231 lines
12 KiB
Markdown
231 lines
12 KiB
Markdown
---
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title: "The \"Deeply Embedded Expression\" Trick in Agda"
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date: 2024-03-11T14:25:52-07:00
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tags: ["Agda"]
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description: "In this post, I talk about a trick I developed to simplify certain Agda proofs."
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---
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I've been working on a relatively large Agda project for a few months now,
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and I'd like to think that I've become quite proficient. Recently, I came
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up with a little trick to help simplify some of my proofs, and it seems like
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this trick might have broader applications.
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In my head, I call this trick 'Deeply Embedded Expressions'. Before I introduce
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it, let me explain the part of my work that motivated developing the trick.
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### Proofs about Map Operations
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A part of my Agda project is the formalization of simple key-value maps.
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I model key-value maps as lists of key-value pairs. On top of this, I implement
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two operations: `join` and `meet`, which in my code are denoted using `⊔` and `⊓`.
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When "joining" two maps, you create a new map that has the keys from both input ones.
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If a key is only present in one of the input maps, then the new "joined" map
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has the same value for that key as the original. On the other hand, if the key
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is present in both maps, then its value in the new map is the result of "joining"
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the original values. The "meet" operation is similar, except instead of taking
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keys from either map, the result only has keys that were present in both maps,
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"meeting" their values. In a way, "join" and "meet" are similar to set union
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and intersection --- but they also operate on the values in the map.
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Given these operations, I need to prove certain properties of these operation.
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The most inconvenient to prove is probably associativity:
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 752 752 >}}
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This property is, in turn, proven using two 'subset' relations on maps, defined
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in the usual way.
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 755 755 >}}
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 774 774 >}}
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The reason this property is so inconvenient to prove is that there are a
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lot of cases to consider. That's because your claim, in words, is something
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like:
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> Suppose a key-value pair `k , v` is present in `(m₁ ⊔ m₂) ⊔ m₃`. Show
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> that `k , v` is also in `m₁ ⊔ (m₂ ⊔ m₃)`.
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The only thing you can really do with `k , v` is figure out how it got into
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the three-way union map: did it come from `m₁`, `m₂`, or `m₃`, or perhaps
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several of them? The essence of the proof boils down to repeated uses
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of the fact that for a key to be in the union, it must be in at least one
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of the two maps. You end up with witnesses, repeated application of the same
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lemmas, lots of `let`-expressions or `where` clauses. It's relatively tedious
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and, what's more frustrating, __driven entirely by the structure of the
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map operations__. It seems like one shouldn't have to mimic that structure
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using boilerplate lemmas. So I started looking at other ways.
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### Case Analysis using GADTs
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A "proof by cases" in a dependently typed language like Agda usually brings
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to mind pattern matching. So, here's an idea: what if for each expression
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involving `⊔` and `⊓`, we had some kind of data type, and that data type
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had exactly as many inhabitants as there are cases to analyze? A data type
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corresponding to `m₁ ⊔ m₂` might have three cases, and the one for
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`(m₁ ⊔ m₂) ⊔ m₃` might have seven. Each case would contain the information
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necessary to perform the proof.
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A data type whose "shape" depends on an expression in the way I described above
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is said to be _indexed by_ that expression. In Agda, GADTs are used to create
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indexed types. My initial attempt was something like this:
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```Agda
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data Provenance (k : A) : B → Map → Set (a ⊔ℓ b) where
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single : ∀ {v : B} {m : Map} → (k , v) ∈ m → Provenance k v m
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in₁ : ∀ {v : B} {m₁ m₂ : Expr} → Provenance k v e₁ → ¬ k ∈k m₂ → Provenance k v (e₁ ⊔ e₂)
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in₂ : ∀ {v : B} {m₁ m₂ : Expr} → ¬ k ∈k m₁ → Provenance k v m₂ → Provenance k v (e₁ ⊔ e₂)
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bothᵘ : ∀ {v₁ v₂ : B} {m₁ m₂ : Expr} → Provenance k v₁ m₁ → Provenance k v₂ m₂ → Provenance k (v₁ ⊔ v₂) (e₁ ⊔ e₂)
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bothⁱ : ∀ {v₁ v₂ : B} {m₁ m₂ : Expr} → Provenance k v₁ m₁ → Provenance k v₂ m₂ → Provenance k (v₁ ⊓ v₂) (e₁ ⊓ e₂)
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```
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I was planning on a proof of associativity (in one direction) that looked
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something like the following --- pattern matching on cases from the new
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`Provenance` type.
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```Agda
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⊔-assoc₁ : ((m₁ ⊔ m₂) ⊔ m₃) ⊆ (m₁ ⊔ (m₂ ⊔ m₃))
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⊔-assoc₁ k v k,v∈m₁₂m₃
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with get-Provenance k,v∈m₁₂m₃
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... | in₂ k∉km₁₂ (single v∈m₃) = ...
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... | in₁ (in₂ k∉km₁ (single v∈m₂)) k∉km₃ = ...
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... | bothᵘ (in₂ k∉km₁ (single {v₂} v₂∈m₂)) (single {v₃} v₃∈m₃) = ...
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... | in₁ (in₁ (single v₁∈m₁) k∉km₂) k∉km₃ = ...
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... | bothᵘ (in₁ (single {v₁} v₁∈m₁) k∉km₂) (single {v₃} v₃∈m₃) = ...
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... | in₁ (bothᵘ (single {v₁} v₁∈m₁) (single {v₂} v₂∈m₂)) k∉ke₃ = ...
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... | bothᵘ (bothᵘ (single {v₁} v₁∈m₁) (single {v₂} v₂∈m₂)) (single {v₃} v₃∈m₃) = ...
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```
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However, this doesn't work. Agda has trouble figuring out which cases of
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the `Provenance` GADT are allowed, in which aren't. Is `m₁ ⊔ m` a single map,
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fit for the `single` case, or should it be broken up into more cases like
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`in₁` and `in₂`? In general, is some expression of type `Map` the "bottom"
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of our recursion, or should it be analyzed further?
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The above hints at what's wrong. The mistake here is requiring Agda to infer
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the shape of our "join" and "meet" expressions from arbitrary terms.
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The set of expressions that we want to reason about is much more restricted --
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each expression will always be of three components: "meet", "join", and base-case maps being
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combined using these operations.
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### Defining an Expression Data Type
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If you're like me, and have spent years of your life around programming language
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theory and domain specific languages (DSLs), the last sentence of the previous
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section may be ringing a bell. In fact, it's eerily similar to how we describe
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recursive grammars:
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> An expression of interest is either,
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> * A map
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> * The "join" of two expressions
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> * The "meet" of two expressions
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Mathematically, we might write this as follows:
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{{< latex >}}
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\begin{array}{rcll}
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e & ::= & m & \text{(maps)} \\
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& | & e \sqcup e & \text{(join)} \\
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& | & e \sqcap e & \text{(meet)}
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\end{array}
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{{< /latex >}}
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And in Agda,
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 543 546 >}}
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In the code, I used the set union and intersection operators
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to avoid overloading the `⊔` and `⊓` more than they already are.
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We have just defined a very small expression language. In computer science,
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a language is called _deeply embedded_ if a data type (or class hierarchy, or
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other 'explicit' representation) is defined for its syntax in the _host_
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language (Agda, in our case). This is in contrast to a _shallow embedding_, in
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which expressions in the (new) language are just expressions in the host
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language.
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In this sense, our `Expr` is deeply embedded --- we defined new container for it,
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and `_∪_` is a distinct entity from `_⊔_`. Our first attempt was a shallow
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embedding. That fell through because the Agda language is much broader than
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our expression language, which makes case analysis very difficult.
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An obvious thing to do with an expression is to evaluate it. This will be
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important for our proofs, because it will establish a connection between
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expressions (created via `Expr`) and actual Agda objects that we need to
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reason about at the end of the day. The notation \(\llbracket e \rrbracket\)
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is commonly used in PL circles for evaluation (it comes from
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[Denotational Semantics](https://en.wikipedia.org/wiki/Denotational_semantics)).
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Thus, my Agda evaluation function is written as follows:
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 586 589 >}}
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On top of this, here is my actual implementation of the `Provenance` data type.
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This time, it's indexed by expressions in `Expr`, which makes it much easier to
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pattern match on instances:
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 591 596 >}}
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Note that we have to use the evaluation function to be able to use
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operators such as `∈`. That's because these are still defined on maps,
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and not expressions.
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With this, I was able to write my proof in the way that I had hoped. It has
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the exact form of my previous sketch-of-proof.
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{{< codelines "agda" "agda-spa/Lattice/Map.agda" 755 773 "" "**(click here to see the full example, including each case's implementation)**" >}}
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### The General Trick
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So far, I've presented a problem I faced in my Agda proof and a solution for
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that problem. However, it may not be clear how useful the trick is beyond
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this narrow case that I've encountered. The way I see it, the "deeply embedded
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expression" trick is applicable whenever you have data that is constructed
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from some fixed set of cases, and when proofs about that data need to follow
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the structure of these cases. Thus, examples include:
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* **Proofs about the origin of keys in a map (this one):** the "data" is the
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key-value map that is being analyzed. The enumeration of cases for this
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map is driven by the structure of the "join" and "meet" operations used
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to build the map.
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* **Automatic derivation of function properties:** suppose you're interested
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in working with continuous functions. You also know that the addition,
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subtraction, and multiplication of two functions preserves continuity.
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Of course, the constant function \(x \mapsto c\) and the identity function
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\(x \mapsto x\) are continuous too. You may define an expression data type
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that has cases for these operations. Then, your evaluation function could
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transform the expression into a plain function, and a proof on the
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structure of the expression can be used to verify the resulting function's
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continuity.
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* **Proof search for algebraic expressions:** suppose that you wanted to
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automatically find solutions for certain algebraic (in)equalities. Instead
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of using some sort of reflection mechanism to inspect terms and determine
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how constraints should be solved, you might represent the set of operations
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in you equation system as cases in a data type. You can then use regular
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Agda code to manipulate terms; an evaluation function can then be used
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to recover the equations in Agda, together with witnesses justifying the
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solution.
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There are some pretty clear commonalities about examples above, which
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are the ingredients to this trick:
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* __The expression:__ you create a new expression data type that encodes all
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the operations (and bases cases) on your data. In my example, this is
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the `Expr` data type.
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* __The evaluation function__: you provide a way to lower the expression
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you've defined back into a regular Agda term. This connects your (abstract)
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operations to their interpretation in Agda. In my example, this is the
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`⟦_⟧` function.
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* __The proofs__: you write proofs that consider only the fixed set of cases
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encoded by the data type (`Expr`), but state properties about the
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_evaluated_ expression. In my example, this is `Provenance` and
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the `Expr-Provenance` function. Specifically, the `Provenance` data type connects
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expressions and the terms they evaluate to, because it is indexed by expressions,
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but contains data in the form `k ∈k ⟦ e₂ ⟧`.
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### Conclusion
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I'll be the first to admit that this trick is quite situational, and may
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not be as far-reaching as the ["Is Something" pattern](https://danilafe.com/blog/agda_is_pattern/)
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I wrote about before, which seems to occur far more in the wild. However, there
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have now been two times when I personally reached for this trick, which seems
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to suggest that it may be useful to someone else.
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I hope you've found this useful. Happy (dependently typed) programming!
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