Internal Parametricity for Cubical Type Theory Evan Cavallo Robert - - PowerPoint PPT Presentation

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internal parametricity for cubical type theory
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Internal Parametricity for Cubical Type Theory Evan Cavallo Robert - - PowerPoint PPT Presentation

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Internal Parametricity for Cubical Type Theory Evan Cavallo Robert Harper Carnegie Mellon University CSL 2020 JAN 16 0 Mar n-Lf type theory Mar n-Lf 82: judgments explained by as speci fi ca ons on untyped programs ( A ,

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CSL 2020 · JAN 16

Internal Parametricity for Cubical Type Theory

Evan Cavallo Robert Harper

Carnegie Mellon University

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CSL 2020 · JAN 16 1

Marn-Löf type theory

Marn-Löf ’82: judgments explained by as specificaons on untyped programs (A, B, M, N) follow work of Angiuli, Favonia, & Harper ’18 on computaonal cubical type theory e.g. “M ∈ nat” is “M computes a natural number”

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CSL 2020 · JAN 16 2

Cubical type theories

coercion operaon ensures everything respects paths univalence: type paths are equivalences

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CSL 2020 · JAN 16 3

intervals might be given by:

De Morgan cubes

Cohen, Coquand, Huber, & Mörtberg 2015

Cartesian cubes

Angiuli, Favonia, & Harper 2018 Angiuli, Brunerie, Coquand, Favonia, Licata, & Harper 2018

structural

Substructural cubes

Bezem, Coquand, & Huber 2013&2017

no contracon (diagonals)

Cubical type theories

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CSL 2020 · JAN 16 4

Cartesian cubical type theory

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CSL 2020 · JAN 16 4

Cartesian cubical type theory

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CSL 2020 · JAN 16 5

Higher inducve types

combine inducve definions and quoents

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CSL 2020 · JAN 16 6

Higher inducve types

define higher-dimensional objects (synthec homotopy theory)

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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2-d

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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2-d

is the associator an isomorphism? 3-d

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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2-d

is the associator an isomorphism? 3-d

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Mac Lane's pentagon? 4-d

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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2-d

is the associator an isomorphism? 3-d

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Mac Lane's pentagon? 4-d

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(van Doorn 2018, Brunerie 2018)

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CSL 2020 · JAN 16 7

Higher inducve types

when combined, require >1-d reasoning e.g., smash product (HoTT Book §6.8)

associave?

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2-d

is the associator an isomorphism? 3-d

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Mac Lane's pentagon? 4-d

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(van Doorn 2018, Brunerie 2018)

can we avoid this complexity?

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CSL 2020 · JAN 16 8

Parametricity

“Parametric” funcons are uniform in type variables: Contrast with “ad-hoc” polymorphic funcons:

Restrict ourselves to write only parametric funcons Parametric funcons sasfy many properes “automacally”

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CSL 2020 · JAN 16 9

Reynolds’ abstracon theorem (’83)

Def: A family of (set-theorec) funcons is parametric when it acts on relaons. e.g., Abstracon theorem: the denotaon of any term in simply-typed λ-calculus (with ×, bool) is parametric. Key idea: λ-calculus has a relaonal interpretaon.

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CSL 2020 · JAN 16 10

Reynolds’ abstracon theorem (’83)

Def: A family of (set-theorec) funcons is parametric when it acts on relaons. e.g., “theorems for free” (Wadler ’89)

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CSL 2020 · JAN 16 11

Make relaonal interp. visible inside type theory

Internal parametricity (Bernardy & Moulin ’12)

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CSL 2020 · JAN 16 11

Make relaonal interp. visible inside type theory

Internal parametricity (Bernardy & Moulin ’12)

cubical type theory parametric type theory construcons act on isomorphisms construcons act on relaons equal over iso x.A related by rel x.A univalence: relavity:

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CSL 2020 · JAN 16 12

Parametric Cubical Type Theory

bridge dimensions path dimensions structural substructural

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CSL 2020 · JAN 16 12

Parametric Cubical Type Theory

bridge dimensions path dimensions structural substructural

Use parametricity to prove results about HITs Use good properes of cubical type theory to get beer results from / simplify internal parametricity Compare and contrast internal parametricity and cubical type theory

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CSL 2020 · JAN 16 13

Parametric Cubical Type Theory

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CSL 2020 · JAN 16 13

Parametric Cubical Type Theory

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CSL 2020 · JAN 16 13

Parametric Cubical Type Theory

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CSL 2020 · JAN 16 13

Parametric Cubical Type Theory

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CSL 2020 · JAN 16 13

Parametric Cubical Type Theory

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CSL 2020 · JAN 16 14

Internal parametricity: affine dimensions

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CSL 2020 · JAN 16 14

Internal parametricity: affine dimensions

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Internal parametricity: affine dimensions

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CSL 2020 · JAN 16 15

Internal parametricity: Bridge-types

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Internal parametricity: “relavity”

Forward: Backward:

▰ Parallels structural Glue/V

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CSL 2020 · JAN 16 17

Internal parametricity: funcon types

paths: funcon extensionality bridges: relaonal interpretaon (difference invisible for paths because of coercion)

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CSL 2020 · JAN 16 18

Internal parametricity: funcon types

Forward: Backward:

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CSL 2020 · JAN 16 18

Internal parametricity: funcon types

Forward: Backward:

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CSL 2020 · JAN 16 18

Internal parametricity: funcon types

Forward: Backward:

“case analysis for interval terms”

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CSL 2020 · JAN 16 19

Internal parametricity: funcon types

Stability of abstracon under substuon relies on absence of diagonals:

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CSL 2020 · JAN 16 19

Internal parametricity: funcon types

Stability of abstracon under substuon relies on absence of diagonals:

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CSL 2020 · JAN 16 19

Internal parametricity: funcon types

Stability of abstracon under substuon relies on absence of diagonals:

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CSL 2020 · JAN 16 19

Internal parametricity: funcon types

Stability of abstracon under substuon relies on absence of diagonals:

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CSL 2020 · JAN 16 20

Cubical equality for internal parametricity

Funcon extensionality & univalence

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CSL 2020 · JAN 16 21

Internal parametricity for cubical equality

Movang example: smash product

In the paper: any map

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is constant or the polymorphic identy. Implies any non-constant map

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is an isomorphism. Key: scales to characterize maps

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CSL 2020 · JAN 16 22

Conclusions

Combine internal parametricity & cubical type theory

Parametricity is especially useful for cubical type theory because it contains inducve types with complex algebraic properes

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As with ordinary type theory, using cubical equality smooths rough edges

▰ Push internal parametricity further

Bridge-discrete types for identy extension lemma

▰ Theories with interval variables

When are different kinds of intervals appropriate?

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