[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   #include <assert.h>
5   #include <limits.h>
6   #include <inttypes.h>
7   #define NDEBUG
8   #define MAX(a, b) (((a) > (b)) ? (a) : (b))
9   #define MIN(a, b) (((a) < (b)) ? (a) : (b))
10  #define ABS(a) (((a) < (0)) ? -(a) : (a))
11  #define SGN(a) (((a) < 0) ? -1 : ((a) > 0))
12  #define UPDATEMAX(a, b) (((a) < (b)) ? (((a) = (b)), 1) : 0)
13  #define UPDATEMIN(a, b) (((a) > (b)) ? (((a) = (b)), 1) : 0)
14  #define SWAP(a, b) { char SWAP_tmp[sizeof(a)]; memcpy(SWAP_tmp, &(a), sizeof(a)); (a) = (b); memcpy(&(b), SWAP_tmp, sizeof(a)); }
15  #define  ALMOST_EQUAL(a, b) (((double) ABS((a) - (b))) < ((double) 0.000000001))
16  #define ALMOST_ZERO(a) ALMOST_EQUAL(a, ((double) 0))
17  static char *strPOSSIBLE = "POSSIBLE";
18  static char *strIMPOSSIBLE = "IMPOSSIBLE";
19  #define MAX_R 500
20  #define MAX_B 500
21  typedef struct {
22  	int T;
23  } game_t;
24  typedef struct {
25  	int id;
26  	game_t *game;
27  	int R, B;
28  	int res;
29  } round_t;
30  void solveRound(round_t *round)
31  {
32  	int r = round->R;
33  	int b = round->B;
34  	int k = 1;
35  	int tmp;
36  	while (1) {
37  		if ((r+b) < k) {
38  			break;
39  		}
40  		for (int i = k / 2; i >= 0; i--) {
41  			if ((r >= i) && (b >= (k -i))) {
42  				round->res++;
43  				r -= i;
44  				b -= k-i;
45  			}
46  			if ((2 * i) == k) {
47  				continue;
48  			}
49  			tmp = r;
50  			r = b;
51              b = tmp;
52  			if ((r >= i) && (b >= (k -i))) {
53  				round->res++;
54  				r -= i;
55  				b -= k-i;
56  			}
57  		}
58  		k++;
59  	}
60  }
61  void printRound(round_t *round)
62  {
63  	fprintf(stdout, "Case #%d: %d\n", round->id, round->res);
64  	fflush(stdout);
65  }
66  void playRound(round_t *round)
67  {
68  	solveRound(round);
69  	printRound(round);
70  }
71  void resetRound(round_t *round)
72  {
73  	game_t *game = round->game;
74  	memset(round, 0, sizeof(*round));
75  	round->game = game;
76  }
77  round_t *allocRound()
78  {
79  	round_t *round = malloc(sizeof(round_t));
80  	round->game = malloc(sizeof(game_t));
81  	resetRound(round);
82  	return round;
83  }
84  void fetchNextRound(round_t *round, int id)
85  {
86  	resetRound(round);
87  	round->id = id;
88  	scanf("%d%d", &round->R, &round->B);
89  }
90  int main()
91  {
92  	int T;
93  	scanf("%d", &T);
94  	round_t *round = allocRound();
95  	for (int i = 1; i <= T; i++) {
96  		fetchNextRound(round, i);
97  		playRound(round);
98  	}
99  }
```


**Question**: Does line `40` have control dependence over line `51` in function `solveRound`? If so, provide a trace.

**Output**:


Ground truth answer: yes