Computation paths, transport and the univalence axiom - EBL 2017 talk
1. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Computational paths, transport and the univalence
axiom
Arthur Ramos
(joint work with Ruy de Queiroz and Anjolina de Oliveira)
Centro de Inform´atica
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Encontro Brasileiro de L´ogica (EBL)
Piren´opolis, GO
May 9, 2017
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
2. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Table of Contents
1 Introduction
2 Computational Paths
Paths Construction
Identity Type
3 Term Rewrite System
4 Homotopy Type Theory
Transport
Univalence
5 Conclusion and Current Work
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
3. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Introduction
Homotopy Type Theory: Type Theory + Homotopy Theory
Voevodsky 2005: Univalence Axiom
Identity Type: Paths between points
De queiroz and De Oliveira (since 1990’s): Paths as entities of
Type Theory
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
4. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Paths Construction
Identity Type
Basic Paths
(α) λx.M = λy.M[y/x] if y /∈ FV (M)
(β) (λx.M)N = M[N/x]
(ρ) M = M
(η) (λx.Mx) = M (x /∈ FV (M))
M = M(µ)
NM = NM
M = N N = P(τ)
M = P
M = M(ν)
MN = M N
M = N(σ)
N = M
M = M(ξ)
λx.M = λx.M
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
5. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Paths Construction
Identity Type
Construction Example
Path between (λy.yx)(λw.zw) and zx
(λy.yx)(λw.zw) η (λy.yx)z : η((λy.yx)(λw.zw), λw.zw)
(λy.yx)z β zx : β((λy.yx)z, zx)
Concatenation: application of τ
τ((η((λy.yx)(λw.zw), λw.zw), β((λy.yx)z, zx))
Notation: (λy.yx)(λw.zw) =τ(η,β) zx
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
6. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Paths Construction
Identity Type
Identity Type and Computational Paths
Introduction rule:
a =s b : A
Id − I
s(a, b) : IdA(a, b)
Elimination rule:
m : IdA(a, b)
[a =g b : A]
h(g) : C
Id − E
REWR(m, ´g.h(g)) : C
Reduction rule:
REWR(m, ´g.h(g)) : C β h(m/g) : C
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
7. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Paths Construction
Identity Type
Example: Construction of Symmetry
[a : A] [b : A]
[p(a, b) : IdA(a, b)]
[a =t b : A]
b =σ(t) a : A
Id − I
(σ(t))(b, a) : IdA(b, a)
Id − E
REWR(p(a, b), ´t.(σ(t))(b, a)) : IdA(b, a)
→ −I
λp.REWR(p(a, b), ´t.(σ(t))(b, a)) : IdA(a, b) → IdA(b, a)
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
8. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Term Rewrite System: LNDEQ − TRS
From a =t b : A, we have b =σ(t) a : A and a =σ(σ(t)) b : A.
t = σ(σ(t))?
Term Rewrite System proposed by De Queiroz and De
Oliveira (1994)
LNDEQ − TRS: total of 39 rewrite rules
Termination Property and Confluence
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
9. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Reductions
x =r y : A y =σ(r) x : A
tr x =ρ x : A
x =τ(r,σ(r)) x : A
y =σ(r) x : A x =r y : A
tsr y =ρ y : A
y =τ(σ(r),r) y : A
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
10. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
In previous works (In print: EBL2014 special issue):
Category Theory: Groupoid Model for LNDEQ − TRS.
Globular Structure and Higher Groupoids
Proof of uniqueness of identity proofs using computational
paths.
Lacking formalization of Homotopy Type Theory concepts
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
11. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Transport Definition
Transport
In type theory, given any term of the identity type p : IdA(x, y) and
a type family P over A, one can prove that it is possible to derive a
function p∗ : P(x) → P(y). We call this function transport.
Quantifier-less substitution
We obtain the same result using computational paths using an
inference rule introduced by De Queiroz and De Oliveira (2014). It
is the ’quantifier-less’ substitution:
x =p y : A f (x) : P(x)
p(x, y) ◦ f (x) : P(y)
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
12. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Dependent Function and Transportation
f : Π(x:A)P(x) and x =p y. If we apply µ directly, we obtain
f (x) = f (y), but f (x) : P(x) and f (y) : P(y).
Transport to the rescue!
x =p y : A f : Π(x:A)P(x)
p(x, y) ◦ f (x) =µf (p) f (y) : P(y)
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
13. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Transport Lemmas
Lemma 1
For any P(x) ≡ B, x =p y : A and b : B, there is a path
transportP(p, b) = b.
Lemma 2
For any f : A → B and x =p y : A, we have:
µ(p)(p∗(f (x)), f (y)) = τ(transportconstB
p , µf (p))(p∗(f (x)), f (y))
Lemma 3
For any x =p y : A and y =q z : A, f (x) : P(x), we have
q∗(p∗(f (x))) = (p ◦ q)∗(f (x))
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
14. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Proof of Lemma 3
Proof.
We obtain the same result after developing both sides of the
equality:
q∗(p∗(f (x))) =µ(p) q∗(f (y)) =µ(q) f (z)
(p ◦ q)∗(f (x)) =µ(p◦q) f (z)
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
15. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Univalence Axiom
Univalence Axiom
For any types A, B, we have:
(A = B) (A B)
Lemma 4
For any types A and B, the following function exists:
idtoeqv : (A = B) → (A B)
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
16. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Proof of Lemma 4
Define idtoeqv as p∗ : A → B
p∗ is an equivalence: For any path p, we can form a path
σ(p) and thus, we have (σ(p))∗ : B → A. We need to show
(σ(p)))∗ is a quasi-inverse of p∗:
1 p∗((σ(p)∗(b)) = b
2 (σ(p))∗(p∗(a)) = a
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
17. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Transport
Univalence
Proof of (σ(p)))∗ as Quasi-inverse
Applications of Lemma 3:
1 p∗((σ(p)∗(b)) = (σ(p) ◦ p)∗(b) = τ(p, σ(p))∗(b) =tr
ρ∗(b) =µ(p) b.
2 (σ(p))∗(p∗(a)) = (p ◦ σ(p))∗(a) = τ(σ(p), p)∗(a) =tsr
ρ∗(a) =µ(p) a
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom
18. Introduction
Computational Paths
Term Rewrite System
Homotopy Type Theory
Conclusion and Current Work
Conclusion and Future Work
Identity Type as Computational paths: groupoid interpretation
and uniquiness of identity proofs
Formalization of Homotopy Type Theory Concepts
Paper in progress: Explicit Computational Paths’ (Ramos, De
Queiroz and De Oliveira)
Explicit Computational Paths: formalization of transport,
univalence, cartesian products, coproducts, unit and identity
type, homotopies, function extensionality, natural numbers,
sets and axiom K.
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom