notG φ)) end /-- Public entry: compile closed formula to R1CS rows. -/ def compileFormulaToR1CS (φ : Formula 0) (domain : List Rat := []) : List Constraint := let env : Env 0 := ⟨fun _ => 0, domain⟩ ((compileFormula φ env).run {}).2.rows.reverse /-! ### Pretty printer & validation -------------------------------------- -/ namespace Constraint def termStr (w : Wire) (c : Rat) : String := let wS := if w = Wire.X then "X" else if w = Wire.one then "1" else if w = Wire.zero then "0" else s!"w{w.id}" if c = 1 then wS else if c = -1 then s!"-{wS}" else s!"{c}·{wS}" def sideStr (m : RBMap Wire Rat compare) : String := if m.isEmpty then "0" else String.intercalate " + " (m.toList.map (fun (w,c) => termStr w c)) def toString (c : Constraint) : String := s!"({sideStr c.A}) × ({sideStr c.B}) = {sideStr c.C}" end Constraint def validateR1CS (rows : List Constraint) (w : Wire → Rat) : Bool := rows.all fun c => let eval (m : RBMap Wire Rat compare) := m.fold (init:=0) fun acc i coeff => acc + coeff * w i eval c.A * eval c.B = eval c.C ------------------- Give my regards to Gabbay> I enjoyed reading the paper and wish you guys well with your project