Lecture 6: Dependent type theory

Teacher: Gilles Dowek

20 years ago: dependent type theory was thought to be going to replace simple type theory. But not quite true, today:

  • Dependent Type Theory:

    • Coq
    • Adga
    • Lean
    • Matita

  • Simple Type Theory:

    • HOL Light (Hales’ theorem, aka Kepler theorem (conjectured in 1610, proved in HOL in 1999))
    • Isabelle/HOL
    • PVS (used by NASA)

The term $λx.x: ι → ι$ and the proof $λα \, α: A ⇒ A$ have nothing to do with one another ⟶ Why not merge the two notions? (we would have one notion of substitution, type checking, etc…)

If everything is mixed: some formulations of the axiom of choice become a theorem!

Dependently typed $λ$-calculus

Example: types array n, where n is the size of the array

NB: In Pascal, arrays had their size in their type, but also their range of indices (array 0 99: indices ranging from 0 to 99)

\[\prod\limits_{ n: ℕ } array \; n \quad \text{ or } \quad (n: ℕ) ⟶ (array \; n)\]

$\prod$ binds the variable $n$ in the target type.

When the set of types is given by a context-free grammar, as in

\[A \; ≝ \; ι \; \mid \; o \; \mid \; A → A\]

you don’t need typing rules (the grammar is sufficient), but here the grammar is not context-free ⟹ we need typing rules (as for term formation)

But types are now considered as terms, and all types have type $Type$

Judgments:

\[Γ ⊢ t:A \qquad Γ ⊢ t:A\\ Γ \text{ well-formed}\]

Typing rules of $λ\Pi$-calculus (also called LF)

\[\cfrac{}{[] \text{ well-formed}}\] \[\cfrac{Γ \text{ well-formed}}{Γ, A: Type \text{ well-formed}}\] \[\cfrac{Γ \text{ well-formed}}{Γ ⊢ x: A} \quad x:A ∈ Γ\] \[\cfrac{Γ ⊢ A: Type \qquad Γ ⊢ B: Type}{Γ ⊢ A → B: Type}\]

These definitions replace simple types. Ground types ($ι, o$) are variables of type $Type$.

ST $λ$-calculus rules

\[\cfrac{Γ ⊢ A: Type}{Γ ⊢ x: A \text{ well-formed}} \quad x:A ∈ Γ\] \[\cfrac{Γ ⊢ A: Type \qquad Γ, x:A ⊢ B: Type \qquad Γ, x:A ⊢ t: B}{Γ ⊢ λx:A. t: A → B}\] \[\cfrac{Γ ⊢ t: A → B \qquad Γ ⊢ t': A}{Γ ⊢ (t \, t'): B}\]

Example:

\[\infer[λ]{⊢ λx:nat. x: nat → nat}{ \infer[var]{x:nat ⊢ x:nat}{\phantom{x:nat ⊢ x:nat}} }\]

now turned into

\[\infer{nat:Type ⊢ λx:nat.x: nat → nat}{ & nat:Type ⊢ nat:Type & \infer{nat: Type, x:nat ⊢ nat: Type}{ \infer{nat: Type, x:nat \text{ well-formed}}{\phantom{nat: Type, x:nat \text{ well-formed}}} } & \infer{nat: Type, x:nat ⊢ x:nat}{ \infer{nat: Type, x: nat \text{ well-formed}}{ \infer{nat: Type ⊢ nat: Type}{ \infer{nat: Type \text{ well-formed}}{ \infer{[] \text{ well-formed}}{\phantom{[] \text{ well-formed}}} } } } } }\]

To come back to the previous example: array has type $nat → Type$. So $nat → Type$ is a type, hence of type $Type$

Similarly, $array \; 0$ has type $Type$, hence $Type$ must have type $Type$ ⟹ Girard’s paradox (based on Burali-Forti’s paradox)

⟶ New constant: $Kind$ for the types $Type$, $nat → Type$, etc… Then, two solutions:

  • either we stop there and $Kind$ has no type
  • or $Kind$ has a type ⟶ universe hierarchy

Beware of naming conventions:

Type Kind
Prop Type
Set Type
$\ast$ $\square$

Warning!

  • $λx:nat. array \; x$ is of type $nat → Type$
  • $\prod\limits_{ x:nat } array \; x$ is of type $Type$

So can form products:

  • from $Type$ to $Kind$
  • from $Type$ to $Type$

but what about $Kind → Kind$ and $Kind → Type$? ⟹ Calculus of Constructions

$=$ is translated into $nat → nat → Type$, that is: $n = n$ is a $Type$, whose inhabitants are proofs of $n=n$

Similarly: $∀x \; (P \, x) ⇒ (P \, x)$ is translated into $\prod\limits_{ n: nat } ((P \, x) → (P \, x))$, a inhabitant of which is $λx: nat. λ y: P(x). y$

Logical Framework

  • 1879: Frege’s Begriffschrift, 1902: Russel’s Principia Mathematica, ZF set theory, Coquand’s CoC, etc… ⟶ theories defined from scratch (e.g. you can define natural numbers but not Peano arithmetic)
  • 1928: Hilbert’s and Ackermann’s Predicate logic / 1991: Plotkin’s $λ\Pi$-calculus (Edinghburgh Logical Framework): formalism not defined from scratch ⟶ there’s one framework, and theories/logics are defined based on this framework (e.g. you can define Peano arithmetic)

$λ\Pi$ + inductive types ⟶ MLTT $λ\Pi$ + polymorphism ⟶ CoC

Example: A proof build by induction applied to a particular number amounts to a direct proof of the property for that number.

Example: slide 10 of p11.pdf

If you don’t have $N(x)$, you can axiomatize the induction scheme, but with two extra reduction rules:

\[Rec(c,π, π', S(S(0))): S(S(0)) ∈ c\]

but also:

\[Rec(c,π, π', S(0)): S(0) ∈ c\\ π' \, S(0) \, Rec(c,π, π', S(0)): S(S(0)) ∈ c\]

but also:

\[π: 0 ∈ c\\ π' \, 0 \, π: S(0) ∈ c\\ π' \, S(0) \, \underbrace{(π' \, 0 \, π)}_{= j (⟨1, σ''⟩)}: S(S(0)) ∈ c\]

⟶ Gödel System T: (Gödel (1956), Tait (1967))

In ST $λ$-calculus with $0$ and successor: you can only define constant functions and the ones which add a constant to one of its arguments.

Primitive recursive functions:

At the beginning:

  • $λ x_1 ⋯ λ x_n. 0$: constant $0$
  • $λ x_1 ⋯ λ x_n. x$: adding the constant $0$
  • $λ x. S(x)$: adding the constant $1$
  • closed by composition

And if you add $Rec$ ⟶ you get all the primitive recursive functions (not Turing complete, but they all terminate), and even more.

\[Rec^{nat} (0, λx,y. SSy, SS0) \\ ⟶ (λx,y. SSy) \, (S0) \, Rec^{nat} (0, λx,y. SSy, S0) \\ ⟶ SS Rec^{nat} (0, λx,y. SSy, S0)\\ ⟶ SS ((λx,y. SSy) \, 0 \, Rec^{nat} (0, λx,y. SSy, 0))\\ ⟶ SS SS Rec^{nat} (0, λx,y. SSy, 0))\\ ⟶ SS SS 0\]

But you don’t have all recursive functions: you can’t program an interpreter of Gödel System T.

Termination of Gödel System T ⟹ every proof in arithmetic has an irreducible form, that terminates with an introduction rule ⟹ consistency of arithmetic

Tait introduced the sets $R_T$ to prove termination of Gödel System T (1967).

MLTT = $λ\Pi$ + explicit reduction rule + System T

  • $refl$: introduction of equality
  • $Leibniz$: elimination of equality

Add a new rewriting rule to eliminate Leibniz cut.

But with

\[x=y ⟶ ∀c \; (x ∈ x ⇒ y ∈ c)\]

⟹ $β$-reduction is doing the job for us

Why do Martin-Löf want to avoid the rewriting rules and use aximatization instead?

\[N(x) ⟶ ∀c, … x=y ⟶ ∀c \; (x ∈ x ⇒ y ∈ c)\]

⟹ Because Martin-Löf (and the Swedish school) wants to avoid impredicativity at all costs

One example where you don’t have the

\[x ∈ c ⟶ x=0 ∨ ∃ y \; (x= S y)\]

Then, we can prove:

\[π: 0 ∈ c\\ π': ∀ x \, (x ∈ c ⇒ S(x) ∈ c)\\ Rec(c, π, π', SS0)\]

If instead of $SS0$, we plug $w$: $Rec(c, π, π’, w)$, the term cannot be reduced. It irreducible but doesn’t end with an introduction rule.

⟹ In MLTT: you don’t have full witness property, but only for closed terms.

Lists:

\[Rec (P, b, g, cons(a,x)) ⟶ g \, b \, x \, Rec(P,b,g,x)\]

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