Composability

Smartink's decentralized exchange follows the following rules, with 0.28% overall fees, when the initial reserves are xxx
 of token XTZ and yyy
 of tokens and the user sells aaa
 of  XTZ to receive bbb
 , the following can be observed with kkk
 being invariant: 
(x+0.9972∗a)∗(y−b)=x∗y=k(x + 0.9972*a)*(y-b)=x*y=k(x+0.9972∗a)∗(y−b)=x∗y=k
By symmetry, the following formula holds with a sale of bbb
 tokens to receive aaa
  : 
(x−a)∗(y+0.9972∗b)=x∗y=k(x-a)*(y+0.9972*b)=x*y=k(x−a)∗(y+0.9972∗b)=x∗y=k
Definition
Composability of such a decentralized exchange means that a trader gets the same price from participating all at once as in a set of small trades. Therefore, traders do not need to strategize how they make trades, which can be complex to achieve in a public blockchain context. They are therefore financially incentivized to quickly perform the largest possible trades, which has a positive impact on the gains of liquidity providers. 
Proof
Let a1,a2>0a_1,a_2>0a1​,a2​>0
 and b1,b2b_1,b_2b1​,b2​
 such that, with a=a1+a2a=a_1+a_2a=a1​+a2​
 and a=b1+b2a = b_1+b_2a=b1​+b2​
:
(x+0.9972∗a1)∗(y−b1)=k=(x+0.9972∗(a1+a2))∗(y−(b1+b2))=(x+0.9972∗a)∗(y−b)(x+0.9972∗a_1)∗(y−b_1)=k=\Biggl(x+0.9972*(a_1+a_2)\Biggr)*(y-(b_1+b_2))
\\=(x+0.9972*a)*(y-b) (x+0.9972∗a1​)∗(y−b1​)=k=(x+0.9972∗(a1​+a2​))∗(y−(b1​+b2​))=(x+0.9972∗a)∗(y−b)
That means that with initial reserves of xxx
 of token XTZ and yyy
 of tokens a user may sell a1a_1a1​
 of XTZ to receive b1b_1b1​
 of tokens and then a2a_2a2​
 of XTZ to receive b2b_2b2​
 of tokens. 
Or a user may “directly” sell a=a1+a2a = a_1 + a_2 a=a1​+a2​
 of XTZ to receive b=b1+b2b = b_1 + b_2b=b1​+b2​
 of tokens.
Let b2b_2b2​
 such that
(x+0.9972∗a2)∗(y−b~2)=k(x+0.9972*a_2)*\Biggl(y-\tilde b_2\Biggr)=k(x+0.9972∗a2​)∗(y−b~2​)=k
Note that b~2\tilde b_2b~2​
 is not necessarily equal to b2b_2b2​
.
Recall that bbb
 is the only real such that:
(x+0.9972∗a)∗(y−b)=k(x+0.9972*a)*(y-b)=k(x+0.9972∗a)∗(y−b)=k

Hence, with initial reserves xxx
 of token XTZ and y of tokens a user may sell a2a_2 a2​
 of XTZ to receive b2b_2b2​
 of tokens and then a2a_2 a2​
 of XTZ to receive b−b2b-b_2b−b2​
 of tokens. 
Or a user may "directly" sell a=a1+a2a=a_1+a_2a=a1​+a2​
 of XTZ to receive b=b1+b2b=b_1+b_2b=b1​+b2​
 of tokens.
As a consequence, a trader who sells aaa
 of XTZ receives the same amount bbb
 he would have received if he had either first sold a1a_1a1​
 and then a2a_2a2​
 or first a2a_2a2​
 and then a1a_1a1​
 . This is composability.

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Definition

Proof