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A Functional Programmer’s Guideto Homotopy Type Theory
Dan LicataWesleyan University
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Homotopy type theory
highercategory theory homotopy theory
dependent type theory
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Semantics
highercategory theory homotopy theory
dependent type theory
Awodey,van den Berg,Gambino,Garner,Hofmann,Lumsdaine,Streicher,Voevodsky,Warren 1994-2010
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Semantics
highercategory theory homotopy theory
dependent type theory
Awodey,van den Berg,Gambino,Garner,Hofmann,Lumsdaine,Streicher,Voevodsky,Warren 1994-2010
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Principles
highercategory theory homotopy theory
dependent type theory
Univalence [Voevodsky, 2006]Higher inductive types [Bauer,Lumsdaine,Shulman,Warren, 2011]
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Principles
highercategory theory homotopy theory
dependent type theory
Univalence [Voevodsky, 2006]Higher inductive types [Bauer,Lumsdaine,Shulman,Warren, 2011]
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FP as a language for objects of higher homotopy type
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FP as a language for objects of higher homotopy type
Mechanized proofsπ1(S1) = ℤ
πk<n(Sn) = 0
π2(S2) = ℤ
Hopf fibrations
π3(S2) = ℤ
πn(Sn) = ℤ
Freudenthal
K(G,n)
Blakers-Massey
Van Kampen
Covering spaces
Whiteheadfor n-types
Cohomology axioms
[Brunerie,Buchholtz,Cavallo,Finster, Hou,Licata,Lumsdaine,Rilke,Shulman]
T2 = S1 × S1
Mayer-VietorisProjective Spaces
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highercategory theory homotopy theory
dependent type theory
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What does this all mean in programming terms?
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In a worldthat’s poweredby violence...
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In a worldthat’s poweredby violence...
In a worldwhere owning aradio was strictly
forbidden…
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In a worldwhere freedom
is history…
In a worldthat’s poweredby violence...
In a worldwhere owning aradio was strictly
forbidden…
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In a world where all functions are monotoneand preserve least upper bounds…
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In a world where all functions are monotoneand preserve least upper bounds…
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In a world whereall functions are continuous…
λ-terms
CPOs
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In a world whereall functions are continuous…
λf.λx.λy.f x yλ-terms
CPOs
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In a world whereall functions are continuous…
λf.λx.λy.f x yλ-terms
CPOs function with the propertyof being continuous
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In a world whereall functions are continuous…
Y(f) = f(Y(f))λ-terms
CPOs something thatonly exists in that world
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In a world where all functionssecretly something… are
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In a world where all functionssecretly something… do
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In a world where all functionssecretly something… do
get “code for free” / generic programs
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In a world where all functionssecretly something… do
get “code for free” / generic programs
can add new principles that depend on them
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Homotopy type theory
highercategory theory homotopy theory
dependent type theory
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In a world wheretypes are spaces
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In a world wheretypes are spaces
each type is a space, with points and paths
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programs are points
In a world wheretypes are spaces
each type is a space, with points and paths
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programs are points
points can be “literally the same” orconnected by a path
In a world wheretypes are spaces
each type is a space, with points and paths
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Many types are discrete (Nat)
0 1 2
3 4 5
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Many types are discrete (Nat)
0 1 2
3 4 52+2
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Many types are discrete (Nat)
0 1 2
3 4 52+26-2
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NM
α
Paths look like equality
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Nid
Mid : Path M M
α
Paths look like equalityreflexivity
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Nid
Mid : Path M M
α-1 : Path N M
αα-1
Paths look like equalityreflexivity
symmetry
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N
P
β
id
Mid : Path M M
α-1 : Path N M
β ◦ α : Path M P
αα-1
Paths look like equalityreflexivity
symmetry
transitivity
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N
P
β
M β ◦ α : Path M Pα
But are data
β ◦ α
γ
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N
P
β
M β ◦ α : Path M Pα
But are data
γγ : Path M P
β ◦ α
γ
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N
P
β
M β ◦ α : Path M Pα
But are data
γγ : Path M P
(β ◦ α) ≠ γβ ◦ α
γ
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N
P
β
M β ◦ α : Path M Pα
But are data
γγ : Path M P
(β ◦ α) ≠ γ
¬ PathPath M P (β ◦ α) γ
β ◦ α
γ
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N
P
β
M β ◦ α : Path M Pα
But are data
γγ : Path M P
(β ◦ α) ≠ γ
¬ PathPath M P (β ◦ α) γ
β ◦ α
γ
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N
P
β
M β ◦ α : Path M Pα
But are data
γγ : Path M P
(β ◦ α) ≠ γ
¬ PathPath M P (β ◦ α) γ
β ◦ α
γ
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Functions “secretly” act on paths
f
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Functions “secretly” act on paths
f
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Path x y →
Path f(x) f(y)
Functions “secretly” act on paths
f
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A
B
ua(f,g,…)A B
f
g
Voevodsky’s univalence axiom
bijections induce paths between types
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Monad interface (classic)[Godemont,Moggi,Wadler]
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Monad interface (classic)[Godemont,Moggi,Wadler]
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Applicative interface[McBride,Patterson]
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Applicative interface
effects influence value but not structure
[McBride,Patterson]
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Applicative interface
effects influence value but not structure
[McBride,Patterson]
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Monad interface (new)
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Monad interface (new)
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv) instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = return a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
pure a = return a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’)
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv) instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = return a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv) instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = return a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
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Instance of classic → instance of new
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Instance of new → instance of classic
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fg
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fg
h = f … g
extend interface with an operation that is determined by the others (convenience, efficiency)
fg
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fg
h = f … g
extend interface with an operation that is determined by the others (convenience, efficiency)
the (default implementation, forget)-bijectioncan be used to dynamically convert between them
fg
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fg
h = f … g
extend interface with an operation that is determined by the others (convenience, efficiency)
the (default implementation, forget)-bijectioncan be used to dynamically convert between them
it’s “obvious” how to apply this in context
fg
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fg
h = f … g
extend interface with an operation that is determined by the others (convenience, efficiency)
the (default implementation, forget)-bijectioncan be used to dynamically convert between them
it’s “obvious” how to apply this in context
partially evaluate to modify source code
fg
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Paths between types
Monad T
App⇒Monad T
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Paths between types
Monad T
App⇒Monad T
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Path-related types do not have same elements
A
B
p
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Path-related types do not have same elements
A
B
p
but paths between types induce bijections
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Path-related types do not have same elements
A
B
coe p : A # Bp
but paths between types induce bijections
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Path-related types do not have same elements
A
B
coe p : A # Bp
coe p-1 : B # A
but paths between types induce bijections
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Path-related types do not have same elements
A
B
coe p : A # Bp
coe p-1 : B # A
(mutually inverse)
but paths between types induce bijections
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Path-related types do not have same elements
A
B
coe p : A # Bp
coe p-1 : B # A
(mutually inverse)
moving along a path might do some work
but paths between types induce bijections
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Voevodsky’s univalence axiom
bijections induce paths between types*
Monad T
App⇒Monad T
Nat × String
String × Nat
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Coercing along univalence
A Bf
g
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A
B
ua(f,g,…)
Coercing along univalence
A Bf
g
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A
B
ua(f,g,…)
Coercing along univalence
A Bf
g
coe ua(f,g,…)
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A
B
ua(f,g,…)
Coercing along univalence
A Bf
g
coe ua(f,g,…)
coe ua(f,g,…)-1
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A
B
ua(f,g,…)
Coercing along univalence
A Bf
g
coe ua(f,g,…)
coe ua(f,g,…)-1
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A
B
ua(f,g,…)
Coercing along univalence
A Bf
g
coe ua(f,g,…)
coe ua(f,g,…)-1
A type B typeA # B type
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Type constructorsact on points
α : Path A A’ β : Path B B’α # β : Path (A # B) (A’ # B’)
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And “secretly” act on paths
α : Path A A’ β : Path B B’α # β : Path (A # B) (A’ # B’)
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coe (α # β) (h :A # B) = coe β ◦ h ◦ coe α-1
And “secretly” act on paths
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instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
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ua(d)
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C : ((Type # Type) # Type) # Type C Mon = {instance : Mon Maybe, sequenceM : ∀ {T A} {Mon T} # List (T A) # T(List A)}
ua(d)
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C : ((Type # Type) # Type) # Type C Mon = {instance : Mon Maybe, sequenceM : ∀ {T A} {Mon T} # List (T A) # T(List A)}
C
C[Monad]
C[App⇒Monad]
ua(d) C[ua(d)]
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C : ((Type # Type) # Type) # Type C Mon = {instance : Mon Maybe, sequenceM : ∀ {T A} {Mon T} # List (T A) # T(List A)}
C
C[Monad]
C[App⇒Monad]
ua(d) C[ua(d)]
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C : ((Type # Type) # Type) # Type C Mon = {instance : Mon Maybe, sequenceM : ∀ {T A} {Mon T} # List (T A) # T(List A)}
C
C[Monad]
C[App⇒Monad]
ua(d) C[ua(d)]
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C[Monad]
C[App⇒Monad]
coe C[ua(d)]
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
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C[Monad]
C[App⇒Monad]
coe C[ua(d)]
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = Some a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
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C[Monad]
C[App⇒Monad]
coe C[ua(d)]
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = Some a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
coe ua(d)
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C[Monad]
C[App⇒Monad]
coe C[ua(d)]
instance Monad Maybe where return a = Some a None >>= k = None (Some a) >>= k = k a sequenceM : ∀ {T A} {Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
instance App⇒Monad Maybe where return a = Some a None >>= k = None (Some x) >>= k = k x pure a = Some a f <*> a = f >>= λf’ # a >>= λ a’ # return (f’ a’) sequenceM : ∀ {T A} {App⇒Monad T} # List (T A) # T(List A) sequenceM [] = return [] sequenceM (x :: xs) = x >>= λ xv # (sequenceM xs) >>= λ xsv # return (xv :: xsv)
coe ua(d)-1
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In a world where all functionssecretly something… do
get “code for free” / generic programs
can add new principles that depend on them
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in a world where all functions act on paths,… and paths between types induce bijectionsyou can allow bijections to induce paths… and ∴ lift any bijection by a generic program
Univalence
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Π,Σ,+,Path,(co)inductives
Which types act on paths? Works for:
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intersection types A ∩ B
intensional type analysis case A of B × C ⇒ …
made explicit as × of predicates
can define non-univalentinductive codes for types
Π,Σ,+,Path,(co)inductives
Doesn’t work for:
Which types act on paths? Works for:
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Other sources of bijectionsList A ≃ Tree/{assoc,unit} A
List and Tree/{assoc,unit} implementations of ordered collections, if coercion of operations agree:treemap f = fromlist ◦ listmap f ◦ tolist (parametricity for graphs of bijections)
(Σ n:Nat.Vec A n) ≃ List A
Everywhere P xs ≃ (x : A) # x ∊ xs # P x
Lots more in libraries/formalizations
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Paths are data
β
α
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Paths are data
β
α
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Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
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Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
48
Cubical type theories
Bezem,Coquand,Huber; Cohen,Coquand,Huber,Mörtberg; Polonsky; Altenkirch,Kaposi; Isaev; Brunerie,Licata;
Angiuli,Harper,Wilson; Pitts,Orton
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Datatypes with paths
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Patches as PathsRepos:Type
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
doc[n]
points describe repository contents
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
a↔b@i
doc[n]
points describe repository contents
paths are patches
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
a↔b@i
doc[n]
points describe repository contents
paths are patches
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
a↔b@i
doc[n]
points describe repository contents
paths are patches
doc[n]
doc[n]
doc[n]
doc[n]
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
a↔b@i
doc[n]
points describe repository contents
paths are patches
doc[n]
doc[n]
doc[n]
doc[n]a↔b@i
c↔d@j
c↔d@j
a↔b@i
[Angiuli,Morehouse,L.,Harper]
50
Patches as PathsRepos:Type
a↔b@i
doc[n]
points describe repository contents
paths are patches
doc[n]
doc[n]
doc[n]
doc[n]a↔b@i
c↔d@j
c↔d@j
a↔b@ii≠jpaths between paths are
equations between patches
[Angiuli,Morehouse,L.,Harper]
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space Repos where doc[n:Nat] : Repos a↔b@i : Path doc[n] doc[n] commute : (i<n, j<n, i≠j) # Square (a↔b@i) (c↔d@j) (c↔d@j) (a↔b@i)
Higher inductive type
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A
Vec Char n
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A
Vec Char n
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A
Vec Char n
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A
Vec Char n
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interp : Repos # Type interp(doc[n]) = Vec Char n interp(a↔b@i) = ua(… actual swap code …) interp(commute) = … proof about above …
Interpreter
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highercategory theory homotopy theory
dependent type theory
55
In a world where all functionssecretly something… are
55
In a world where all functionssecretly something… do
