Dependent Type Theory Lecture 1

Nicola Gambino

MGS - Leicester 2009

Nicola Gambino Dependent Type Theory References

B. Nordstöm, K. Petersson, and J. M. Smith Martin-Löf’s Type Theory Handbook of Logic in Computer Science, Vol. 5 Oxford University Press, 2001.

B. Nordström, K. Petersson, and J. M. Smith Programming in Martin-Löf’s Type Theory Oxford University Press, 1990.

Nicola Gambino Dependent Type Theory Outline

1 Preliminaries

2 The dependent type theory ML

Nicola Gambino Dependent Type Theory Categorical judgements

There are four forms of categorical judgements:

A ∈ Type A = B ∈ Type a ∈ A a = b ∈ A

These express, respectively, that: A is a type A and B are deﬁnitionally equal types a is an element of A a and b are deﬁnitionally equal elements of A.

Nicola Gambino Dependent Type Theory Hypothetical judgements (I)

There are four forms of hypothetical judgements:

(Γ) A ∈ Type (Γ) A = B ∈ Type (Γ) a ∈ A (Γ) a = b ∈ A.

Here Γ is a context of variable declarations of the form (Γ) = x0 ∈ A0, x1 ∈ A1(x0),..., xn ∈ An(x0,..., xn−1) .

Nicola Gambino Dependent Type Theory Hypothetical judgements (II)

For the context Γ to be well-formed we need to know that the judgements

A0 ∈ Type

(x0 ∈ A0) A1(x0) ∈ Type . . x0 ∈ A0,..., xn−1 ∈ An−1(x0,..., xn−2) An(x0,..., xn−1) ∈ Type

are derivable.

Nicola Gambino Dependent Type Theory Deduction rules

Deduction rules have the form

(Γ1) J1 ··· (Γn) Jn (Γ) J

We often omit contexts that are common to premisses and conclusion.

Nicola Gambino Dependent Type Theory Outline

√ 1 Preliminaries

2 The dependent type theory ML

Nicola Gambino Dependent Type Theory The dependent type theory ML (I)

Two groups of rules:

General deduction rules, e.g.

(Γ) A ∈ Type (Γ, x ∈ A) x ∈ A

Deduction rules for the following forms of type:

0 , 1 , Nat , A + B ,

IdA(a, b) , (Σx ∈ A)B(x) , (Πx ∈ A)B(x) .

Nicola Gambino Dependent Type Theory The dependent type theory ML (II)

For each form of type, we give four groups of deduction rules:

Formation rules Introduction rules Elimination rules Computation rules

Nicola Gambino Dependent Type Theory Deduction rules for the type of natural numbers (I)

Formation rule:

Nat ∈ Type

Introduction rules:

n ∈ Nat 0 ∈ Nat succ(n) ∈ Nat

Nicola Gambino Dependent Type Theory Deduction rules for the type of natural numbers (II)

Elimination rule:

c ∈ Nat d ∈ C(0)(x ∈ Nat, y ∈ C(x)) e(x, y) ∈ C(succ(x)) natrec(c, d, e) ∈ C(c)

Computation rules:

d ∈ C(0)(x ∈ Nat, y ∈ C(x)) e(x, y) ∈ C(succ(x)) natrec(0, d, e) = d ∈ C(0)

n ∈ Nat d ∈ C(0)(x ∈ Nat, y ∈ C(x)) e(x, y) ∈ C(succ(x)) natrec(succ(n), d, e) = e(n, natrec(n, d, e)) ∈ C(succ(n))

Nicola Gambino Dependent Type Theory Deduction rules for the unit type (I)

Formation rules:

1 ∈ Type

Introduction rule:

∗ ∈ 1

Nicola Gambino Dependent Type Theory Deduction rules for the unit type (II)

Elimination rule:

c ∈ 1 d ∈ C(∗) rec(c, d) ∈ C(c)

Computation rule:

d ∈ C(∗) rec(∗, d) = d ∈ C(∗)

Nicola Gambino Dependent Type Theory Deduction rules for the empty type (I)

Formation rule:

0 ∈ Type

Introduction rules:

Nicola Gambino Dependent Type Theory Deduction rules for the empty type (II)

Elimination rule:

c ∈ 0 rec(c) ∈ C(c) Computation rules:

Nicola Gambino Dependent Type Theory Deduction rules for sum types (I)

Formation rules:

A ∈ Type B ∈ Type A + B ∈ Type

Introduction rules:

a ∈ A b ∈ B inl(a) ∈ A + B inr(b) ∈ A + B

Nicola Gambino Dependent Type Theory Deduction rules for sum types (II)

Elimination rule: c ∈ A + B (x ∈ A) d(x) ∈ C(inl(x)) (y ∈ B) e(y) ∈ C(inr(y)) case(c, d, e) ∈ C(c)

Computation rules:

a ∈ A (x ∈ A) d(x) ∈ C(inl(x)) (y ∈ B) e(y) ∈ C(inr(y)) case(inl(a), d, e) = d(a) ∈ C(inl(a))

b ∈ B (x ∈ A) d(x) ∈ C(inl(x)) (y ∈ B) e(y) ∈ C(inr(y)) case(inr(b), d, e) = e(b) ∈ C(inr(b))

Nicola Gambino Dependent Type Theory Deduction rules for identity types (I)

Formation rule:

A ∈ Type a ∈ A b ∈ A

IdA(a, b) ∈ Type

Introduction rule:

a ∈ A

r(a) ∈ IdA(a, a)

Nicola Gambino Dependent Type Theory Deduction rules for identity types (II)

Elimination rule:

p ∈ IdA(a, b)(x ∈ A) d(x) ∈ C(x, x, r(x)) J(a, b, p, d) ∈ C(a, b, p)

Computation rule:

a ∈ A (x ∈ A) d(x) ∈ C(x, x, r(x)) J(a, a, r(a), d) = d(a) ∈ C(a, a, r(a))

Nicola Gambino Dependent Type Theory Deduction rules for Σ-types (I)

Formation rule:

(x ∈ A) B(x) ∈ Type (Σx ∈ A)B(x) ∈ Type

Introduction rule:

a ∈ A b ∈ B(a) pair(a, b) ∈ (Σx ∈ A)B(x)

Nicola Gambino Dependent Type Theory Deduction rules for Σ-types (II)

Elimination rule:

c ∈ (Σx ∈ A)B(x)(x ∈ A, y ∈ B(x)) d(x, y) ∈ C(pair(x, y)) split(c, d) ∈ C(c)

Computation rule:

a ∈ A b ∈ B(a)(x ∈ A, y ∈ B(x)) d(x, y) ∈ C(pair(x, y)) split(pair(a, b), d) = d(a, b) ∈ C(pair(a, b))

Nicola Gambino Dependent Type Theory Deduction rules for Π-types (I)

Formation rule:

(x ∈ A) B(x) ∈ Type (Πx ∈ A)B(x) ∈ Type

Introduction rule:

(x ∈ A) b(x) ∈ B(x) (λx ∈ A)b(x) ∈ (Πx ∈ A)B(x)

Nicola Gambino Dependent Type Theory Deduction rules for Π-types (II)

Elimination rule:

b ∈ (Πx ∈ A)B(x) ∈ Type a ∈ A app(b, a) ∈ B(a)

Computation rule:

(x ∈ A) b(x) ∈ B(x) a ∈ A app((λx ∈ A)b(x), a) = b(a) ∈ B(a)

Nicola Gambino Dependent Type Theory Remarks

We used ‘Type’ for what is called ‘Set’ in the Handbook paper. We did not use explicitly any logical framework.

Nicola Gambino Dependent Type Theory