Lecture 3: Inductive/co-Inductive types

Examples of inductive types:

$Nat$:

• $zero: Nat$
• $succ: Nat ⟶ Nat$

$Bool$:

• $true: Bool$
• $false: Bool$

Inductive	Co-inductive types
Positive connectives	Negative connectives
built from constructor	built from destructors
initial algebra	final co-algebra
co-limits	limits
ex: $Nat, Bool, \oplus, \otimes$	ex: record (defined by its projections), streams (we can take the first or the rest), in linear logic: $\&$
recursor	“cofix point” (to recursively build objects from the destructor)

NB: In linear logic, there are two conjunctions: one is positive, the other negative.

Inductive types

“Rec” is an eliminator for the constructor.

Natural numbers

In Coq:

Inductive Nat: U :=
    zero: Nat
  | succ: Nat -> Nat

or, by just giving the argument of the type:

Inductive Nat: U :=
    zero
  | succ (n: Nat)

With the system T operator:

\[\cfrac{Γ ⊢ t: Nat \quad Γ ⊢ C: U \quad Γ ⊢ u: C \quad Γ, n:Nat, a: C ⊢ v: C}{Γ ⊢ \texttt{rec } \, t \, [zero → u \mid succ \, n/a → v]: C}\]

• $\texttt{rec } \, zero \, [zero → u \mid succ \, n/a → v] ⟶ u$

• \[\texttt{rec } \, (succ \, n') \, [zero → u \mid succ \, n /a→ v] ⟶ v[n:=n']\left[a:= \texttt{rec } \, n' \, [zero → u \mid succ \, n/a → v]\right]\]

let rec Rec n = match n with
      zero -> u
    | suc n' -> v[n := n'][a := Rec n']

Dependent version:

\[\cfrac{Γ ⊢ t: Nat \quad Γ, n:Nat ⊢ C: U \quad Γ ⊢ u: C[0/n] \quad Γ, n:Nat, a: C[n/n] ⊢ v: C[succ \, n, n]}{Γ ⊢ \texttt{rec } \, t \, [zero → u \mid succ \, n/a → v]: C[t/n]}\]

NB: it’s an “annotated version” of Peano arithmetic induction scheme, where the annotations are the proofs.

Lists

Inductive list (A: U): U :=
      nil : list A
    | cons: A -> list A

\[\cfrac{Γ, A:U ⊢ l: list \, A \quad Γ, A:U ⊢ C: U \quad Γ, A:U ⊢ u: C \quad Γ, A:U, a:A, l: list \, A, b:C ⊢ v: C}{Γ, A:U ⊢ \texttt{rec } \, l \, [nil → u \mid cons \, a \, l/b → v]: C}\]

• $\texttt{rec } \, nil \, [nil → u \mid cons \, a \, l/b → v] ⟶ u$

• \[\texttt{rec } \, (cons \, t \, l) \, [nil → u \mid cons \, a \, l /b→ v] ⟶ v[t/a][l/l]\left[\texttt{rec } \, l \, [nil → u \mid cons \, a \, l /b→ v]/b\right]\]

Dependent version:

\[\cfrac{Γ, A:U ⊢ l: list \, A \quad Γ, A:U, l:list \, A ⊢ C: U \quad Γ, A:U ⊢ u: C[nil/l] \quad Γ, A:U, a:A, l: list \, A, b:C[l/l] ⊢ v: C[cons \, a \, l/l]}{Γ, A:U ⊢ \texttt{rec } \, l \, [nil → u \mid cons \, a \, l/b → v]: C[l/l]}\]

NB: if $C$ is a proposition $hProp$, then the proof doesn’t matter!

Booleans

Inductive Bool: U :=
      true : Bool
    | false: Bool

\[\cfrac{Γ ⊢ b: Bool \quad Γ ⊢ C: U \quad Γ ⊢ t: C \quad Γ ⊢ u: C}{Γ ⊢ \texttt{rec } \, b \, [true → t \mid false → u]: C}\]

“if b then t else u”

Trees

Inductive tree (A: Ui): Uj≥i :=
      leaf : tree A
    | node : A -> tree A -> tree A -> tree A

Co-inductive types

Co-inductive types are defined by their destructors.

Products

Record Prod (A: Ui) (B: Ui'): Uj≥i,i' := {fst: A, snd: B}

• $fst: Prod \, A \, B ⟶ A$
• $snd: Prod \, A \, B ⟶ B$

Now, in let’s build the (almost) dual version of rec: a constructor for the destructors.

\[\cfrac{Γ, A:U_i, B:U_{i'} ⊢ t: A \quad Γ, A:U_i, B:U_{i'} ⊢ u: B}{Γ ⊢ \lbrace fst = t ; snd = u \rbrace: Prod \, A \, B}\]

• $fst \, \lbrace fst=t; snd=u \rbrace ⟶ t$
• $snd \, \lbrace fst=t; snd=u \rbrace ⟶ u$

NB: it looks like the categorical product

Streams

They represent infinite lists.

CoInductive Stream (A: Ui) : Uj≥i := {hd: A; tl: Stream A}

• $hd: Stream \, A ⟶ A$
• $tl: Stream \, A ⟶ Stream \, A$

\[\cfrac{Γ, A:U ⊢ t: A \quad Γ, A:U, a: Stream \, A ⊢ u: Stream \, A}{Γ, A:U ⊢ \lbrace hd = t ;\; tl/a = u \rbrace: Stream \, A}\]

• $hd \, \lbrace hd=t; \; tl/a=u \rbrace ⟶ t$
• $tl \, \lbrace hd=t; \; tl/a=u \rbrace ⟶ u[\lbrace hd=t;\; tl/a =u \rbrace/a]$

Not that interesting, since we have always the same tail.

Notation by Abel (inventor of Agda), Pientka (Beluga), and Setzer.

NB: Warning! the “co-inductive” keyword in Coq isn’t what we’re talking about. Negative connectives are called “primitive projections” in Coq.