Add tips page to the book 2

data/tableOfContents.yml

@@ -48,5 +48,7 @@ part:
   chapter:
   - include: src/plfa/backmatter/Acknowledgements.md
     epub-type: acknowledgements
+  - include: src/plfa/backmatter/Tips.md
+    epub-type: appendix
   - include: src/plfa/backmatter/Fonts.lagda.md
     epub-type: appendix

src/plfa/backmatter/Tips.md

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+---
+title     : "Tips"
+permalink : /Tips/
+---
+
+# Interactive development
+
+## Equational reasoning skeleton
+
+When starting to work on an equational reasoning problem, you can begin like this:
+```agda
+begin
+    ?
+  ≡⟨ ? ⟩
+    ?
+  ≡⟨ ? ⟩
+    ?
+  ≡⟨ ? ⟩
+    ?
+∎
+```
+
+This sets up a series of reasoning steps with holes for both the expressions at each step, and the step itself.
+Since the reasoning steps are holes, Agda will always accept this, so you can make incremental progress while keeping your file compiling.
+
+Now call "solve" (`C-c C-s`).
+You can call this outside a hole to have it try solving all holes in the file, or in a hole to try solving only that hole.
+This will solve the expression holes at the beginning and the end so you don't have to fill them in.
+You can do this again if you fix a step, so if you fill in the reasoning you want to do for a step, you can use "solve" and Agda will fill in the next term.
+
+You can prefix the solve command such that it computes terms and types (`C-u C-c C-s`). This `C-u` prefix also works for many other commands.
+
+Alternatively you can use "auto" (`C-c C-a`) which tries to fill in holes using anything it can think of.
+This can work in many more situations, but you do risk it filling in junk like a genie twisting your wish if you haven't constrained the hole is enough.
+
+## Goal information will show computed terms
+
+Suppose you have a partial equational reasoning proof:
+```agda
+begin
+    1 + (x + y)
+  ≡⟨ ?2 ⟩
+    ?
+  ≡⟨ ? ⟩
+    ?
+  ≡⟨ ? ⟩
+    ?
+∎
+```
+
+The first term can be computed more.
+To see this, you can ask for the goal type (`C-t`) for `?2` and it will show this:
+```
+suc (x + y) = _y_10
+```
+i.e. your LHS and an unknown RHS (because the next step is still a hole), but it will show the LHS fully computed, which you can copy into your file directly.
+
+## Quickly setting up `with` clauses
+
+If you have a function like this:
+```agda
+foo a = ?
+```
+and you want to introduce a `with`-clause, there is no direct hotkey for this.
+The fastest thing to do is to write
+```agda
+foo a with my-thing
+... | x = ?
+```
+and then ask to case-split on `x`.
+
+## Quickly setting up record skeletons
+
+"Refine" (`C-c C-r`) will set up a skeleton for a record constructor if Agda knows a hole must be a record type.
+This is useful for e.g. isomorphisms, products, sigmas, or existentials.
+
+# General tips
+
+## Try and make your definitions compute as much as possible
+
+For example, compare:
+```agda
+2 * foo b
+```
+vs
+```agda
+foo b * 2
+```
+
+The definition of `*` does case analysis on its left operand.
+This means that if the left operand to `*` is a constructor application, Agda can do some computation for you.
+The first example computes to `foo b + (foo b + zero)`, but the second one does not compute at all: Agda can't pattern match on `foo b`, it could be anything!
+
+Aligning your expressions with with the computation order like this lets Agda do more definitional simplification, which means you have to do less work.
+Unfortunately it does require you to know _which_ arguments are the one that Agda can evaluate.
+Binary operators conventionally match on their left operand first if the choice is free.
+For example, `_+_`, `_*_`, and `_++_` all match first on their left operand, but `_^_` has to match first on the right operand, the exponent.
+You can always look it up yourself by going to the function definition (`M-.` or middle mouse click).
+
+## `x + x` is better than `2 * x`
+
+An expression like `2 * x` will compute to `x + (x + zero)`.
+You now need to get rid of the `x + zero` with a rewrite or similar.
+You don't have this problem if you just write `x + x`.
+
+This is probably less true for larger constants: writing out `8 * x` would be tedious.
+
+## The three styles of proof all tend to use the same assumptions
+
+Consider these three proofs of the same theorem:
+```agda
+*-distrib-+ : ∀ (m n p : ℕ) → (m + n) * p ≡ m * p + n * p
+*-distrib-+ zero n p = refl
+*-distrib-+ (suc m) n p =
+  begin
+    (suc m + n) * p
+  ≡⟨⟩
+    suc (m + n) * p
+  ≡⟨⟩
+    p + (m + n) * p
+  ≡⟨ cong (p +_) (*-distrib-+ m n p) ⟩
+    p + (m * p + n * p)
+  ≡⟨ sym (+-assoc p (m * p) (n * p)) ⟩
+    (p + m * p) + n * p
+  ≡⟨⟩
+    (suc m * p) + n * p
+  ∎
+
+*-distrib-+‵ : ∀ (m n p : ℕ) → (m + n) * p ≡ m * p + n * p
+*-distrib-+‵ zero n p = refl
+*-distrib-+‵ (suc m) n p = trans ( cong (p +_) (*-distrib-+‵ m n p)) ( sym ( +-assoc p (m * p) (n * p)))
+
+*-distrib-+‶ : ∀ (m n p : ℕ) → (m + n) * p ≡ m * p + n * p
+*-distrib-+‶ zero n p = refl
+*-distrib-+‶ (suc m) n p rewrite +-assoc p (m * p) (n * p) | *-distrib-+‶ m n p = refl
+```
+
+They use three approaches for doing "multiple steps":
+1. Equational reasoning
+2. `trans`
+3. `rewrite`
+
+However, notably they all use the same supporting proofs!
+That means you can often write the overall proof however you find easiest and then rewrite it into another form if you want.
+Do keep in mind that compact proofs are not always the most readable.
+Strive to find a happy medium that shows each important step clearly without too many trivial steps in between.
+
+## Avoid mutual recursion in proofs by recursing outside the lemma
+
+Look at the second part of the proof of commutativity of `+`:
+```agda
++-comm m (suc n) =
+  begin
+    m + suc n
+  ≡⟨ +-suc m n ⟩
+    suc (m + n)
+  ≡⟨ cong suc (+-comm m n) ⟩
+    suc (n + m)
+  ≡⟨⟩
+    suc n + m
+  ∎
+```
+We use two equalities: the `+-suc` lemma, and a recursive use of `+-comm`.
+
+If you were doing this for the first time, you might be tempted to make _one_ lemma for both those steps:
+```agda
++-suc-comm : m + suc n ≡ suc (n + m)
+```
+But then you would need to call `+-comm` from within this lemma, which would make `+-comm` and `+-suc-comm` mutually recursive.
+
+Instead, follow this example and use the recursive call before or after your lemma.
+
+## When to use implicit parameters
+
+Roughly: if it can be inferred easily from the result, e.g.
+```
+≤-refl : ∀ { n : ℕ } → n ≤ n
+```
+Agda will be able to infer the `n` from the `n ≤ n` in the result.
+
+Note that the situation is different for constructors and functions.
+Generally, Agda can infer `x` from `f x` if `f` is a constructor (like `≤`) but _not_ if it is a function (like `_+_`).
+
+## Avoiding "green slime"
+
+"Green slime" is when you use a function in the index of a returned data type value.
+For example:
+```agda
+data Tree (A : Set) : ℕ → Set where
+  leaf : A → Tree A 1
+  node : ∀ {s : ℕ} → Tree A s → Tree A s → Tree A (s + s)
+```
+Here we end up with a `Tree (s + s)` as the result of `node`.
+This means that if we pattern match on a value of type `Tree A n`, Agda may get confused about making `n` and `s + s` the same.
+
+There are a few ways to avoid green slime, but one way that often works is to use a variable and also include a proof that it has the value that you want. So:
+
+```agda
+data Tree (A : Set) : ℕ → Set where
+  leaf : A → Tree A 1
+  node :
+    ∀ {s : ℕ} {s′ : ℕ}
+    → Tree A s
+    → Tree A s
+    → s′ ≡ s + s
+    → Tree A s′
+```
+
+Now you can still get the information about the sum by matching on the proof, but Agda has an easier time.
+
+Conor McBride coined the term green slime in reference to the boom of hazardous green slime in 1970s science fiction due to the advent of colour television.