[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis 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. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

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

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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**: Does `(i,15)` have data dependence over `(P,26)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes