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3.2.2 Normalization judgment "the above rule is a shorthand for: statEnv |- [Object]_Subscript == Mapped Object" I would say there's no need to retain '==' here. The standard '=' of inference rules seems to have the right semantics. See, e.g., 4.8.2 / STA / rule 3, or 5.14 / SCP / rule 2. Moreover, a mapping rule is not just a shorthand for a judgment, but for a whole inference rule, with the given judgment as the conclusion, and generally no premises. There are a few normalization rules (in 4.7.1 and 4.7.1.1) that *do* have a premise, but there's nothing here to support that syntax.
Changed "Normalization judgments" to "Normalization rules" and corresponding text to indicate that normalization rules are an abbreviated form of inference rule. Changed "==" to "="