[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” 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. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

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


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Is there information flow from `(varA,lineA)` to `(varB,lineB)`? If so, provide one feasible trace.", respond with a JSON object:

- If information flow exists:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["varOrExpr", lineNumber            /*, "use" optional */],
      "to":   ["varOrExpr", lineNumber            /*, "use" optional */],
      "type": "data" | "control"
    },
    // … additional edges
  ]
}
```
- If no information flow exists, omit the Trace field:
```json
{
  "InformationFlow": false
}
```

- Every trace edge must specify "from", "to" (each may include a third "use" element when the line is a conditional statement that only reads without redefining/updating the value), and "type".
- **We do not consider loop in the trace**.


A **trace** represents a **transitive chain of direct flows** arise from compositions of **direct** implicit or explicit flows, where each edge represents a **direct implicit flow** (through control dependence) or a **direct explicit flow** edge (through data dependence).


You only need to provide **one** valid trace if you conclude a information flow 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  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Is there information flow from `(balance,4)` to `(transaction,10)`? If so, provide a trace.

**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. Here we focus on `(status,7)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  
Hence, A transitive of flow of direct implicit or explicit flows forms a information flow trace.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["balance", 4],
      "to":   ["balance", 5],
      "type": "data"
    },
    {
      "from": ["balance", 5],
      "to":   ["balance", 6, "use"],
      "type": "data"
    },
    {
      "from": ["balance", 6, "use"],
      "to":   ["status", 7],
      "type": "control"
    },
    {
      "from": ["status", 7],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```

 
- The edge from line 6 to `(status,7)` is through control dependence and therefore with a type of "control". 
- Line 6 only reads the value of `balance` without defining/updating it. So it is marked with `"use"`. 
- You only need to provide **one** valid chain if you conclude a the information flow exists. 


#### Example Question 1.2

Is there information flow from  `(flag,3)` to `(limit,9)`? If so, provide a trace.

**Analysis**
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from either `(status,7)` (if line 6’s condiation evaluates to `true`) or on the original `(status,2)` (if not) to `(limit,9)`. 
- Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 6 is treated as being evaluated to both possibly true and possibly false.
  - If line 6 is true, `(limit,9)` depends on `(status,7)`. 
    - Line 7: `(status,7)` doesn't have any data dependence as it is assigned with a constant number. The execution of line 7 is control-dependent on the condition in line 6.
    - Line 6: the condition at line 6 directly reads from `(balance,5)`. 
    - Line 5: `(balance,5)` is updated based on the previous value of `balance` in line 4.
    - Line 4: `(balance,4)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
  - If line 6 is not true, `(limit,9)` depends on `(status,2)`. 
    - Line 2: `(status,2)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
- Therefore, among all the possible paths, there is no transitive (indirect) chain of implicit or explicit flow from `(flag,3)` to `(limit,9)`. The information flow doesn't exist.


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

#### Example Question 1.3

Is there information flow from  `(status,2)` to `(transaction,10)`? If so, provide a trace.

**Analysis**
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`.
- Line 9: `(limit,9)` reads `status`, o there is a direct explicit flow from either `(status,7)`, or the original `(status,2)`, to `(limit,9)`.
- Similar to the above example, line 6 is possibly evaluated to true or false.
- If line 6 evaluates to `true`, there is a direct explicit flow from `(status,7)` to `(limit,9)`, which directly data-dependent on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (status,7) -> (limit,9) -> (transaction,10)`.
- If line 6 evaluates to `false`, `(limit,9)` has a direct data dependence on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (limit,9) -> (transaction,10)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["status", 2],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```


### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Is there information flow from  `(size,2)` to `(summary,17)`? If so, provide a trace.

**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. We focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`.
- Line 2: `size` is initialized with a constant number. It is not guarded with any conditions. 
- Therefore, there is an transitive trace of direct implicit/excplit flows from `(size,2)` to `(summary,17)`.

Only one valid trace is required for the output. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["size", 2],
      "to":   ["size", 9, "use"],
      "type": "data"
    },
    {
      "from": ["size", 9, "use"],
      "to":   ["j", 14],
      "type": "control"
    },
    {
      "from": ["j", 14],
      "to":   ["summary", 17],
      "type": "data"
    }
  ]
}
```


#### Example Question 2.2:
Is there information flow from  `(j,5)` to `(score,11)`? If so, provide a trace. 

**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`.
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop).
- If line 11 reads `j` from line 5, it forms a tansitive explicit flow trace: `(j,5) -> (score,11)`.
- If line 11 reads `j` from line 14, it forms a tansitive explicit flow trace: `(j,5) -> (j,14) -> (score,11)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["j", 5],
      "to":   ["score", 11],
      "type": "data"
    }
  ]
}
```



#### Example Question 2.3:
Is there information flow from  `(j,1)` to `(i,15)`? If so, provide a trace.

**Analysis**: 
- Line 15: `(i,15)` updates `i` based on its previous value. So there is a direct explicit flow from either `(i,4)` (initialization) or itself (loop-carried) to `(i,15)`. But we do not consider loop-carried dependencies across iterations in this analysis. So we focus on `(i,4)`.
- The execution of line 15 is conditional on the while loop at line 9.
- Line 9: `while i < size` depends on `i` and `size`. It directly reads from `(i,4)` (or possibly `(i,15)`) and `(size,2)`.
- Both `(i,4)` and `(size,2)` are initialized with constants and are not themselves control-dependent on any earlier values or branches.
- There is no use of `(j,1)` in the computation of `(i,15)`, nor does `(j,1)` influence whether line 15 executes.
- Therefore, there is no transitive flow, explicit or implicit, from `(j,1)` to `(i,15)`.


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


#### Example Question 2.4:
Is there information flow from  `(val,1)` to `(last,16)`? If so, provide a trace.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8 (or by the loop updates in line 10). Under static analysis, multiple update sites may contribute. We conservatively pick one valid trace (line 8).
- Line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7.
- Therefore, there exists a trace from `(val,1)` to `(last,16)`, and the edge from line 7 to `(arr,8)` is through implicit and explicit flows. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["val", 1],
      "to":   ["val", 7, "use"],
      "type": "data"
    },
    {
      "from": ["val", 7, "use"],
      "to":   ["arr", 8],
      "type": "control"
    },
    {
      "from": ["arr", 8],
      "to":   ["last", 16],
      "type": "data"
    }
  ]
}
```

---

[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   #include<assert.h>
7   #define inf 1072114514
8   #define llinf 4154118101919364364
9   #define mod 1000000007
10  #define pi 3.1415926535897932384
11  int max(int a,int b){if(a>b){return a;}return b;}
12  int min(int a,int b){if(a<b){return a;}return b;}
13  int zt(int a,int b){return max(a,b)-min(a,b);}
14  int round(int a,int b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
15  int ceil(int a,int b){if(a%b==0){return a/b;}return (a/b)+1;}
16  int gcd(int a,int b){int c;while(b!=0){c=a%b;a=b;b=c;}return a;}
17  int lcm(int a,int b){int c=gcd(a,b);a/=c;return a*b;}
18  int nCr(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=(a+1-i);r/=i;}return r;}
19  int nHr(int a,int b){return nCr(a+b-1,b);}
20  int fact(int a){int i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
21  int pow(int a,int b){int i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
22  int dsum(int x){int r=0;while(x){r+=(x%10);x/=10;}return r;}
23  int dsumb(int x,int b){int r=0;while(x){r+=(x%b);x/=b;}return r;}
24  int sankaku(int x){return ((1+x)*x)/2;}
25  void swap(int *a,int *b){int c;c=(*a);(*a)=(*b);(*b)=c;}
26  long long llmax(long long a,long long b){if(a>b){return a;}return b;}
27  long long llmin(long long a,long long b){if(a<b){return a;}return b;}
28  long long llzt(long long a,long long b){return llmax(a,b)-llmin(a,b);}
29  long long llround(long long a,long long b){if((a%b)*2 >= b){return (a/b)+1;}return a/b;}
30  long long llceil(long long a,long long b){if(a%b==0){return a/b;}return (a/b)+1;}
31  long long llgcd(long long a,long long b){long long c;while(b!=0){c=a%b;a=b;b=c;}return a;}
32  long long lllcm(long long a,long long b){long long c=llgcd(a,b);a/=c;return a*b;}
33  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;}
34  long long llnHr(long long a,long long b){return llnCr(a+b-1,b);}
35  long long llfact(long long a){long long i,r=1;for(i=1;i<=a;i++){r*=i;}return r;}
36  long long llpow(long long a,long long b){long long i,r=1;for(i=1;i<=b;i++){r*=a;}return r;}
37  long long lldsum(long long x){long long r=0;while(x){r+=(x%10);x/=10;}return r;}
38  long long lldsumb(long long x,long long b){long long r=0;while(x){r+=(x%b);x/=b;}return r;}
39  long long llsankaku(long long x){return ((1+x)*x)/2;}
40  void llswap(long long *a,long long *b){long long c;c=(*a);(*a)=(*b);(*b)=c;}
41  double dbmax(double a,double b){if(a>b){return a;}return b;}
42  double dbmin(double a,double b){if(a<b){return a;}return b;}
43  double dbzt(double a,double b){return dbmax(a,b)-dbmin(a,b);}
44  void dbswap(double *a,double *b){double c;c=(*a);(*a)=(*b);(*b)=c;}
45  void chswap(char *a,char *b){char c;c=(*a);(*a)=(*b);(*b)=c;}
46  int sortfncsj(const void *a,const void *b){if(*(int *)a>*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
47  int sortfnckj(const void *a,const void *b){if(*(int *)a<*(int *)b){return 1;}if(*(int *)a==*(int *)b){return 0;}return -1;}
48  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;}
49  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;}
50  int dbsortfncsj(const void *a,const void *b){if(*(double *)a>*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
51  int dbsortfnckj(const void *a,const void *b){if(*(double *)a<*(double *)b){return 1;}if(*(double *)a==*(double *)b){return 0;}return -1;}
52  int strsortfncsj(const void *a,const void *b){return strcmp((char *)a,(char *)b);}
53  int strsortfnckj(const void *a,const void *b){return strcmp((char *)b,(char *)a);}
54  int chsortfncsj(const void *a,const void *b){if(*(char *)a>*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
55  int chsortfnckj(const void *a,const void *b){if(*(char *)a<*(char *)b){return 1;}if(*(char *)a==*(char *)b){return 0;}return -1;}
56  void shuffledget(int x[],int n){
57      int i,b[524288],p,c;
58      for(i=0;i<n;i++){
59          b[i]=i;
60      }
61      for(i=n;i>=1;i--){
62          p=rand()%i;
63          c=b[i-1];b[i-1]=b[p];b[p]=c;
64      }
65      for(i=0;i<n;i++){
66          scanf("%d",&x[b[i]]);
67      }
68  }
69  int dx4[4]={1,-1,0,0};
70  int dy4[4]={0,0,1,-1};
71  int dx8[8]={-1,-1,-1,0,0,1,1,1};
72  int dy8[8]={-1,0,1,-1,1,-1,0,1};
73  int search(int x,int a[],int n){
74      int st=0,fi=n-1,te;
75      while(st<=fi){
76          te=(st+fi)/2;
77          if(a[te]<x){st=te+1;}else{fi=te-1;}
78      }
79      return st;
80  }
81  void prarr(int arr[],int n){
82    int i;
83    for(i=0;i<n;i++){
84      if(i){printf(" ");}
85      printf("%d",arr[i]);
86    }
87    printf("\n");
88    return;
89  }
90  void getperm(int a[],int n){
91    int i,p;
92    for(i=0;i<n;i++){
93      a[i]=i;
94    }
95    for(i=n-1;i>=1;i--){
96      p=rand()%(i+1);
97      swap(&a[p],&a[i]);
98    }
99  }
100  typedef struct{
101  int val;
102  int node;
103  }sd;
104  int sdsortfnc(const void *a,const void *b){
105  if(((sd*)a)->val < ((sd*)b)->val){return -1;}
106  if(((sd*)a)->val > ((sd*)b)->val){return 1;}
107  return 0;
108  }
109  void coordinate_comp(int a[],int n){
110    int i,c=0;
111    sd dat[524288];
112    for(i=0;i<n;i++){
113      dat[i].val=a[i];
114      dat[i].node=i;
115    }
116    qsort(dat,n,sizeof(dat[0]),sdsortfnc);
117    a[dat[0].node]=c;
118    for(i=1;i<n;i++){
119      if(dat[i-1].val!=dat[i].val){c++;}
120      a[dat[i].node]=c;
121    }
122  }
123  int main(void){
124    int cs;
125    int i,j,n,m,k,a,b,c,h,w,l,t,sn;
126    int r=0;
127    int fl[64];
128    scanf("%d",&t);
129    for(cs=1;cs<=t;cs++){
130      scanf("%d",&n);
131      sn=n;
132      printf("Case #%d:\n",cs);
133      w=0;
134      for(i=1;i<50;i++){
135        w+=(1<<(i-1));
136        if(w>=n){
137          l=i;
138          break;
139        }
140      }
141      for(i=l;i>=1;i--){
142        if(n>=(1<<(i-1))+(i-1)){
143          fl[i]=1;
144          n-=(1<<(i-1));
145        }
146        else{
147          fl[i]=0;
148          n--;
149          }
150      }
151      a=0;
152      w=0;
153      for(i=1;i<=l;i++){
154        if(fl[i]==1){
155          w+=(1<<(i-1));
156          if(a==0){
157            for(j=1;j<=i;j++){
158              printf("%d %d\n",i,j);
159            }
160          }
161          else{
162            for(j=i;j>=1;j--){
163              printf("%d %d\n",i,j);
164            }
165          }
166          a=1-a;
167        }
168        else{
169          w++;
170          if(a==0)
171            printf("%d %d\n",i,1);
172          else
173            printf("%d %d\n",i,i);
174        }
175      }
176      while(w!=sn){
177        l++;
178        if(a==0)
179          printf("%d %d\n",l,1);
180        else
181          printf("%d %d\n",l,l);
182        w++;
183      }
184    }
185    return 0;
186  }
```


**Question**: Is there information flow from `(w,135)` to `(fl,147)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes