Notation for Yichen's paper
Figure 4 in Yichen's paper uses some notation that might not be familiar.
Here are some notes to help you along:
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The turnstile symbol ⊢, in "G ⊢ x", basically means "x is true in the environment G".
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The horizontal line is called a "rule": if all the things above the line are true, then the thing below the line must be true.
The Wikipedia page on sequents has more details.