diff --git a/amm-v2-docs/formula.md b/amm-v2-docs/formula.md
index 31db604..b8d9dcd 100644
--- a/amm-v2-docs/formula.md
+++ b/amm-v2-docs/formula.md
@@ -70,5 +70,40 @@ $$\Delta y = \frac{\Delta L}{L} * y_{0}$$
 In case users want to zap out to Asset B, $\Delta x$ will be swapped, then total amount users receive is
 $$Out = \Delta y + \frac{(f_{d} - f_{n}) * \Delta x * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * \Delta x}$$
 
-### 5. Withdraw Imbalance
-TODO
\ No newline at end of file
+### 6. Withdraw Imbalance
+
+Users want to withdraw with a ratio $A/B$. 
+
+We have the basic withdrawal formulas:
+
+$$\Delta x = \frac{\Delta L}{L} * x_{0}$$
+$$\Delta y = \frac{\Delta L}{L} * y_{0}$$
+
+Suppose we need to swap some in $\Delta x$ to get $\frac{\Delta x'}{\Delta y'} = \frac{A}{B}$.
+
+So we have the formula:
+$$\frac{\Delta x - swap_{x}}{\Delta y + receive_{y}} =\frac{A}{B} (1)$$
+
+
+$$ receive_{y} = \frac{(f_{d} - f_{n}) * swap_{x} * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * swap_{x}}(2)$$
+
+(With $x_{0}$ and $y_{0}$ being $x$ and $y$ after withdrawal, $swap_{x}$ is the amount need to be swapped to adapt the ratio $A/B$ that users expect).
+
+Combination of formula (1) and (2), we have:
+
+$$a * swap_{x} ^ 2 + b * swap_{x} + c = 0$$
+where 
+$$a = (f_{d} - f_{n}) * B$$
+$$b = A*(f_{d} - f_{n})*(y_{0}+\Delta y) + B *(f_{d} * x_{0} - (f_{d} - f_{n})*\Delta x)$$
+$$ c =f_{d} * x_{0} *(A * \Delta y - B * \Delta x) $$
+
+### 7. Partial Swap
+Allow users swap only if price is exactly matched.
+
+In case users want to swap with price $A/B$
+We have 2 formulas: 
+$$\frac{\Delta x}{\Delta y} = \frac{A}{B}  (1)$$
+$$ \Delta y = \frac{(f_{d} - f_{n}) * \Delta x * y_{0}}{x_{0} * f_{d} + (f_{d} - f_{n}) * \Delta x} (2)$$
+We can calculate $\Delta x$: 
+$$\Delta x = \frac{A * (f_{d} - f_{n}) * y_{0} - B * f_{d} * x_{0}}{(f_{d} - f_{n}) * B}$$
+where $\Delta x$ is the maximum amount can be swapped to adapt $A/B$ ratio
\ No newline at end of file
diff --git a/lib/amm_dex_v2/math.ak b/lib/amm_dex_v2/math.ak
index d2b7bfa..4317096 100644
--- a/lib/amm_dex_v2/math.ak
+++ b/lib/amm_dex_v2/math.ak
@@ -362,8 +362,8 @@ pub fn calculate_zap_out(
 // a * swap_amount_in ^ 2 + b * swap_amount_in + c = 0
 // Where:
 // - a = fee_diff * expect_io_ratio_denominator
-// - b = expect_io_ratio_numerator * fee_diff ( reserve_out + amount_out ) * expect_io_ratio_denominator * (fee_denominator * reserve_in - fee_diff * amount_in)
-// - c = fee_denominator * (expect_io_ratio_numerator * amount_out - expect_io_ratio_denominator * amount_in) 
+// - b = expect_io_ratio_numerator * fee_diff ( reserve_out + amount_out ) + expect_io_ratio_denominator * (fee_denominator * reserve_in - fee_diff * amount_in)
+// - c = fee_denominator * reserve_in* (expect_io_ratio_numerator * amount_out - expect_io_ratio_denominator * amount_in) 
 fn calculate_withdraw_swap_amount(
   amount_in: Int,
   amount_out: Int,
diff --git a/lib/amm_dex_v2/order_validation.ak b/lib/amm_dex_v2/order_validation.ak
index e600262..35c78a0 100644
--- a/lib/amm_dex_v2/order_validation.ak
+++ b/lib/amm_dex_v2/order_validation.ak
@@ -42,7 +42,7 @@ fn validate_swap_exact_in(
   profit_sharing_opt: Option<(Int, Int)>,
   pool_state: PoolState,
   kill_on_failed: Bool,
-  is_slippage_sastified: fn(Int) -> Bool,
+  is_slippage_satisfied: fn(Int) -> Bool,
 ) -> PoolState {
   let (
     datum_reserve_a,
@@ -98,8 +98,8 @@ fn validate_swap_exact_in(
       trading_fee_denominator: trading_fee_denominator,
       fee_sharing_opt: profit_sharing_opt,
     )
-  let slippage_sastified = is_slippage_sastified(amount_out)
-  if slippage_sastified {
+  let slippage_satisfied = is_slippage_satisfied(amount_out)
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -220,8 +220,8 @@ fn validate_swap_exact_out(
       trading_fee_denominator: trading_fee_denominator,
       fee_sharing_opt: profit_sharing_opt,
     )
-  let slippage_sastified = necessary_amount_in <= maximum_swap_amount
-  if slippage_sastified {
+  let slippage_satisfied = necessary_amount_in <= maximum_swap_amount
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -317,8 +317,8 @@ fn validate_deposit(
       trading_fee_b_denominator: trading_fee_b_denominator,
       profit_sharing_opt: profit_sharing_opt,
     )
-  let slippage_sastified = lp_amount >= minimum_lp
-  if slippage_sastified {
+  let slippage_satisfied = lp_amount >= minimum_lp
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -392,11 +392,11 @@ fn validate_withdraw(
       withdrawal_lp_amount,
       total_liquidity,
     )
-  let slippage_sastified = and {
+  let slippage_satisfied = and {
       amount_a_out >= minimum_amount_a,
       amount_b_out >= minimum_amount_b,
     }
-  if slippage_sastified {
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -486,8 +486,8 @@ fn validate_zap_out(
       trading_fee_b_denominator: trading_fee_b_denominator,
       profit_sharing_opt: profit_sharing_opt,
     )
-  let slippage_sastified = amount_out >= minimum_receive
-  if slippage_sastified {
+  let slippage_satisfied = amount_out >= minimum_receive
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -706,8 +706,8 @@ fn validate_withdraw_imbalance(
       total_liquidity: total_liquidity,
       profit_sharing_opt: profit_sharing_opt,
     )
-  let slippage_sastified = amount_a_out >= minimum_amount_a
-  if slippage_sastified {
+  let slippage_satisfied = amount_a_out >= minimum_amount_a
+  if slippage_satisfied {
     let expect_order_value_out =
       order_in_value
         |> value.add(ada_policy_id, ada_asset_name, -used_batcher_fee)
@@ -1060,7 +1060,7 @@ pub fn apply_orders(
           profit_sharing_opt: profit_sharing_opt,
           pool_state: pool_state,
           kill_on_failed: order_killable,
-          is_slippage_sastified: fn(amount_out) {
+          is_slippage_satisfied: fn(amount_out) {
             amount_out >= minimum_receive
           },
         )
@@ -1089,7 +1089,7 @@ pub fn apply_orders(
           profit_sharing_opt: profit_sharing_opt,
           pool_state: pool_state,
           kill_on_failed: False,
-          is_slippage_sastified: fn(amount_out) {
+          is_slippage_satisfied: fn(amount_out) {
             amount_out <= stop_loss_receive
           },
         )
@@ -1119,7 +1119,7 @@ pub fn apply_orders(
           profit_sharing_opt: profit_sharing_opt,
           pool_state: pool_state,
           kill_on_failed: False,
-          is_slippage_sastified: fn(amount_out) {
+          is_slippage_satisfied: fn(amount_out) {
             amount_out >= minimum_receive || amount_out <= stop_loss_receive
           },
         )
diff --git a/plutus.json b/plutus.json
index cc3622f..462cc16 100644
--- a/plutus.json
+++ b/plutus.json
@@ -64,8 +64,8 @@
           }
         }
       ],
-      "compiledCode": 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-      "hash": "1e59d3693711b222d24821cc562b64605949ca666a369c02bbd189b7"
+      "compiledCode": 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+      "hash": "96ac8583651b71682fdb6422cd9abd927c000429a3172e5dffeac47d"
     },
     {
       "title": "order_validator.validate_expired_order_cancel",
diff --git a/validators/authen_minting_policy.ak b/validators/authen_minting_policy.ak
index 1a76c62..5f42ed9 100644
--- a/validators/authen_minting_policy.ak
+++ b/validators/authen_minting_policy.ak
@@ -20,7 +20,6 @@ validator(
     let ScriptContext { transaction, purpose } = context
     expect Mint(authen_policy_id) = purpose
     when redeemer is {
-      // The redeemer can be called once to initialize the whole AMM V2 system
       CreatePool -> {
         let Transaction { inputs, mint, redeemers, .. } = transaction
         // validate that there's a single Factory UTxO in the Transaction Inputs. 
@@ -61,7 +60,6 @@ validator(
           policy_id: asset_b_policy_id,
           asset_name: asset_b_asset_name,
         } = asset_b
-        // todo, consider whether we need this logic or not?
         expect utils.sorted_asset(asset_a, asset_b)
         let lp_asset_name =
           utils.compute_lp_asset_name(
@@ -75,6 +73,7 @@ validator(
           lp_asset_name: lp_asset_name,
         )
       }
+      // The redeemer can be called once to initialize the whole AMM V2 system
       MintFactoryAuthen -> {
         let Transaction { inputs, mint, outputs, .. } = transaction
 
diff --git a/validators/factory_validator.ak b/validators/factory_validator.ak
index 5232d75..5f988b3 100644
--- a/validators/factory_validator.ak
+++ b/validators/factory_validator.ak
@@ -238,6 +238,10 @@ validator(
       // Pool Reserve must be the same between datum and value 
       pool_datum_reserve_a == amount_a,
       pool_datum_reserve_b == amount_b,
+      // Initial Pool Reserves and liquditiy must be positive
+      amount_a > 0,
+      amount_b > 0,
+      total_liquidity > 0,
       // trading_fee_percentage must be between **0.05%** and **10%**
       is_valid_trading_fee,
       // Dynamic Fee is not allowed in Pool Creation