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@ -202,7 +202,6 @@ similar function for our signs. We call this function "[least upper bound](https
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since it is the "least (most specific)
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since it is the "least (most specific)
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element that's greater (less specific) than either `s1` or `s2`". Conventionally,
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element that's greater (less specific) than either `s1` or `s2`". Conventionally,
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this function is written as \(a \sqcup b\) (or in our case, \(s_1 \sqcup s_2\)).
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this function is written as \(a \sqcup b\) (or in our case, \(s_1 \sqcup s_2\)).
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The \((\sqcup)\) symbol is also called the _join_ of \(a\) and \(b\).
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We can define it for our signs so far using the following [Cayley table](https://en.wikipedia.org/wiki/Cayley_table).
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We can define it for our signs so far using the following [Cayley table](https://en.wikipedia.org/wiki/Cayley_table).
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{{< latex >}}
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{{< latex >}}
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@ -265,8 +264,7 @@ and the "least upper bound" function can be constructed from one another.
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As it turns out, the `min` function has very similar properties to `max`:
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As it turns out, the `min` function has very similar properties to `max`:
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it's idempotent, commutative, and associative. For a partial order like
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it's idempotent, commutative, and associative. For a partial order like
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ours, the analog to `min` is "greatest lower bound", or "the largest value
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ours, the analog to `min` is "greatest lower bound", or "the largest value
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that's smaller than both inputs". Such a function is denoted as \(a\sqcap b\),
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that's smaller than both inputs". Such a function is denoted as \(a\sqcap b\).
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and often called the "meet" of \(a\) and \(b\).
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As for what it means, where \(s_1 \sqcup s_2\) means "combine two signs where
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As for what it means, where \(s_1 \sqcup s_2\) means "combine two signs where
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you don't know which one will be used" (like in an `if`/`else`),
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you don't know which one will be used" (like in an `if`/`else`),
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\(s_1 \sqcap s_2\) means "combine two signs where you know
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\(s_1 \sqcap s_2\) means "combine two signs where you know
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@ -449,6 +449,10 @@ The same is true for maps, under certain conditions.
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The finite-height property is crucial to lattice-based static program analysis;
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The finite-height property is crucial to lattice-based static program analysis;
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we'll talk about it in more detail in the next post of this series.
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we'll talk about it in more detail in the next post of this series.
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{{< todo >}}
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I started using 'join' but haven't introduced it before.
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{{< /todo >}}
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### Appendix: Proof of Uniqueness of Keys
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### Appendix: Proof of Uniqueness of Keys
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I will provide sketches of the proofs here, and omit the implementations
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I will provide sketches of the proofs here, and omit the implementations
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@ -1 +1 @@
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Subproject commit 25e9057dd8b1ddc571f3b26d80b72e86bbcbaa72
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Subproject commit a4bff7623dc7c4b05b59714d7b919857a876422c
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