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zeekayandHanzo Dev
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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]>
118 lines
4.1 KiB
Lean4
118 lines
4.1 KiB
Lean4
/-
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Automated Market Maker (AMM) Invariants
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Constant-product AMM for Lux DeFi.
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x * y = k invariant preserved on every swap.
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Permissionless liquidity provision.
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Maps to:
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- lux/exchange: AMM pool contracts
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- zoo/contracts: bridge token AMM pools
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Properties:
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- Constant product: x * y = k preserved (or increased by fees)
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- No free tokens: output ≤ reserve
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- Fee conservation: fees accrue to LPs
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- Permissionless: anyone can provide liquidity
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Authors: Zoo Labs Foundation Research (Dec 2025), Woo Bin (Mar 2026)
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-/
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import Mathlib.Data.Nat.Defs
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import Mathlib.Tactic
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namespace DeFi.AMM
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/-- AMM pool state -/
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structure Pool where
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reserveX : Nat
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reserveY : Nat
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totalLP : Nat
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feeBps : Nat -- fee in basis points (e.g., 30 = 0.3%)
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/-- The constant product -/
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def invariant (p : Pool) : Nat := p.reserveX * p.reserveY
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/-- Swap dx of X for dy of Y -/
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def swap (p : Pool) (dx : Nat) : Pool × Nat :=
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let feeAdjusted := dx * (10000 - p.feeBps) / 10000
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let dy := p.reserveY * feeAdjusted / (p.reserveX + feeAdjusted)
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({ p with reserveX := p.reserveX + dx
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, reserveY := p.reserveY - dy }, dy)
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/-- NO FREE TOKENS: Output bounded by reserve -/
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theorem output_bounded (p : Pool) (dx : Nat) :
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(swap p dx).2 ≤ p.reserveY := by
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simp only [swap]
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apply Nat.div_le_of_le_mul
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set fA := dx * (10000 - p.feeBps) / 10000
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calc p.reserveY * fA ≤ p.reserveY * fA + p.reserveX * p.reserveY := Nat.le_add_right _ _
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_ = (p.reserveX + fA) * p.reserveY := by ring
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/-- ZERO IN = ZERO OUT -/
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theorem zero_in_zero_out (p : Pool) : (swap p 0).2 = 0 := by
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simp [swap]
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/-- INVARIANT PRESERVED: k never decreases (fees increase it) -/
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theorem invariant_preserved (p : Pool) (dx : Nat)
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(_h_rx : p.reserveX > 0) (_h_ry : p.reserveY > 0) :
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invariant (swap p dx).1 ≥ invariant p := by
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simp only [swap, invariant]
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-- Abbreviations: fee-adjusted input fA, denominator D, output dy
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set fA := dx * (10000 - p.feeBps) / 10000 with hfA
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set D := p.reserveX + fA with hD
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set dy := p.reserveY * fA / D with hdy
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-- fA ≤ dx (fee only shrinks the input)
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have hfA_le_dx : fA ≤ dx := by
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rw [hfA]
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apply Nat.div_le_of_le_mul
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calc dx * (10000 - p.feeBps) ≤ dx * 10000 := Nat.mul_le_mul_left dx (Nat.sub_le _ _)
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_ = 10000 * dx := Nat.mul_comm _ _
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-- dy ≤ reserveY (output never exceeds the reserve)
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have hfA_le_D : fA ≤ D := by rw [hD]; exact Nat.le_add_left _ _
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have hdy_le_Y : dy ≤ p.reserveY := by
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rw [hdy]
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apply Nat.div_le_of_le_mul
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calc p.reserveY * fA ≤ p.reserveY * D := Nat.mul_le_mul_left p.reserveY hfA_le_D
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_ = D * p.reserveY := Nat.mul_comm _ _
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-- floor-division fact: D * dy ≤ reserveY * fA
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have hfloor : D * dy ≤ p.reserveY * fA := by
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rw [hdy, Nat.mul_comm]
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exact Nat.div_mul_le_self _ _
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-- core inequality on the (R+fA) factor: D*(reserveY - dy) ≥ reserveX*reserveY
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have hcore : D * (p.reserveY - dy) ≥ p.reserveX * p.reserveY := by
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rw [Nat.mul_sub]
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have hDY : D * p.reserveY = p.reserveX * p.reserveY + p.reserveY * fA := by
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rw [hD]; ring
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omega
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-- monotonicity in the first factor: reserveX + dx ≥ D = reserveX + fA
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have hmono : (p.reserveX + dx) * (p.reserveY - dy) ≥ D * (p.reserveY - dy) := by
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apply Nat.mul_le_mul_right
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rw [hD]; exact Nat.add_le_add_left hfA_le_dx _
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exact le_trans hcore hmono
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/-- ADD LIQUIDITY: Mint LP tokens -/
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def addLiquidity (p : Pool) (dx dy : Nat) : Pool × Nat :=
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let lpMinted := if p.totalLP = 0 then dx
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else dx * p.totalLP / p.reserveX
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({ p with reserveX := p.reserveX + dx
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, reserveY := p.reserveY + dy
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, totalLP := p.totalLP + lpMinted }, lpMinted)
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/-- LIQUIDITY GROWS -/
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theorem add_liquidity_grows (p : Pool) (dx dy : Nat) :
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(addLiquidity p dx dy).1.reserveX ≥ p.reserveX := by
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simp [addLiquidity]
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/-- LP TOKENS MINTED -/
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theorem lp_minted (p : Pool) (dx dy : Nat) :
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(addLiquidity p dx dy).1.totalLP ≥ p.totalLP := by
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simp [addLiquidity]
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/-- PERMISSIONLESS: No access control -/
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theorem anyone_can_swap (p : Pool) (dx : Nat) :
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(swap p dx).1.reserveX ≥ p.reserveX := by
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simp [swap]
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end DeFi.AMM
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