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I do not see any difference between sending cashu coins to a receiver with and without any proof.
The sender can send the same coins with different proofs to many senders. What they can verify offline? Nothing related to double spend.
Only the first receiver going online takes the coins. What is the purpose of such proof then?
Ff the sender creates a token locked to your public key (for untrusted transactions), then no, no one else can reclaim it because only you can redeem it with your private key. The trust there is in the mint.
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But does that mean that the sender must be online to get the token from the mint? Otherwise, how to obtain a token with my pubkey in advance?
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Yes. Or at least must have planned it in advance.
Maybe there is a vending machine at the peak of a mountain where neither the machine nor the users will have Internet access. You could plan in advance and get some tokens for that machine.
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