Depending on equations A proof-relevant framework for unification - - PowerPoint PPT Presentation

Depending on equations

A proof-relevant framework for unification in dependent type theory Jesper Cockx

DistriNet – KU Leuven

3 September 2017

Unification for dependent types

Unification is used for many purposes: logic programming, type inference, term rewriting, automated theorem proving, natural language processing, . . . This talk: checking definitions by dependent pattern matching

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Disclaimer

My work is on dependently typed languages, I know little about unification.

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Disclaimer

My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = ⇒ (x = y)

x:=y

= = ⇒ OK

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Disclaimer

My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = ⇒ (x = y)

x:=y

= = ⇒ OK (suc x = zero) = ⇒ ⊥

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Disclaimer

My work is on dependently typed languages, I know little about unification. This talk is about first-order unification: (suc x = suc y) = ⇒ (x = y)

x:=y

= = ⇒ OK (suc x = zero) = ⇒ ⊥ . . . but there will be types everywhere!

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x
• Indexed datatypes: Vec A n, . . .

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x
• Indexed datatypes: Vec A n, . . .
• Identity types: x ≡A y

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x
• Indexed datatypes: Vec A n, . . .
• Identity types: x ≡A y
• Universes: Typei

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x
• Indexed datatypes: Vec A n, . . .
• Identity types: x ≡A y
• Universes: Typei
• Univalence: (A ≡ B) ≃ (A ≃ B)

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Dependent types: the ‘big five’

During this presentation, we’ll spot:

• Dependent functions: (x : A) → B x
• Indexed datatypes: Vec A n, . . .
• Identity types: x ≡A y
• Universes: Typei
• Univalence: (A ≡ B) ≃ (A ≃ B)

and see how they interact with unification!

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Depending on equations

Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms

Depending on equations

Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms

Why use dependent types?

With dependent types, you can . . .

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Why use dependent types?

With dependent types, you can . . . . . . guarantee that a program matches its specification

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Why use dependent types?

With dependent types, you can . . . . . . guarantee that a program matches its specification . . . use the same language for writing programs and proofs

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Why use dependent types?

With dependent types, you can . . . . . . guarantee that a program matches its specification . . . use the same language for writing programs and proofs . . . develop programs and proofs interactively

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Dependent types

Per Martin-L¨

• f

A dependent type is a family of types, depending

• n a term of a base type.

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Dependent types

Per Martin-L¨

• f

A dependent type is a family of types, depending

• n a term of a base type.

e.g. Vec A n is the type of vectors of length n.

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The Agda language

Agda is a purely functional language

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The Agda language

Agda is a purely functional language . . . with a strong, static type system

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The Agda language

Agda is a purely functional language . . . with a strong, static type system . . . for writing programs and proofs

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The Agda language

Agda is a purely functional language . . . with a strong, static type system . . . for writing programs and proofs . . . with datatypes and pattern matching

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The Agda language

Agda is a purely functional language . . . with a strong, static type system . . . for writing programs and proofs . . . with datatypes and pattern matching . . . with first-class dependent types

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The Agda language

Agda is a purely functional language . . . with a strong, static type system . . . for writing programs and proofs . . . with datatypes and pattern matching . . . with first-class dependent types . . . with support for interactive development

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The Agda language

Agda is a purely functional language . . . with a strong, static type system . . . for writing programs and proofs . . . with datatypes and pattern matching . . . with first-class dependent types . . . with support for interactive development All examples are (mostly) valid Agda code!

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Using dependent types

With dependent types, we can give more precise types to our programs: replicate : (n : N) → A → Vec A n

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Using dependent types

With dependent types, we can give more precise types to our programs: replicate : (n : N) → A → Vec A n ⇒ replicate 10 ‘a’ : Vec Char 10

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Using dependent types

With dependent types, we can give more precise types to our programs: replicate : (n : N) → A → Vec A n tail : (n : N) → Vec A (suc n) → Vec A n

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Using dependent types

With dependent types, we can give more precise types to our programs: replicate : (n : N) → A → Vec A n tail : (n : N) → Vec A (suc n) → Vec A n append : (m n : N) → Vec A m → Vec A n → Vec A (m + n)

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Simple pattern matching

data N : Type where zero : N suc : N → N

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum x y = { }

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum zero y = { } minimum (suc x) y = { }

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum zero y = zero minimum (suc x) y = { }

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum zero y = zero minimum (suc x) zero = { } minimum (suc x) (suc y) = { }

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum zero y = zero minimum (suc x) zero = zero minimum (suc x) (suc y) = { }

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Simple pattern matching

data N : Type where zero : N suc : N → N minimum : N → N → N minimum zero y = zero minimum (suc x) zero = zero minimum (suc x) (suc y) = suc (minimum x y)

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n)

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail k xs = { }

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail k nil = { } -- suc k = zero tail k (cons n x xs) = { } -- suc k = suc n

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail k nil = { } -- impossible tail k (cons n x xs) = { } -- suc k = suc n

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail k (cons n x xs) = { } -- suc k = suc n

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail k (cons n x xs) = { } -- k = n

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail .n (cons n x xs) = { }

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Dependent pattern matching

data Vec (A : Type) : N → Type where nil : Vec A zero cons : (n : N) → A → Vec A n → Vec A (suc n) tail : (k : N) → Vec A (suc k) → Vec A k tail .n (cons n x xs) = xs

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Specialization by unification

Agda uses unification to:

• eliminate impossible cases
• specialize the result type

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Specialization by unification

Agda uses unification to:

• eliminate impossible cases
• specialize the result type

The output of unification can change Agda’s notion of equality!

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Specialization by unification

Agda uses unification to:

• eliminate impossible cases
• specialize the result type

The output of unification can change Agda’s notion of equality! Main question: How to make sure the output of unification is correct?

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Depending on equations

Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms

Q: What is the fastest way to start a fight between type theorists?

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Q: What is the fastest way to start a fight between type theorists? A: Mention the topic of equality.

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The identity type x ≡A y

. . . a dependent type depending on x, y : A.

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The identity type x ≡A y

. . . a dependent type depending on x, y : A. . . . type theory’s built-in notion of equality.

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The identity type x ≡A y

. . . a dependent type depending on x, y : A. . . . type theory’s built-in notion of equality. . . . the type of proofs that x = y.

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Operations on the identity type

refl : x ≡A x

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Operations on the identity type

refl : x ≡A x sym : x ≡A y → y ≡A x

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Operations on the identity type

refl : x ≡A x sym : x ≡A y → y ≡A x trans : x ≡A y → y ≡A z → x ≡A z

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Operations on the identity type

refl : x ≡A x sym : x ≡A y → y ≡A x trans : x ≡A y → y ≡A z → x ≡A z cong f : x ≡A y → f x ≡B f y

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Operations on the identity type

refl : x ≡A x sym : x ≡A y → y ≡A x trans : x ≡A y → y ≡A z → x ≡A z cong f : x ≡A y → f x ≡B f y subst P : x ≡A y → P x → P y

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Unification problems as telescopes

A unification problem consists of

• 1. Flexible variables x1 : A1, x2 : A2, . . .
• 2. Equations u1 = v1 : B1, . . .

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Unification problems as telescopes

A unification problem consists of

• 1. Flexible variables x1 : A1, x2 : A2, . . .
• 2. Equations u1 = v1 : B1, . . .

This can be represented as a telescope: (x1 : A1)(x2 : A2) . . . (e1 : u1 ≡B1 v1)(e2 : u2 ≡B2 v2) . . . e.g. (k : N)(n : N)(e : suc k ≡N suc n)

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Unification problems as telescopes

A unification problem consists of

• 1. Flexible variables Γ
• 2. Equations u1 = v1 : B1, . . .

This can be represented as a telescope: Γ (e1 : u1 ≡B1 v1)(e2 : u2 ≡B2 v2) . . . e.g. (k : N)(n : N)(e : suc k ≡N suc n)

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Unification problems as telescopes

A unification problem consists of

• 1. Flexible variables Γ
• 2. Equations ¯

u = ¯ v : ∆ This can be represented as a telescope: Γ(¯ e : ¯ u ≡∆ ¯ v) e.g. (k : N)(n : N)(e : suc k ≡N suc n)

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Unifiers as telescope maps

A unifier of ¯ u and ¯ v is a substitution σ : Γ′ → Γ such that ¯ uσ = ¯ vσ.

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Unifiers as telescope maps

A unifier of ¯ u and ¯ v is a substitution σ : Γ′ → Γ such that ¯ uσ = ¯ vσ. This can be represented as a telescope map: f : Γ′ → Γ(¯ e : ¯ u ≡∆ ¯ v) e.g. f : () → (n : N)(e : n ≡N zero) f () = zero; refl

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Evidence of unification

A map f : () → (n : N)(e : n ≡N zero) gives us two things:

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Evidence of unification

A map f : () → (n : N)(e : n ≡N zero) gives us two things:

• 1. A value for n such that n ≡N zero

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Evidence of unification

A map f : () → (n : N)(e : n ≡N zero) gives us two things:

• 1. A value for n such that n ≡N zero
• 2. Explicit evidence e of n ≡N zero

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Evidence of unification

A map f : () → (n : N)(e : n ≡N zero) gives us two things:

• 1. A value for n such that n ≡N zero
• 2. Explicit evidence e of n ≡N zero

= ⇒ Unification is guaranteed to respect ≡!

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Three valid unifiers

f1 : (k : N) → (k n : N)(e : k ≡N n) f1 k = k; k; refl f2 : () → (k n : N)(e : k ≡N n) f2 () = zero; zero; refl f3 : (k n : N) → (k n : N)(e : k ≡N n) f3 k n = k; k; refl

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Most general unifiers

A most general unifier of ¯ u and ¯ v is a unifier σ such that for any σ′ with ¯ uσ′ = ¯ vσ′, there is a ν such that σ′ = σ ◦ ν.

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Most general unifiers

A most general unifier of ¯ u and ¯ v is a unifier σ such that for any σ′ with ¯ uσ′ = ¯ vσ′, there is a ν such that σ′ = σ ◦ ν. This is quite difficult to translate to type theory directly. . .

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Most general unifiers

A most general unifier of ¯ u and ¯ v is a unifier σ such that for any σ′ with ¯ uσ′ = ¯ vσ′, there is a ν such that σ′ = σ ◦ ν. This is quite difficult to translate to type theory directly. . . Intuition: if f : Γ′ → Γ(¯ e : ¯ u ≡∆ ¯ v) is MGU, we can go back from Γ(¯ e : ¯ u ≡∆ ¯ v) to Γ′ without losing any information.

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Equivalences

A function f : A → B is an equivalence if it has both a left and a right inverse: isLinv : (x : A) → g1 (f x) ≡A x isRinv : (y : B) → f (g2 y) ≡B y In this case, we write f : A ≃ B.

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Most general unifiers are equivalences!

f : Γ(¯ e : ¯ u ≡∆ ¯ v) ≃ Γ′

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Example of unification

(k n : N)(e : suc k ≡N suc n)

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Example of unification

(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n)

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Example of unification

(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n) ≃ (k : N)

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Example of unification

(k n : N)(e : suc k ≡N suc n) ≃ (k n : N)(e : k ≡N n) ≃ (k : N) f : (k : N) → (k n : N)(e : suc k ≡N suc n) f k = k; k; refl

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The solution rule

solution : (x : A)(e : x ≡A t) ≃ ()

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The deletion rule

deletion : (e : t ≡A t) ≃ ()

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The injectivity rule

injectivitysuc : (e : suc x ≡N suc y) ≃ (e′ : x ≡N y)

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Negative unification rules

A negative unification rule applies to impossible equations, e.g. suc x = zero.

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Negative unification rules

A negative unification rule applies to impossible equations, e.g. suc x = zero. This can be represented by an equivalence: (e : suc x ≡N zero) ≃ ⊥ where ⊥ is the empty type.

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The conflict rule

conflictsuc,zero : (e : suc x ≡N zero) ≃ ⊥

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The cycle rule

cyclen,suc n : (e : n ≡N suc n) ≃ ⊥

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Unifiers as equivalences

By requiring unifiers to be equivalences:

• we exclude bad unification rules
• we can safely introduce new rules

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Unifiers as equivalences

By requiring unifiers to be equivalences:

• we exclude bad unification rules
• we can safely introduce new rules

Next, we’ll explore how this idea can help us. Any questions so far?

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Depending on equations

Checking dependently typed programs Unification in dependent type theory Unification of dependently typed terms

Time for the interesting bits!

• Equations between types
• Heterogeneous equations
• Equations on indexed datatypes
• Equations between equations

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Equations between types

Types are first-class terms of type Type: Bool : Type, N : Type, N → N : Type, . . .

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Equations between types

Types are first-class terms of type Type: Bool : Type, N : Type, N → N : Type, . . . We can form equations between types, e.g. Bool ≡Type Bool.

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Equations between types

Types are first-class terms of type Type: Bool : Type, N : Type, N → N : Type, . . . We can form equations between types, e.g. Bool ≡Type Bool. Q: Can we apply the deletion rule?

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Equations between types

Types are first-class terms of type Type: Bool : Type, N : Type, N → N : Type, . . . We can form equations between types, e.g. Bool ≡Type Bool. Q: Can we apply the deletion rule? A: Depends on which type theory we use!

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The univalence axiom (2009)

Vladimir Voevodsky

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The univalence axiom (2009)

Vladimir Voevodsky “Isomorphic types can be identified.”

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The univalence axiom (2009)

Vladimir Voevodsky “Isomorphic types can be identified.” (A ≡ B) ≃ (A ≃ B)

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The univalence axiom (2009)

Bool is equal to Bool in two ways: true false Bool

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The univalence axiom (2009)

Bool is equal to Bool in two ways: true false Bool true false Bool

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The univalence axiom (2009)

Bool is equal to Bool in two ways: true false Bool true false Bool

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The univalence axiom (2009)

Bool is equal to Bool in two ways: true false Bool true false Bool

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Limiting the deletion rule

The deletion rule does not always hold: there might be multiple proofs of x ≡A x. E.g. Bool ≡Type Bool has two elements.

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Limiting the deletion rule

The deletion rule does not always hold: there might be multiple proofs of x ≡A x. E.g. Bool ≡Type Bool has two elements. We cannot use deletion in this case!

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Heterogeneous equations

Σn:NVec A n is the type of pairs (n, xs) where n : N and xs : Vec A n.

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Heterogeneous equations

Σn:NVec A n is the type of pairs (n, xs) where n : N and xs : Vec A n. (e : (0, nil) ≡Σn:NVec A n (1, cons 0 x xs)) ≃ (e1 : 0 ≡N 1)(e2 : nil ≡Vec A ??? cons 0 x xs)

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Heterogeneous equations

Σn:NVec A n is the type of pairs (n, xs) where n : N and xs : Vec A n. (e : (0, nil) ≡Σn:NVec A n (1, cons 0 x xs)) ≃ (e1 : 0 ≡N 1)(e2 : nil ≡Vec A ??? cons 0 x xs) What is the type of e2?

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Heterogeneous equations

Solution: use equation variables as placeholders for their solutions: (e : (0, nil) ≡Σn:NVec A n (1, cons 0 x xs)) ≃ (e1 : 0 ≡N 1)(e2 : nil ≡Vec A e1 cons 0 x xs)

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Heterogeneous equations

Solution: use equation variables as placeholders for their solutions: (e : (0, nil) ≡Σn:NVec A n (1, cons 0 x xs)) ≃ (e1 : 0 ≡N 1)(e2 : nil ≡Vec A e1 cons 0 x xs) This is called a telescopic equality.

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeA (Bool, false))

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeA (Bool, false)) ≃ (e1 : Bool ≡Type Bool)(e2 : true ≡e1 false)

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeA (Bool, false)) ≃ (e1 : Bool ≡Type Bool)(e2 : true ≡e1 false) ≃ ⊥

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeA (Bool, false)) ≃ (e1 : Bool ≡Type Bool)(e2 : true ≡e1 false) ≃ ⊥ The conflict rule does not apply!

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeBool (Bool, false))

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeBool (Bool, false)) ≃ (e1 : Bool ≡Type Bool)(e2 : true ≡Bool false)

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Be careful with heterogeneous equations!

(e : (Bool, true) ≡ΣA:TypeBool (Bool, false)) ≃ (e1 : Bool ≡Type Bool)(e2 : true ≡Bool false) ≃ ⊥ Whether a unification rule can be applied depends on the type of the equation!

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Injectivity for indexed data

Do standard unification rules apply to constructors of indexed datatypes? (e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ ???

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Injectivity for indexed data

Idea: simplify equations between indices together with equation between constructors: (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys)

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Injectivity for indexed data

Idea: simplify equations between indices together with equation between constructors: (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys) ≃ (e′

1 : k ≡N n)(e′ 2 : x ≡A y)

(e′

3 : xs ≡Vec A e1 ys)

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Injectivity for indexed data

Idea: simplify equations between indices together with equation between constructors: (e1 : suc k ≡N suc n) (e2 : cons k x xs ≡Vec A e1 cons n y ys) ≃ (e′

1 : k ≡N n)(e′ 2 : x ≡A y)

(e′

3 : xs ≡Vec A e1 ys)

Length of the Vec must be fully general: must be an equation variable.

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The image datatype

The type Im f y consists of elements image x such that f x = y: data Im (f : A → B) : B → Type where image : (x : A) → Im f (f x)

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Solving unsolvable equations

(x1 x2 : A)(e1 : f x1 ≡B f x2) (e2 : image x1 ≡Im f e1 image x2)

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Solving unsolvable equations

(x1 x2 : A)(e1 : f x1 ≡B f x2) (e2 : image x1 ≡Im f e1 image x2) ≃ (x1 x2 : A)(e : x1 ≡A x2)

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Solving unsolvable equations

(x1 x2 : A)(e1 : f x1 ≡B f x2) (e2 : image x1 ≡Im f e1 image x2) ≃ (x1 x2 : A)(e : x1 ≡A x2) ≃ (x1 : A)

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What if the indices are not fully general?

(e : cons n x xs ≡Vec A (suc n) cons n y ys)

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What if the indices are not fully general?

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl)

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What if the indices are not fully general?

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl) ≃ (e′

1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′

1 ys)

(p : cong suc e′

1 ≡suc n≡Nsuc n refl)

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What if the indices are not fully general?

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e1 : suc n ≡N suc n) (e2 : cons n x xs ≡Vec A e1 cons n y ys) (p : e1 ≡suc n≡Nsuc n refl) ≃ (e′

1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′

1 ys)

(p : cong suc e′

1 ≡suc n≡Nsuc n refl)

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Higher-dimensional equations

(e′

1 : n ≡N n)(e′ 2 : x ≡A y)(e′ 3 : xs ≡Vec A e′

1 ys)

(p : cong suc e′

1 ≡suc n≡Nsuc n refl)

We call an equation between equality proofs (e.g. p) a higher-dimensional equation.

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How to solve higher-dimensional equations?

Existing unification rules do not apply. . .

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How to solve higher-dimensional equations?

Existing unification rules do not apply. . . We solve the problem in three steps:

• 1. lower the dimension of equations
• 2. solve lower-dimensional equations
• 3. lift unifier to higher dimension

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Step 1: lower the dimension of equations

We replace all equation variables by regular variables: instead of (e1 : n ≡N n)(e2 : x ≡A y)(e3 : xs ≡Vec A e1 ys) (p : cong suc e1 ≡suc n≡Nsuc n refl) let’s first consider (k : N)(u : A)(us : Vec A k) (e : suc k ≡N suc n)

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Step 2: solve lower-dimensional equations

This gives us an equivalence f of type (k : N)(u : A)(us : Vec A k) (e : suc k ≡N suc n) ≃ (u : A)(us : Vec A n)

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Step 3: lift unifier to higher dimension

We lift f to an equivalence f ↑ of type (e1 : n ≡N n)(e2 : x ≡A y) (e3 : xs ≡Vec A e1 ys) (p : cong suc e1 ≡suc n≡Nsuc n refl) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Final result of steps 1-3

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys)

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Final result of steps 1-3

(e : cons n x xs ≡Vec A (suc n) cons n y ys) ≃ (e2 : x ≡A y)(e3 : xs ≡Vec A n ys) This is the forcing rule for cons.

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Lifting equivalences: (mostly) general case

• Theorem. If we have an equivalence f of type

(x : A)(e : b1 x ≡B x b2 x) ≃ C we can construct f ↑ of type (e : u ≡A v)(p : cong b1 e ≡r≡B es cong b2 e) ≃ (e′ : f u r ≡C f v s)

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Implementation in Agda

This is all used by Agda to check definitions by dependent pattern matching.

• More general than before
• Fixed many bugs
• Implementation matches theory

You can try it for yourself: wiki.portal.chalmers.se/agda

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Conclusion

Unification rules should return evidence

• f their correctness.

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Conclusion

Unification rules should return evidence

• f their correctness.

A most general unifier can be represented internally as an equivalence.

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Conclusion

Unification rules should return evidence

• f their correctness.

A most general unifier can be represented internally as an equivalence. Unification cannot ignore the types!

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Questions?

If you want to know more, you can:

• Try out Agda:

wiki.portal.chalmers.se/agda

• Look at the source:

github.com/agda/agda

• Read my thesis:

Dependent pattern matching and proof-relevant unification (2017)

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Two applications of unification

Filling in implicit arguments Checking definitions by pattern matching

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Two applications of unification

Filling in implicit arguments

• Higher order

Checking definitions by pattern matching

• First order

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Two applications of unification

Filling in implicit arguments

• Higher order
• ‘Syntactic’

Checking definitions by pattern matching

• First order
• ‘Semantic’

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Two applications of unification

Filling in implicit arguments

• Higher order
• ‘Syntactic’
• MGU optional

Checking definitions by pattern matching

• First order
• ‘Semantic’
• MGU required

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Two applications of unification

Filling in implicit arguments

• Higher order
• ‘Syntactic’
• MGU optional

Checking definitions by pattern matching

• First order
• ‘Semantic’
• MGU required

Focus of this talk

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Two notions of equality

Definitional equality x = y : A

• Weaker

Propositional equality e : x ≡A y

• Stronger

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Two notions of equality

Definitional equality x = y : A

• Weaker
• Decidable

Propositional equality e : x ≡A y

• Stronger
• Undecidable

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Two notions of equality

Definitional equality x = y : A

• Weaker
• Decidable
• Meta-theoretic

Propositional equality e : x ≡A y

• Stronger
• Undecidable
• Internal to theory

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Two notions of equality

Definitional equality x = y : A

• Weaker
• Decidable
• Meta-theoretic
• Implicit

Propositional equality e : x ≡A y

• Stronger
• Undecidable
• Internal to theory
• Explicit

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