[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   #define NDEBUG
3   #include <assert.h>
4   #include <stdlib.h>
5   #include <string.h>
6   #include <math.h>
7   #ifdef NDEBUG
8   #define DEBUG_PRINT(fmt, ...) ((void)0)
9   #else
10  #define DEBUG_PRINT(fmt, ...) fprintf(stderr, fmt, ##__VA_ARGS__)
11  #endif
12  #define ASSERT_RANGE(min,x,max) assert(((x)>=(min)) && ((x)<=(max)))
13  int sort_int(int *a, int N) {
14  	int i;
15  	for (i = 0; i < N-1; i++) {
16  		int j;
17  		for (j = i + 1; j < N; j++) {
18  			if (a[i] > a[j]) {
19  				int tmp = a[i];
20  				a[i] = a[j];
21  				a[j] = tmp;
22  			}
23  		}
24  	}
25  	return 0;
26  }
27  void merge_int(int *a, int *b, int lower, int sep, int upper) {
28  	int i_lower = lower;
29  	int i_upper = sep;
30  	int i_b = 0;
31  	while ((i_lower < sep) && (i_upper < upper)) {
32  		if (a[i_lower] < a[i_upper]) {
33  			b[i_b++] = a[i_lower++];
34  		} else {
35  			b[i_b++] = a[i_upper++];
36  		}
37  	}
38  	if (i_lower >= sep) {
39  		while (i_upper < upper) {
40  			b[i_b++] = a[i_upper++];
41  		}
42  	} else {
43  		while (i_lower < sep) {
44  			b[i_b++] = a[i_lower++];
45  		}
46  	}
47  	for (i_lower = 0; i_lower < i_b; i_lower++) {
48  		a[i_lower+lower] = b[i_lower];
49  	}
50  }
51  void merge_sort_int(int *a, int *b, int lower, int upper) {
52  	int sep;
53  	if ((lower == upper) || (lower + 1 == upper)) {
54  		return;
55  	}
56  	sep = (lower + upper) / 2;
57  	merge_sort_int(a,b,lower,sep);
58  	merge_sort_int(a,b,sep,upper);
59  	merge_int(a,b,lower,sep,upper);
60  }
61  #define S_MAX 30000
62  int main(int argc, char **argv) {
63  	long long t1, t2;
64  	long long a1, a2;
65  	long long b1, b2;
66  	fscanf(stdin,"%lld%lld",&t1,&t2) > 0;
67  	fscanf(stdin,"%lld%lld",&a1,&a2) > 0;
68  	fscanf(stdin,"%lld%lld",&b1,&b2) > 0;
69  	long long diff1 = (a1-b1)*t1;
70  	long long diff2 = (a2-b2)*t2;
71  	long long diff3 = diff1 + diff2;
72  	DEBUG_PRINT("%lld %lld %lld\n",diff1, diff2, diff3);
73  	if ((diff1 > 0) && (diff2 > 0)) {
74  		fprintf(stdout,"0\n");
75  		return 0;
76  	}
77  	if ((diff1 < 0) && (diff2 < 0)) {
78  		fprintf(stdout,"0\n");
79  		return 0;
80  	}
81  	if ((diff1 > 0) && (diff2 < 0)) {
82  		if (diff3 > 0) {
83  			fprintf(stdout,"0\n");
84  			return 0;
85  		}
86  		if (diff3 == 0) {
87  			fprintf(stdout,"infinity\n");
88  			return 0;
89  		}
90  		assert(diff3 < 0);
91  		long long  n = diff1 / llabs(diff3);
92  		if ((diff1 * n) == llabs(diff3)) {
93  			fprintf(stdout,"%lld\n",2*n);
94  		} else {
95  			fprintf(stdout,"%lld\n",1+2*n);
96  		}
97  		return 0;
98  	}
99  	if ((diff1 < 0) && (diff2 > 0)) {
100  		if (diff3 < 0) {
101  			fprintf(stdout,"0\n");
102  			return 0;
103  		}
104  		if (diff3 == 0) {
105  			fprintf(stdout,"infinity\n");
106  			return 0;
107  		}
108  		assert(diff3 > 0);
109  		long long n = llabs(diff1) / diff3;
110  		if ((llabs(diff1) * n) == diff3) {
111  			fprintf(stdout,"%lld\n",2*n);
112  		} else {
113  			fprintf(stdout,"%lld\n",1+2*n);
114  		}
115  		return 0;
116  	}
117  	return 0;
118  }
```


**Question**: Does line `31` have control dependence over line `44` in function `merge_int`? If so, provide a trace.

**Output**:


Ground truth answer: no