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sqlite3_stmt_readonly is dependent on the current database
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I personally think that's the correct behavior.

Statement classification should not depend on the current schema at all in fact.  
And DDL statements are all basically non read-only, by design in my opinion.  
And for DMLs, only SELECT can be read-only.

Unless I misunderstand what that API is supposed to return, just from its name (haven't read the doc).