[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   def solution(number_of_signs):
2       a_to_b = []
3       for i in range(number_of_signs):
4           sign = raw_input()
5           d, a, b = sign.split()
6           d = int(d)
7           a = int(a)
8           b = int(b)
9           a_to_b.append((d + a, d - b))
10      N = None
11      M = None
12      maximum_set_size = 1
13      number_of_maximum_set_sized = 0
14      current_set_size = 1
15      a_for_comparision, b_for_comparision = a_to_b.pop(0)
16      is_finish = False
17      while not is_finish and len(a_to_b) > 0:
18          for index, tup in enumerate(a_to_b):
19              if index == len(a_to_b) - 1:
20                  is_finish = True
21              a, b = tup
22              if N is not None and M is None:
23                  if a != a_for_comparision:
24                      M = b
25                      b_for_comparision = b
26                  current_set_size += 1
27              elif M is not None and N is None:
28                  if b != b_for_comparision:
29                      N = a
30                      a_for_comparision = a
31                  current_set_size += 1
32              elif M is not None and N is not None:
33                  if b != b_for_comparision and a != a_for_comparision:
34                      if current_set_size > maximum_set_size:
35                          maximum_set_size = current_set_size
36                          number_of_maximum_set_sized = 1
37                      elif current_set_size == maximum_set_size:
38                          number_of_maximum_set_sized += 1
39                      current_set_size = 1
40                      N = None
41                      M = None
42                      a_to_b = a_to_b[index:]
43                      break
44                      a_for_comparision = a
45                      b_for_comparision = b
46                  else:
47                      current_set_size+=1
48              elif a == a_for_comparision and b == b_for_comparision:
49                  if (N is None):
50                      a_for_comparision = a
51                  if (M is None):
52                      b_for_comparision = b
53                  current_set_size += 1
54                  continue
55              elif a == a_for_comparision and b != b_for_comparision:
56                  if (N is None):
57                      N = a
58                      a_for_comparision = N
59                  b_for_comparision = b
60                  current_set_size += 1
61              elif b == b_for_comparision and a != a_for_comparision:
62                  if (M is None):
63                      M = b
64                      b_for_comparision = b
65                  a_for_comparision = a
66                  current_set_size += 1
67              elif b != b_for_comparision and a != a_for_comparision:
68                  if len(a_to_b) > index:
69                      if a == a_to_b[index+1][0]:
70                          N = a_to_b[index +1][0]
71                          a_for_comparision = a_to_b[index +1][0]
72                          M = b_for_comparision
73                          current_set_size+=1
74                      else:
75                          M = a_to_b[index +1][1]
76                          b_for_comparision = a_to_b[index +1][1]
77                          N = a_for_comparision
78                          current_set_size+=1
79                  else:
80                      current_set_size += 1
81      return maximum_set_size, number_of_maximum_set_sized
82  def read_input():
83      number_of_tests = raw_input()
84      for test_number in range(int(number_of_tests)):
85          number_of_signs = raw_input()
86          try:
87              print_output(solution(int(number_of_signs)), test_number + 1)
88          except Exception as exc:
89              pass
90  def print_output(answer, test_case):
91      print "Case #{0}: {1} {2}".format(test_case, answer[0], answer[1])
92  read_input()
```


**Question**: Does `(b,5)` have data dependence over `(a_to_b,42)` in function `solution`? If so, provide a trace.

**Output**:


Ground truth answer: yes