Git as a HIT Dan Licata Wesleyan University 1 1 Darcs Git as a - - PowerPoint PPT Presentation

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Git as a HIT

Dan Licata Wesleyan University

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Git as a HIT

Dan Licata Wesleyan University

Darcs

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Generator for

HITs

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Generator for

HITs

Homotopy Type Theory is an extension of Agda/Coq based on connections with homotopy theory

[Hofmann&Streicher,Awodey&Warren,Voevodsky,Lumsdaine,Garner&van den Berg]

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Generator for

HITs

Homotopy Type Theory is an extension of Agda/Coq based on connections with homotopy theory

[Hofmann&Streicher,Awodey&Warren,Voevodsky,Lumsdaine,Garner&van den Berg]

Higher inductive types (HITs) are a new type former!

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Generator for

HITs

Homotopy Type Theory is an extension of Agda/Coq based on connections with homotopy theory

[Hofmann&Streicher,Awodey&Warren,Voevodsky,Lumsdaine,Garner&van den Berg]

Higher inductive types (HITs) are a new type former! They were originally invented[Lumsdaine,Shulman,…] to model basic spaces (circle, spheres, the torus, …) and constructions in homotopy theory

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2

Generator for

HITs

Homotopy Type Theory is an extension of Agda/Coq based on connections with homotopy theory

[Hofmann&Streicher,Awodey&Warren,Voevodsky,Lumsdaine,Garner&van den Berg]

Higher inductive types (HITs) are a new type former! They were originally invented[Lumsdaine,Shulman,…] to model basic spaces (circle, spheres, the torus, …) and constructions in homotopy theory But they have many other applications, including some programming ones!

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Patches

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a b c diff 2c2 < b

• > d

a d c

=

Patch

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a b c id a b c

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a b c id a b c a b c p a d c q a d e

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a b c id a b c a b c p a d c q a d e q o p

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a b c id a b c a b c p a d c q a d e q o p a b c p a d c

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a b c id a b c a b c p a d c q a d e q o p a b c p a d c !p

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a b c id a b c a b c p a d c q a d e q o p a b c p a d c !p

undo/rollback

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a b c id a b c a b c p a d c q a d e q o p a b c p a d c !p

undo/rollback [Yorgey,Jacobson,…]

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Simple Setup

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a u s s

• i

s a u s b u s

a ↔ b at 0 a ↔ b at 0

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Simple Setup

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“Repository” is a char vector of fixed length n

a u s s

• i

s

Basic patch is a ↔ b at i where i<n

a u s b u s

a ↔ b at 0 a ↔ b at 0

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Domain-Specific Language

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Domain-Specific Language

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Domain-Specific Language

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swapat a b i v permutes a and b at position i in v

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Domain-Specific Language

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Spec: ∀ p. interp p is a bijection: ∀ v. g (f v) = v where (f,g)=interp p ∀ v. f (g v) = v

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Domain-Specific Language

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Spec: ∀ p. interp p is a bijection: ∀ v. g (f v) = v where (f,g)=interp p ∀ v. f (g v) = v

undo really un-does

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Domain-Specific Language

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Spec: ∀ p. interp p is a bijection: ∀ v. g (f v) = v where (f,g)=interp p ∀ v. f (g v) = v

undo really un-does

Can package this as:

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Merging

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a b c p a d c q a b e

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Merging

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a b c p a d c q a b e a d e q’ p’

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Merging

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a b c p a d c q a b e a d e q’ p’ p=b↔d at 1 q=c↔e at 2

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Merging

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a b c p a d c q a b e a d e q’ p’ p=b↔d at 1 q=c↔e at 2 p’=p q’=q

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Merging

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a b c p a d c q a b e a d e q’ p’

=

p=b↔d at 1 q=c↔e at 2 p’=p q’=q

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Merging

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merge : (p q : Patch) Σq’,p’:Patch. Maybe(q’ o p = p’ o q)

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Merging

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merge : (p q : Patch) Σq’,p’:Patch. Maybe(q’ o p = p’ o q) When are two patches equal?

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Patch Equality

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(a↔b at i)o(c↔d at j) = (c↔d at j)o(a↔b at i) if i≠j

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Patch Equality

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(a↔b at i)o(c↔d at j) = (c↔d at j)o(a↔b at i) if i≠j (a↔a at i) = id !(a↔b at i) = (a↔b at i) (a↔b at i) = (b↔a at i)

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Patch Equality

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(a↔b at i)o(c↔d at j) = (c↔d at j)o(a↔b at i) if i≠j (a↔a at i) = id !(a↔b at i) = (a↔b at i) (a↔b at i) = (b↔a at i) Basic Axioms:

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Patch Equality

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(a↔b at i)o(c↔d at j) =(c↔d at j)o(a↔b at i)

Basic axioms:

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Patch Equality

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p

(a↔b at i)o(c↔d at j) =(c↔d at j)o(a↔b at i)

Basic axioms: Group laws:

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Patch Equality

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’

(a↔b at i)o(c↔d at j) =(c↔d at j)o(a↔b at i)

Basic axioms: Group laws: Congruence:

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Patch as Quotient Type

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id o p ~ p ~ p o id po(qor) ~ (poq)or !p o p ~ id ~ p o !p p~p p~q if q~p p~r if p~q and q~r !p ~ !p’ if p ~ p’ p o q ~ p’ o q’ if p ~ p’ and q ~ q’ (a↔b at i)o(c↔d at j)~ (c↔d at j)o(a↔b at i) ...

Elements: Equality:

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Patch as Quotient Type

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id o p ~ p ~ p o id po(qor) ~ (poq)or !p o p ~ id ~ p o !p p~p p~q if q~p p~r if p~q and q~r !p ~ !p’ if p ~ p’ p o q ~ p’ o q’ if p ~ p’ and q ~ q’ (a↔b at i)o(c↔d at j)~ (c↔d at j)o(a↔b at i) ...

Elements: Equality: Quotient Type: Patch := Patch’/~

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Patch as Quotient Type

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id o p ~ p ~ p o id po(qor) ~ (poq)or !p o p ~ id ~ p o !p p~p p~q if q~p p~r if p~q and q~r !p ~ !p’ if p ~ p’ p o q ~ p’ o q’ if p ~ p’ and q ~ q’ (a↔b at i)o(c↔d at j)~ (c↔d at j)o(a↔b at i) ...

Elements: Equality: Elimination rule: define on Patch’ as before, then prove p ~ q implies interp p = interp q for all 14+ rules for ~ Quotient Type: Patch := Patch’/~

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Patches as a HIT

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1.How do you define Patch using a higher inductive type? 2.What is the elimination rule? 3.How do you use the elim. rule to define interp?

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Patches as a HIT

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1.How do you define Patch using a higher inductive type? 2.What is the elimination rule? 3.How do you use the elim. rule to define interp?

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Higher Inductive Type

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Generator for

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … Generator for

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … RepoDesc : Type Generator for

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … RepoDesc : Type vec : RepoDesc Generator for

generator for element

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … RepoDesc : Type vec : RepoDesc (a↔b at i) : vec = vec Generator for

generator for element generator for equality

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … RepoDesc : Type vec : RepoDesc (a↔b at i) : vec = vec Generator for

proof-relevant! generator for element generator for equality

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Higher Inductive Type

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Type freely generated by constructors for elements, equalities, equalities between equalities, … RepoDesc : Type vec : RepoDesc (a↔b at i) : vec = vec commute: (a↔b at i)o(c↔d at j) =(c↔d at j)o(a↔b at i) Generator for

proof-relevant! generator for element generator for equality generator for equality between equalities

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Type: Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Element: vec : RepoDesc Type: Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Element: vec : RepoDesc Equality: a↔b at i : vec = vec Type: Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Element: vec : RepoDesc Equality: a↔b at i : vec = vec Type: Patch

{

Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Element: vec : RepoDesc Equality: a↔b at i : vec = vec Equality between equalities:

commute : (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i)

… basic axioms only! Type: Patch

{

Patch

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id o p = p = p o id po(qor) = (poq)or !p o p = id = p o !p p=p p=q if q=p p=r if p=q and q=r !p = !p’ if p = p’ p o q = p’ o q’ if p = p’ and q = q’ (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i) ...

Elements: Equality: Type: RepoDesc Element: vec : RepoDesc Equality: a↔b at i : vec = vec Equality between equalities:

commute : (a↔b at i)o(c↔d at j)= (c↔d at j)o(a↔b at i)

… basic axioms only! Type: Patch

{

Patch

Everything else comes “for free” from the equality type!

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Typed Patches

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RepoDesc : Type vec : RepoDesc a↔b at i : vec = vec Generator for compressed : RepoDesc gzip : vec = compressed

generators for elements generators for equalities

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Typed Patches

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RepoDesc : Type vec : RepoDesc a↔b at i : vec = vec Generator for compressed : RepoDesc gzip : vec = compressed

generators for elements generators for equalities

{

Patch vec compressed

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Patches as a HIT

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1.How do you define Patch using a higher inductive type? 2.What is the elimination rule for RepoDesc? 3.How do you use the elim. rule to define interp?

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RepoDesc A To define a function it suffices to Generator for

RepoDesc recursion

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RepoDesc A To define a function it suffices to Generator for map the element generators of RepoDesc to elements of A

RepoDesc recursion

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RepoDesc A To define a function it suffices to Generator for map the element generators of RepoDesc to elements of A map the equality generators of RepoDesc to equalities between the corresponding elements of A

RepoDesc recursion

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RepoDesc A To define a function it suffices to Generator for map the element generators of RepoDesc to elements of A map the equality generators of RepoDesc to equalities between the corresponding elements of A map the equality-between-equality generators to equalities between the corresponding equalities in A

RepoDesc recursion

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec)

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

You only specify f on generators, not id,o,!,group laws,congruence,… (1 patch and 4 basic axioms, instead of 4 and 14!)

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

Type-generic equality rules say that functions act homomorphically on id,o,!,…

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

Type-generic equality rules say that functions act homomorphically on id,o,!,…

=f1(a↔b at i)o f1(c↔d at j)

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

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RepoDesc recursion

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f : RepoDesc A To define a function it suffices to give Generator for

f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose a b c d i j i≠j) := … : f1((a↔b at i)o(c↔d at j)) = f1((c↔d at j)o(a↔b at j))

All functions on RepoDesc respect patches All functions on patches respect patch equality

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Patches as a HIT

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1.How do you define Patch using a higher inductive type? 2.What is the elimination rule for RepoDesc? 3.How do you use the elim. rule to define interp?

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Interp

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Goal is to define: Generator for

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i interp(q o p) = (interp q) ob (interp p) interp(id) = (λx.x, …) interp(!p) = !b (interp p)

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Interp

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Goal is to define: Generator for But only tool available is RepoDesc recursion: no direct recursion over proofs of equality

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i interp(q o p) = (interp q) ob (interp p) interp(id) = (λx.x, …) interp(!p) = !b (interp p)

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Generator for Need to pick A and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) f(vec) := … : A f1(a↔b at i) := … : f(vec) = f(vec) f2(compose) := … interp(a↔b at i) = swapat a b i

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) f(vec) := … : Type f1(a↔b at i) := … : f(vec) = f(vec) f2(compose) := … interp(a↔b at i) = swapat a b i

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) f(vec) := Vec Char n : Type f1(a↔b at i) := … : f(vec) = f(vec) f2(compose) := … interp(a↔b at i) = swapat a b i

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i f(vec) := Vec Char n : Type f1(a↔b at i) := … : Vec Char n = Vec Char n f2(compose) := …

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i f(vec) := Vec Char n : Type f1(a↔b at i) := ua(swapat a b i) : Vec Char n = Vec Char n f2(compose) := …

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i f(vec) := Vec Char n : Type f1(a↔b at i) := ua(swapat a b i) : Vec Char n = Vec Char n f2(compose) := … Voevodky’s univalence axiom ⊃ bijective types are equal

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i f(vec) := Vec Char n : Type f1(a↔b at i) := ua(swapat a b i) : Vec Char n = Vec Char n f2(compose) := <proof about swapat as before>

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(a↔b at i) = swapat a b i I(vec) := Vec Char n : Type I1(a↔b at i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := <proof about swapat as before>

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Generator for Key idea: pick A = Type and define

interp : vec = vec Bijection (Vec Char n) (Vec Char n) I(vec) := Vec Char n : Type I1(a↔b at i) := ua(swapat a b i) : Vec Char n = Vec Char n I2(compose) := <proof about swapat as before> interp(p) = ua-1(I1(p))

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Generator for

interp : vec = vec Bijection (Vec Char n) (Vec Char n) interp(p) = ua-1(I1(p)) interp(a↔b at i) = swapat a b i interp(q o p) = (interp q) ob (interp p) interp(id) = (λx.x, …) interp(!p) = !b (interp p) Satisfies the desired equations (as propositional equalities):

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Generator for

Summary

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Generator for

Summary

I : RepoDesc Type interprets RepoDesc’s as Types, patches as bijections, satisfying patch equalities

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Generator for

Summary

I : RepoDesc Type interprets RepoDesc’s as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,...

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Generator for

Summary

I : RepoDesc Type interprets RepoDesc’s as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,... Univalence lets you give a computational model of equality proofs (here, patches); guaranteed to satisfy laws

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Generator for

Summary

I : RepoDesc Type interprets RepoDesc’s as Types, patches as bijections, satisfying patch equalities Higher inductive elim. defines functions that respect equality: you specify what happens on the generators; homomorphically extended to id,o,!,... Univalence lets you give a computational model of equality proofs (here, patches); guaranteed to satisfy laws Shorter definition and code than using quotients: 1 basic patch & 4 basic axioms of equality, instead of 4 patches & 14 equations

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Generator for

Where does this programming technique come from?

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Homotopy type theory

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a b p

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Homotopy type theory

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a b p

a space is a type A

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Homotopy type theory

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a b p

points are elements a:A a space is a type A

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Homotopy type theory

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a b p

points are elements a:A a space is a type A paths are proofs of equality p : a =A b

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Homotopy type theory

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a b p

points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

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Homotopy type theory

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a b p id

points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

id : a = a (refl)

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Homotopy type theory

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a b p id !p

points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

id : a = a (refl) !p : b = a (sym)

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Homotopy type theory

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a b p c q id !p

points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

id : a = a (refl) !p : b = a (sym) q o p : a = c (trans)

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Homotopy type theory

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a b p c q id !p

points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

id : a = a (refl) !p : b = a (sym) q o p : a = c (trans)

homotopies

id o p = p !p o p = id r o (q o p) = (r o q) o p

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Homotopy type theory

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points are elements a:A a space is a type A paths are proofs of equality p : a =A b path operations

id : a = a (refl) !p : b = a (sym) q o p : a = c (trans)

homotopies

id o p = p !p o p = id r o (q o p) = (r o q) o p

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Equality elimination rule

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Type of equalities between a and -

a id a

is inductively generated by

y3 y1 y2 p1 p3 p2

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Equality elimination rule

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Fix a type A with element a:A. For a family of types C(y:A, p:a=y), to give an element of C(y,p) for all y and p:a=y, suffices to give an element of C(a,id) Type of equalities between a and -

a id a

is inductively generated by

y3 y1 y2 p1 p3 p2

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Composition and Assoc

_o_ : a = b b = c a = c id o p = p

• -assoc : (p : a=b)(q : b=c)(r : c=d)

p o (q o r) = (p o q) o r

• -assoc id id id = id

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Functions are functors

f : A B has action at all levels f1 : (a1 a2 : A) a1 =A a2 f(a1) =B f(a2) f2 : (a1 a2 : A)(p p’ : a1 =A a2)

p =a1=a2 p’

f1(p) =f(a1)=f(a2) f1(p’) and so on

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The Circle

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Circle S1 is HIT generated by

loop base

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The Circle

41

Circle S1 is HIT generated by base : S1 loop : base = base

loop base

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The Circle

41

Circle S1 is HIT generated by base : S1 loop : base = base

loop base

Free type: equipped with

id loop-1

inv : loop o loop-1 = id id loop-1 loop o loop ...

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The Circle

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Circle recursion: function S1 X determined by base’ : X loop’ : base’ = base’

loop base loop’ base’

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Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id loop

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id loop loop-1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id loop loop-1 loop o loop

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id loop loop-1 loop o loop loop-1 o loop-1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1 1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1 1

• 1

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1 1

• 1

2

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1 1

• 1

2

• 2

43

Fundamental group of circle

43

How many different loops are there on the circle, up to homotopy?

loop base

= id id loop loop-1 loop o loop loop-1 o loop-1 loop o loop-1 1

• 1

2

• 2

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Fundamental group of circle

44

• Theorem. Group of loops on the circle

is isomorphic to ℤ Proof: Define universal cover

44

Fundamental group of circle

44

• Theorem. Group of loops on the circle

is isomorphic to ℤ Proof: Define universal cover Cover : S1 Type Cover(base) := ℤ Cover1(loop) := ua(successor) : ℤ = ℤ

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Fundamental group of circle

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• Theorem. Group of loops on the circle

is isomorphic to ℤ Proof: Define universal cover Cover : S1 Type Cover(base) := ℤ Cover1(loop) := ua(successor) : ℤ = ℤ

interpret loop as “add 1” bijection

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Homotopy in HoTT

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π1(S1) = ℤ πk<n(Sn) = 0 π2(S2) = ℤ Hopf fibration π3(S2) = ℤ πn(Sn) = ℤ Freudenthal π4(S3) = ℤ? James Construction K(G,n) Blakers-Massey Van Kampen Covering spaces Whitehead for n-types Cohomology axioms [Brunerie, Finster, Hou, Licata, Lumsdaine, Shulman]

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Generator for

What’s next?

Operational semantics of HITs and univalence is still an

• pen problem in general, though some special cases

are known Have just started exploring programming applications Extensions to this example: more realistic basic patches, patches that can fail (partial bijections), implement merge

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