Constant Function Market Makers，about The trade function

[muzzby]

I read about constant function market makers and saw a formula in the 'Trading function', (x+rΔx)(y−Δy)=k. In point 5 of the explanation, it says 'the quantity of token0 in the pool has changed (x+rΔx), and the quantity of token1 has also changed (y-Δy)'. I have a doubt that shouldn't the change in quantity of token0 be (x+rΔx+Δx)? Please advise, thank you!

[Jeiwan]

The formula is correct. The initial reserves are x and y, and swapping increases x and decreases y (or the other way around). Thus, $\Delta x$ is added and $\Delta y$ is removed. Since fees in Uniswap are taken from input amounts (the token being sold), r is applied to $\Delta x$. r in the formula is $1 - \text{fee}$, where fee is 0.003 (Uniswap V3 has different fee tiers). So, $r \Delta x$ is $0.997 * \Delta x$.

[bluenights004]

agree with muzzby, it seems + ∆x is missing in formula which represents adding the increase cause by swap, the liquidity provider fee (rΔx) is the only item added in previous reserve x ?

in other words, previous reserve (x) + liquidity provider fee (r∆x) + increase in x (∆x) due to swap = new reserve balance of x, should be included in the formula?

[Brivan-26]

@bluenights004 You're wrong about something:

in other words, previous reserve (x) + liquidity provider fee (r∆x) + increase in x (∆x) due to swap

(r∆x) is not the liquidity provider fee. The liquidity provider fee is constant, that is taken from the sold token.
If you are about to sell ∆x tokens0, the pool takes fee from it, so the real amount of token 0 being added to the pool is (1- fee)*∆x.
In the context of Uniswap, consider the following example: We have a pool of token0/token1. Initially, we have 100 of tokens each.
supposing you are selling 40 of token0 to get x amount of token1. After the trade happens, the pool will not have 100 + 40 of token0 as the protocol takes seme fees, in uniswap it is 0.3%. So, 0.3% of 40 token0 you're selling will be reducted, which means the pool will have 100 + 0.97*40 of token0.
I hope this clears the confusion
cc @muzzby