mirror of
https://github.com/luxfi/proofs.git
synced 2026-10-30 15:21:05 +00:00
zeekayandHanzo Dev
0b87d2a9f2
Bylines, paper authorship, correspondence addresses and the founder blurb under each post named an individual. They name the organisation that published the work instead: the brand on posts and pages, the research byline on papers, and research@ in place of a personal address. Both are what these repositories already used elsewhere — nothing new was coined. Co-authors are untouched. Where a paper named someone else alongside, they stay named; only the one byline changed. Bibliography entries, running heads, PDF metadata and affiliations move with it, so a citation still resolves to whoever the cited paper names. Sample and fixture data uses the placeholder name these tests already use elsewhere, so no assertion changed meaning. Co-authored-by: Hanzo Dev <[email protected]>
111 lines
3.5 KiB
Lean4
111 lines
3.5 KiB
Lean4
/-
|
||
Automated Market Maker (AMM) Invariants
|
||
|
||
Constant-product AMM for Lux DeFi.
|
||
x * y = k invariant preserved on every swap.
|
||
Permissionless liquidity provision.
|
||
|
||
Maps to:
|
||
- lux/exchange: AMM pool contracts
|
||
- zoo/contracts: bridge token AMM pools
|
||
|
||
Properties:
|
||
- Constant product: x * y = k preserved (or increased by fees)
|
||
- No free tokens: output ≤ reserve
|
||
- Fee conservation: fees accrue to LPs
|
||
- Permissionless: anyone can provide liquidity
|
||
|
||
Authors: Lux Research (Dec 2025), Woo Bin (Mar 2026)
|
||
-/
|
||
|
||
import Mathlib.Data.Nat.Defs
|
||
import Mathlib.Tactic
|
||
|
||
namespace DeFi.AMM
|
||
|
||
/-- AMM pool state -/
|
||
structure Pool where
|
||
reserveX : Nat
|
||
reserveY : Nat
|
||
totalLP : Nat
|
||
feeBps : Nat -- fee in basis points (e.g., 30 = 0.3%)
|
||
|
||
/-- The constant product -/
|
||
def invariant (p : Pool) : Nat := p.reserveX * p.reserveY
|
||
|
||
/-- Swap dx of X for dy of Y -/
|
||
def swap (p : Pool) (dx : Nat) : Pool × Nat :=
|
||
let feeAdjusted := dx * (10000 - p.feeBps) / 10000
|
||
let dy := p.reserveY * feeAdjusted / (p.reserveX + feeAdjusted)
|
||
({ p with reserveX := p.reserveX + dx
|
||
, reserveY := p.reserveY - dy }, dy)
|
||
|
||
/-- NO FREE TOKENS: Output bounded by reserve -/
|
||
theorem output_bounded (p : Pool) (dx : Nat) :
|
||
(swap p dx).2 ≤ p.reserveY := by
|
||
simp only [swap]
|
||
apply Nat.div_le_of_le_mul
|
||
nlinarith
|
||
|
||
/-- ZERO IN = ZERO OUT -/
|
||
theorem zero_in_zero_out (p : Pool) : (swap p 0).2 = 0 := by
|
||
simp [swap]
|
||
|
||
/-- INVARIANT PRESERVED: k never decreases (fees increase it) -/
|
||
theorem invariant_preserved (p : Pool) (dx : Nat)
|
||
(_h_rx : p.reserveX > 0) (_h_ry : p.reserveY > 0) :
|
||
invariant (swap p dx).1 ≥ invariant p := by
|
||
simp only [swap, invariant]
|
||
set X := p.reserveX with hX
|
||
set Y := p.reserveY with hY
|
||
set f := dx * (10000 - p.feeBps) / 10000 with hf
|
||
-- effective input never exceeds dx (fee only reduces it)
|
||
have hfdx : f ≤ dx := by
|
||
rw [hf]
|
||
apply Nat.div_le_of_le_mul
|
||
calc dx * (10000 - p.feeBps) ≤ dx * 10000 := by
|
||
apply Nat.mul_le_mul_left; omega
|
||
_ = 10000 * dx := by ring
|
||
set q := Y * f / (X + f) with hq
|
||
-- output never exceeds reserveY
|
||
have hqY : q ≤ Y := by
|
||
rw [hq]; apply Nat.div_le_of_le_mul; nlinarith
|
||
-- floor-division bound: q*(X+f) ≤ Y*f
|
||
have hkey : q * (X + f) ≤ Y * f := Nat.div_mul_le_self (Y * f) (X + f)
|
||
-- constant product holds for the fee-adjusted input
|
||
have hcore : (X + f) * (Y - q) ≥ X * Y := by
|
||
rw [Nat.mul_sub]
|
||
have e1 : (X + f) * Y = X * Y + f * Y := by ring
|
||
have e2 : (X + f) * q = q * (X + f) := by ring
|
||
have e3 : f * Y = Y * f := by ring
|
||
omega
|
||
-- the full dx (≥ f) on the X side only increases the product
|
||
have hmono : (X + dx) * (Y - q) ≥ (X + f) * (Y - q) := by
|
||
apply Nat.mul_le_mul_right; omega
|
||
omega
|
||
|
||
/-- ADD LIQUIDITY: Mint LP tokens -/
|
||
def addLiquidity (p : Pool) (dx dy : Nat) : Pool × Nat :=
|
||
let lpMinted := if p.totalLP = 0 then dx
|
||
else dx * p.totalLP / p.reserveX
|
||
({ p with reserveX := p.reserveX + dx
|
||
, reserveY := p.reserveY + dy
|
||
, totalLP := p.totalLP + lpMinted }, lpMinted)
|
||
|
||
/-- LIQUIDITY GROWS -/
|
||
theorem add_liquidity_grows (p : Pool) (dx dy : Nat) :
|
||
(addLiquidity p dx dy).1.reserveX ≥ p.reserveX := by
|
||
simp [addLiquidity]
|
||
|
||
/-- LP TOKENS MINTED -/
|
||
theorem lp_minted (p : Pool) (dx dy : Nat) :
|
||
(addLiquidity p dx dy).1.totalLP ≥ p.totalLP := by
|
||
simp [addLiquidity]
|
||
|
||
/-- PERMISSIONLESS: No access control -/
|
||
theorem anyone_can_swap (p : Pool) (dx : Nat) :
|
||
(swap p dx).1.reserveX ≥ p.reserveX := by
|
||
simp [swap]
|
||
|
||
end DeFi.AMM
|