Do I need to write all intended properties before I start the Fuzzer
No, you can choose if you want to check for one property at a time or rather check multiple properties at the same time. You can run several Fuzzing campaigns.
Important:
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More is not necessarily better! Reflect on which properties are important.
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Start with two or three simple properties just to get started. Make sure to add a property that should not hold. This allows you to verify if the fuzzer is set up correctly.
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You can always add more properties, just like you can always add more unit tests.