[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

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

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(i_lower,28)` have data dependence over `(b,44)` in function `merge_int`? If so, provide a trace.

**Output**:


Ground truth answer: yes