### Work some more on lattices 2

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
parent 84f28ae5ce
commit 4938cdaecd
2 changed files with 37 additions and 1 deletions

#### 2 content/blog/01_spa_agda_lattices.md View File

 @ -422,7 +422,7 @@ only thing we needed is to be able to check and see if two elements are equal or not; this is called _decidable equality_. Since that's the only thing we used, this means that we can define an "above/below" lattice like this for any type for which we can check if two elements are equal. In Agda, I encoded this using a parameterized module: this using a [parameterized module](https://agda.readthedocs.io/en/latest/language/module-system.html#parameterised-modules):   {{< codelines "Agda" "agda-spa/Lattice/AboveBelow.agda" 5 8 >}}  

#### 36 content/blog/02_spa_agda_combining_lattices.md View File

 @ -29,3 +29,39 @@ challenging, but for for a two-level map like $$\text{info}$$ above, we'd need to do a lot more work. We need tools to build up such complicated lattices!   The way to do this, it turns out, is by using simpler lattices as building blocks. To start with, let's take a look at a very simple way of combining lattices: taking the Cartesian product.   ### The Cartesian Product Lattice   Suppose you have two lattices $$L_1$$ and $$L_2$$. As I covered in the previous post, each lattice comes equipped with a "least upper bound" operator $$(\sqcup)$$ and a "greatest lower bound" operator $$(\sqcap)$$. Since we now have two lattices, let's use numerical suffixes to disambiguate between the operators of the first and second lattice: $$(\sqcup_1)$$ will be the LUB operator of the first lattice $$L_1$$, and $$(\sqcup_2)$$ of the second lattice $$L_2$$.   Then, let's take the Cartesian product of the elements of $$L_1$$ and $$L_2$$; mathematically, we'll write this as $$L_1 \times L_2$$, and in Agda, we can just use the standard [Data.Product](https://agda.github.io/agda-stdlib/master/Data.Product.html) module. In Agda, I'll define the lattice as another [parameterized module](https://agda.readthedocs.io/en/latest/language/module-system.html#parameterised-modules). Since both $$L_1$$ and $$L_2$$ are lattices, this parameterized module will require IsLattice instances for both types:   {{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 1 7 >}}   Elements of $$L_1 \times L_2$$ are in the form $$(l_1, l_2)$$, where $$l_1 \in L_1$$ and $$l_2 \in L_2$$. The first thing we can get out of the way is define what it means for two such elements to be equal. That's easy enough: we have an equality predicate _≈₁_ that checks if an element of $$L_1$$ is equal to another, and we have _≈₂_ that does the same for $$L_2$$. It's reasonably to say that _pairs_ of elements are equal if their respective first and second elements are equal:   {{< latex >}}  (l_1, l_2) \approx (j_1, j_2) \iff l_1 \approx_1 j_1 \land l_2 \approx_2 j_2 {{< /latex >}}   In Agda:   {{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 39 40 >}}