Computation paths, transport and the univalence axiom - EBL 2017 talk

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 Informática
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Encontro Brasileiro de Lógica (EBL)
Pirenópolis, GO
May 9, 2017
Arthur Ramos (joint work with Ruy de Queiroz and Anjolina de Oliveira)Computational paths, transport and the univalence axiom

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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)

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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)

Introduction
Computational Paths
Term Rewrite System
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)

Introduction
Computational Paths
Term Rewrite System
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))

Introduction
Computational Paths
Term Rewrite System
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)

Introduction
Computational Paths
Term Rewrite System
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)

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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

Introduction
Computational Paths
Term Rewrite System
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.

Computation paths, transport and the univalence axiom - EBL 2017 talk

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