[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 <stdlib.h>
3   #include <string.h>
4   #include <math.h>
5   int A[101][101];
6   int B[101][101];
7   int r, c, h, v;
8   int guests, chips, share;
9   int fr(int rRemaining, int i, int j) {
10  	int k;
11  	if( (rRemaining + 1)*share != (B[r][j] - B[i][j]) )
12  		return 0;
13  	if(rRemaining == 0) 
14  		return 1;
15  	k = i+1;
16  	while(k < r) {
17  		if(B[k][j] - B[i][j] == share) {
18  			return fr(rRemaining - 1, k, j);
19  		}
20  		k++;
21  	}
22  	return 0;
23  }
24  int fc(int cRemaining, int i, int j) {
25  	int k;
26  	if( (cRemaining + 1)*share != (B[i][c] - B[i][j]) )
27  		return 0;
28  	if(cRemaining == 0) 
29  		return 1;
30  	k = i+1;
31  	while(k < c) {
32  		if(B[i][k] - B[i][j] == share) {
33  			return fr(cRemaining - 1, i, k);
34  		}
35  		k++;
36  	}
37  	return 0;
38  }
39  int main() {
40  	int t, tc;
41  	int i, j, possible;
42  	char str[101];
43  	scanf("%d", &t);
44  	tc = 1;
45  	while(tc <= t) {
46  		scanf("%d", &r);
47  		scanf("%d", &c);
48  		scanf("%d", &h);
49  		scanf("%d", &v);
50  		guests = (h+1)*(v+1);
51  		chips = 0;
52  		for (i = 0; i <= r; ++i) {
53  			if(i != 0) scanf("%s", str);
54  			for (j = 0; j <= c; ++j) {
55  				if(i == 0 || j == 0) 
56  					A[i][j] = B[i][j] = 0;
57  				else {
58  					A[i][j] = (str[j-1] == '@') ? 1 : 0; 
59  					if(A[i][j]) 
60  						chips++;
61  				}
62  			}
63  		}
64  		if(chips % guests != 0) {
65  			printf("Case #%d: IMPOSSIBLE\n", tc);
66  		} else {
67  			for (i = 1; i <= r; ++i) {
68  				for (j = 1; j <= c; ++j) {
69  					B[i][j] = A[i][j] + B[i-1][j] + B[i][j-1] - B[i-1][j-1];
70  				}
71  			}
72  			share = chips / guests;
73  			possible = 0;
74  			for (i = 1; i < r && !possible; ++i) {
75  				for (j = 1; j < c && !possible; ++j) {
76  					if(B[i][j] == share) {
77  						possible = 1;
78  						if(!fr(h - 1, i, j)) {
79  							possible = 0;
80  						}
81  						if(!fc(v - 1, i, j)) {
82  							possible = 0;
83  						}
84  						if(possible) {
85  							printf("Case #%d: POSSIBLE\n", tc);
86  						}
87  					}
88  				}
89  			}
90  			if(!possible) printf("Case #%d: IMPOSSIBLE\n", tc);
91  		}
92  		tc++;
93  	}
94  	return 0;
95  }
```


**Question**: Does line `45` have control dependence over line `56` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes