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To Do

Changes I would like to make to the book:

  • Relations, exercise o+o≡e --> (recommended)
  • Relations, exercise ≤-iff-< --> ≤→<, <→≤
  • Relations, exercise Bin-predicates --> Bin-predicate change canonical form of zero to be empty string, <>. Similar change to subsequent exercise.

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Suggestion from Conor for Inference

Conor McBride [email protected]

29 Oct 2018, 09:34

Hi Phil

In a rush, but...

data Tag : Set where
  tag-ℕ : Tag
  tag-⇒ : Tag

...that's just Bool. Bool is almost never your friend.

Get evidence!

-- yer types
data Type : Set where
  nat : Type
  _=>_ : Type -> Type -> Type

-- logic
data Zero : Set where
record One : Set where constructor <>

-- evidence of not being =>
Not=> : Type -> Set
Not=> (_ => _) = Zero
Not=> _ = One

-- constructing the "=> or not" view
data Is=>? : Type -> Set where
  is=> : (S T : Type) -> Is=>? (S => T)
  not=> : {T : Type} -> Not=> T -> Is=>? T

-- this will need all n cases, but you do it once
is=>? : (T : Type) -> Is=>? T
is=>? nat = not=> <>
is=>? (S => T) = is=> _ _

-- worked example: domain
data Maybe (X : Set) : Set where
  yes : X -> Maybe X
  no : Maybe X

-- only two cases
dom : Type -> Maybe Type
dom T with is=>? T
dom .(S => T) | is=> S T = yes S
dom T | not=> p = no

-- addendum: in the not=> p case, if we subsequently inspect T, we can rule out the => case using p
{- with T
dom T | not=> p | nat = no
dom T | not=> () | q => q₁
-}

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Where to put Lists?

Three possible orders:

  • (a) As current
  • (b) Put Lists immediately after Induction.
    • requires moving composition & extensionality earlier
    • requires moving parameterised modules earlier for monoids
    • add material to relations: lexical ordering, subtype ordering, All, Any, All-++ iff
    • add material to isomorphism: All-++ isomorphism
    • retain material on decidability of All, Any in Decidable
  • (c) Put Lists after Decidable
    • requires moving Any-decidable from Decidable to Lists
  • (d) As (b) but put parameterised modules in a separate chapter

Tradeoffs:

  • (b) Distribution of exercises near where material is taught
  • (b) Additional reinforcement for simple proofs by induction
  • (a,c) Can drop material if there is lack of time
  • (a,c) Earlier emphasis on induction over evidence
  • (c) More consistent structuring principle

Set up lists of exercises to do

  • Use md rather than HTML
  • Tell students to do all exercises, and just mark some as stretch?
  • Make a list of exercises to do, with some marked as stretch?
  • Compare with previous set of exercises to discover some holes?
  • Add ==N as an exercise to Relations?

Other questions

  • Resolve any issues with modules to define properties of orderings?
  • Resolve any issues with equivalence and Setoids?

Old questions

Possible structures for the book

  • One possible development

    • raw terms
    • scoped terms (is conversion from raw to scoped a function?)
    • typed terms (via bidirectional typing)

  • The above could be developed either for

    • pure lambda terms with full normalisation
    • PCF with top-level reduction to value

  • If I follow raw-scoped-typed then:

    • might want to have reductions for completely raw terms later in the book rather than earlier
    • full normalisation requires substitution of open terms

  • Today's task (Tue 8 May)

    • consider lambda terms to values (not PCF)

    • raw, scoped, typed

    • Note that substitution for open terms is not hard, it is proving it correct that is difficult!

    • can put each development in a separate module to support reuse of names

  • Today's thoughts (Thu 10 May)

    • simplify TypedFresh
      • Does it become easier once I have suitable lemmas about free in place?
    • still need a chain of development
      • raw -> scoped -> typed
      • raw -> typed and typed -> raw needed for examples
      • look again at raw to scoped
      • look at scoped to typed
      • typed to raw requires fresh names
      • fresh name strategy: primed or numbers?
      • ops on strings: show, read, strip from end
    • trickier ideas
      • factor TypedFresh into Barendregt followed by substitution? This might actually lead to a much longer development
      • would be cool if Barendregt never required renaming in case of substitution by closed terms, but I think this is hard

  • Today's achievements and next steps (Thu 10 May)

    • defined break, make to extract a prefix and count primes at end of an id. But hard to do corresponding proofs. Need to figure out how to exploit abstraction to make terms readable.
    • Conversion of raw to scoped and scoped to raw is easy if I use impossible
    • Added conversion of TypedDB to PHOAS in extra/DeBruijn-agda-list-4.lagda
    • Next: try adding bidirectional typing to convert Raw or Scoped to TypedDB
    • Next: Can proofs in Typed be simplified by applying suitable lemmas about free?
    • updated Agda from: Agda version 2.6.0-4654bfb-dirty to: Agda version 2.6.0-2f2f4f5 Now TypedFresh.lagda computes 2+2 in milliseconds (as opposed to failing to compute it in one day).

STLC

PHOAS

The following comments were collected on the Agda mailing list.

Untyped lambda calculus

Relevant papers

Agda resources

Syntax for lambda calculus

  • ƛ \Gl-
  • ∙ .

Adrian's comments from MeetUp

Adrian King I think we've finally gone through the whole book now up through chapter Properties, and we've come up with just three places where we thought the book could have done a better job of preparing us for the exercises.

Starting from the end of the book:

  • Chapter Quantifiers, exercise Bin-isomorphism:

In the to∘from case, we want to show that:

⟨ to (from x) , toCan {from x} ⟩ ≡ ⟨ x , canX ⟩

I found myself wanting to use a general lemma like:

exEq :
∀ {A : Set} {x y : A} {p : A → Set} {px : p x} {py : p y} →
x ≡ y →
px ≡ py →
⟨ x , px ⟩ ≡ ⟨ y , py ⟩
exEq refl refl = refl

which is in some sense true, but in Agda it doesn't typecheck, because Agda can't see that px's type and py's type are the same. My solution made explicit use of heterogeneous equality.

I realize there are ways to solve this that don't explicitly use heterogeneous equality, but the surprise factor of the exercise might have been lower if heterogeneous equality had been mentioned by that point, perhaps in the Equality chapter.

  • Chapter Quantifiers, exercise ∀-×:

I assume the intended solution looks something like:

∀-× : ∀ {B : Tri → Set} → ((x : Tri) → B x) ≃ (B aa × B bb × B cc)
∀-× = record {
to = λ f → ⟨ f aa , ⟨ f bb , f cc ⟩ ⟩ ;
from = λ{ ⟨ baa , ⟨ bbb , bcc ⟩ ⟩ → λ{ aa → baa ; bb → bbb ; cc → bcc } } ;
from∘to = λ f → extensionality λ{ aa → refl ; bb → refl ; cc → refl } ;
to∘from = λ{ y → refl } }

but it doesn't typecheck; in the extensionality presented earlier (in chapter Isomorphism), type B is not dependent on type A, but it needs to be here. Agda's error message is sufficiently baffling here that you might want to warn people that they need a dependent version of extensionality, something like:

Extensionality : (a b : Level) → Set _
Extensionality a b =
{A : Set a} {B : A → Set b} {f g : (x : A) → B x} →
(∀ x → f x ≡ g x) → f ≡ g

postulate
exten : ∀ {a b} → Extensionality a b
  • Chapter Relations, exercise Bin-predicates:

I made use here of the inspect idiom, in Aaron Stump's variant, which is syntactically more convenient:

keep : ∀ {ℓa} → {a : Set ℓa} → (x : a) → Σ a (λ y → y ≡ x)
keep x = x , refl

Pattern matching on keep m, where m is some complicated term, lets you keep around an equality between the original term and the pattern matched, which is often convenient, as in:

roundTripTwice {x} onex with keep (from x)
roundTripTwice {x} onex | zero , eq with oneIsMoreThanZero onex
roundTripTwice {x} onex | zero , eq | 0<fromx rewrite sym eq =
⊥-elim (not0<0 0<fromx)
roundTripTwice {x} onex | suc n , eq rewrite sym eq | toTwiceSuc {n} |
cong x0_ (sym (roundTrip (one onex))) | sym eq = refl

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