In my youth I read a the wonderful story about a car manufacturer that made a very fancy car which had superb traction and when you went through a corner you would not feel any vibration or other problems while steering it through even the most sharp corners. It was amazing.
And it was deadly.
When the car got tested by people that were not professional drivers, those people got confused with the total lack of vibrations, slippage and other things that happen when you go through a corner too fast. They just kept on going faster until the car eventually could not compensate, slipped and crashed.
The car maker ended up artificially added vibrations to the steering wheel when the car noticed it was getting close to its limits. This made the people realize they were getting to the limits and they slowed down.
No system is perfect and the average human operator will use it within limits. The problem with the car was that it didn't make clear what its limits were.
A system that is hiding all its inner working may forget that showing weakness and limits are features that we, humans, require in order to understand how to use it. Having added vibrations to the steering column made the car a very popular type which is still sold to this day.
Bitcoin Cash is a similar system that hides most of its inner workings.
In Bitcoin Cash the Double Spend Proofs (DSPs) create notifications when bad things happen. So we don't have an illusion of the system being perfect. And by getting accurate feedback of where the limits are, we will be more comfortable in using the system to its fullest.
Double Spend Proofs identify people and systems that are faulty or intentionally trying to break the system for their own gain.
This identification of such bad actors comes with a cryptographic proof of the deed and allows users to take action to avoid being stolen from.
When in 2009 Satoshi rolled out the Bitcoin system he solved, for the first time in history, the decentralized avoidance of double spends in a chain of transactions.
We have since then started to explore the safety of transactions that are not yet part of the chain. The so called "zero confirmations" transactions.
We can say that the risk of accepting a payment is directly related to how many confirmations it has. Where 1 confirmation is when it first gets mined in a block. As such the greatest risk is when it has not yet appeared in a block.
Before DSPs the risk for the thief to try and defraud a merchant was zero. The merchant would not find out and not be notified in all the cases where the attempt to defraud failed. Allowing the thief to keep on trying with zero risk.
Empirical research has been done on the current network. People tried different ways to double spend and realized that the risk of a double spend is actually quite low. They concluded its not easy to pull a successful double spend which defrauds a merchant.
What this research also concluded is that the mono-culture of the network is very important in keeping the double spends from being accepted. All miners run with practically the same settings which makes timing the main tool for the attacker to defraud a merchant.
As the network grows it will become inevitable that miners will change their miner settings in order to become more profitable with their specific setup. The loss of a mono-culture in Bitcoin Cash mining is inevitable and likely healthy. The side-effect is that it will become easier to work around the "first seen" principle that is our main defense against double spends. And this in turn means that the double spend proof will end up being a increasingly valuable tool in the toolbox to keep the network healthy and the instant payment feature low risk.
The basic Bitcoin Cash transaction takes an existing "output" and spends it. The value embedded in that output we are spending is then send elsewhere in our transaction, itself in turn creating a new output. This triple entry accounting where your transaction removes one output and creates a new one assumes that there is only one entry that spends any single output. Should there be two independent transactions which both spend the same output and send it to different places, that is what we call a double spend.
Because in Bitcoin Cash all transactions are cryptographically sealed, the attempt to move one output twice is a standalone proof that the owner is responsible for those two transactions. This follows from the simple fact that the owner of the output is the only one that can move the funds. A double spend proof is not much more than a pointer to the output it tries to double spend and essential parts of the transactions doing the spending.
Including both cryptographic signatures, of both spends, that only the owner could have made. It is then just a matter of validating both signatures that allows us to establish without a doubt that the output has been spent twice.
The full specification of the Double Spent Proof is here.