``` module FastExp where ``` # Imports ``` open import Data.Nat open import Data.Nat.Properties open import Data.Nat.Induction hiding (rec) open import Data.Product using (_×_; _,_; Σ; Σ-syntax; ∃; ∃-syntax; proj₁; proj₂) open import Data.Sum using (_⊎_; inj₁; inj₂) open import Induction open import Relation.Binary.PropositionalEquality open import Exponents open import Parity ``` I've been putting off learning how to use Agda's `Induction` library for some time because I knew it would take a serious effort. (The documentation is incomplete.) However, I ran into yet another situation that calls for it, so I finally decided to dive in! The purpose of the `Induction` library is to provide alternate forms of induction and recursion, such as complete induction, and it helps you build your own forms of induction. (Induction and recursion are the same thing in Agda.) Recall that Agda provides built-in support for structural recursion, but sometimes you want to define a function that doesn't fit into that mold. For example, suppose I wanted to write down the fast exponentiation function. That is, I want to define a function `fast-exp` such that fast-exp n x ≡ x ^ n Here's a naive attempt to write the function in Agda. I'm using an auxilliary function `parity` that determines whether a number `n` is even (`n ≡ 2 * k`) or odd (`n ≡ 1 + 2 * k`). fast-exp : ℕ → ℕ → ℕ fast-exp zero x = 1 fast-exp (suc n) x with parity n ... | inj₁ (k , refl) = x * fast-exp k (x * x) fast-exp (suc n) x | inj₂ (k , refl) = x * x * (fast-exp k (x * x)) Agda's termination checker rejects this program because it can't tell that the argument `k` in the recursive call to `fast-exp` is smaller than the input parameter `suc n`. We'll use the `Induction` library to work around this problem. There are two layers to the `Induction` library, there's a lower-level layer that resides in the `src/Induction.agda` file of the Agda standard library and there is a higher-level layer that is specific to induction on natural numbers in the `src/Data/Nat/Induction.agda` file. We'll start by using the higher-level layer to define `fast-exp` and then we'll take a look at the lower-level layer and build a custom induction principle for natural numbers and use it to redo our definition of `fast-exp`. The `Nat.Induction` library provides support for complete induction (aka. strong induction), which allows you to make a recursive call with any natural number smaller than the current one. The main ingredient that you need to define is a "step" function. This function looks a lot like the recursive function that you're trying to define, but it takes an extra parameter, let's name it `rec`, that will give you access to the recursive call. Here's a first (erroneous) attempt to define such a step function for `fast-exp`. Notice how the recursive calls to `fast-exp` above have been replaced by calls to `rec`. fe-step zero rec x = 1 fe-step (suc n) rec x with parity n ... | inj₁ (k , refl) = x * rec k (x * x) fe-step (suc n) rec x | inj₂ (k , refl) = x * x * (rec k (x * x)) ## The `CRec` Type Operator The above doesn't quite work because the `rec` given to us by the `Nat.Induction` library is not a function, it is a big tuple with one element for every natural number smaller than the current one. The type of this tuple is given by `CRec` in the `Nat.Induction` library. The `CRec` type operator takes three parameters: a universe level parameter (ignore that for now), a function that produces the type for each element in the tuple (given its index from the back), and the size of the tuple. For the purposes of defining `fast-exp`, we want each element in the tuple to be a function, in particular, the fast exponentiation function that's been partially applied to its first parameter. So the type of each element should be `ℕ → ℕ`. We define the following `FERec` abbreviation for the use of `CRec` that matches our needs. ``` FERec : ℕ → Set FERec n = CRec _ (λ i → ℕ → ℕ) n ``` The next thing we'll need is a way to access the nth element of the tuple. The Agda standard library has a `projₙ` function for this purpose, but to use it we'd need to prove that the `CRec` type operator produces a `Product` type. Instead we'll roll our own `projₙ` function for `CRec`. It takes a tuple of length `suc k` (so that it's non-empty), an index `n`, and a proof that `n` is less than `suc k`. ``` projₙ : ∀{ℓ P k} → CRec ℓ P (suc k) → (n : ℕ) → n ≤′ k → P n projₙ {l} rec n ≤′-refl = proj₁ rec projₙ {l} rec n (≤′-step n≤k) = projₙ (proj₂ rec) n n≤k ``` ## A Step Function for Fast Exponentiation Next we define the step function for fast exponentiation. The type for the `rec` parameter is `FERec n`. To make the recursive call, we use `projₙ` to access the appropriate partially-applied version of fast exponentiation (for a smaller natural number) from the `rec` tuple. However, to do so, we have to prove that `k` is less than `n`, the length of the tuple. (More about this below.) ``` fe-step : (n : ℕ) → FERec n → ℕ → ℕ fe-step zero rec x = 1 fe-step (suc n′) rec x with parity n′ ... | inj₁ (k , refl) = x * projₙ rec k lt (x * x) where lt : k ≤′ 2 * k lt = ≤⇒≤′ (m≤m+n k _) fe-step (suc n′) rec x | inj₂ (k , refl) = x * (x * (projₙ rec k lt (x * x))) where lt : k ≤′ 1 + (2 * k) lt = ≤⇒≤′ (≤-step (m≤m+n k _)) ``` Regarding `fe-step`, the case for `n ≡ zero` is straightforward. In the case for `n ≡ suc n′`, we have two subcases to consider, when `n′` is even (`n′ ≡ 2 * k`) and when `n′` is odd (`n′ ≡ 1 + 2 * k`). For the even subcase, to call `projₙ` we need to show that `k ≤ 2 * k`, which we do with the theorem `m≤m+n` from `Nat.Properties`. For the odd subcase, to call `projₙ` we need to show that `k ≤ 1 + 2 * k`, which we do using `≤-step` and then `m≤m+n`. ## Use `cRec` to Define Fast Exponentiation The final step to defining `fast-exp` is to apply the `cRec` function from `Nat.Induction` to our step function, `fe-step`. Similar to the `CRec` type operator, the `cRec` function also needs to know the type of the elements in the `rec` tuple, which is `ℕ → ℕ`. ``` fast-exp : ℕ → ℕ → ℕ fast-exp = cRec (λ _ → (ℕ → ℕ)) fe-step ``` ## Proof that Fast Exponentiation is Correct Of course, the whole point of programming in a proof assistant like Agda is to prove the correctness of our programs. Let's prove that fast-exp n x ≡ x ^ n which will give us an opportunity to 1) use `Nat.Induction` in an inductive proof and 2) reason about a function that was defined using `Nat.Induction`. We're going to refer to the correctness condition many times, so we define the following abbreviation for it. ``` fe-ok : ℕ → Set fe-ok n = ∀ x → fast-exp n x ≡ x ^ n ``` When reasoning about `fast-exp`, we'll need to reason about the tuple that gets passed to the `rec` parameter of `fe-step`. It turns out that the `cRec` function builds that tuple using an auxilliary function named `cRecBuilder`. So we define the following abbreviation named `fe-rec` for applying the `cRecBuilder` function to our step function. ``` fe-rec : (n : ℕ) → CRec _ (λ _ → (ℕ → ℕ)) n fe-rec n = cRecBuilder (λ _ → (ℕ → ℕ)) fe-step n ``` The nth function in the tuple produced by `fe-rec` is the `fast-exp n` function. ``` projₙ-fe : ∀ n k x (lt : n ≤′ k) → projₙ (fe-rec (suc k)) n lt x ≡ fast-exp n x projₙ-fe n n x ≤′-refl = refl projₙ-fe n (suc k) x (≤′-step lt) = projₙ-fe n k x lt ``` (This proof goes through easily because we do induction on `n ≤′ k`, which is using the alternate form of less-than. If we had instead used the normal less-than `≤` in the definition of `projₙ`, this proof would be more difficult.) We prove that `fast-exp` is correct using complete induction. Since induction and recursion are the same thing in Agda, this means the proof is a (dependetly typed) recursive function defined using `cRec`. Recall that the first argument to `cRec` is the type for the elements of the `rec` tuple. However, because we are now doing induction, we should instead think of the `rec` tuple as the induction hypothesis. In our step function for the proof we will use the parameter name `IH` instead of `rec`. Furthermore, because this tuple serves as the induction hypothesis, its elements will need to be proofs that `fast-exp` is correct for particular (smaller) exponents. So the type of the element at position `n` should be the proposition `fe-ok n`. Also, recall that the `CRec` type operator produces the type of the tuple. So for our current purposes, `CRec _ fe-ok n` should be the type of `IH`. The second argument to `cRec` is a step function, which in this case needs to construct a proof that `fast-exp n x ≡ x ^ n` for an arbitrary `n`, given the induction hypothesis `IH`. We define the `step` function in the `where` clause of our theorem below. The `step` function mimics the structure of the `fe-step` function, doing case analysis on the result of `parity n`. We then do the appropriate equational reasoning. The one unusual step is the first one, which uses the `projₙ-fe` lemma to replace the "raw" recursive call via `projₙ` (as it appears in the body of `fe-step`) with a call to `fast-exp`. ``` fast-exp-is-correct : ∀ n x → fast-exp n x ≡ x ^ n fast-exp-is-correct = cRec fe-ok step where step : (n : ℕ) → CRec _ fe-ok n → fe-ok n step zero IH x = refl step (suc n) IH x with parity n ... | inj₁ (k , refl) = begin x * projₙ (fe-rec (1 + 2 * k)) k lt (x * x) ≡⟨ cong (λ X → x * X) (projₙ-fe k (k + (k + zero)) (x * x) lt) ⟩ x * fast-exp k (x * x) ≡⟨ cong (λ X → x * X) (projₙ IH k (≤⇒≤′ (m≤m+n _ _)) (x * x)) ⟩ x * ((x * x) ^ k) ≡⟨ cong (λ X → x * X) (*-distribˡ-^ x x k) ⟩ x * (x ^ k * x ^ k) ≡⟨ cong (λ X → x * (x ^ k * x ^ X)) (sym (+-identityʳ k)) ⟩ x * (x ^ k * x ^ (k + 0)) ≡⟨ cong (λ X → x * X) (sym (^-distribˡ-+-* x k (k + zero))) ⟩ x * (x ^ (2 * k)) ∎ where open ≡-Reasoning lt = ≤⇒≤′ (m≤m+n k (k + zero)) ... | inj₂ (k , refl) = begin x * (x * (projₙ (fe-rec (2 + (2 * k))) k lt (x * x))) ≡⟨ cong (λ X → x * (x * X)) (projₙ-fe k (1 + (2 * k)) (x * x) lt) ⟩ x * (x * (fast-exp k (x * x))) ≡⟨ cong (λ X → x * (x * X)) (projₙ IH k (≤⇒≤′ (≤-step (m≤m+n _ _))) (x * x)) ⟩ x * (x * (x * x) ^ k) ≡⟨ cong (λ X → x * (x * X)) (*-distribˡ-^ x x k) ⟩ x * (x * ((x ^ k) * (x ^ k))) ≡⟨ cong (λ X → x * (x * ((x ^ k) * (x ^ X))))(sym(+-identityʳ k)) ⟩ x * (x * (x ^ k * x ^ (k + 0))) ≡⟨ cong (λ X → x * (x * X)) (sym (^-distribˡ-+-* x k (k + zero))) ⟩ x * (x * (x ^ (2 * k))) ∎ where open ≡-Reasoning lt = ≤⇒≤′ (≤-step (m≤m+n k (k + zero))) ``` ## Diving Deeper, Definition Induction using the `Induction` Library We have seen how to use the facilities in `Nat.Induction` for complete induction. Next we explore how to use the lower-level library in `src/Induction.agda` to build our own recursion/induction principle. In particular, we'll build an alternative form of complete induction that should be more familiar to everyone, one in which the `rec` parameter is simply a function, not a tuple of functions. ## The `RecStruct` Type Operator and `SRec` The `RecStruct` type operator in `src/Induction.agda` produces the type for the operators like `CRec`. The parameter `A` of `RecStruct` is for the things you're doing induction on (e.g. `ℕ`) and the universe levels `ℓ₁` and `ℓ₂` can be ignored for now. RecStruct : ∀ {a} → Set a → (ℓ₁ ℓ₂ : Level) → Set _ RecStruct A ℓ₁ ℓ₂ = Pred A ℓ₁ → Pred A ℓ₂ Recall that `Pred A ℓ₁` is a function from an element of `A` to a type (an instance of `Set`). In `FERec` above, `(λ i → ℕ → ℕ)` is an example of something of type `Pred ℕ _`, and we applied `CRec` to this predicate. Let us take a look at the definition of `CRec`. It generates a tuple type of length `n` where each element is of type `P i` (for `i < n`) except for `0`. The last element is just unit. CRec : ∀ ℓ → RecStruct ℕ ℓ ℓ CRec ℓ P zero = ⊤ CRec ℓ P (suc n) = P n × CRec ℓ P n For our alternative form of complete induction, we replace the tuple with a function. In particular, a function that takes a number `k`, a proof that `k` is smaller than the current `n`, and produces the function's result for `k`. So the type of our `rec` parameter is given by the following `SRec` operator. ``` SRec : ∀ ℓ → RecStruct ℕ ℓ ℓ SRec ℓ P = λ n → ∀ k → k <′ n → P k ``` (We use `<′` instead of `<` to make it easier to define `sRecBuilder` and prove `fe-rec-fast-exp₂` in the following.) ## Building the Arguments for the `rec` Parameter The next step is to build a value of type `SRec` for every natural number, so that we can pass these values into the `rec` parameter of the client's step function. Thus, we have to define a function analogous to the `cRecBuilder` we discussed above. Let us name our builder function `sRecBuilder`. As a guide, the `Induction` library defines the `RecursorBuilder` type operator that specifies the type for builder functions. Its input parameter `Rec` has type `RecStruct` (e.g. `SRec` is a valid argument to `RecursorBuilder`). RecursorBuilder : ∀ {a ℓ₁ ℓ₂} {A : Set a} → RecStruct A ℓ₁ ℓ₂ → Set _ RecursorBuilder Rec = ∀ P → (Rec P ⊆′ P) → Universal (Rec P) A builder function takes 1) a predicate `P` that specifies the result type of the recursive function (just like the `P` in `SRec`), and 2) a step function that produces a `P` given a `rec` parameter of type `Rec P`. The result of the builder function is a value of type `Rec P`, that is, a value that can be passed into the `rec` parameter of the client's step function. So our `sRecBuilder` function has type `RecursorBuilder (SRec ℓ)`. So it takes a predicate `P`, a step function, and its output is of type `SRec ℓ P`, so the output is a function with parameters `n` and `k` and a proof of `k <′ n`. The function produces an element of `P k`. We define `sRecBuilder` using induction on `k <′ n`. We discuss the two cases below. ``` sRecBuilder : ∀ {ℓ} → RecursorBuilder (SRec ℓ) sRecBuilder P step .(suc k) k ≤′-refl = step k rec where rec = sRecBuilder P step k sRecBuilder P step (suc n) k (≤′-step lt) = sRecBuilder P step n k lt ``` * In the case of `≤′-refl`, we have `n = suc k`. We need to produce `P k`, which we can do with the call `step k rec`, but we need to fill in the second argument `rec`. This we do by the recursive call to `sRecBuilder`. * The case for `≤′-step` is even easier. We simply call `sRecBuilder` recursively. ## The `build` Function to Finish `sRec` The final step in creating our custom induction/recursion principle is to invoke the `build` function in `src/Induction.agda` to produce `sRec`. The `build` function takes a builder function, such as `sRecBuilder`, and produces a `Recursor`, which is a function that, given a step function, produces a recursive function. ``` sRec : ∀{ℓ} → Recursor (SRec ℓ) sRec = build sRecBuilder ``` ## Revisiting Fast Exponentiation with Strong Recursion/Induction Analogous to `FERec`, we define the type for the `rec` parameter of our step function with the below `FERec₂`, but this time use `SRec` instead of `CRec`. ``` FERec₂ : ℕ → Set FERec₂ n = SRec _ (λ i → ℕ → ℕ) n ``` The step function `fe-step₂` is similar to `fe-step`, but this time the `rec` parameter is easier to work with. It's a function that we can call. It just requires an extra argument with the proof that the argument `k` is less than `n`. ``` fe-step₂ : (n : ℕ) → FERec₂ n → (ℕ → ℕ) fe-step₂ zero rec x = 1 fe-step₂ (suc n′) rec x with parity n′ ... | inj₁ (k , refl) = x * rec k lt (x * x) where lt : k <′ 1 + (2 * k) lt = ≤⇒≤′ (s≤s (m≤m+n k _)) ... | inj₂ (k , refl) = x * x * rec k lt (x * x) where lt : k <′ 2 + (2 * k) lt = ≤⇒≤′ (s≤s (≤-step (m≤m+n k _))) ``` We define our second fast exponentiation using `sRec` and our new step function `fe-step₂`. ``` fast-exp₂ : ℕ → ℕ → ℕ fast-exp₂ = sRec (λ _ → (ℕ → ℕ)) fe-step₂ ``` ## Revisiting the Proof that Fast Exponentiation is Correct Let us see how this alternate definition of fast exponentiation affects our proof of correctness. (TLDR: it only changes one lemma.) We define an abbreviation for the correctness condition as follows. ``` fe-ok₂ : ℕ → Set fe-ok₂ n = ∀ x → fast-exp₂ n x ≡ x ^ n ``` And another appreviation for calling `sRecBuilder` with the new step function `fe-step₂`. ``` fe-rec₂ : (n : ℕ) → SRec _ (λ _ → (ℕ → ℕ)) n fe-rec₂ n = sRecBuilder (λ _ → (ℕ → ℕ)) fe-step₂ n ``` Last time we proved the lemma `projₙ-fe` to relate the "raw" recursive call to a call to `fast-exp`. Here we need a similar lemma, but it is somewhat simpler because we no longer need to use `projₙ`. So we just need to relate `fe-rec₂` to `fast-exp₂`. ``` fe-rec-fast-exp₂ : ∀ k n x (lt : k <′ suc n) → fe-rec₂ (suc n) k lt x ≡ fast-exp₂ k x fe-rec-fast-exp₂ k .k x ≤′-refl = refl fe-rec-fast-exp₂ k (suc n′) x (≤′-step lt) = fe-rec-fast-exp₂ k n′ x lt ``` We prove that `fast-exp₂` is correct, again with a proof by complete induction. The only difference is that the first step in the equational reasoning is to use `fe-rec-fast-exp₂` instead of `projₙ-fe`. ``` fast-exp₂-is-correct : ∀ n x → fast-exp₂ n x ≡ x ^ n fast-exp₂-is-correct = cRec fe-ok₂ step where step : (n : ℕ) → CRec _ fe-ok₂ n → fe-ok₂ n step zero IH x = refl step (suc n′) IH x with parity n′ ... | inj₁ (k , refl) = begin x * fe-rec₂ (suc n′) k lt (x * x) ≡⟨ cong(λ X → x * X) (fe-rec-fast-exp₂ k n′ (x * x) lt) ⟩ x * fast-exp₂ k (x * x) ≡⟨ cong(λ X → x * X) (projₙ IH k lt₂ (x * x)) ⟩ x * (x * x) ^ k ≡⟨ cong(λ X → x * X) (*-distribˡ-^ x x k) ⟩ x * ((x ^ k) * (x ^ k)) ≡⟨ cong(λ X → x * (x ^ k * x ^ X)) (sym(+-identityʳ k)) ⟩ x * (x ^ k * x ^ (k + 0)) ≡⟨ cong(λ X → x * X) (sym (^-distribˡ-+-* x k (k + zero))) ⟩ x * x ^ (2 * k) ∎ where open ≡-Reasoning lt = s≤′s (≤⇒≤′ (m≤m+n k (k + zero))) lt₂ = ≤⇒≤′ (m≤m+n k (k + zero)) ... | inj₂ (k , refl) = begin (x * x) * fe-rec₂ (2 + 2 * k) k lt (x * x) ≡⟨ cong(λ X → (x * x) * X)(fe-rec-fast-exp₂ k (1 + (2 * k)) (x * x) lt) ⟩ (x * x) * fast-exp₂ k (x * x) ≡⟨ cong(λ X → (x * x) * X)(projₙ IH k lt₂ (x * x)) ⟩ (x * x) * (x * x) ^ k ≡⟨ cong(λ X → (x * x) * X) (*-distribˡ-^ x x k) ⟩ (x * x) * (x ^ k * x ^ k) ≡⟨ cong(λ X → (x * x) * (x ^ k * x ^ X)) (sym(+-identityʳ k)) ⟩ (x * x) * (x ^ k * x ^ (k + 0)) ≡⟨ cong(λ X → (x * x) * X) (sym (^-distribˡ-+-* x k (k + zero))) ⟩ (x * x) * (x ^ (2 * k)) ≡⟨ *-assoc x _ _ ⟩ x * (x * (x ^ (2 * k))) ∎ where open ≡-Reasoning lt = s≤′s (≤⇒≤′ (≤-step (m≤m+n k (k + zero)))) lt₂ = (≤⇒≤′ (≤-step (m≤m+n k (k + zero)))) ``` ## A Parting Thought As I was fumbling around and bumping into deadends on my way to writing the above definitions and proofs, perhaps the trickiest part of using the `Induction` library was stating and proving the lemmas that relate the "raw" recursive call to the recursive function, e.g., `projₙ-fe` and `fe-rec-fast-exp₂`. I wonder whether the `Induction` library could somehow also provide a general lemma for that, perhaps with help from the client.

## Thursday, April 07, 2022

### Using Agda's Induction/Recursion Library

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