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# Pool Process Flows

Automating Tried And Tested Procedures

## Pool unlocking/initialization​​

### Overview​​

No action (minting / burning / swaps) can be performed prior to pool initialization.
In addition to setting the initial sqrt price, a small amount of token0 and token1 is required to be seeded for the initialization of reinvestL to the value of MIN_LIQUIDITY. This is required to prevent division by zero in the calcRMintQty() function when swaps are performed. MIN_LIQUIDITY was chosen to be reasonably small enough to avoid calculation inaccuracies for swaps, and from taking unreasonably large capital amounts from the caller.
As part of this anti-spam feature, Elastic allocates
$10^5$
token weis as liquidity for the reinvestment curve. While this amount has been carefully selected to suit the majority of tokens, there are rare exceptions where token teams decide to implement tokens with less decimals for reasons of their own. For reference, the majority of ERC20 tokens (LINK, MATIC, ANKR, SHIB, etc.) are created with
$10^{18}$
decimals, WBTC has
$10^8$
decimals, major stablecoin tokens like USDC/USDT have
$10^6$
decimals.
In cases where the token has a low decimal value and the per unit value of the token high, the amount taken might be of significant value (i.e. a token is created with
$10^2$
decimals and each unit has a 1USD value, which results in
$10^5$
tokens with a value of 100USD being taken as an anti-spam feature). As a permissionless platform, KyberSwap supports the listing of all tokens which meet the ERC20 standard and as such, users are responsible for checking if their token falls into the aforementioned category.

## Minting and burning (add/remove liquidity)​​

### Overview​​

Adding and removing liquidity have very similar flows. One of the main differences is that mint() is possibly a permissioned function, but burn() is not. More information relating to the requirement for this can be found in this section on whitelisting position managers.

#### Implementation details​​

• A simple check is performed to ensure that the requested liquidity amount to mint / burn is non-zero
• _tweakPosition() is called, which does the following:
• Load the pool state into memory poolData (current price, tick and liquidity values)
• Call _syncFeeGrowth() to update fee growth data. Mints reinvestment tokens if necessary
• Call _syncSecondsPerLiquidity() to update seconds per liquidity data
• The updated global values and poolData is passed into _updatePosition()
• Updates (initializes) the lower and upper position ticks. Will insert or remove the tick from the linked list whenever necessary
• Calculates feeGrowthInside and returns the amount of reinvestment tokens claimable by the position
• Transfers the claimable reinvestment tokens to the position owner, if any
• Calculates the token0 and token1 quantity required to be collected from (add liquidity) or sent to (remove liquidity) msg.sender. Will apply liquidity changes to pool liquidity if the specified position is active
• In the case of adding liquidity, a callback is made to collect the tokens
• Emit event

## Swap

### Overview​​

Like KyberSwap Classic, there are 4 different types of swaps available that a user can specify.
1. 1.
Swap from a specified amount of token 0 (exactInput0)
2. 2.
Swap from a specified amount of token 1 (exactInput1)
3. 3.
Swap to a specified amount of token 0 (exactOutput0)
4. 4.
Swap to a specified amount of token 1 (exactOutput1)
Swapping token 0 for token 1 (cases 1 and 4) cause the pool price and tick to move downwards, while swapping token 1 for token 0 (cases 2 and 3) cause the pool price and tick to move upwards.
In addition, the user can specify a price limit that the swap can reach. The minimum and maximum price limits a user can specify is MIN_SQRT_RATIO + 1 and MAX_SQRT_RATIO - 1.
The algorithm exits when either the specified amount has been fully used, or if the price limit has been reached.

#### Implementation details​​

The swap amount is a int256 to implicitly suggest whether it is exact input (> 0) or exact output (< 0).
1. 1.
Fetch the initial pool state
• $L_{base​}$
:= pool.baseL (liquidity provided by positions)
• $L_{reinvest​}$
:= pool.reinvestL (liquidity from fees collected)
• $\sqrt{P_{current}}$
:= pool.sqrtP (current sqrt price of token1/token0)
• $t_c$
:= pool.currentTick (tick associated with pool price)
• $t_n$
:= pool.nextTick (next initialized tick from current tick)
2. 2.
Verify specified price limit
$\sqrt{P_{lim}}$
• Cases 1 & 4: MIN_SQRT_RATIO <
$\sqrt{P_{lim}}$
<
$\sqrt{P_{current}}$
• Cases 2 & 3:
$\sqrt{P_{current}}$
<
$\sqrt{P_{lim}}$
< MAX_SQRT_RATIO
3. 3.
While specified amount
$delta_{remaining}$
not used up or price limit not reached,
• Calculate temp next tick
$t_{tmp}$
and next sqrt price
$\sqrt{P_{next}}$
. The temporary next tick is to ensure that the next tick does not exceed the MAX_TICK_DISTANCE cap from the current tick, so as not to violate the 5% price difference requirement.
• $\sqrt{P_{next}}$
= TickMath.getSqrtRatioAtTick(
$t_{tmp}$
)
• Check if
$\sqrt{P_{next}}$
exceeds
$\sqrt{P_{lim}}$
,
• If true then
$\sqrt{P_{target}}$
=
$\sqrt{P_{lim}}$
• If false then
$\sqrt{P_{target}}$
=
$\sqrt{P_{next}}$
• Call SwapMath.computeSwapStep() to calculate the actual swap input and output amounts to be used, swap fee amount and next pool price
• Subtract amount to be used (usedAmount) to swapData.specifiedAmount
• Add amount to be sent to user (returnedAmount) to swapData.returnedAmount
$\Delta{L}$
to
$L_{reinvest}$
• Check if swap will reach next tick
• If true, set swapData.currentTick = willUpTick ? tempNextTick : tempNextTick - 1 and continue
• If false, recalculate the current tick based on current price and break the loop
• If
$t_{tmp}$
==
$t_n$
, we are crossing tick
$t_n$
:
• Calculate amount of reinvestment tokens to be minted for fees to be sent to government and to for LP contributions, and update feeGrowthGlobal
• Cross tick
$t_n$
: updates the tick outside values and apply tick.liquidityNet to pool liquidity whilst fetching the next tick
$t_n$
4. 4.
Perform actual minting of reinvestment tokens if necessary
5. 5.
Update pool state (price, ticks, liquidity, feeGrowth, reinvestment variables)
6. 6.
Send token to caller, execute swap callback to collect token
• Negative quantity = transfer to caller
• Positive quantity = collect from caller

#### computeSwapStep() Flow​​

Inputs
Field
Type
Explanation
liquidity
uint256
active base liquidity + reinvestment liquidity
currentSqrtP
uint160
current sqrt price
targetSqrtP
uint160
sqrt price limit nextSqrtP can take
feeInBps
uint256
swap fee in basis points
specifiedAmount
int256
amount remaining to be used for the swap
isExactInput
bool
true if specifiedAmount refers to input amount, false if specifiedAmount refers to output amount
isToken0
bool
true if specifiedAmount is in token0, false if specifiedAmount is in token1
Outputs
Field
Type
Explanation
usedAmount
int256
actual amount to be used for the swap. >= 0 if isExactInput = true, <= 0 if isExactInput = false
returnedAmount
int256
output qty (<= 0) to be accumulated if isExactInput = true, input qty (>= 0) if isExactInput = false
deltaL
uint256
collected swap fee, to be incremented to reinvest liquidity
nextSqrtP
uint160
new sqrt price after the computed swap step
1. 1.
Calculate the amount required to reach targetSqrtP from currentSqrtP by calling calcReachAmount().
2. 2.
If amount required exceeds specifiedAmount, then the targetPrice will not be reached, and we expect the resulting price nextSqrtP to not exceed targetSqrtP.
• usedAmount := specifiedAmount
• Estimate
$\Delta{L}$
, the swap fee to be collected by calling estimateIncrementalLiquidity()
• Calculate the final price nextSqrtP by calling calcFinalPrice()
3. 3.
Otherwise, the temporary next tick will be crossed.
• usedAmount will be the amount calculated in step 1
• calculate
$\Delta{L}$
by calling calcIncrementalLiquidity()
• set the resulting price nextSqrtP = targetSqrtP
4. 4.
Finally, calculate returnedAmount by calling calcReturnedAmount().

### Swapping formula​​

Assume that:
• x1, x2: the amount of token0 before/after swap
• y1, y2: the amount of token1 before/after swap
• L1, L2: the liquidity before/after swap
• p1, p2: the price before/after swap

#### Swap exact input from token0 -> token1​​

• Given L1, p1, fee and
$\Delta y$
, calculate
$\Delta L$
and p2
$\Delta L = {L_1 * {\Large {\Delta y * fee \over 2 * y1}}}$
(1)
$\Delta L = {\Large {\Delta y * fee \over 2 * \sqrt p_1}}$
Finally calculate new
$\sqrt{p_2}$
$\sqrt{p_2} = \Large {y_1 + \Delta y \over L_2}$
(2)
$\sqrt{p_2} = \Large {L1 * \sqrt p1 + \Delta y \over L1 + \Delta L}$
• Given L1, p1 and p2 calculate the
$\Delta L$
and
$\Delta y$
From (2) we have:
$\sqrt p_2 * (L1 + \Delta x * \sqrt p_1) = \sqrt p_1 * (L1 + \Delta L)$
combine with (1)
$2 * \sqrt p_2 * (L1 + \Delta x * \sqrt p_1) = \sqrt p_1 * ( 2 * L1 + \Delta x * fee * \sqrt p_1$
=>
$\Delta x * \sqrt p1 * (2 * \sqrt p_2 - fee * \sqrt p_1) = 2 * L1 * (\sqrt p_1 - \sqrt p_2)$
=> (3)
$\Delta x = \Large {\frac{2 * L1 * (\sqrt p_1 - \sqrt p_2)}{\sqrt p_1 * (2 * \sqrt p_2 - fee * \sqrt p_1)} }$

#### Swap exact input from token1 -> token0​​

• Given L1, p1, fee and
$\Delta y$
, calculate
$\Delta L$
and p2
$\Delta L = {L_1 * {\Large {\Delta y * fee \over 2 * y1}}}$
(1)
$\Delta L = {\Large {\Delta y * fee \over 2 * \sqrt p_1}}$
Finally calculate new
$\sqrt{p_2}$
$\sqrt{p_2} = \Large {y_1 + \Delta y \over L_2}$
(2)
$\sqrt{p_2} = \Large {L1 * \sqrt p1 + \Delta y \over L1 + \Delta L}$
• Given L1, p1 and p2 calculate the
$\Delta L$
and
$\Delta y$
From (1) and (2)
$\sqrt p_2 * (L1 + {\Large {\Delta y * fee \over 2 * \sqrt p_1}}) = L1 * \sqrt p1 + \Delta y$
=>
$\Delta y * (2 * \sqrt p_1 - fee * \sqrt p_2) = 2 * \sqrt p_1 * L1 * (\sqrt p_2 - \sqrt p_1)$
=> (3)
$\Delta y = \Large {2 * \sqrt p_1 * L1 * (\sqrt p_2 - \sqrt p_1) \over (2 * \sqrt p_1 - fee * \sqrt p_2)}$

#### Swap exact output from token0 -> token1 (isExactInput = false, isToken0 = false)​​

• Given L1, p1 and p2, calculate the
$\Delta y$
$y1 - \Delta y = L2 * \sqrt p_2$
=>
$\Delta y = \Delta L * \sqrt p_2 + L1 * (\sqrt p_2 - \sqrt p_1)$
=>
$\Delta y = {\Large \frac{\Delta x * fee * \sqrt p_1}{2}} * \sqrt p_2 + L1 * (\sqrt p_2 - \sqrt p_1)$
=>
$\Delta y = {\Large \frac{fee * \sqrt p_1 * L1 * (\sqrt p_1 - \sqrt p_2)}{\sqrt p_1 * (2 * \sqrt p_2 - fee * \sqrt p_1)}} * \sqrt p_2 + L1 * (\sqrt p_2 - \sqrt p_1)$
=>
$\Delta y = {\Large \frac{L1 (\sqrt p_1 - \sqrt p_2) (2 * \sqrt p_2 - fee * \sqrt p_1 - fee * \sqrt p_2)}{2 * \sqrt p_2 - fee * \sqrt p_1}}$
• Given L1, p1 and
$\Delta y$
, calculate
$\Delta L$
$(L1 + \Delta L)^2 = (y1 - \Delta y) * (x1 + \Delta x)$
=>
$(L1 + \Delta L)^2 = (y1 - \Delta y) * (x1 + {\Large \frac{2 * \Delta L}{\sqrt p_1 * fee}})$
=>
$(L1 + \Delta L)^2 * fee = L1^2 * fee - fee * \Delta y * L1 / \sqrt p_1 + {\Large \frac{2 * \Delta L}{\sqrt p_1}} * (L1 / \sqrt p_1 - \Delta y)$
=>
$fee * \Delta L^2 - 2 * (L1 - fee * L1 - \Delta y / \sqrt p_1) * \Delta L + L1 * fee * \Delta y / \sqrt p_1 = 0$
This can be transformed into
$a * \Delta L ^2 - 2 * b * \Delta L + c = 0$
$\Delta L$
will be the smaller solution of this equation

#### Swap exact output token1 -> token0 (isExactInput = false, isToken0 = true)​​

• Given L1, p1 and p2, calculate the
$\Delta x$
$x1 - \Delta x = L2 / \sqrt p_2$
=>
$\Delta x = {\Large \frac{L1}{\sqrt p_1} - \frac{\Delta L}{\sqrt p_2} - \frac{L1}{\sqrt p_2}}$
=>
$\Delta x = {\Large \frac{L1}{\sqrt p_1} - \frac{L1}{\sqrt p_2} - \frac{\Delta y * fee}{ 2 * \sqrt p_1 * \sqrt p_2}}$
=>
$\Delta x = {\Large \frac{L1}{\sqrt p_1} - \frac{L1}{\sqrt p_2} - {2 * \sqrt p_1 * L1 * (\sqrt p_2 - \sqrt p_1) \over (2 * \sqrt p_1 - fee * \sqrt p_2)} * \frac{fee}{2 * \sqrt p_1 * \sqrt p_2}}$
=>
$\Delta x = {\Large \frac{L1 * (\sqrt p_2 - \sqrt p_1)}{\sqrt p_2 * \sqrt p_1} * (1 - \frac{fee * \sqrt p_1}{2 * \sqrt p_1 - fee * \sqrt p_2})}$
=>
$\Delta x = {\Large \frac{L1 * (\sqrt p_2 - \sqrt p_1) * (2 * \sqrt p_1 - fee * \sqrt p_2 - fee * \sqrt p_1)}{\sqrt p_2 * \sqrt p_1 * (2 * \sqrt p_1 - fee * \sqrt p_2)}}$
• Given L1, p1 and
$\Delta x$
, calculate
$\Delta L$
$(L1 + \Delta L)^2 = (x1 - \Delta x)(y1 + \Delta y)$
=>
$(L1 + \Delta L)^2 = L1 ^ 2 - \Delta x * L1 * \sqrt p_1 + \Delta y * (L1 / \sqrt p_1 - \Delta x)$
=>
$(L1 + \Delta L)^2 = L1 ^ 2 - \Delta x * L1 * \sqrt p_1 + \Delta L * 2 * \sqrt p_1 /fee * (L1 / \sqrt p_1 - \Delta x)$
=>
$fee * \Delta L^2 - 2 * (L1 * (1 - fee) - \Delta x * \sqrt p_1) * \Delta L + \Delta x * L1 * \sqrt p_1 * fee=0$