At the time moment \(t_0\), a user deposits \(x\) amount of the Target token. He gets: \(y = x \cdot s(t_0)\) units of Zero token, and the same amount of Claim token.
In case \(s(t_0) < S(t_0)\) the user may immediately collect a certain amount of the Target tokens from just issued Claim token. It would be reasonable to consider this addition amount of the Target token to be added to the original deposited amount. The exact additional amount could be fond from the following equation:
\[ \Delta x = (x + \Delta x) \cdot s(t_0) \cdot \left( \frac{1}{s(t_0)} - \frac{1}{S(t_0)} \right) = (x + \Delta x) \cdot \left( 1 - \frac{s(t_0)}{S(t_0)} \right) \\ \Delta x \left( 1 - \left( 1 - \frac{s(t_0)}{S(t_0)} \right) \right) = x \cdot \left( 1 - \frac{s(t_0)}{S(t_0)} \right) \\ \Delta x \cdot \frac{s(t_0)}{S(t_0)} = x \cdot \left( 1 - \frac{s(t_0)}{S(t_0)} \right) \\ \Delta x = \frac{x \cdot \left( 1 - \frac{s(t_0)}{S(t_0)} \right)}{\frac{s(t_0)}{S(t_0)}} = x \cdot \left( \frac{S(t_0)}{s(t_0)} - 1 \right) \]
So the full deposited amount is:
\[ X = x + \Delta x = x + x \cdot \left( \frac{S(t_0)}{s(t_0)} - 1 \right) = x \cdot \frac{S(t_0)}{s(t_0)} \]
and the full amount of issued tokens (the same for Zero and Claim tokens) is:
\[ Y = X \cdot s(t_0) = x \cdot \frac{S(t_0)}{s(t_0)} \cdot s(t_0) = x \cdot S(t_0) \]
Note, that the outcome for the user is as if the current scale was \(S(t_0)\) rather than $s(t_0). Thus is would be reasonable for the protocol to use the current max scale instead of the current scale when calculating the number of Zeros and Claim to be issued.
In case the user already have some amount of Claim token when issuing more tokens, earnings should be collected from these Claim token amount before issuing more Claim token. Let's assume that the user already has the amount \(Y_l\) of Claim token and earnings wele last colected at the time \(t_l\). Then the user may collect the following amount \(x_l\) of Target token from his existing Claim holdings:
\[ x_l = Y_l \cdot \left( \frac{1}{s(t_l)} - \frac{1}{S(t_0)} \right) \]
Thus the full amount of issued tokens, that takes into accountthe collected earnings from existing Claim holdings, is:
\[ Y = \left( x + Y_l \cdot \left( \frac{1}{s(t_l)} - \frac{1}{S(t_0)} \right) \right) \cdot S(t_0) \]
In the ideal case, when a user redeems \(y\) amount of Zero token, he gets the following amount of the Target token:
\[ x_z = y\frac{(1 - \theta)}{s(t_m)} \]
In the ideal case, when a user redeems \(y\) amount of Claim token, he gets the following total amount of the Target token:
\[ x_c = y\left( \frac{\theta}{s(t_m)} + \frac{1}{s(t_0)} - \frac{1}{s(t_m)} \right) = y \left( \frac{1}{s(t_0)} - \frac{(1 - \theta)}{s(t_m)} \right) \]
Note that:
\[ x_z + x_c = \frac{y}{s(t_0)} = \frac{x \cdot s(t_0)}{s(t_0)} = x \]
i.e. the whole deposited amount is redeemed.
The following amount could be collected by Claim token holders before maturity:
\[ x'_c = y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) \]
So the following amount is to be collected after maturity:
\[ x''_c = x_c - x'_c = y\left( \frac{1}{s(t_0)} - \frac{(1 - \theta)}{s(t_m)} \right) - y\left( \frac{1}{s(t_0)} - \frac{y}{(t_m)} \right) = y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \]
As it is impossible to collect a negative amount, the actual amount collected after maturity is:
\[ X''_c = \max (0, x''_c) = \max \left( 0, y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) \]
Thus the actual total amount of the Target tokens got by the Claim token hoder is:
\[ X_c = x'_c + X''_c = y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) + \max \left( 0, y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) \]
and the actual amount of the Target tokens got by the Zero token holder is:
\[ X_z = x - X_c = \frac{y}{s(t_0)} - y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) - \max \left( 0, y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) =\\= y\left( \frac{1}{S(t_m)} - \max \left( 0, \left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) \right) \]
When
\[ y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \geqslant 0 \]
we have:
\[ X_c = y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) + \max \left( 0, y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) =\\= y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) + y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) =\\= y\left( \frac{1}{s(t_0)} - \frac{(1 - \theta)}{s(t_m)} \right) = x_c \]
\[ X_z = y\left( \frac{1}{S(t_m)} - \max \left( 0, \left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) \right) =\\= y\left( \frac{1}{S(t_m)} - \left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) =\\= y \frac{(1 - \theta)}{s(t_m)} = x_z \]
So, in a summy day scenario, \(X_z = x_z\) and \(X_c = x_c\).
When the day is not so sunny, i.e.
\[ y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) < 0 \]
we have:
\[ X_c = y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) + \max \left( 0, y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) =\\= y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) + 0 =\\= y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) \]
\[ X_z = y\left( \frac{1}{S(t_m)} - \max \left( 0, \left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \right) \right) =\\= y\left( \frac{1}{S(t_m)} - 0 \right) =\\= \frac{y}{S(t_m)} \]
Lets transform the summy day condition a bit:
\[ y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) \geqslant 0 \\ \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \geqslant 0 \\ s(t_m) - S(t_m) \cdot (1 - \theta) \geqslant 0 \\ s(t_m) \geqslant S(t_m) \cdot (1 - \theta) \\ \frac{s(t_m)}{S(t_m)} \geqslant 1 - \theta \]
So, the final formulas are:
Zero token holders get the following amount of the Target tokens after maturity:
\[ X_z = \begin{cases} y \frac{(1 - \theta)}{s(t_m)} & \text{if} & \frac{s(t_m)}{S(t_m)} \geqslant 1 - \theta, \\ \frac{y}{S(t_m)} & \text{if} & \frac{s(t_m)}{S(t_m)} < 1 - \theta \end{cases} \]
Claim token holders get the following mount of the Target token after maturity:
\[ X''_c = \begin{cases} y\left( \frac{1}{S(t_m)} - \frac{(1 - \theta)}{s(t_m)} \right) & \text{if} & \frac{s(t_m)}{S(t_m)} \geqslant 1 - \theta, \\ 0 & \text{if} & \frac{s(t_m)}{S(t_m)} < 1 - \theta \end{cases} \]
Claim token holders get the following total amount of the Target tokens:
\[ X_c = \begin{cases} y\left( \frac{1}{s(t_0)} - \frac{(1 - \theta)}{s(t_m)} \right) & \text{if} & \frac{s(t_m)}{S(t_m)} \geqslant 1 - \theta, \\ y\left( \frac{1}{s(t_0)} - \frac{1}{S(t_m)} \right) & \text{if} & \frac{s(t_m)}{S(t_m)} < 1 - \theta \end{cases} \]
\[ X''_c = \max \left( 0, y \cdot \frac{\theta}{S(t_m)} - \left( y \cdot \frac{1 - \theta}{s(t_m)} - y \cdot \frac{1 - \theta}{S(t_m)} \right) \right) \]
A user wants to provide the amount \(x\) of a Target token as liquidity to a Balancer pool that trades this Target token against a Zero token derived from this Target token.
The user splits the oritinal Target amount into two parts: \(x = x' + x''\). The first part \(x'\) goes directly to the pool. The second part \(x''\) is used to issue Zero that will go to the pool. The amount of Zero issued is:
\[ y = x'' \cdot S(t) \]
Let \(X\) be the pool reserves of the Target token, and \(Y\) be the pool reserves of Zero. The user needs to provide the tokens at the same proportion, so:
\[ \frac{x'}{y} = \frac{X}{Y} \\ \frac{x'}{x'' \cdot S(t)} = \frac{X}{Y} \\ \frac{x - x''}{x'' \cdot S(t)} = \frac{X}{Y} \\ (x - x'') \cdot Y = x'' \cdot S(t) \cdot X \\ x \cdot Y = x'' \cdot \left( S(t) \cdot X + Y \right) \\ x'' = x \cdot \frac{Y}{S(t) \cdot X + Y} \]
A user wants to provide the amount \(x\) of a Target token as liquidity to a Balancer pool that trades the Underlying token of this Target token to a Zero token derived from this Target token.
The user splits the oritinal amount into two parts: \(x = x' + x''\). The first part \(x'\) is redeemed for the Underlying token. The amount of obtained underlying token is:
\[ z = x' \cdot s(t) \]
The second part \(x''\) is used to issue Zero that will go to the pool. The amount of Zero issued is:
\[ y = x'' \cdot S(t) \]
Let \(Z\) be the pool reserves of the Underlying token, and \(Y\) be the pool reserves of Zero. The user needs to provide the tokens at the same proportion, so:
\[ \frac{z}{y} = \frac{Z}{Y} \\ \frac{x' \cdot s(t)}{x'' \cdot S(t)} = \frac{Z}{Y} \\ \frac{(x - x'') \cdot s(t)}{x'' \cdot S(t)} = \frac{Z}{Y} \\ (x - x'') \cdot s(t) \cdot Y = x'' \cdot S(t) \cdot Z \\ x \cdot s(t) \cdot Y = x'' \cdot \left( S(t) \cdot Z + Y \right) \\ x'' = x \cdot \frac{s(t) \cdot Y}{S(t) \cdot Z + s(t) \cdot Y} \]
\[ x'' = x \cdot \frac{Y}{X + Y} \]
The whole schema is more or less equivalent to this one:
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