## Step 1: Define the symbolic representation of the problem

Let's define the symbolic variables:
- $x_1$ represents the number of AA batteries
- $x_2$ represents the number of D batteries

The objective function to maximize profit is: $0.50x_1 + 1x_2$

The constraints are:
1. Budget constraint: $1x_1 + 3x_2 \leq 1000$
2. Demand constraint: $x_1 + x_2 \leq 1000$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Step 2: Convert the problem into a Gurobi-compatible format

We need to express the problem in a format that can be solved by Gurobi. This involves defining the variables, the objective function, and the constraints using Gurobi's Python API.

## 3: Write the Gurobi code

```python
import gurobi as gp

# Define the model
model = gp.Model("Battery_Store_Optimization")

# Define the variables
x1 = model.addVar(name="AA_batteries", lb=0, ub=1000, vtype=gp.GRB.CONTINUOUS)
x2 = model.addVar(name="D_batteries", lb=0, ub=1000, vtype=gp.GRB.CONTINUOUS)

# Define the objective function
model.setObjective(0.50 * x1 + 1 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(x1 + 3 * x2 <= 1000, name="budget_constraint")
model.addConstr(x1 + x2 <= 1000, name="demand_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"AA batteries: {x1.varValue}")
    print(f"D batteries: {x2.varValue}")
    print(f"Max Profit: {model.objVal}")
else:
    print("No optimal solution found.")
```

## 4: Provide the symbolic representation in JSON format

```json
{
    "sym_variables": [
        ["x1", "AA batteries"],
        ["x2", "D batteries"]
    ],
    "objective_function": "0.50x1 + 1x2",
    "constraints": [
        "x1 + 3x2 <= 1000",
        "x1 + x2 <= 1000",
        "x1 >= 0",
        "x2 >= 0"
    ]
}
```