[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

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


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

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


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

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

---

[YOUR TURN]

Below is **your target snippet**. 

```C
1   #include<stdio.h>
2   #include<math.h>
3   int min(int a, int b)
4   {
5       if (a < b)
6           return a;
7       else
8           return b;
9   }
10  int main()
11  {
12      int n, m;
13      scanf("%d %d", &n, &m);
14      int a[1003], b[1003];
15      int i;
16      for (i = 0; i < n; i++)
17          scanf("%d", &a[i]);
18      for (i = 0; i < m; i++)
19          scanf("%d", &b[i]);
20      int x;
21      for(i=0;i<n-1;i++)
22          if (a[i] <= a[i + 1])
23          {
24              if (a[i] == a[i + 1])
25              {
26                  printf("0\n");
27                  return 0;
28              }
29              x = a[i];
30              a[i] = a[i + 1];
31              a[i + 1] = x;
32              if (i > 0)
33                  i -= 2;
34          }
35      for(i=0;i<m-1;i++)
36          if (b[i] <= b[i + 1])
37          {
38              if (b[i] == b[i + 1])
39              {
40                  printf("0\n");
41                  return 0;
42              }
43              x = b[i];
44              b[i] = b[i + 1];
45              b[i + 1] = x;
46              if (i > 0)
47                  i -= 2;
48          }
49      int p = 1000000007;
50      double ans = 1;
51      int k = 0;
52      int y;
53      int j;
54      int q;
55      for (i = n * m; i >= min(a[n - 1], b[m - 1]); i--)
56      {
57          x = y = q = 0;
58          j = 0;
59          while (a[j] > i)
60          {
61              j++;
62              x++;
63          }
64          if (a[j] == i)
65          {
66              x = 1;
67              q++;
68          }
69          j = 0;
70          while (b[j] > i)
71          {
72              y++;
73              j++;
74          }
75          if (b[j] == i)
76          {
77              y = 1;
78              q++;
79          }
80          if (q==0)
81              ans *= x * y - k;
82          else
83              ans *= x * y;
84          ans = round(fmod(ans, p));
85          k++;
86      }
87      for (j = 0; j < i; j++)
88      {
89          ans *= j + 1;
90          ans = round(fmod(ans, p));
91      }
92      printf("%.0lf\n", round(ans));
93      return 0;
94  }
```


**Question**: Does line `55` have control dependence over line `81` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes