How are Harry Potter's children half-bloods?

For some pure-bloods, any known Muggle ancestry at all is enough to prevent someone from being classed as a pure-blood. So Lily's Muggle parentage would prevent her grandchildren from being pure-bloods, in their eyes.

However, another method of classification looks at a person's grandparents. If all 4 are magical you're a pure-blood, if some are magical and some are Muggle あなた count as a half-blood, and if all 4 are Muggles you're Muggle-born (even if your parents were also a Muggle-born witch and a Muggle-born wizard). This method would judge Harry and Ginny's children to be pure-blood since James, Lily, Arthur and Molly were all magical.