math refactor (#20)

src/ConstantSum/README.md

@@ -1,5 +1,5 @@
 # Constant Sum Market Maker
-This will be all the background needed to understand the `GeometricMean` DFMM.
+This will be all the background needed to understand the `ConstantSum` DFMM.
 
 ## Conceptual Overview
 The `ConstantSum` DFMM gives the LP a portfolio that will allow exchange of a pair of tokens at a single price.
@@ -63,7 +63,7 @@ Allocations and deallocations should not change the price of a pool and since th
 We need only compute the new $L$.
 Specifically:
 $$
-L + \Delta_L = x+\Delta_X + \frac{y+\Delta_Y}{P}
+\Delta_L = \Delta_X + \frac{\Delta_Y}{P}
 $$
 
 

src/GeometricMean/README.md

@@ -78,17 +78,30 @@ $$
 
 
 ## Allocations and Deallocations
-Allocations and deallocations should not change the price of a pool, so the ratio of reserves cannot change:
+**Input $\delta_X$:** If a user wants to allocate a specific amount of $\delta_X$, then it must be that:
 $$
-P = \frac{w_x}{w_y} \frac{y}{x}  = \frac{w_x}{w_y} \frac{y+\Delta_y}{x+\Delta_x}.
+\frac{x}{L} = \frac{x+\Delta_X}{L+\Delta_L}
 $$
-**Input $\Delta_X$:** If a user wants to allocate a specific amount of $\Delta_X$, then they must also allocate:
+which yields:
 $$
-\boxed{\Delta_Y = \frac{y}{x}(x+\Delta_X)-y}
+\Delta_L = L \frac{\Delta_X}{x}
 $$
-**Input $\Delta_Y$:** For a given $\Delta_Y$, then they must have:
+Then it must be that since the ratio of reserves cannot change.
 $$
-\boxed{\Delta_X = \frac{x}{y}(y+\Delta_Y)-x}
+\Delta_Y = y\frac{\Delta_X}{x} 
+$$
+
+**Input $\Delta_Y$:** To allocate a specific amount of $\Delta_Y$, then it must be that:
+$$
+\frac{y}{\mu L} = \frac{y+\Delta_Y}{\mu(L+\Delta_L)}
+$$
+which yields:
+$$
+\Delta_L = L \frac{\Delta_Y}{y}
+$$
+and we likewise get
+$$
+\Delta_X = x\frac{\Delta_Y}{y}
 $$