[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   #include<string.h>
3   #include<stdlib.h>
4   #include<stdbool.h>
5   #include<time.h>
6   #define inf 1072114514
7   #define llinf 4154118101919364364
8   #define mod 1000000007
9   #define pi 3.1415926535897932384
10  int max(int a,int b){if(a>b){return a;}return b;}
11  int min(int a,int b){if(a<b){return a;}return b;}
12  int zt(int a,int b){return max(a,b)-min(a,b);}
13  int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
14  int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
15  int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
16  int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
17  int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
18  int nHr(int a,int b){return nCr(a+b-1,b);}
19  int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
20  int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
21  int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
22  int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
23  int sankaku(int x){return ((1+x)*x)/2;}
24  long long llmax(long long a,long long b){if(a>b){return a;}return b;}
25  long long llmin(long long a,long long b){if(a<b){return a;}return b;}
26  long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
27  long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
28  long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
29  long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
30  long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
31  long long llnCr(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
32  long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
33  long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
34  long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
35  long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
36  long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
37  long long llsankaku(long long x){return ((1+x)*x)/2;}
38  double dbmax(double a,double b){if(a>b){return a;}return b;}
39  double dbmin(double a,double b){if(a<b){return a;}return b;}
40  double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
41  int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
42  int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
43  int llsortfncsj(const void *a,const void *b){if(*(long long *)a>*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
44  int llsortfnckj(const void *a,const void *b){if(*(long long *)a<*(long long *)b){return 1;}if(*(long long *)a==*(long long *)b){return 0;}return -1;}
45  int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
46  int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
47  int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
48  int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
49  int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
50  int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
51  void shuffledget(int x[],int n){
52      srand(time(0));
53      int i,b[524288],p,c;
54      for(i=0;i<n;i++){
55          b[i]=i;
56      }
57      for(i=n;i>=1;i--){
58          p=rand()%i;
59          c=b[i-1];b[i-1]=b[p];b[p]=c;
60      }
61      for(i=0;i<n;i++){
62          scanf("%d",&x[b[i]]);
63      }
64  }
65  int dx4[4]={1,-1,0,0};
66  int dy4[4]={0,0,1,-1};
67  int dx8[8]={-1,-1,-1,0,0,1,1,1};
68  int dy8[8]={-1,0,1,-1,1,-1,0,1};
69  int search(int x,int a[],int n){
70      int st=0,fi=n-1,te;
71      while(st<=fi){
72          te=(st+fi)/2;
73          if(a[te]<x){st=te+1;}else{fi=te-1;}
74      }
75      return st;
76  }
77  void prarr(int arr[],int n){
78    int i;
79    for(i=0;i<n;i++){
80      if(i){printf(" ");}
81      printf("%d",arr[i]);
82    }
83    printf("\n");
84    return;
85  }
86  typedef struct{
87  int val;
88  int node;
89  }sd;
90  int sdsortfnc(const void *a,const void *b){
91  if(((sd*)a)->val < ((sd*)b)->val){return -1;}
92  if(((sd*)a)->val > ((sd*)b)->val){return 1;}
93  return 0;
94  }
95  int main(void){
96      long long i,j,w,fl=0,dg,st,fi,te;
97      bool mem[16][2]={0};
98      char s[8],t[8];
99      for(i=9;i>=0;i--){
100        w=llpow(10,i);
101        printf("? %lld\n",w);
102        fflush(stdout);
103        scanf("%s",s);
104        if(s[0]=='Y')
105          {mem[i][0]=1;}
106        else
107          {fl|=2;}
108        w=llpow(10,i+1)-1;
109        printf("? %lld\n",w);
110        fflush(stdout);
111        scanf("%s",t);      
112        if(t[0]=='Y')
113          {mem[i][1]=1;}
114        else
115          {fl|=1;}
116      }
117      if(fl==1){
118        for(i=9;i>=0;i--){
119          if(mem[i][1]==0){
120            dg=i;
121            break;
122          }
123          if(i==0){
124            printf("! 1\n");
125            fflush(stdout);
126            return 0;
127          }
128        }
129      }
130      if(fl==2){
131        for(i=0;i<=9;i++){
132          if(mem[i][0]==0){
133            dg=i;
134            break;
135          }
136        }
137      }
138      if(fl==3){
139        for(i=0;i<=9;i++){
140          if((mem[i][0]&mem[i][1])==1){
141            dg=i;
142            break;
143          }
144        }
145      }
146      st = 1;
147      fi = 1;
148      for (i = 1; i <= dg; i++) 
149        st *= 10;
150      for (i = 1; i <= dg + 1; i++) 
151        fi *= 10;
152      fi -= 1;
153      while(st<=fi){
154        te=(st+fi)/2;
155        printf("? %lld\n",te*10);
156        fflush(stdout);
157        scanf("%s",s);
158        if(s[0]=='N'){
159          fi=te-1;
160        }
161        else{st=te+1;}
162      }
163      printf("! %lld\n",fi);
164      fflush(stdout);
165      return 0;
166  }
```


**Question**: Does `(t,111)` have data dependence over `(mem,113)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no