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Appendix

Appendix A: Range Mechanism​

Objective​

The goal is to accurately account a position's accured fees accured and duration for which it is active. A position is defined to be active if the current pool tick lies within the lower and upper ticks of the position.

Global value​

The feeGrowthGlobal and secondsPerLiquidityGlobal variables represent the total amount for 1 unit of unbounded liquidity.

Outside value​

The outside value (feeGrowthOutside and secondsPerLiquidityOutside) for a tick represents the accumulated value relative to the currentTick. By definition, we can visually represent it to be as such:

  • tick <= currentTick
  • tick > currentTick

Initialization​

When a tick is initialized, all growth is assumed to happen below it. Hence, the outside value is initialized to the following values under these cases:

  • tick <= currentTick: outside value := global value
  • tick > currentTick: outside value := 0

Crossing ticks​

Due to the definition of the outside value, a tick's outside value is reversed whenever the pool tick crosses it. Specifically, outside value := global value - outside value.

Note​

When swapping downtick, the current tick is further decremented by 1. This is to ensure a tick's outside value is interpreted correctly. swapData.currentTick = willUpTick ? tempNextTick : tempNextTick - 1;

Calculating value inside ticks​

The current tick can be

  1. below a position (currentTick < lowerTick) The value inside can therefore be calculated as tickLower's outside value - tickUpper's outside value
  2. within a position (lowerTick <= currentTick < upperTick) The value inside can therefore be calculated as global value - (lower + upper tick's outside values)
  3. above a position (upperTick <= currentTick) The value inside can therefore be calculated as tickUpper's outside value - tickLower's outside value

Appendix B: Tick-range Mechanism​

  • To represent liquidity mapping, LyfeblocDMM uses linked list, the linked list always starts by MIN_TICK and ends by MAX_TICK.

  • The pool also need to track to the highest initialized tick which is lower than or equal to currentTick (aka nearestCurrentTick).

For example:

  • the pool is initalized at tick 5, the linked list will be [MIN_TICK, MAX_TICK] andnearestCurrentTick will be MIN_TICK.
  • A adds liquidity at tick [-5, 10], the linked list will be [MIN_TICK, -5, 10, MAX_TICK] and nearestCurrentTick will be -5.
  • C adds liquidity at tick [0, 100], the linked list will be [MIN_TICK, -5, 0, 10, 100, MAX_TICK] and nearestCurrentTick will be 0.
  • B swaps and currentTick = 15, then nearestCurrentTick will be 10
  • A remove all liquidity at tick [-5, 10], the linked list will be [MIN_TICK, 0, 100, MAX_TICK] andnearestCurrentTick will be 0.

Appendix C: Reinvest Mechanism​

LyfeblocDMM does not take fee in forms of token0 and token1, the fee will be both tokens and reinvested into tick range from [0, infinity]

  • baseL is the liquidity from lp provider and reinvestL is the liquidity from the fee
  • User can still earn fee from reinvestL, so we create reinvestment token to represent the proportion of user in reinvestL.
  • Each time, users add/remove liquidity or cross a tick, we have to mint the reinvestment token equavalent to the increment of reinvestL by baseL. The formula to calculate mintQty is at `ReinvestmentMath.calcrMintQty
  • reinvestment token: after London hardfork, it is [costly](https://hackmd.io/@fvictorio/ gas-costs-after-berlin) to warm up another contract. Therfore, we decided to merge the reinvestment token and pool contract and the pool contract will be inherited from ERC20.