[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 <string.h>
3   #include <stdlib.h>
4   #define lli long long int
5   #define BIG 1000000007
6   void debug_c(char **dp, int h, int w)
7   {
8   	printf("----debug\n");
9   	int i = 0;
10  	int j = 0;
11  	while (i < h)
12  	{
13  		j = 0;
14  		while (j < w)
15  		{
16  			printf("%d,",dp[i][j]);
17  			j++;
18  		}
19  		printf("\n");
20  		i++;
21  	}
22  	printf("----debug\n");
23  }
24  void debug(int **dp, int h, int w)
25  {
26  	printf("----debug\n");
27  	int i = 0;
28  	int j = 0;
29  	while (i < h)
30  	{
31  		j = 0;
32  		while (j < w)
33  		{
34  			printf("%d",dp[j][i]);
35  			j++;
36  		}
37  		printf("\n");
38  		i++;
39  	}
40  	printf("----debug\n");
41  }
42  int main(void)
43  {	
44  	int h, w;
45  	scanf("%d %d", &h, &w);
46  	int i = 0;
47  	int j = 0;
48  	char **a;
49  	int **dp;
50  	a = (char **)malloc(sizeof(char *)*(h));
51  	dp = (int **)malloc(sizeof(int *)*(w));
52  	while (i < w)
53  	{
54  		dp[i] = (int *)malloc(sizeof(int)*(h));
55  		i++;
56  	}
57  	while (j < h)
58  	{
59  		a[j] = (char *)malloc(sizeof(char)*(w));
60  		j++;
61  	}
62  	i = 0;
63  	j = 0;
64  	while (i < h)
65  	{
66  		j = 0;
67  		scanf("%s",a[i]);
68  		while (j < w)
69  		{
70  			dp[j][i] = 0;
71  			if(a[i][j] == '#')
72  			{
73  				dp[j][i] = -1;
74  			}
75  			j++;
76  		}
77  		i++;
78  	}
79  	dp[0][0] = 1;
80  	i = 1;
81  	j = 1;
82  	while (i < h)
83  	{
84  		if(dp[0][i] == -1)
85  		{
86  			dp[0][i] = 0;
87  		}
88  		else
89  		{
90  			dp[0][i] = dp[0][i-1];
91  		}
92  		i++;
93  	}
94  	while (j < w)
95  	{
96  		if(dp[j][0] == -1)
97  		{
98  			dp[j][0] = 0;
99  		}
100  		else
101  		{
102  			dp[j][0] = dp[j-1][0];
103  		}
104  		j++;
105  	}
106  	i = 1;
107  	j = 1;
108  	while (i< h)
109  	{
110  		j = 1;
111  		while (j < w)
112  		{
113  			if(dp[j][i] == -1)
114  			{
115  				dp[j][i] = 0;
116  			}
117  			else
118  			{
119  				dp[j][i] = (dp[j-1][i]+dp[j][i-1])%BIG;
120  			}
121  			j++;
122  		}
123  		i++;
124  	}
125  	printf("%d\n",dp[w-1][h-1]);
126  	return (0);
127  }
```


**Question**: Does line `84` have control dependence over line `86` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes