[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**. 

```Python
1   import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time,copy,functools
2   sys.setrecursionlimit(10**7)
3   def LI(): return [int(x) for x in sys.stdin.readline().split()]
4   def LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]
5   def LF(): return [float(x) for x in sys.stdin.readline().split()]
6   def LS(): return sys.stdin.readline().split()
7   def I(): return int(sys.stdin.readline())
8   def F(): return float(sys.stdin.readline())
9   def S(): return input()
10  def pf(s): return print(s, flush=True)
11  def main():
12      inf = 10**20
13      eps = 1.0 / 10**10
14      mod = 10**9+7
15      ddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]
16      dd = [(-1,0),(0,1),(1,0),(0,-1)]
17      rr = []
18      while True:
19          n = I()
20          if n == 0:
21              break
22          a = []
23          for _ in range(n):
24              a.append(LS())
25          d = collections.defaultdict(int)
26          ed = collections.defaultdict(lambda: None)
27          for i in range(n):
28              x, y, di = a[i]
29              x = int(x)
30              y = int(y)
31              d[(x,y)] = i + 1
32              if di == 'x':
33                  d[(x+1,y)] = i + 1
34                  ed[(x,y)] = (x+1,y)
35                  ed[(x+1,y)] = (x,y)
36              else:
37                  d[(x,y+1)] = i + 1
38                  ed[(x,y)] = (x,y+1)
39                  ed[(x,y+1)] = (x,y)
40          ee = collections.defaultdict(set)
41          dka = list(d.keys())
42          for x,y in list(d.keys()):
43              dt = d[(x,y)]
44              for di,dj in dd:
45                  ni = x + di
46                  nj = y + dj
47                  if d[(ni,nj)] > 0 and d[(ni,nj)] != dt:
48                      ee[(x,y)].add((ni,nj))
49          v = collections.defaultdict(bool)
50          f = True
51          for k in dka:
52              if v[k]:
53                  continue
54              s1 = set()
55              s2 = set()
56              ns1 = set([k])
57              ns2 = set()
58              while ns1 or ns2:
59                  s1 |= ns1
60                  na = list(ns1)
61                  ns1 = set()
62                  for k in na:
63                      ns1 |= ee[k]
64                      ns2.add(ed[k])
65                  ns2 -= s2
66                  na = list(ns2)
67                  s2 |= ns2
68                  ns2 = set()
69                  for k in na:
70                      ns2 |= ee[k]
71                      ns1.add(ed[k])
72                  ns1 -= s1
73                  ns2 -= s2
74                  s1 |= ns1
75                  s2 |= ns2
76                  if s1 & s2:
77                      f = False
78                      break
79              if f:
80                  for k in s1:
81                      v[k] = 1
82                  for k in s2:
83                      v[k] = 1
84              else:
85                  break
86          if f:
87              rr.append('Yes')
88          else:
89              rr.append('No')
90      return '\n'.join(map(str,rr))
91  print(main())
```


**Question**: Does `(d,31)` have data dependence over `(s1,59)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes