slides - Dan Licata - Wesleyan University

Git as a HIT
Dan Licata
Wesleyan University
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Darcs
Git as a HIT
Dan Licata
Wesleyan University
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HITs
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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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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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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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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
Patch
diff
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=
2c2
<b
-->d
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id
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id
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p
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q
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id
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qop
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id
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p
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qop
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id
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p
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!p
qop
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id
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p
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!p
undo/rollback
qop
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id
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!p
undo/rollback
qop
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[Yorgey,Jacobson,…]
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Simple Setup
a
u
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o
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b at 0
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b at 0
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Simple Setup
“Repository” is a char vector of fixed length n
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s
Basic patch is a
u
o
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s
b at i where i<n
a
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b at 0
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b at 0
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Domain-Specific Language
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Domain-Specific Language
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Domain-Specific Language
swapat a b i v permutes a and b at position i in v
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Domain-Specific Language
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
undo really un-does
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
undo really un-does
Spec: ∀ p. interp p is a bijection:
∀ v. g (f v) = v
where (f,g)=interp p
∀ v. f (g v) = v
Can package this as:
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Merging
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Merging
p
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q’
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p’
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Merging
p=b d at 1
p
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q
q=c e at 2
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q’
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p’
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Merging
p=b d at 1
p
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q
q=c e at 2
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q’
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p’
p’=p
q’=q
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Merging
p=b d at 1
p
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q
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=
q’
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q=c e at 2
p’
p’=p
q’=q
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Merging
merge : (p q : Patch)
! Σq’,p’:Patch.
Maybe(q’ o p =
p’ o q)
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Merging
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
(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
(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
Basic Axioms:
(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
Basic axioms:
(a b at i)o(c d at j)
=(c d at j)o(a b at i)
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Patch Equality
(a b at i)o(c d at j)
Group laws:
id o p = p = p o id
=(c d at j)o(a b at i)
po(qor) = (poq)or
Basic axioms:
!p o p = id = p o !p
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Patch Equality
(a b at i)o(c d at j)
Group laws:
id o p = p = p o id
=(c d at j)o(a b at i)
po(qor) = (poq)or
Basic axioms:
!p o p = id = p o !p
Congruence:
p=p
!p = !p’ if p = p’
p=q if q=p
p o q = p’ o q’ if
p = p’ and q = q’
p=r if p=q and q=r
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Patch as Quotient Type
Elements:
Equality:
(a b at i)o(c d at j)~
(c d at j)o(a b at i)
...
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’
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Patch as Quotient Type
Elements:
Quotient Type:
Patch := Patch’/~
Equality:
(a b at i)o(c d at j)~
(c d at j)o(a b at i)
...
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’
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Patch as Quotient Type
Elements:
Quotient Type:
Patch := Patch’/~
Equality:
Elimination rule:
(a b at i)o(c d at j)~
(c d at j)o(a b at i)
...
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
define on Patch’ as before,
then prove p ~ q implies
interp p = interp q
for all 14+ rules for ~
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’
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Patches as a HIT
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
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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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
RepoDesc : Type
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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
RepoDesc : Type
vec : RepoDesc
generator for element
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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
RepoDesc : Type
vec : RepoDesc
generator for element
(a b at i) : vec = vec
generator for equality
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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
proof-relevant!
RepoDesc : Type
vec : RepoDesc
generator for element
(a b at i) : vec = vec
generator for equality
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Higher Inductive Type
Type freely generated by constructors for elements,
equalities, equalities between equalities, …
proof-relevant!
RepoDesc : Type
vec : RepoDesc
generator for element
(a b at i) : vec = vec
generator for equality
commute:
(a b at i)o(c d at j)
generator for equality
between equalities
=(c d at j)o(a b at i)
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Type: Patch
Elements:
Equality:
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
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’
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Type: Patch
Type: RepoDesc
Elements:
Equality:
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
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’
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Type: Patch
Type: RepoDesc
Element: vec : RepoDesc
Elements:
Equality:
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
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’
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Type: Patch
Type: RepoDesc
Elements:
Element: vec : RepoDesc
Equality:
a b at i : vec = vec
Equality:
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
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’
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Type: Patch
Type: RepoDesc
Elements:
Element: vec : RepoDesc
Equality:
Patch
{
a b at i : vec = vec
Equality:
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
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’
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Type: Patch
Type: RepoDesc
Elements:
Element: vec : RepoDesc
Equality:
Patch
{
a b at i : vec = vec
Equality:
Equality between equalities:
(a b at i)o(c d at j)=
commute :
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
id o p = p = p o id
(c d at j)o(a b at i)
po(qor) = (poq)or
… basic axioms only!
!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’
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Type: Patch
Type: RepoDesc
Elements:
Element: vec : RepoDesc
Equality:
Patch
{
a b at i : vec = vec
Equality:
Equality between equalities:
(a b at i)o(c d at j)=
commute :
(a b at i)o(c d at j)=
(c d at j)o(a b at i)
...
id o p = p = p o id
(c d at j)o(a b at i)
po(qor) = (poq)or
… basic axioms only!
!p o p = id = p o !p
p=p
Everything else comes
“for free” from
the equality type!
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’
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Typed Patches
RepoDesc : Type
vec : RepoDesc
compressed : RepoDesc
generators for elements
a b at i : vec = vec
generators for equalities
gzip : vec = compressed
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Typed Patches
RepoDesc : Type
vec : RepoDesc
compressed : RepoDesc
generators for elements
a b at i : vec = vec
generators for equalities
{
gzip : vec = compressed
Patch vec compressed
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Patches as a HIT
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 recursion
To define a function RepoDesc ! A
it suffices to
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RepoDesc recursion
To define a function RepoDesc ! A
it suffices to
map the element generators of RepoDesc
to elements of A
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RepoDesc recursion
To define a function RepoDesc ! A
it suffices to
map the element generators of RepoDesc
to elements of A
map the equality generators of RepoDesc
to equalities between the corresponding elements of A
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RepoDesc recursion
To define a function RepoDesc ! A
it suffices to
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
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RepoDesc recursion
To define a function f : RepoDesc ! A
it suffices to give
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RepoDesc recursion
To define a function f : RepoDesc ! A
it suffices to give
f(vec) := … : A
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RepoDesc recursion
To define a function f : RepoDesc ! A
it suffices to give
f(vec) := … : A
f1(a b at i) := … : f(vec) = f(vec)
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RepoDesc recursion
To define a function f : RepoDesc ! A
it suffices to give
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
To define a function f : RepoDesc ! A
it suffices to give
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
To define a function f : RepoDesc ! A
it suffices to give
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
To define a function f : RepoDesc ! A
it suffices to give
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
To define a function f : RepoDesc ! A
it suffices to give
=f1(a b at i)o
f(vec) := … : A
f1(c d at j)
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
To define a function f : RepoDesc ! A
it suffices to give
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
To define a function f : RepoDesc ! A
it suffices to give
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
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
Goal is to define:
interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(id) = (λx.x, …)
interp(q o p) = (interp q) ob (interp p)
interp(!p) = !b (interp p)
interp(a b at i) = swapat a b i
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Interp
Goal is to define:
interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(id) = (λx.x, …)
interp(q o p) = (interp q) ob (interp p)
interp(!p) = !b (interp p)
interp(a b at i) = swapat a b i
But only tool available is RepoDesc recursion:
no direct recursion over proofs of equality
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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Need to pick A and define
f(vec) := … : A
f1(a b at i) := … : f(vec) = f(vec)
f2(compose) := …
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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
f(vec) := … : Type
f1(a b at i) := … : f(vec) = f(vec)
f2(compose) := …
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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
f(vec) := Vec Char n : Type
f1(a b at i) := … : f(vec) = f(vec)
f2(compose) := …
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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
f(vec) := Vec Char n : Type
f1(a b at i) := … : Vec Char n = Vec Char n
f2(compose) := …
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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
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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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
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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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
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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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(a b at i) = swapat a b i
Key idea: pick A = Type and define
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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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(p) = ua-1(I1(p))
Key idea: pick A = Type and define
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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interp : vec = vec
" Bijection (Vec Char n) (Vec Char n)
interp(p) = ua-1(I1(p))
Satisfies the desired equations (as propositional equalities):
interp(id) = (λx.x, …)
interp(q o p) = (interp q) ob (interp p)
interp(!p) = !b (interp p)
interp(a b at i) = swapat a b i
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Summary
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Summary
I : RepoDesc ! Type interprets RepoDesc’s as Types,
patches as bijections, satisfying patch equalities
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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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35
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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35
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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Where does this
programming technique
come from?
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Homotopy type theory
p
a
b
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Homotopy type theory
a space is a type A
p
a
b
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Homotopy type theory
a space is a type A
p
a
b
points are
elements
a:A
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Homotopy type theory
a space is a type A
p
a
points are
elements
a:A
b
paths are
proofs of equality
p : a =A b
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Homotopy type theory
a space is a type A
path operations
p
a
points are
elements
a:A
b
paths are
proofs of equality
p : a =A b
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Homotopy type theory
a space is a type A
id
a
points are
elements
a:A
path operations
id
p
: a = a (refl)
b
paths are
proofs of equality
p : a =A b
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Homotopy type theory
a space is a type A
id
a
points are
elements
a:A
p
!p
path operations
id
!p
b
: a = a (refl)
: b = a (sym)
paths are
proofs of equality
p : a =A b
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Homotopy type theory
a space is a type A
id
a
p
!p
path operations
id
: a = a (refl)
!p
: b = a (sym)
q o p : a = c (trans)
b
q
c
points are
elements
a:A
paths are
proofs of equality
p : a =A b
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Homotopy type theory
a space is a type A
id
a
p
!p
path operations
id
: a = a (refl)
!p
: b = a (sym)
q o p : a = c (trans)
b
q
c
homotopies
points are
elements
a:A
id o p = p
!p o p = id
paths are
proofs of equality
p : a =A b
r o (q o p)
= (r o q) o p
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Homotopy type theory
a space is a type A
path operations
id
: a = a (refl)
!p
: b = a (sym)
q o p : a = c (trans)
homotopies
points are
elements
a:A
id o p = p
!p o p = id
paths are
proofs of equality
p : a =A b
r o (q o p)
= (r o q) o p
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Equality elimination rule
Type of equalities
between a and -
is inductively
generated by
y2
p2
y1
p1
a
p3
id
y3
a
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Equality elimination rule
Type of equalities
between a and -
is inductively
generated by
y2
p2
y1
p1
a
p3
id
y3
a
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)
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Composition and Assoc
_o_ : a = b " b = c " a = c
id o p = p
o-assoc : (p : a=b)(q : b=c)(r : c=d)
" p o (q o r) = (p o q) o r
o-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
Circle S1 is HIT generated by
loop
base
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The Circle
Circle S1 is HIT generated by
base :
loop : base = base
loop
S1
base
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The Circle
Circle S1 is HIT generated by
S1
base :
loop : base = base
loop
-1
id
loop
base
Free type: equipped with
id
inv : loop o loop-1 = id
loop-1
...
loop o loop
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The Circle
Circle recursion:
function S1 ! X determined by
base’ : X
loop’ : base’ = base’
loop
base
base’
loop’
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
loop
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
loop
loop-1
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
loop
loop-1
loop o loop
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Fundamental group of circle
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
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Fundamental group of circle
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
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Fundamental group of circle
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
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
0
id
loop
loop-1
loop o loop
loop-1 o loop-1
loop o loop-1
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
0
loop
1
loop-1
loop o loop
loop-1 o loop-1
loop o loop-1
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
0
loop
1
loop-1
-1
loop o loop
loop-1 o loop-1
loop o loop-1
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
0
loop
1
loop-1
-1
loop o loop
2
loop-1 o loop-1
loop o loop-1
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
0
loop
1
loop-1
-1
loop o loop
2
loop-1 o loop-1
-2
loop o loop-1
= id
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Fundamental group of circle
How many different loops are there on
the circle, up to homotopy?
loop
base
id
0
loop
1
loop-1
-1
loop o loop
2
loop-1 o loop-1
-2
loop o loop-1
0
= id
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Fundamental group of circle
Theorem. Group of loops on the circle
is isomorphic to ℤ
Proof: Define universal cover
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Fundamental group of circle
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
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
π1(S1) = ℤ
Freudenthal
Van Kampen
πk<n(Sn) = 0
πn(Sn) = ℤ
Covering spaces
Hopf fibration
K(G,n)
π2(S2) = ℤ
Cohomology
axioms
π3(S2) = ℤ
James
Construction
π4(S3) = ℤ?
Whitehead
for n-types
Blakers-Massey
[Brunerie, Finster, Hou,
Licata, Lumsdaine, Shulman]
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What’s next?
Operational semantics of HITs and univalence is still an
open 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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