One of the oldest and most famous examples of an invariant is the accounting equation, which is the foundation of double-entry accounting. The accounting equation says, loosely, that the total amount of debits on a company's ledger must equal the total amount of credits. It's easy to prove that double-entry accounting, when done correctly, preserves this invariant: for every transaction, all increases (or decreases) to credit accounts must be equal to all increases (or decreases) to debit accounts. It's easy to see that if debits and credits balance before a transaction, they balance after the transaction. Thus, the invariant is always preserved.
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