internal parametricity and cubical type theory€¦ · hottest 2019 23 examples: bridge-discrete...
TRANSCRIPT
![Page 1: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/1.jpg)
HoTTEST 2019 0
Internal Parametricity
and Cubical Type Theory
Evan Cavallo
& Robert Harper
Carnegie Mellon University
![Page 2: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/2.jpg)
HoTTEST 2019 1
Internal Parametricity
Bernardy & Moulin.
A Computa�onal Interpreta�on of Parametricity. 2012.
Type Theory in Color. 2013.
Bernardy, Coquand, & Moulin.
A Presheaf Model of Parametric Type Theory. 2015.
Nuyts, Vezzosi, & Devriese.
Parametric Quan�fiers for Dependent Type Theory. 2017.
C & Harper. [arXiv:1901.00489]
Parametric Cubical Type Theory. 2019.
![Page 3: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/3.jpg)
HoTTEST 2019 1
Internal Parametricity
Bernardy & Moulin.
A Computa�onal Interpreta�on of Parametricity. 2012.
Type Theory in Color. 2013.
Bernardy, Coquand, & Moulin.
A Presheaf Model of Parametric Type Theory. 2015.
Nuyts, Vezzosi, & Devriese.
Parametric Quan�fiers for Dependent Type Theory. 2017.
C & Harper. [arXiv:1901.00489]
Parametric Cubical Type Theory. 2019.
![Page 4: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/4.jpg)
HoTTEST 2019 2
THIS TALK:
What is internal parametricity,
and how does it relate to
higher-dimensional type theory?
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HoTTEST 2019 3
Parametric polymorphism, intui�vely
Parametric func�ons are “uniform” in type variables:
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HoTTEST 2019 3
Parametric polymorphism, intui�vely
Parametric func�ons are “uniform” in type variables:
Compare with “ad-hoc” polymorphism:
![Page 7: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/7.jpg)
HoTTEST 2019 4
Reynolds’ abstrac�on theorem (1983)
DEF: A family of (set-theore�c) func�ons is
parametric when it preserves all rela�ons.
![Page 8: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/8.jpg)
HoTTEST 2019 4
Reynolds’ abstrac�on theorem (1983)
DEF: A family of (set-theore�c) func�ons is
parametric when it preserves all rela�ons. e.g.,
![Page 9: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/9.jpg)
HoTTEST 2019 4
Reynolds’ abstrac�on theorem (1983)
DEF: A family of (set-theore�c) func�ons is
parametric when it preserves all rela�ons. e.g.,
Abstrac�on theorem: the denota�on of any term in
simply-typed λ-calculus (with ×, bool) is parametric.
![Page 10: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/10.jpg)
HoTTEST 2019 5
Reynolds’ abstrac�on theorem (1983)
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HoTTEST 2019 5
Reynolds’ abstrac�on theorem (1983)
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HoTTEST 2019 6
Reynolds’ abstrac�on theorem (1983)
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HoTTEST 2019 6
Reynolds’ abstrac�on theorem (1983)
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HoTTEST 2019 6
Reynolds’ abstrac�on theorem (1983)
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HoTTEST 2019 7
Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
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HoTTEST 2019 7
Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
![Page 17: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/17.jpg)
HoTTEST 2019 7
Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
![Page 18: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/18.jpg)
HoTTEST 2019 7
Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
![Page 19: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/19.jpg)
HoTTEST 2019 7
Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
Abstrac�on theorem extends interpreta�on to terms
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HoTTEST 2019 8
Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
![Page 21: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/21.jpg)
HoTTEST 2019 8
Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
i.e., can we give an opera�onal account of
parametricity?
![Page 22: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/22.jpg)
HoTTEST 2019 8
Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
i.e., can we give an opera�onal account of
parametricity?
![Page 23: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/23.jpg)
HoTTEST 2019 8
Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
i.e., can we give an opera�onal account of
parametricity?
![Page 24: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/24.jpg)
HoTTEST 2019 8
Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
i.e., can we give an opera�onal account of
parametricity?
![Page 25: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/25.jpg)
HoTTEST 2019 9
An interlude: cubical type theory
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HoTTEST 2019 9
An interlude: cubical type theory
category of dimension contexts could be:
faces, degeneracies, and permuta�ons [BCH]
+ diagonals [AFH, ABCFHL]
+ connec�ons [CCHM]
![Page 27: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/27.jpg)
HoTTEST 2019 9
An interlude: cubical type theory
category of dimension contexts could be:
faces, degeneracies, and permuta�ons [BCH]
+ diagonals [AFH, ABCFHL]
+ connec�ons [CCHM]
respect for equality ensured by Kan opera�ons
![Page 28: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/28.jpg)
HoTTEST 2019 9
An interlude: cubical type theory
category of dimension contexts could be:
faces, degeneracies, and permuta�ons [BCH]
+ diagonals [AFH, ABCFHL]
+ connec�ons [CCHM]
respect for equality ensured by Kan opera�ons
univalence via G / Glue / V types
![Page 29: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/29.jpg)
HoTTEST 2019 10
Interlude: cubical type theory
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HoTTEST 2019 10
Interlude: cubical type theory
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HoTTEST 2019 10
Interlude: cubical type theory
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HoTTEST 2019 10
Interlude: cubical type theory
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HoTTEST 2019 10
Interlude: cubical type theory
Can we do the same for rela�ons?
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HoTTEST 2019 11
Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
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HoTTEST 2019 11
Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
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HoTTEST 2019 11
Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
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HoTTEST 2019 11
Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
Degeneracies:
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HoTTEST 2019 11
Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
Degeneracies:
Permuta�ons:
![Page 39: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/39.jpg)
HoTTEST 2019 12
Bridge types
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HoTTEST 2019 12
Bridge types
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HoTTEST 2019 12
Bridge types
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HoTTEST 2019 12
Bridge types
etc.
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HoTTEST 2019 13
Bridges in func�on types
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HoTTEST 2019 13
Bridges in func�on types
In cartesian cubical type theories:
![Page 45: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/45.jpg)
HoTTEST 2019 13
Bridges in func�on types
In cartesian cubical type theories:
In BCH, this violates freshness requirements!
![Page 46: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/46.jpg)
HoTTEST 2019 13
Bridges in func�on types
In cartesian cubical type theories:
In BCH, this violates freshness requirements!
Rela�onal interpreta�on of func�on types:
![Page 47: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/47.jpg)
HoTTEST 2019 13
Bridges in func�on types
In cartesian cubical type theories:
In BCH, this violates freshness requirements!
Rela�onal interpreta�on of func�on types:
equivalent in the presence of J
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HoTTEST 2019 14
Bridges in func�on types
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HoTTEST 2019 14
Bridges in func�on types
Forward:
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HoTTEST 2019 14
Bridges in func�on types
Forward:
Backward:
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HoTTEST 2019 14
Bridges in func�on types
Forward:
Backward:Backward:
![Page 52: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/52.jpg)
HoTTEST 2019 14
Bridges in func�on types
Forward:
Backward:Backward:
![Page 53: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/53.jpg)
HoTTEST 2019 14
Bridges in func�on types
Forward:
Backward:Backward:
“case analysis for dimension terms”
![Page 54: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/54.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 55: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/55.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 56: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/56.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 57: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/57.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 58: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/58.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 59: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/59.jpg)
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
![Page 60: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/60.jpg)
HoTTEST 2019 16
Bridges in the universe (“rela�vity”)
Want:
![Page 61: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/61.jpg)
HoTTEST 2019 16
Bridges in the universe (“rela�vity”)
Want:
Forward:
![Page 62: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/62.jpg)
HoTTEST 2019 16
Bridges in the universe (“rela�vity”)
Want:
Forward:
Backward:
![Page 63: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/63.jpg)
HoTTEST 2019 16
Bridges in the universe (“rela�vity”)
Want:
Forward:
Backward:
BCH G-types for rela�ons
![Page 64: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/64.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
![Page 65: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/65.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
![Page 66: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/66.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
![Page 67: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/67.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
![Page 68: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/68.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
![Page 69: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/69.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
![Page 70: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/70.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
![Page 71: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/71.jpg)
HoTTEST 2019 17
Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
![Page 72: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/72.jpg)
HoTTEST 2019 18
Want:
Bridges in the universe (“rela�vity”)
![Page 73: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/73.jpg)
HoTTEST 2019 18
Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
![Page 74: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/74.jpg)
HoTTEST 2019 18
Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
Validated by interpreta�on in refined presheaves
![Page 75: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/75.jpg)
HoTTEST 2019 18
Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
Validated by interpreta�on in refined presheaves
Alterna�ve: use univalence?
![Page 76: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/76.jpg)
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
![Page 77: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/77.jpg)
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
Structural variables for paths,
substructural variables for bridges
![Page 78: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/78.jpg)
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
Structural variables for paths,
substructural variables for bridges
Extend Kan opera�ons to make Bridge types Kan
![Page 79: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/79.jpg)
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
Structural variables for paths,
substructural variables for bridges
Extend Kan opera�ons to make Bridge types Kan
w/ computa�onal seman�cs
![Page 80: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/80.jpg)
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
![Page 81: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/81.jpg)
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
![Page 82: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/82.jpg)
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
![Page 83: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/83.jpg)
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
![Page 84: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/84.jpg)
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
(and the other inverse condi�on)
![Page 85: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/85.jpg)
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
![Page 86: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/86.jpg)
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
Given define a rela�on.
![Page 87: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/87.jpg)
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
Given define a rela�on.
Apply at the Gel type for in a fresh .
![Page 88: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/88.jpg)
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
Given define a rela�on.
Apply at the Gel type for in a fresh .
![Page 89: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/89.jpg)
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
Given define a rela�on.
Apply at the Gel type for in a fresh .
Extract a witness.
![Page 90: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/90.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
![Page 91: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/91.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
![Page 92: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/92.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
Map from to by case analysis
![Page 93: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/93.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
Map from to by case analysisMap from to by case analysis
![Page 94: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/94.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
Map from to by case analysisMap from to by case analysis
![Page 95: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/95.jpg)
HoTTEST 2019 22
Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
Map from to by case analysisMap from to by case analysis
Prove this is an equivalence! (iterated parametricity)
![Page 96: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/96.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
![Page 97: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/97.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
![Page 98: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/98.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
Plays the role of the iden�ty extension lemma
![Page 99: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/99.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
Plays the role of the iden�ty extension lemma
e.g., any func�on from to a bridge-discrete type is
constant
![Page 100: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/100.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
Plays the role of the iden�ty extension lemma
e.g., any func�on from to a bridge-discrete type is
constant
The sub-universe of bridge-discrete types is closed
under all type formers except , including
![Page 101: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/101.jpg)
HoTTEST 2019 23
Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
Plays the role of the iden�ty extension lemma
e.g., any func�on from to a bridge-discrete type is
constant
The sub-universe of bridge-discrete types is closed
under all type formers except , including
THM: is rela�vis�c
![Page 102: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/102.jpg)
HoTTEST 2019 24
Examples: excluded middle
![Page 103: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/103.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
![Page 104: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/104.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
![Page 105: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/105.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
![Page 106: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/106.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
Thus, .
![Page 107: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/107.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
Thus, .
Corollary: , where
![Page 108: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/108.jpg)
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
Thus, .
Corollary: , where
(see also: Booij, Escardó, Lumsdaine, & Shulman,
Parametricity, automorphisms of the universe, and excluded middle)
![Page 109: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/109.jpg)
HoTTEST 2019 25
Examples: suspension
![Page 110: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/110.jpg)
HoTTEST 2019 25
Examples: suspension
What are the terms ?
![Page 111: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/111.jpg)
HoTTEST 2019 25
Examples: suspension
What are the terms ?
Completely determined by .
![Page 112: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/112.jpg)
HoTTEST 2019 25
Examples: suspension
What are the terms ?
Completely determined by .
Key Lemma:
![Page 113: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/113.jpg)
HoTTEST 2019 25
Examples: suspension
What are the terms ?
Completely determined by .
Key Lemma:
(case of graph lemma in ordinary parametricity)
![Page 114: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/114.jpg)
HoTTEST 2019 26
Future work
![Page 115: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/115.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
![Page 116: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/116.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
Seman�c: for in cubical type theory
without func�on types, a proof with bridges
gives an element in cubical sets, +??
-
![Page 117: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/117.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
Seman�c: for in cubical type theory
without func�on types, a proof with bridges
gives an element in cubical sets, +??
-
- Syntac�c?
![Page 118: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/118.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
Seman�c: for in cubical type theory
without func�on types, a proof with bridges
gives an element in cubical sets, +??
-
- Syntac�c?
Proving algebraic proper�es of HITs
![Page 119: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/119.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
Seman�c: for in cubical type theory
without func�on types, a proof with bridges
gives an element in cubical sets, +??
-
- Syntac�c?
Proving algebraic proper�es of HITs
?
e.g., what are the terms of type
![Page 120: Internal Parametricity and Cubical Type Theory€¦ · HoTTEST 2019 23 Examples: bridge-discrete types We always have a map from (homogeneous) paths to bridges is bridge-discrete](https://reader034.vdocument.in/reader034/viewer/2022042921/5f6bdce5175baa429459e683/html5/thumbnails/120.jpg)
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
Seman�c: for in cubical type theory
without func�on types, a proof with bridges
gives an element in cubical sets, +??
-
- Syntac�c?
Proving algebraic proper�es of HITs
?
e.g., what are the terms of type
conjecture: must be constant or iden�ty
Use to prove pentagon, hexagon, etc