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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
3.9 KiB
Lean4
118 lines
3.9 KiB
Lean4
/-
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Authors: Antje Worring, Zoo Labs Foundation Research
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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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set fa := dx * (10000 - p.feeBps) / 10000 with hfa
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apply Nat.div_le_of_le_mul
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calc p.reserveY * fa ≤ p.reserveY * (p.reserveX + fa) :=
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Nat.mul_le_mul_left _ (by omega)
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_ = (p.reserveX + fa) * p.reserveY := Nat.mul_comm _ _
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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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set f := dx * (10000 - p.feeBps) / 10000 with hf
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set X := p.reserveX
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set Y := p.reserveY
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-- fee-adjusted input never exceeds dx
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have hfdx : f ≤ dx := by
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apply Nat.div_le_of_le_mul
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calc dx * (10000 - p.feeBps) ≤ dx * 10000 :=
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Nat.mul_le_mul_left _ (Nat.sub_le _ _)
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_ = 10000 * dx := Nat.mul_comm _ _
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set s := X + f with hs
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set dy := Y * f / s with hdy
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-- dy * s ≤ Y * f (floor division bound)
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have hbound : s * dy ≤ Y * f := by
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rw [hdy, Nat.mul_comm]; exact Nat.div_mul_le_self _ _
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-- the fee-adjusted swap preserves the product, fee only helps
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have key : X * Y ≤ (X + f) * (Y - dy) := by
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have expand : (X + f) * (Y - dy) = (X + f) * Y - (X + f) * dy := by
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rw [Nat.mul_sub]
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rw [expand]
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have h1 : (X + f) * dy ≤ Y * f := by rw [← hs]; exact hbound
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have h2 : (X + f) * Y - Y * f = X * Y := by
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have : (X + f) * Y = X * Y + Y * f := by ring
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omega
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calc X * Y = (X + f) * Y - Y * f := h2.symm
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_ ≤ (X + f) * Y - (X + f) * dy := Nat.sub_le_sub_left h1 _
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-- larger actual input dx ≥ f only increases the product
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have hmono : (X + f) * (Y - dy) ≤ (X + dx) * (Y - dy) :=
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Nat.mul_le_mul_right _ (by omega)
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exact le_trans key 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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