Write a bit more, enable support for paragraph links

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-05-21 19:04:08 -07:00
parent 4938cdaecd
commit ee8b1f5dc0
3 changed files with 46 additions and 3 deletions

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@ -33,6 +33,10 @@ defaultContentLanguage = 'en'
[markup.goldmark.extensions.passthrough.delimiters] [markup.goldmark.extensions.passthrough.delimiters]
block = [['\[', '\]'], ['$$', '$$']] block = [['\[', '\]'], ['$$', '$$']]
inline = [['\(', '\)']] inline = [['\(', '\)']]
[markup.goldmark.parser]
[markup.goldmark.parser.attribute]
block = true
title = true
[languages] [languages]
[languages.en] [languages.en]

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@ -249,6 +249,7 @@ It turns out to be convenient, however, to not require definitional equality
would force us to consider lists with the same elements but a different would force us to consider lists with the same elements but a different
order to be unequal. Instead, we parameterize our definition of `IsSemilattice` order to be unequal. Instead, we parameterize our definition of `IsSemilattice`
by a binary relation `_≈_`, which we ask to be an [equivalence relation](https://en.wikipedia.org/wiki/Equivalence_relation). by a binary relation `_≈_`, which we ask to be an [equivalence relation](https://en.wikipedia.org/wiki/Equivalence_relation).
{#definitional-equality}
{{< codelines "Agda" "agda-spa/Lattice.agda" 23 39 >}} {{< codelines "Agda" "agda-spa/Lattice.agda" 23 39 >}}

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@ -52,10 +52,13 @@ for both types:
Elements of \(L_1 \times L_2\) are in the form \((l_1, l_2)\), where 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 \(l_1 \in L_1\) 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 way is define what it means for two such elements to be equal. Recall that
enough: we have an equality predicate `_≈₁_` that checks if an element we opted for a [custom equivalence relation]({{< relref "01_spa_agda_lattices#definitional-equality" >}})
instead of definitional equality to allow similar elements to be considered
equal; we'll have to define a similar relation for our new product lattice.
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 \(L_2\). 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 It's reasonable to say that _pairs_ of elements are equal if their respective
first and second elements are equal: first and second elements are equal:
{{< latex >}} {{< latex >}}
@ -65,3 +68,38 @@ first and second elements are equal:
In Agda: In Agda:
{{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 39 40 >}} {{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 39 40 >}}
Verifying that this relation has the properties of an equivalence relation
boils down to the fact that `_≈₁_` and `_≈₂_` are themselves equivalence
relations.
{{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 42 48 >}}
In fact, defining \((\sqcup)\) and \((\sqcap)\) by simply applying the
corresponding operators from \(L_1\) and \(L_2\) seems quite natural as well.
{{< latex >}}
(l_1, l_2) \sqcup (j_1, j_2) \triangleq (l_1 \sqcup_1 j_1, l_2 \sqcup_2 j_2) \\
(l_1, l_2) \sqcap (j_1, j_2) \triangleq (l_1 \sqcap_1 j_1, l_2 \sqcap_2 j_2)
{{< /latex >}}
In Agda:
{{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 50 54 >}}
All that's left is to prove the various (semi)lattice properties. Intuitively,
we can see that since the "combined" operator `_≈_` just independently applies
the element operators `_≈₁_` and `_≈₂_`, as long as they are idempotent,
commutative, and associative, so is the "combined" operator itself.
Moreover, the proofs that `_⊔_` and `_⊓_` form semilattices are identical
up to replacing \((\sqcup)\) with \((\sqcap)\). Thus, in Agda, we can write
the code once, parameterizing it by the binary operators involved (and proofs
that these operators obey the semilattice laws).
{{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 56 82 >}}
Similarly to the semilattice properties, proving lattice properties boils
down to applying the lattice properties of \(L_1\) and \(L_2\) to
individual components.
{{< codelines "Agda" "agda-spa/Lattice/Prod.agda" 84 96 >}}