Higher Induc ve Types in Computa onal Cubical Type Theory Evan - - PowerPoint PPT Presentation

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Higher Induc ve Types in Computa onal Cubical Type Theory Evan Cavallo & Robert Harper Carnegie Mellon University POPL 2019 0 cubical type theory dependent type theory with a univalent, proof-relevant internal equality POPL 2019

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POPL 2019

Higher Inducve Types in Computaonal Cubical Type Theory

Evan Cavallo & Robert Harper

Carnegie Mellon University

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POPL 2019 1

dependent type theory with a univalent, proof-relevant internal equality

cubical type theory

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POPL 2019 1

dependent type theory with a univalent, proof-relevant internal equality

cubical type theory indexed higher inducve types

■ quoent types for this equality ■ indexed inducve types that respect it

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

" "

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

" "

isomorphism ⇒ equal types

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

" "

isomorphism ⇒ equal types

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POPL 2019 2

dependent type theory with a univalent, proof-relevant internal equality

higher type theory:

[Awodey & Warren; Voevodsky]

" "

isomorphism ⇒ equal types (axiomazed by homotopy type theory)

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POPL 2019 3

cubical type theory:

computaonal higher type theory via dimension variables

[Cohen, Coquand, Huber & Mörtberg; Angiuli, Favonia & Harper]

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POPL 2019 3

cubical type theory:

computaonal higher type theory via dimension variables

[Cohen, Coquand, Huber & Mörtberg; Angiuli, Favonia & Harper]

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POPL 2019 3

cubical type theory:

computaonal higher type theory via dimension variables

[Cohen, Coquand, Huber & Mörtberg; Angiuli, Favonia & Harper]

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POPL 2019 4

cubical type theory:

path types

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POPL 2019 4

cubical type theory:

path types

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POPL 2019 5

cubical type theory:

coercion

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POPL 2019 5

cubical type theory:

coercion

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POPL 2019 5

cubical type theory:

coercion

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POPL 2019 6

cubical type theory:

composion

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POPL 2019 6

cubical type theory:

composion

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POPL 2019 6

cubical type theory:

composion

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POPL 2019 6

cubical type theory:

composion

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POPL 2019 6

cubical type theory:

composion

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POPL 2019 6

cubical type theory:

general case:

composion

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POPL 2019 7

cubical type theory: + univalence

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POPL 2019 8

higher inducve types

quoents for proof-relevant equality

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POPL 2019 8

higher inducve types

quoents for proof-relevant equality

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POPL 2019 8

higher inducve types

quoents for proof-relevant equality

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POPL 2019 8

higher inducve types

quoents for proof-relevant equality

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POPL 2019 8

higher inducve types

quoents for proof-relevant equality

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POPL 2019 9

higher inducve types

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POPL 2019 10

higher inducve types

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POPL 2019 10

higher inducve types

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POPL 2019 10

higher inducve types

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POPL 2019 10

higher inducve types

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POPL 2019 10

higher inducve types

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POPL 2019 10

higher inducve types

etc.

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POPL 2019 11

higher inducve types

in generality

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POPL 2019 11

higher inducve types

in generality

■ axiomac type theory:

  • Sojakova: W-quoents
  • Basold, Geuvers, & van der Weide; Dybjer &

Moeneclaey; Kaposi & Kovács ■ semancs:

  • Dybjer & Moeneclaey
  • Lumsdaine & Shulman: cell monads

■ cubical type theory:

  • Coquand, Huber, & Mörtberg: examples, schema sketch
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POPL 2019 11

higher inducve types

in generality

■ axiomac type theory:

  • Sojakova: W-quoents
  • Basold, Geuvers, & van der Weide; Dybjer &

Moeneclaey; Kaposi & Kovács ■ semancs:

  • Dybjer & Moeneclaey
  • Lumsdaine & Shulman: cell monads

■ cubical type theory:

  • Coquand, Huber, & Mörtberg: examples, schema sketch
  • ur contribuon:

cubical schema with computaonal semancs

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POPL 2019 11

higher inducve types

in generality

■ axiomac type theory:

  • Sojakova: W-quoents
  • Basold, Geuvers, & van der Weide; Dybjer &

Moeneclaey; Kaposi & Kovács ■ semancs:

  • Dybjer & Moeneclaey
  • Lumsdaine & Shulman: cell monads

■ cubical type theory:

  • Coquand, Huber, & Mörtberg: examples, schema sketch
  • ur contribuon:

cubical schema with computaonal semancs

(including indexed inducve types)

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POPL 2019 12

  • 1. schema
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POPL 2019 12

  • 1. schema

eliminaon principle

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? can we implement coercion and composion?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? can we implement coercion and composion?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? can we implement coercion and composion?

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POPL 2019 13

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? can we implement coercion and composion?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon? can we implement coercion and composion? and eliminaon?

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon? can we implement coercion and composion? and eliminaon?

＊ more complicated in general case ＊

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon? can we implement coercion and composion? and eliminaon?

＊ more complicated in general case ＊

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POPL 2019 14

  • 2. semancs

what are the values of an inducve type? can we implement coercion and composion? and eliminaon? can we implement coercion and composion? and eliminaon?

＊ more complicated in general case ＊

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POPL 2019 15

indexed inducve types

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POPL 2019 15

indexed inducve types

identy type (subject of HoTT axioms)

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POPL 2019 15

indexed inducve types

identy type (subject of HoTT axioms)

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POPL 2019 15

indexed inducve types

identy type (subject of HoTT axioms)

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POPL 2019 15

indexed inducve types

identy type (subject of HoTT axioms)

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POPL 2019 15

indexed inducve types

identy type (subject of HoTT axioms)✓

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POPL 2019 16

all in all

■ schema for indexed higher inducve types

  • torus, higher truncaons, localizaons, etc.
  • identy types

■ computaonal semancs

  • PERs on untyped operaonal semancs
  • canonicity theorem

■ fragment implemented in redtt proof assistant

github.com/RedPRL/redtt

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POPL 2019 17

thank you!

Angiuli, Hou (Favonia), & Harper. Cartesian cubical computaonal type theory: Construcve bbreasoning with paths and equalies. CSL 2018. Awodey & Warren. Homotopy theorec models of identy types. Mathemacal Proceedings of bbthe Cambridge Philosophical Society, 2009. Basold, Geuvers, & van der Weide. Higher Inducve Types in Programming. J. UCS, 2017. Cohen, Coquand, Huber, & Mörtberg. Cubical type theory: A construcve interpretaon of the bbunivalence axiom. TYPES 2015. Coquand, Huber, & Mörtberg. On higher inducve types in cubical type theory. LICS 2018. Dybjer & Moeneclaey. Finitary Higher Inducve Types in the Groupoid Model. MFPS 2017. Kaposi & Kovács. A syntax for higher inducve-inducve types. FSCD 2018. Lumsdaine & Shulman. Semancs of higher inducve types. arXiv 1705.07088, 2017.

  • Sojakova. Higher inducve types as homotopy-inial algebras. POPL 2015.
  • Swan. An algebraic weak factorisaon system on 01-substuon sets: a construcve proof. J.

bbLogic & Analysis, 2016.