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Using these definitions of soft fork and hard fork:

soft fork – tightening of the consensus rules, new blocks remain valid for old nodes

hard fork – loosening of the consensus rules, new blocks might not be valid for old nodes

It seems to me that if someone tried to deploy a pure hard fork that only loosens some consensus rules but doesn't tighten others, their new network would need to have more hashpower than the original network, or the previous chain (that remains valid for their new network) would always outpace it and no actual fork would take place. In other words, for a hard fork to be successful it should also include a soft fork. Is this correct?

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Yes, a hard fork needs wipeout protection to prevent nodes from reorganizing to a chaintip with more work following the old rules, if the hard fork only loosens rules. (I.e. I would say that a hard fork needs to include some rule tightening to achieve wipeout protection, but I wouldn't say that a hard fork also needs to include a soft fork.)

Past hard fork proposal did usually prescribe rules that exclude a potentially surviving unupgraded chaintip from being considered. For example, Bitcoin Cash changed the sighash algorithm at the fork height, making Bitcoin transactions invalid on Bitcoin Cash and vice versa, and the segwit2x proposal would have required the block at fork height to have a weight of more than 4,000,000 WU.

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