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HoTTEST 2019 0
Internal Parametricity
and Cubical Type Theory
Evan Cavallo
& Robert Harper
Carnegie Mellon University
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.
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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.
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THIS TALK:
What is internal parametricity,
and how does it relate to
higher-dimensional type theory?
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Parametric polymorphism, intui�vely
Parametric func�ons are “uniform” in type variables:
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Parametric polymorphism, intui�vely
Parametric func�ons are “uniform” in type variables:
Compare with “ad-hoc” polymorphism:
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Reynolds’ abstrac�on theorem (1983)
DEF: A family of (set-theore�c) func�ons is
parametric when it preserves all rela�ons.
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Reynolds’ abstrac�on theorem (1983)
DEF: A family of (set-theore�c) func�ons is
parametric when it preserves all rela�ons. e.g.,
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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.
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Reynolds’ abstrac�on theorem (1983)
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Reynolds’ abstrac�on theorem (1983)
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Reynolds’ abstrac�on theorem (1983)
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Reynolds’ abstrac�on theorem (1983)
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Reynolds’ abstrac�on theorem (1983)
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Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
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Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
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Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
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Reynolds’ abstrac�on theorem (1983)
Key idea: λ-calculus has a rela�onal interpreta�on.
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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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Internal Parametricity (Bernardy et al)
Can we internalize the rela�onal interpreta�on in
dependent type theory?
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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?
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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?
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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?
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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?
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An interlude: cubical type theory
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An interlude: cubical type theory
category of dimension contexts could be:
faces, degeneracies, and permuta�ons [BCH]
+ diagonals [AFH, ABCFHL]
+ connec�ons [CCHM]
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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
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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
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Interlude: cubical type theory
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Interlude: cubical type theory
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Interlude: cubical type theory
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Interlude: cubical type theory
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Interlude: cubical type theory
Can we do the same for rela�ons?
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Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
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Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
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Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
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Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
Degeneracies:
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Internal Parametricity (Bernardy et al)
Context of bridge variables (colors)
Faces, degeneracies, and permuta�ons [BCH]
Faces:
Degeneracies:
Permuta�ons:
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Bridge types
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Bridge types
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Bridge types
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Bridge types
etc.
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Bridges in func�on types
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Bridges in func�on types
In cartesian cubical type theories:
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Bridges in func�on types
In cartesian cubical type theories:
In BCH, this violates freshness requirements!
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Bridges in func�on types
In cartesian cubical type theories:
In BCH, this violates freshness requirements!
Rela�onal interpreta�on of func�on types:
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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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Bridges in func�on types
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Bridges in func�on types
Forward:
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Bridges in func�on types
Forward:
Backward:
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Bridges in func�on types
Forward:
Backward:Backward:
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Bridges in func�on types
Forward:
Backward:Backward:
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Bridges in func�on types
Forward:
Backward:Backward:
“case analysis for dimension terms”
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Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
HoTTEST 2019 15
Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
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Bridges in func�on types
Stability of capture under subs�tu�on relies on
absence of diagonals:
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Bridges in the universe (“rela�vity”)
Want:
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Bridges in the universe (“rela�vity”)
Want:
Forward:
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Bridges in the universe (“rela�vity”)
Want:
Forward:
Backward:
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Bridges in the universe (“rela�vity”)
Want:
Forward:
Backward:
BCH G-types for rela�ons
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Bridges in the universe (“rela�vity”)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
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Bridges in the universe (“rela�vity”)
cartesian (Glue/V) substructural (Gel)
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Want:
Bridges in the universe (“rela�vity”)
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Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
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Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
Validated by interpreta�on in refined presheaves
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Want:
Bridges in the universe (“rela�vity”)
Bernardy, Coquand, & Moulin: add equali�es
Validated by interpreta�on in refined presheaves
Alterna�ve: use univalence?
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
HoTTEST 2019 19
Parametric cubical type theory (C & Harper)
Structural variables for paths,
substructural variables for bridges
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
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
HoTTEST 2019 20
Parametric cubical type theory (C & Harper)
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Parametric cubical type theory (C & Harper)
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Parametric cubical type theory (C & Harper)
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Parametric cubical type theory (C & Harper)
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Parametric cubical type theory (C & Harper)
(and the other inverse condi�on)
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Examples: Church booleans
What can we say about ?
HoTTEST 2019 21
Examples: Church booleans
What can we say about ?
Given define a rela�on.
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Examples: Church booleans
What can we say about ?
Given define a rela�on.
Apply at the Gel type for in a fresh .
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 .
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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.
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Examples: actual booleans
What are the bridges in ?
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Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
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Examples: actual booleans
What are the bridges in ?
Define the Gel type corresponding to paths in .
Map from to by case analysis
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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
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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
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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)
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Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
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Examples: bridge-discrete types
We always have a map from (homogeneous) paths to
bridges
is bridge-discrete when this is an equivalence
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
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
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
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
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Examples: excluded middle
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Examples: excluded middle
Consider the weak excluded middle:
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Examples: excluded middle
Consider the weak excluded middle:
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Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
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Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
Thus, .
HoTTEST 2019 24
Examples: excluded middle
Consider the weak excluded middle:
Any func�on must be constant,
because is bridge-discrete.
Thus, .
Corollary: , where
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)
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Examples: suspension
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Examples: suspension
What are the terms ?
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Examples: suspension
What are the terms ?
Completely determined by .
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Examples: suspension
What are the terms ?
Completely determined by .
Key Lemma:
HoTTEST 2019 25
Examples: suspension
What are the terms ?
Completely determined by .
Key Lemma:
(case of graph lemma in ordinary parametricity)
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Future work
HoTTEST 2019 26
Future work
Connec�on to ordinary cubical type theory
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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, +??
-
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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?
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
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
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
