An Introduction to Type Theory Part 2 Tallinn, September 2003 - - PowerPoint PPT Presentation

An Introduction to Type Theory

Part 2

Tallinn, September 2003 http://www.cs.nott.ac.uk/˜txa/tallinn/

Thorsten Altenkirch University of Nottingham

An Introduction to Type Theory – p.1/46

Plan of the course

• 1. Intuitionistic logic from a type theoretic perspective
• 2. Basic constructions of Type Theory
• 3. Programming with dependent types

An Introduction to Type Theory – p.2/46

Today

Basic constructions of Type Theory

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

.

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• Proof or program?

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• Proof or program?

• Pattern matching and elimination

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• Proof or program?

• Pattern matching and elimination

Uniqueness of equality proofs

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• Proof or program?

• Pattern matching and elimination

Uniqueness of equality proofs Inductive families

An Introduction to Type Theory – p.3/46

Today

Basic constructions of Type Theory From Logic to Type Theory:

. Example: Decidability of

• Proof or program?

• Pattern matching and elimination

Uniqueness of equality proofs Inductive families Loose ends

An Introduction to Type Theory – p.3/46

From Logic to Type Theory

An Introduction to Type Theory – p.4/46

From Logic to Type Theory

In intuitionistic logic constructing proofs is very much like writing functional programs.

An Introduction to Type Theory – p.4/46

From Logic to Type Theory

In intuitionistic logic constructing proofs is very much like writing functional programs. In Type Theory we go one step further: Proofs = Programs Propositions = Types

An Introduction to Type Theory – p.4/46

From Logic to Type Theory

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications:

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Function types (

• )

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Pi-types (

• )

Function types (

• )

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Pi-types (

• )

Function types (

• )

Conjunction (

• )

Existential quantification (

) Cartesian product (

)

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Pi-types (

• )

Function types (

• )

Conjunction (

• )

Existential quantification (

) Sigma-types (

) Cartesian product (

)

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Pi-types (

• )

Function types (

• )

Conjunction (

• )

Existential quantification (

) Sigma-types (

) Cartesian product (

) Disjunction Disjoint union (

)

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

We will also make the following indentifications: Implication (

• )

Universal quantifications (

) Pi-types (

• )

Function types (

• )

Conjunction (

• )

Existential quantification (

) Sigma-types (

) Cartesian product (

) Disjunction Disjoint union (

) Disjoint union (

)

An Introduction to Type Theory – p.5/46

From Logic to Type Theory

Elimination becomes more subtle:

An Introduction to Type Theory – p.6/46

From Logic to Type Theory

Elimination becomes more subtle:

Pattern matching We have to keep track of the steps in

the type.

An Introduction to Type Theory – p.6/46

From Logic to Type Theory

Elimination becomes more subtle:

Pattern matching We have to keep track of the steps in

the type.

Elimination constants Get replaced by their dependent

versions.

An Introduction to Type Theory – p.6/46

Set theoretic encodings

An Introduction to Type Theory – p.7/46

Set theoretic encodings

Let

• ✁

be sets, then:

• ✂

• ✂✄

• ✆

• ✁
• ✂

• ✂

• ☞

An Introduction to Type Theory – p.7/46

Set theoretic encodings

Let

• ✁

be sets, then:

• ✂

• ✂✄

• ✆

• ✁
• ✂

• ✂

• ☞

Let

• be a set and for each

• let

be a set, then:

• ☞

• ✂✄

• ✆

• ☎

• ☞

• ✂

• ☞

• ☞

An Introduction to Type Theory – p.7/46

Example: Decidability of

• An Introduction to Type Theory – p.8/46

Example: Decidability of

• In intuitionistic arithmetic we can prove

• ✁

• ✁

• ✁

• ✄
• ✁

where

• ✄
• ✁
• ☎

• ✁

• ✄
• ✁

• ✆

An Introduction to Type Theory – p.8/46

Example: Decidability of

• In intuitionistic arithmetic we can prove

• ✁

• ✁

• ✁

• ✄
• ✁

where

• ✄
• ✁
• ☎

• ✁

• ✄
• ✁

• ✆

We will use this example to motivate the idea that proofs = programs.

An Introduction to Type Theory – p.8/46

Natural Numbers

An Introduction to Type Theory – p.9/46

Natural Numbers

How to form ?

An Introduction to Type Theory – p.9/46

Natural Numbers

How to form ?

An Introduction to Type Theory – p.9/46

Natural Numbers

How to form ?

How to construct ?

An Introduction to Type Theory – p.9/46

Natural Numbers

How to form ?

How to construct ?

• ✟
• ✁

• ✁

An Introduction to Type Theory – p.9/46

Equality

An Introduction to Type Theory – p.10/46

Equality

How to form ?

An Introduction to Type Theory – p.10/46

Equality

How to form ?

• ✟

• ☎
• ✆

An Introduction to Type Theory – p.10/46

Equality

How to form ?

• ✟

• ☎
• ✆

How to prove ?

An Introduction to Type Theory – p.10/46

Equality

How to form ?

• ✟

• ☎
• ✆

How to prove ?

• ✟

• ✠

• ☎

An Introduction to Type Theory – p.10/46

Proving decidablity

An Introduction to Type Theory – p.11/46

Proving decidablity

We will present a proof

• ✟
• ✁

• ✁

• ✄
• ✁

using pattern matching.

An Introduction to Type Theory – p.11/46

Proving decidablity

We will present a proof

• ✟
• ✁

• ✁

• ✄
• ✁

using pattern matching. We will discuss the rules for pattern matching later.

An Introduction to Type Theory – p.11/46

Peano’s axioms

An Introduction to Type Theory – p.12/46

Peano’s axioms

• ✁

• ✟
• ✁

• ✞

• ✆

An Introduction to Type Theory – p.12/46

Peano’s axioms

• ✁

• ✟
• ✁

• ✞

• ✆

where empty pattern

An Introduction to Type Theory – p.12/46

Peano’s axioms

• ✁

• ✟
• ✁

• ✞

• ✆

where empty pattern

• ✁

• ✟
• ✁

• ✝
• ✆

An Introduction to Type Theory – p.12/46

Peano’s axioms

• ✁

• ✟
• ✁

• ✞

• ✆

where empty pattern

• ✁

• ✟
• ✁

• ✝
• ✆

where empty pattern

An Introduction to Type Theory – p.12/46

Peano’s axioms

An Introduction to Type Theory – p.13/46

Peano’s axioms

• ✠

• ✁

• ✁

• ✁

• ✞
• ✞

An Introduction to Type Theory – p.13/46

Peano’s axioms

• ✠

• ✁

• ✁

• ✁

• ✞
• ✞

where

• ✠

• ✄
• ✠

• ✝
• ✠

• ✝

An Introduction to Type Theory – p.13/46

Peano’s axioms

• ✠

• ✁

• ✁

• ✁

• ✞
• ✞

where

• ✠

• ✄
• ✠

• ✝
• ✠

• ✝
• ✂

• ✁

• ✞

• ✁

An Introduction to Type Theory – p.13/46

Peano’s axioms

• ✠

• ✁

• ✁

• ✁

• ✞
• ✞

where

• ✠

• ✄
• ✠

• ✝
• ✠

• ✝
• ✂

• ✁

• ✞

• ✁

where

• ✂

• ✄
• ✠

• ✝

• ✠

• An Introduction to Type Theory – p.13/46

Proving decidability

An Introduction to Type Theory – p.14/46

Proving decidability

• ✞

• ✁

• ✁

• ✁

• ✄
• ✁

• ✄

• ✞

• ✄
• ✞

An Introduction to Type Theory – p.14/46

Proving decidability

• ✞

• ✁

• ✁

• ✁

• ✄
• ✁

• ✄

• ✞

• ✄
• ✞

where

• ✞
• ✁

• ✂

• ✝
• ✂

• ✠

• ✁
• ✝

• ✞
• ✁

• ✂
• ✡

• ✂
• ✄

• ✂

• ✁

An Introduction to Type Theory – p.14/46

Proving decidability

An Introduction to Type Theory – p.15/46

Proving decidability

• ✟
• ✁

• ✁

• ✁

• ✄
• ✁

An Introduction to Type Theory – p.15/46

Proving decidability

• ✟
• ✁

• ✁

• ✁

• ✄
• ✁

where

• ✂

• ✠

• ✝

• ✄

• ✂
• ✄
• ✁

• ✁

• ✄

• ✝
• ✂
• ✄
• ✁

• ✝

• ✄

• ✝

• ✠
• ✞
• ✁

• ✁

An Introduction to Type Theory – p.15/46

Proof or program ?

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form. If it is

• ✂

• then the numbers are equal and
• ✟
• ✁

proves this.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form. If it is

• ✂

• then the numbers are equal and
• ✟
• ✁

proves this. If it is

• ✂
• ✡

then the numbers are not equal and

• ✄
• ✁

proves this.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form. If it is

• ✂

• then the numbers are equal and
• ✟
• ✁

proves this. If it is

• ✂
• ✡

then the numbers are not equal and

• ✄
• ✁

proves this.

• is a program whose specification is in its type.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form. If it is

• ✂

• then the numbers are equal and
• ✟
• ✁

proves this. If it is

• ✂
• ✡

then the numbers are not equal and

• ✄
• ✁

proves this.

• is a program whose specification is in its type.

Equality proofs contain no information, hence they do not have to be calculated at run time.

An Introduction to Type Theory – p.16/46

Proof or program ?

We can use

• to effectively decide whether two

numbers

• ✁

are equal. Reduce

• ✁

to its canonical form. If it is

• ✂

• then the numbers are equal and
• ✟
• ✁

proves this. If it is

• ✂
• ✡

then the numbers are not equal and

• ✄
• ✁

proves this.

• is a program whose specification is in its type.

Equality proofs contain no information, hence they do not have to be calculated at run time. Hence

• is not less efficient than an ordinary program

to determine equality of natural numbers.

An Introduction to Type Theory – p.16/46

Proof or program ?

The same principle can be applied to other problems, e.g. once we have specified

• ✁

• ✁

• ✂✁

An Introduction to Type Theory – p.17/46

Proof or program ?

The same principle can be applied to other problems, e.g. once we have specified

• ✁

• ✁

• ✂✁

we can implement a primality checker as

• ✞
• ✁

• ✁

• ✁

• ✁

• ✁

An Introduction to Type Theory – p.17/46

Pattern matching for

• An Introduction to Type Theory – p.18/46

Pattern matching for

• If a pattern variable

has type

• ✁

we can split the pattern into two, replacing

by

• in the first line and by

• in the second, where
• ✟

is a fresh variable.

An Introduction to Type Theory – p.18/46

Pattern matching for

• If a pattern variable

has type

• ✁

we can split the pattern into two, replacing

by

• in the first line and by

• in the second, where
• ✟

is a fresh variable. Since

may appear in the type we have to substitute

by

• and

• respectively.

An Introduction to Type Theory – p.18/46

Pattern matching for

• If a pattern variable

has type

• ✁

we can split the pattern into two, replacing

by

• in the first line and by

• in the second, where
• ✟

is a fresh variable. Since

may appear in the type we have to substitute

by

• and

• respectively.

We may use the function

we are defining recursively

• n a subpattern, (e.g.
• above).

An Introduction to Type Theory – p.18/46

Pattern matching for

• If a pattern variable

has type

• ✁

we can split the pattern into two, replacing

by

• in the first line and by

• in the second, where
• ✟

is a fresh variable. Since

may appear in the type we have to substitute

by

• and

• respectively.

We may use the function

we are defining recursively

• n a subpattern, (e.g.
• above).

The precise rules governing structural recursion in the presence of other variables and mutual recursive definitions are more involved.

An Introduction to Type Theory – p.18/46

Pattern matching for

• An Introduction to Type Theory – p.19/46

Pattern matching for

• ✄
• ✟
• ☞

has the same canonical constant

as

hence the same rules for pattern matching apply.

An Introduction to Type Theory – p.19/46

Pattern matching for

• ✄
• ✟
• ☞

has the same canonical constant

as

hence the same rules for pattern matching apply. Similarily

• ☎

has canonical constants

• ✂

• ✂
• as

and hence the same rules for pattern matching apply.

An Introduction to Type Theory – p.19/46

Pattern matching for

• ✄
• ✟
• ☞

has the same canonical constant

as

hence the same rules for pattern matching apply. Similarily

• ☎

has canonical constants

• ✂

• ✂
• as

and hence the same rules for pattern matching apply. As a consequence of

• ✁

• ✂✁

variables ranging

• ver

and

types may occur in the type and have to be substituted.

An Introduction to Type Theory – p.19/46

Elimination constants

An Introduction to Type Theory – p.20/46

Elimination constants

As a special instance of the pattern matching rules we will derive elimination constants.

An Introduction to Type Theory – p.20/46

Elimination constants

As a special instance of the pattern matching rules we will derive elimination constants. The principle Equivalence of pattern matching and elimination still holds.

An Introduction to Type Theory – p.20/46

Elimination constants

As a special instance of the pattern matching rules we will derive elimination constants. The principle Equivalence of pattern matching and elimination still holds. That is every pattern matching proof can be replaced by

• ne only using elimination constants.

An Introduction to Type Theory – p.20/46

Elimination for

• An Introduction to Type Theory – p.21/46

Elimination for

• ✟
• ✁

• ✂✁

• ✂

• ✁

• ✁
• ✄

• ✟

• ✁
• ✁

• ✟
• An Introduction to Type Theory – p.21/46

Elimination for

• ✟
• ✁

• ✂✁

• ✂

• ✁

• ✁
• ✄

• ✟

• ✁
• ✁

• ✟
• where

• ✁

• ✁

• ✁

• ✂

• ✁

An Introduction to Type Theory – p.21/46

One stone, two birds

An Introduction to Type Theory – p.22/46

One stone, two birds

Note that

• ✁

unifies two different principles:

An Introduction to Type Theory – p.22/46

One stone, two birds

Note that

• ✁

unifies two different principles:

primitive recursion We obtain simply typed primitive

recursion if the motive

• is constant.

An Introduction to Type Theory – p.22/46

One stone, two birds

Note that

• ✁

unifies two different principles:

primitive recursion We obtain simply typed primitive

recursion if the motive

• is constant.

induction When reading

as

• ✁

we obtain the principle of induction.

An Introduction to Type Theory – p.22/46

Elimination for

An Introduction to Type Theory – p.23/46

Elimination for

• ✁

• ✟

• ☎

• ✂✁

• ✟
• ☎

• ☞
• ✄
• ✂

• ✆

• ✄
• ✂
• ✆

• ✟
• ☎

• ✁
• ✁
• ✟
• An Introduction to Type Theory – p.23/46

Elimination for

• ✁

• ✟

• ☎

• ✂✁

• ✟
• ☎

• ☞
• ✄
• ✂

• ✆

• ✄
• ✂
• ✆

• ✟
• ☎

• ✁
• ✁
• ✟
• where

• ✁
• ✁

• ✂

• ☎

• ✁
• ✁

• ✂
• ✆

• ✁

An Introduction to Type Theory – p.23/46

A little quiz

An Introduction to Type Theory – p.24/46

A little quiz

What is the construct corresponding to

• ✁

in programming?

An Introduction to Type Theory – p.24/46

A little quiz

What is the construct corresponding to

• ✁

in programming? The type corresponding to

• ✂

is called Unit, written

• .

We didn’t need an elimination constant for

• ✂

, do we need one for

• ?

An Introduction to Type Theory – p.24/46

Elimination for

An Introduction to Type Theory – p.25/46

Elimination for

• ✟

• ✂✁

• ✟

• ☞

• ✂✁

• ☎

• ☞
• ✆

• ✄

• ✟

• ☞

• ✁

• ✁

• ✟
• An Introduction to Type Theory – p.25/46

Elimination for

• ✟

• ✂✁

• ✟

• ☞

• ✂✁

• ☎

• ☞
• ✆

• ✄

• ✟

• ☞

• ✁

• ✁

• ✟
• where

• ✁

• ✁

• ✡

An Introduction to Type Theory – p.25/46

Alternative: projections

An Introduction to Type Theory – p.26/46

Alternative: projections

There is an alternative form of elimination for

using projections.

An Introduction to Type Theory – p.26/46

Alternative: projections

There is an alternative form of elimination for

using projections.

• ✟

• ✂✁

• ✟

• ☞

• ✞

• ✟
• ✞

• ✟

• ✞

• ✝

An Introduction to Type Theory – p.26/46

Alternative: projections

There is an alternative form of elimination for

using projections.

• ✟

• ✂✁

• ✟

• ☞

• ✞

• ✟
• ✞

• ✟

• ✞

• ✝

where

• ✞

• ☎

• ✆

An Introduction to Type Theory – p.26/46

Comparing

• ✁

vs.

• ,
• ✄

An Introduction to Type Theory – p.27/46

Comparing

• ✁

vs.

• ,
• ✄

Which form of elimination is better?

An Introduction to Type Theory – p.27/46

Comparing

• ✁

vs.

• ,
• ✄

Which form of elimination is better? Can we use

• ✁

• ✁

to implement

• ✞

and

?

An Introduction to Type Theory – p.27/46

The axiom of choice

An Introduction to Type Theory – p.28/46

The axiom of choice

We can use

• ✞

and

to implement the axiom of choice.

An Introduction to Type Theory – p.28/46

The axiom of choice

We can use

• ✞

and

to implement the axiom of choice.

• ✁

• ✟
• ✁
• ✂✁

• ☎

• ☞

• ☎

• ✁
• ✠

• ✁

• ☎

• ☞
• ☎

An Introduction to Type Theory – p.28/46

The axiom of choice

We can use

• ✞

and

to implement the axiom of choice.

• ✁

• ✟
• ✁
• ✂✁

• ☎

• ☞

• ☎

• ✁
• ✠

• ✁

• ☎

• ☞
• ☎

where

• ✁
• ✠

• ✄

• ☞
• ✞

• ☞

An Introduction to Type Theory – p.28/46

The axiom of choice

An Introduction to Type Theory – p.29/46

The axiom of choice

This shows that the axiom of choice is justified constructively.

An Introduction to Type Theory – p.29/46

The axiom of choice

This shows that the axiom of choice is justified constructively. However, in the presence of the principle of excluded middle it is a sign of non-constructive reasoning.

An Introduction to Type Theory – p.29/46

Pattern matching for

• An Introduction to Type Theory – p.30/46

Pattern matching for

• The rules for pattern matching for equality proofs

involve unification problems.

An Introduction to Type Theory – p.30/46

Pattern matching for

• The rules for pattern matching for equality proofs

involve unification problems. Given a pattern variable

• ✆

, there are the following cases:

An Introduction to Type Theory – p.30/46

Pattern matching for

• The rules for pattern matching for equality proofs

involve unification problems. Given a pattern variable

• ✆

, there are the following cases: The unification problem

• ✆

is unsolvable, in this cas we can eliminate the pattern.

An Introduction to Type Theory – p.30/46

Pattern matching for

• The rules for pattern matching for equality proofs

involve unification problems. Given a pattern variable

• ✆

, there are the following cases: The unification problem

• ✆

is unsolvable, in this cas we can eliminate the pattern. The unification problem

• ✆

has a most general solution which is given by a substitution

• . Then

can be replaced by

• ✠

• and the substitution
• has

to be applied to the type as well.

An Introduction to Type Theory – p.30/46

Pattern matching for

• The rules for pattern matching for equality proofs

involve unification problems. Given a pattern variable

• ✆

, there are the following cases: The unification problem

• ✆

is unsolvable, in this cas we can eliminate the pattern. The unification problem

• ✆

has a most general solution which is given by a substitution

• . Then

can be replaced by

• ✠

• and the substitution
• has

to be applied to the type as well. The unification problem

• ✆

is irreducible, in this case we cannot reduce the pattern.

An Introduction to Type Theory – p.30/46

Reducing unification problems

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here.

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here. Problems of the form

• can be solved trivially.

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here. Problems of the form

• can be solved trivially.

Problems of the

• ✞
• r

• ✞

are unsolvable.

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here. Problems of the form

• can be solved trivially.

Problems of the

• ✞
• r

• ✞

are unsolvable. Problems of the form

• , where
• does not occur in
• can be solved and give rise to the substitution
• ✄
• ✝
• .

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here. Problems of the form

• can be solved trivially.

Problems of the

• ✞
• r

• ✞

are unsolvable. Problems of the form

• , where
• does not occur in
• can be solved and give rise to the substitution
• ✄
• ✝
• .

The problem

• ✞

can be reduced to

• ✁

.

An Introduction to Type Theory – p.31/46

Reducing unification problems

We only consider the special case of terms over

• ✁

here. Problems of the form

• can be solved trivially.

Problems of the

• ✞
• r

• ✞

are unsolvable. Problems of the form

• , where
• does not occur in
• can be solved and give rise to the substitution
• ✄
• ✝
• .

The problem

• ✞

can be reduced to

• ✁

. All other problems are irreducible.

An Introduction to Type Theory – p.31/46

Question

An Introduction to Type Theory – p.32/46

Question

Can we generalize our proof

• ✂

to

• ✁

• ✁
• ✂

• ☎

• ☞

• ✡

• ☎
• ✆

An Introduction to Type Theory – p.32/46

Question

Can we generalize our proof

• ✂

to

• ✁

• ✁
• ✂

• ☎

• ☞

• ✡

• ☎
• ✆

where

• ✂

• ✠

• ✠

• An Introduction to Type Theory – p.32/46

Elimination for

• An Introduction to Type Theory – p.33/46

Elimination for

• ✟

• ✟
• ☎

• ☞

• ✆

• ✂✁

• ☎

• ☞
• ☎

• ✠

• ✟

• ✆

• ✄

• ✁

• ✟
• ☎

• An Introduction to Type Theory – p.33/46

Elimination for

• ✟

• ✟
• ☎

• ☞

• ✆

• ✂✁

• ☎

• ☞
• ☎

• ✠

• ✟

• ✆

• ✄

• ✁

• ✟
• ☎

• where

• ✄

• ✁

• ✠

• ✡

An Introduction to Type Theory – p.33/46

Pattern matching vs. elimination ?

Does the Equivalence of pattern matching and elimination still hold?

An Introduction to Type Theory – p.34/46

Uniqueness of equality proofs.

An Introduction to Type Theory – p.35/46

Uniqueness of equality proofs.

• ✟

• ✁

• ✆

• ☎

• ✂

• ✂

An Introduction to Type Theory – p.35/46

Uniqueness of equality proofs.

• ✟

• ✁

• ✆

• ☎

• ✂

• ✂

where

• ☎

• ✠

• ✠

• ✠

• ✠

An Introduction to Type Theory – p.35/46

Uniqueness of equality proofs.

In the early 90ies it was an open problem wether

• could be derived from

• ✄

• ✁

.

An Introduction to Type Theory – p.36/46

Uniqueness of equality proofs.

In the early 90ies it was an open problem wether

• could be derived from

• ✄

• ✁

. In 1993 Hofmann and Streicher showed that

• does

not hold in the groupoid model of Type Theory, although

• ✄

• ✁

can be interpreted.

An Introduction to Type Theory – p.36/46

Uniqueness of equality proofs.

In the early 90ies it was an open problem wether

• could be derived from

• ✄

• ✁

. In 1993 Hofmann and Streicher showed that

• does

not hold in the groupoid model of Type Theory, although

• ✄

• ✁

can be interpreted. However, this can be fixed by introducing another elimination constant.

An Introduction to Type Theory – p.36/46

Another elimination for

• An Introduction to Type Theory – p.37/46

Another elimination for

• ✟

• ✟
• ☎

• ☞

• ☎

• ✂✁

• ☎

• ☞
• ✄
• ✠

• ✟

• ☎

• ✄

• ✁
• ✡

• ✟
• ☎
• An Introduction to Type Theory – p.37/46

Another elimination for

• ✟

• ✟
• ☎

• ☞

• ☎

• ✂✁

• ☎

• ☞
• ✄
• ✠

• ✟

• ☎

• ✄

• ✁
• ✡

• ✟
• ☎
• where

• ✄

• ✁
• ✡

• ✠

• ✡

An Introduction to Type Theory – p.37/46

Uniqueness of equality proofs.

An Introduction to Type Theory – p.38/46

Uniqueness of equality proofs.

• ✟

• ✁

• ✆

• ☎

• ✂

• ✂

An Introduction to Type Theory – p.38/46

Uniqueness of equality proofs.

• ✟

• ✁

• ✆

• ☎

• ✂

• ✂

where

• ☎

• ✂
• ✠
• ✄

• ✁

• ✄

• ✁
• ☎

• ✠

• ✠

• An Introduction to Type Theory – p.38/46

Conor’s result

An Introduction to Type Theory – p.39/46

Conor’s result

In 1999 Conor McBride showed as part of his PhD that Equiva- lence of pattern matching and elimination holds, when using

• ✄

• ✁
• .

An Introduction to Type Theory – p.39/46

Conor’s result

In 1999 Conor McBride showed as part of his PhD that Equiva- lence of pattern matching and elimination holds, when using

• ✄

• ✁
• .

In fact he showed this in the presence of inductive families, of which

• is a special case.

An Introduction to Type Theory – p.39/46

in logic

An Introduction to Type Theory – p.40/46

in logic

How to define

• ✟
• ✁

• ✂✁

• ✁

?

An Introduction to Type Theory – p.40/46

in logic

How to define

• ✟
• ✁

• ✂✁

• ✁

?

• ✁
• ✁
• ✟
• ✁

• ☎
• ✁

An Introduction to Type Theory – p.40/46

in logic

How to define

• ✟
• ✁

• ✂✁

• ✁

?

• ✁
• ✁
• ✟
• ✁

• ☎
• ✁

There is an alternative inductive definition.

• ✁
• ✁

• ✞

An Introduction to Type Theory – p.40/46

in Type Theory

An Introduction to Type Theory – p.41/46

in Type Theory

How to form ?

An Introduction to Type Theory – p.41/46

in Type Theory

How to form ?

• ✁

• ✁

An Introduction to Type Theory – p.41/46

in Type Theory

How to form ?

• ✁

• ✁

How to construct?

An Introduction to Type Theory – p.41/46

in Type Theory

How to form ?

• ✁

• ✁

How to construct?

• ✁

• ✁

• ✁
• ✟
• ✁

• ✟

• ✝
• ✄

An Introduction to Type Theory – p.41/46

Pattern matching for

An Introduction to Type Theory – p.42/46

Pattern matching for

• ✁

• ✠

• ✁

• ✁

• ✁

• ✄

• ✂

• ✂

An Introduction to Type Theory – p.42/46

Pattern matching for

• ✁

• ✠

• ✁

• ✁

• ✁

• ✄

• ✂

• ✂

where

• ✁

• ✠
• ✁

• ✁

• ✝

• ✂

• ✁

• ✠

• ✝

• ✝

• ✂

• ✝

• ✄

• ✁

• ✠
• ✁

• ✂

An Introduction to Type Theory – p.42/46

Leq in LEGO

Inductive [Leq : Nat

• Nat
• Set]

Constructors [le0 :

n:Nat

Leq ze n] [leS :

m,n|Nat

(Leq m n)

• (Leq (su m) (su n))];

An Introduction to Type Theory – p.43/46

Elimination for Leq

decl Leq_elim :

• C_Leq:
• x1,x2|Nat

(Leq x1 x2)

TYPE

(

• n:Nat

C_Leq (le0 n))

(

• m,n|Nat

• x1:Leq m n

(C_Leq x1)

C_Leq (leS x1))

• x1,x2|Nat

• z:Leq x1 x2

C_Leq z [[C_Leq:

• x1,x2|Nat

(Leq x1 x2)

TYPE][f_le0:

• n1:Nat

C_Leq (le0 n1)] [f_leS:

• m,n|Nat

• x1:Leq m n

(C_Leq x1)

C_Leq (leS x1)][n1:Nat][m,n|Nat] [x1:Leq m n] Leq_elim C_Leq f_le0 f_leS (le0 n1)

f_le0 n1 || Leq_elim C_Leq f_le0 f_leS (leS x1)

f_leS x1 (Leq_elim C_Leq f_le0 f_leS x1)]

An Introduction to Type Theory – p.44/46

Inductive definitions

An Introduction to Type Theory – p.45/46

Inductive definitions

Inductive definitions are a basic concept of Type Theory

An Introduction to Type Theory – p.45/46

Inductive definitions

Inductive definitions are a basic concept of Type Theory Inductive types can be imagined as defining a collection

• f trees.

An Introduction to Type Theory – p.45/46

Inductive definitions

Inductive definitions are a basic concept of Type Theory Inductive types can be imagined as defining a collection

• f trees.

We can have infinitary constructors, but they have to be strictly positive.

An Introduction to Type Theory – p.45/46

Inductive definitions

Inductive definitions are a basic concept of Type Theory Inductive types can be imagined as defining a collection

• f trees.

We can have infinitary constructors, but they have to be strictly positive. There are also size restrictions: there is no type of all types.

An Introduction to Type Theory – p.45/46

Loose ends

An Introduction to Type Theory – p.46/46

Loose ends

The role of equality in Type Theory extensional vs intensional

An Introduction to Type Theory – p.46/46

Loose ends

The role of equality in Type Theory extensional vs intensional Universes and reflection predicative impredicative inconsistent

An Introduction to Type Theory – p.46/46