Simplify :
`a - [a - bar(b + a) - {a - (a - bar(b-a))}]`

Answer

To simplify the expression \( a - [a - \overline{(b + a)} - \{a - (a - \overline{(b - a)})\}] \), we will follow the order of operations and simplify step by step.

### Step 1: Identify and simplify the innermost expressions
Start with the innermost expression \( a - \overline{(b - a)} \).

- The expression \( b - a \) is negated, so it becomes \( -b + a \) or \( a - b \).
- Therefore, \( \overline{(b - a)} = a - b \).

Now, substitute this back into the expression:

\[ a - [a - \overline{(b + a)} - \{a - (a - (a - b))\}] \]

### Step 2: Simplify \( a - (a - (a - b)) \)
Now simplify the expression inside the curly brackets:

1. \( a - (a - b) = a - a + b = b \).

Now substitute this back into the expression:

\[ a - [a - \overline{(b + a)} - b] \]

### Step 3: Simplify \( \overline{(b + a)} \)
Next, we need to simplify \( \overline{(b + a)} \):

1. The expression \( b + a \) is negated, so it becomes \( -b - a \).

Now substitute this back into the expression:

\[ a - [a - (-b - a) - b] \]

### Step 4: Simplify the brackets
Now simplify the expression inside the brackets:

1. \( a - (-b - a) = a + b + a = 2a + b \).

So now we have:

\[ a - [2a + b - b] \]

### Step 5: Simplify further
Now simplify the expression inside the brackets:

1. \( 2a + b - b = 2a \).

So now we have:

\[ a - 2a \]

### Step 6: Final simplification
Now simplify \( a - 2a \):

1. \( a - 2a = -a \).

Thus, the simplified expression is:

\[ \boxed{-a} \]