[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.

## 1. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` is **defined or updated** at `lineNumber`.


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Which variable instances have information flow over `(targetVar, targetLine)`? List all such variables.", respond:, respond with a JSON object:

```json
   {
    "InfomationFlowSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

If you believe there is no variables have information flow on `(targetVar, targetLine)`, respond:

```json
{
  "InfomationFlowSources": false
}
```


## 3. **Intraprocedural Data Dependence**  
   All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.


## 4. **Example Code Snippet**:


### Example 1
```python
1  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Which variable instances have information flow over `(transaction,10)`? List all such variables.


**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["limit", 9],
      ["status", 2],
      ["status", 7],
      ["balance", 4],
      ["balance", 5],
    ]
   }
```

 
#### Example Question 1.2
Which variable instances have information flow over `(flag,3)`? List all such variables.

**Analysis**
Line 3 initializes `flag` without any data dependency or control dependency. 


**Output**:
```json
{
  "InfomationFlowSources": []
}
```



### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Which variable instances have information flow over `(summary,17)`? List all such variables.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
    ]
   }
```


**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. `diff` is from line 13. Next, we focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 never evaluates to `true`, `(summary,17)` reads `j` from `(j,5)`. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9. There is also an explicit flow from `size` to `j`.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`. It also reads `i` from line 15 or line 4.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have information flow over `(score,11)`? List all such variables.


**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`. `arr` may come from line 3, line 8, or line 10 given the index `j+1` is with an unknown value. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop). Similar for `arr`, it could be read from line 3, line 8, or line 10.
- If `arr` is read from line 10, there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.
- If `arr` is read from line 8, there is an explicit flow from `(j,5)` and `(size,2)` to `arr`. There is also implicit flow from `val` (as in the condition in line 7) read from line 1.
- If `arr` is read from line 3, there is an explicit flow from `(size,2)` to `arr`.
- Line 11 is control-dependent on line 9, therefore there is a implicit flow from `i` (possibly from `(i,15)` or `(i,4)`) and `size` (possibly from `(size,2)`) to line 11.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```



#### Example Question 2.3:
Which variable instances have information flow over `(last,16)`? List all such variables.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8, by the loop updates in line 10, or by initialization at line 3. Under static analysis, multiple update sites may contribute. 
- If from Line 3: there is an explicit flow from `(size,2)` to it.
- If from line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7. There is also an explicit flow from `(j,5)` and `(size,2)` to `arr`.
- If from line 10:  there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.


**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.ByteArrayInputStream;
2   import java.io.IOException;
3   import java.io.InputStream;
4   import java.io.PrintWriter;
5   import java.util.Arrays;
6   import java.util.InputMismatchException;
7   public class Main {
8   	static InputStream is;
9   	static PrintWriter out;
10  	static String INPUT = "";
11  	static void solve()
12  	{
13  		int n = ni(), A = ni(), B = ni();
14  		int[] P = new int[n];
15  		for(int i = 0;i < n;i++)
16  			P[i] = ni();
17  		for(int i = 0;i < n;i++)
18  			P[i]--;
19  		long[][] dp = new long[n+1][n+1];
20  		for(int i = 0;i <= n;i++)
21  			Arrays.fill(dp[i], Long.MAX_VALUE / 2);
22  		for(int i = 0;i <= n;i++){
23  			for(int j = 0;j <= n;j++){
24  				if(i == 0 && j == 0){
25  					dp[i][j] = 0;
26  				}else if(i-1 >= 0 && j-1 >= 0 && P[i-1] == j-1){
27  					dp[i][j] = dp[i-1][j-1];
28  				}else{
29  					if(i-1 >= 0)
30  						dp[i][j] = Math.min(dp[i][j], dp[i-1][j] + (P[i-1] <= j ? B : A));
31  					if(j-1 >= 0)
32  						dp[i][j] = Math.min(dp[i][j], dp[i][j-1]);
33  				}
34  			}
35  		}
36  		out.println(dp[n][n]);
37  	}
38  	public static void main(String[] args) throws Exception
39  	{
40  		long S = System.currentTimeMillis();
41  		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
42  		out = new PrintWriter(System.out);
43  		solve();
44  		out.flush();
45  		long G = System.currentTimeMillis();
46  		tr(G-S+"ms");
47  	}
48  	private static boolean eof()
49  	{
50  		if(lenbuf == -1)return true;
51  		int lptr = ptrbuf;
52  		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
53  		try {
54  			is.mark(1000);
55  			while(true){
56  				int b = is.read();
57  				if(b == -1){
58  					is.reset();
59  					return true;
60  				}else if(!isSpaceChar(b)){
61  					is.reset();
62  					return false;
63  				}
64  			}
65  		} catch (IOException e) {
66  			return true;
67  		}
68  	}
69  	private static byte[] inbuf = new byte[1024];
70  	static int lenbuf = 0, ptrbuf = 0;
71  	private static int readByte()
72  	{
73  		if(lenbuf == -1)throw new InputMismatchException();
74  		if(ptrbuf >= lenbuf){
75  			ptrbuf = 0;
76  			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
77  			if(lenbuf <= 0)return -1;
78  		}
79  		return inbuf[ptrbuf++];
80  	}
81  	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
82  	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
83  	private static double nd() { return Double.parseDouble(ns()); }
84  	private static char nc() { return (char)skip(); }
85  	private static String ns()
86  	{
87  		int b = skip();
88  		StringBuilder sb = new StringBuilder();
89  		while(!(isSpaceChar(b))){
90  			sb.appendCodePoint(b);
91  			b = readByte();
92  		}
93  		return sb.toString();
94  	}
95  	private static char[] ns(int n)
96  	{
97  		char[] buf = new char[n];
98  		int b = skip(), p = 0;
99  		while(p < n && !(isSpaceChar(b))){
100  			buf[p++] = (char)b;
101  			b = readByte();
102  		}
103  		return n == p ? buf : Arrays.copyOf(buf, p);
104  	}
105  	private static char[][] nm(int n, int m)
106  	{
107  		char[][] map = new char[n][];
108  		for(int i = 0;i < n;i++)map[i] = ns(m);
109  		return map;
110  	}
111  	private static int[] na(int n)
112  	{
113  		int[] a = new int[n];
114  		for(int i = 0;i < n;i++)a[i] = ni();
115  		return a;
116  	}
117  	private static int ni()
118  	{
119  		int num = 0, b;
120  		boolean minus = false;
121  		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
122  		if(b == '-'){
123  			minus = true;
124  			b = readByte();
125  		}
126  		while(true){
127  			if(b >= '0' && b <= '9'){
128  				num = num * 10 + (b - '0');
129  			}else{
130  				return minus ? -num : num;
131  			}
132  			b = readByte();
133  		}
134  	}
135  	private static long nl()
136  	{
137  		long num = 0;
138  		int b;
139  		boolean minus = false;
140  		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
141  		if(b == '-'){
142  			minus = true;
143  			b = readByte();
144  		}
145  		while(true){
146  			if(b >= '0' && b <= '9'){
147  				num = num * 10 + (b - '0');
148  			}else{
149  				return minus ? -num : num;
150  			}
151  			b = readByte();
152  		}
153  	}
154  	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
155  }
```


**Question**: Which variable instances have information flow over `(dp,27)` in function `solve`? List all such variables.

**Output**:


Ground truth answer: yes