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125/**
126 * @notice Yellow Paper Def. 6.2 (Withdrawal Fee)
127 * @dev When covBefore >= 1, fee is 0
128 * @dev When covBefore < 1, we apply a fee to prevent withdrawal arbitrage
129 * @param k K slippage parameter in WAD
130 * @param n N slippage parameter
131 * @param c1 C1 slippage parameter in WAD
132 * @param xThreshold xThreshold slippage parameter in WAD
133 * @param cash cash position of asset in WAD
134 * @param liability liability position of asset in WAD
135 * @param amount amount to be withdrawn in WAD
136 * @return The final fee to be applied
137 */
138function _withdrawalFee(
139    uint256 k,
140    uint256 n,
141    uint256 c1,
142    uint256 xThreshold,
143    uint256 cash,
144    uint256 liability,
145    uint256 amount
146) internal pure returns (uint256) {
147    uint256 covBefore = cash.wdiv(liability);
148    if (covBefore >= WAD) {
149        return 0;
150    }
151
152    if (liability <= amount) {
153        return 0;
154    }
155
156    uint256 cashAfter;
157    // Cover case where cash <= amount
158    if (cash > amount) {
159        cashAfter = cash - amount;
160    } else {
161        cashAfter = 0;
162    }
163
164    uint256 covAfter = (cashAfter).wdiv(liability - amount);
165    uint256 slippageBefore = _slippageFunc(k, n, c1, xThreshold, covBefore);
166    uint256 slippageAfter = _slippageFunc(k, n, c1, xThreshold, covAfter);
167    uint256 slippageNeutral = _slippageFunc(k, n, c1, xThreshold, WAD); // slippage on cov = 1
168
169    // fee = [(Li - Di) * SlippageAfter] + [g(1) * Di] - [Li * SlippageBefore]
170    return
171        ((liability - amount).wmul(slippageAfter) + slippageNeutral.wmul(amount)) - liability.wmul(slippageBefore);
172}
173
174/**
175 * @notice Yellow Paper Def. 7.2 (Deposit Fee)
176 * @dev When covBefore <= 1, fee is 0
177 * @dev When covBefore > 1, we apply a fee to prevent deposit arbitrage
178 * @param k K slippage parameter in WAD
179 * @param n N slippage parameter
180 * @param c1 C1 slippage parameter in WAD
181 * @param xThreshold xThreshold slippage parameter in WAD
182 * @param cash cash position of asset in WAD
183 * @param liability liability position of asset in WAD
184 * @param amount amount to be deposited in WAD
185 * @return The final fee to be applied
186 */
187function _depositFee(
188    uint256 k,
189    uint256 n,
190    uint256 c1,
191    uint256 xThreshold,
192    uint256 cash,
193    uint256 liability,
194    uint256 amount
195) internal pure returns (uint256) {
196    // cover case where the asset has no liquidity yet
197    if (liability == 0) {
198        return 0;
199    }
200
201    uint256 covBefore = cash.wdiv(liability);
202    if (covBefore <= WAD) {
203        return 0;
204    }
205
206    uint256 covAfter = (cash + amount).wdiv(liability + amount);
207    uint256 slippageBefore = _slippageFunc(k, n, c1, xThreshold, covBefore);
208    uint256 slippageAfter = _slippageFunc(k, n, c1, xThreshold, covAfter);
209
210    // (Li + Di) * g(cov_after) - Li * g(cov_before)
211    return ((liability + amount).wmul(slippageAfter)) - (liability.wmul(slippageBefore));
212}

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23/**
24 * @notice Yellow Paper Def. 2.4 (Price Slippage Curve)
25 * @dev Calculates g(xr,i) or g(xr,j). This function always returns >= 0
26 * @param k K slippage parameter in WAD
27 * @param n N slippage parameter
28 * @param c1 C1 slippage parameter in WAD
29 * @param xThreshold xThreshold slippage parameter in WAD
30 * @param x coverage ratio of asset in WAD
31 * @return The result of price slippage curve
32 */
33function _slippageFunc(
34    uint256 k,
35    uint256 n,
36    uint256 c1,
37    uint256 xThreshold,
38    uint256 x
39) internal pure returns (uint256) {
40    if (x < xThreshold) {
41        return c1 - x;
42    } else {
43        return k.wdiv((((x * RAY) / WAD).rpow(n) * WAD) / RAY); // k / (x ** n)
44    }
45}