Computations, Paths, Types and Proofs

What is a Proof of an Equality? The Functional Interpretation of Propositional Equality Normal form for equality proofs
Computations, Paths, Types and Proofs
Ruy de Queiroz
(joint work with Anjolina de Oliveira)
Centro de Informática
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
IV Encontro de Teoria da Computaç ão (ETC 2019)
XXXIX CSBC, Belém–PA, 15–16 Julho 2019
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Informática Universidade Federal de Pernambuco (UFPE) Recife, Brazil

Building Bridges! Opening (new) Paths!
Computation– (Algebraic) Topology–Logic–(Higher) Categories–(Higher) Algebra
A single concept may serve as a bridging bond: path
Computation: convertibility between λ-terms
(Algebraic) Topology: homotopy theory
Logic: proofs of equality
(Higher) Categories: polycategories
(Higher) Algebra: ∞-groupoids
Paths as structure-preserving maps.
Path equivalences: homotopy theory

Equality in λ-Calculus: convertibility
Proofs of equality as (reversible) sequences of contractions, i.e. paths
Church’s (1936) original λ-calculus paper:
NB: equality as the reﬂexive, symmetric and transitive closure of
1-step contraction: symmetric closure of rewriting paths.

Equality in λ-Calculus: composition of definitional
contractions
Proofs of equality: paths
Definition (Hindley & Seldin 2008)
P is βη-equal or βη-convertible to Q (notation P =βη Q) iff Q is
obtained from P by a finite (perhaps empty) series of
β-contractions, η-contractions, reversed β-contractions,
reversed η-contractions, or changes of bound variables. That is,
P =β Q iff there exist P0, . . . , Pn (n ≥ 0) such that
P0 ≡ P, Pn ≡ Q,
(∀i ≤ n − 1) (Pi 1β Pi+1 or Pi+1 1β Pi
or Pi 1η Pi+1 or Pi+1 1η Pi
or Pi ≡α Pi+1).

Curry-Howard Interpretation: Intuitionistic Type Theory
‘Formulae-as-Types’, ‘Proofs-as-Programs’
(1934) H. Curry came up with an early version of type inference
for the combinators of Combinatory Logic. The types of
combinators could be seen as axioms of implicational logic:
‘α → β’ could be read as
(1) ‘the type of functions from type α to type β’ ;
(2) ‘the formula “α implies β” ’.
Axioms of Implicational Logic:
α → α
α → β → α
(α → β → γ) → (α → β) → α → γ
Types of Combinators:
I : α → α
K : α → β → α
S : (α → β → γ) → (α → β) → α → γ

Curry-Howard Interpretation: Intuitionistic Type Theory
‘Formulae-as-Types’, ‘Proofs-as-Programs’
(1969) W. Howard came up with an extension of Curry’s
functionality interpretation to full intuitionistic predicate logic:
⊥ as ∅ (empty type)
α ∧ β as α × β (product)
α ∨ β as α + β (sum)
∀xγ as Πxγ (dependent product)
∃xγ as Σxγ (dependent sum)
(1972) P. Martin-L¨of came up with Type Theory extending
Howard’s Formulae-as-Types with Natural Numbers and
Universes.
(1973) P. Martin-L¨of came up with Intuitionistic Type Theory
extending Type Theory with Identity Types.

Type Theory and Derivations-as-Terms
Howard on the so-called Curry-Howard ‘Formulae-as-Types’
“ [de Bruijn] discovered the idea of derivations as
terms, and the accompanying idea of
formulae-as-types, on his own. (...)
Martin-L¨of suggested that the derivations-as-terms
idea would work particularly well in connection with
Prawitz’s theory of natural deduction.”
(W.Howard, Wadler’s Blog, 2014)
1-step contraction originating from logic: contractions in
redundant proofs correspond to contractions in redundant
terms.
Curry-Howard ‘Formulae-as-Types’:
Proof equivalence ←→ Term equivalence

Type Theory
Hausdorff Trimester: Types, Sets and Constructions, May 2 – Aug 24, 2018, Unn Bonn
“Type theory, originally conceived as a bulwark against the paradoxes
of naive set theory, has languished for a long time in the shadow of
axiomatic set theory which became the mainstream foundation of
mathematics. The first renaissance of type theory occurred with the
advent of computer science and Bishop’s development of a
practice-oriented constructive mathematics. It was followed by a
second quite recent one that not only champions type theory as a
central framework for achieving the goal of fully formalized
mathematics amenable to verification by computer-based proof
assistants, but also finds deep and unexpected connections between
type theory and homotopy theory. ”

Type Theory
Hausdorff Trimester: Types, Sets and Constructions, May 2 – Aug 24, 2018, Unn Bonn
“Constructive set theory and mathematics distinguishes itself from its
traditional counterpart, classical set theory and mathematics based
on it, by insisting that proofs of existential theorems must afford
means for constructing an instance. Constructive reasoning emerges
naturally in core areas of mathematics and in the theory of
computation. The aim of the Hausdorff Trimester is to create a forum
for research on and dissemination of exciting recent developments,
which are of central importance to modern foundations of
mathematics.”

Homotopy Type Theory
Open Source Book
Institute for Advanced Study, Princeton
approx. 600p.
Open-source book: The Univalent Foundations Program
27 main participants. 58 contributors
Available on GitHub. Latest version June 26, 2019

Open Source Book
“Homotopy type theory is a new branch of mathematics that
combines aspects of several different ﬁelds in a surprising way. It is
based on a recently discovered connection between homotopy theory
and type theory. Homotopy theory is an outgrowth of algebraic
topology and homological algebra, with relationships to higher
category theory; while type theory is a branch of mathematical logic
and theoretical computer science. Although the connections between
the two are currently the focus of intense investigation, it is
increasingly clear that they are just the beginning of a subject that will
take more time and more hard work to fully understand. It touches on
topics as seemingly distant as the homotopy groups of spheres, the
algorithms for type checking, and the deﬁnition of weak ∞-groupoids.”

Origins of Research Programme
Vladimir Voevodsky (IAS, Princeton) (b. 4 June 1966 – d. 30
September 2017)
1st
(?) use of term ‘homotopy λ-calculus’: tech report Notes on
homotopy λ-calculus (Started Jan 18, Feb 11, 2006): “In this paper
we suggest a new approach to the foundations of mathematics. (...)
A key development (totally unnoticed by the mathematical
community) occurred in the 70-ies when the typed λ-calculus
was enriched by the concept of dependent types.”
Steve Awodey (Dept Phil, CMU)
1st
(?) use of term ‘homotopy type theory’: Eighty-sixth Peripatetic
Seminar on Sheaves and Logic, Nancy, 8–9 September 2007

Henk Barendregt on the Evolution of Type Theory
“Type theory as coming originally from Whitehead-Russell and
simplified and essentially extended by Ramsey (simplifying), Church
(adding lambda terms), de Bruijn (adding dependent types), Scott
(adding inductive types with recursion), Girard (adding higher order
types), Martin-Löf (showing the natural position and power of
intuitionism) all lead to proof-checking based on type theory with
successes like the full formalization of the 4CT and the
Feit-Thompson theorem by Gonthier and collaborators and the
forthcoming one of the Kepler conjecture by Hales and collaborators.
Now, there are some difficulties with types (...). For this reason there
is work in progress by Voevodsky and collaborators to modify this
theory.”
(Barendregt, 21/02/2014, FOM list)

HoTT Approach to Spaces: New Proofs
“Homotopy type theory: towards Grothendieck’s dream”, M. Shulman, 2013 Int.
Category Theory conference, Sydney
“Progress in synthetic homotopy theory:
• π1(S1
) = Z (Shulman, Licata)
• πk (Sn
) = 0 for k < n (Brunerie, Licata)
• πn(Sn
) = Z (Licata, Brunerie)
• The long exact sequence of a ﬁbration (Voevodsky)
• The Hopf ﬁbration and π3(S2
) = Z (Lumsdaine, Brunerie)
• The Freudenthal suspension theorem (Lumsdaine)
• The Blakers–Massey theorem (Lumsdaine, Finster, Licata)
• The van Kampen theorem (Shulman)
• Whitehead’s theorem for n-types (Licata)
• Covering space theory (Hou) ”
“Homotopy type theory can also serve as a foundational system
for mathematics whose basic objects are ∞-groupoids rather
than sets.”

Algebraic Topology and Fundamental Groups
Calculation of Fundamental Groups
“One of the main ideas of algebraic topology is to consider two
spaces to be equivalent if they have ‘the same shape’ in a sense that
is much broader than homeomorphism.” (...)
“The fundamental group of a space X will be defined so that its
elements are loops in X starting and ending at a fixed basepoint
x0 ∈ X, but two such loops are regarded as determining the same
element of the fundamental group if one loop can be continuously
deformed to the other within the space X”. (...)
“One can often show that two spaces are not homeomorphic by
showing that their fundamental groups are not isomorphic, since it will
be an easy consequence of the definition of the fundamental group
that homeomorphic spaces have isomorphic fundamental groups.”
Algebraic Topology, Allan Hatcher, Cornell Univ, 2001

Computational Paths: New Proofs
Calculation of Fundamental Groups
Calculation of Fundamental Groups of Surfaces:
circle
cylinder
M¨obius band
torus
two-holed torus
real projective plane
Van Kampen theorem
T. M. L. de Veras, A. F. Ramos, R. J. G. B. de Queiroz, A. G. de
Oliveira, On the Calculation of Fundamental Groups in
Homotopy Type Theory by Means of Computational Paths,
arXiv 1804.01413

Geometry and Logic
Alexander Grothendieck
Alexander Grothendieck
b. 28 March 1928, Berlin, Prussia, Germany
d. 13 November 2014 (aged 86), Saint-Girons, Ari`ege, France

Geometry and Logic
Alexander Grothendieck: The Homotopy Hypothesis
“... the study of n-truncated homotopy types (of
semisimplicial sets, or of topological spaces) [should
be] essentially equivalent to the study of so-called
n-groupoids. . . . This is expected to be achieved by
associating to any space (say) X its “fundamental
n-groupoid” Πn(X).... The obvious idea is that
0-objects of Πn(X) should be the points of X,
1-objects should be “homotopies” or paths between
points, 2-objects should be homotopies between
1-objects, etc. ”
(Grothendieck, “Pursuing Stacks” (1983))
homotopy types ←→ ∞-groupoids

∞-Groupoids and polycategories
Voevodsky’s attempt to realize Grothendieck’s dream
“It is known that CW-complexes X such that πi(X) = 0
for i ≥ 2 can be described by groupoids from the
homotopy point of view. In the unpublished paper
“Pursuing stacks” Grothendieck proposed the idea of a
multi-dimensional generalization of this connection
that used polycategories. The present note is devoted
to the realization of this idea.”
(“∞-Groupoids as a model for a homotopy category”, V A
Voevodskii and M M Kapranov, Communications of the Moscow
Mathematical Society, 45:239–240, 1990)

From n-CPO’s to ∞-CPOs
Dana Scott’s D∞ and typed vs untyped λ-Calculus
“This particular paper has, of course, and odd historical
role: in it [October 1969] the author argues against the
type-free calculi of Church and Curry, Kleene and Rosser,
and their later uses by B¨ohm and Strachey. But then in
November of 1969, after writing this report, the author
himself saw that the method of monotone continuous
functions (...) could be applied to posets other than just
those generated from the integers (with bottom) by the very
simple type constructors.”
(“A type-theoretical alternative to ISWIN, CUCH, OWHY”, 1993)

Higher-Order Functions, Complete Partial Orders
Dana Scott’s D∞ and typed vs untyped λ-Calculus
“The process of “completing” spaces is a very general one,
and the full implication of the method are not yet clear. A
second example of the idea concerns function spaces. Let
D be given, and set D0 = D and Dn+1 = (Dn → Dn). The
spaces Dn are (a selection of) the “higher-type” spaces of
functions of functions of functions ... . It turns out there is a
way of naturally embedding each Dn successfully into the
next space Dn+1. These embeddings make it possible to
pass to a limit space D∞ which contains the originally given
D and is such that D∞
∼= (D∞ → D∞).” (“Outline of a
mathematical theory of computation”, 1970)
(D∞ and ∞-groupoids: An extensional λ-model with ∞–grupoid
structure. Daniel Martinez Rivillas, R.J.G.B. de Q.
arXiv:1906.05729 )Ruy de Queiroz (joint work with Anjolina de Oliveira)

Geometry and Logic
Vladimir Voevodsky
“From an observation by Grothendieck:
formalism of higher equivalences (theory of grupoids)
=
homotopy theory (theory of shapes up to a
deformation)
combined with some other ideas leads to an encoding of
mathematics in terms of the homotopy theory. Unlike the usual
encodings in terms of set theory this one respects
equivalences.”

Type Theory and Homotopy Theory
Steve Awodey: a calculus to reason about abstract homotopy
“Homotopy type theory is a new field devoted to a recently discovered
connection between Logic and Topology – more specifically, between
constructive type theory, which was originally invented as a
constructive foundation for mathematics and now has many
applications in the theory of programming languages and formal proof
verification, and homotopy theory, a branch of algebraic topology
devoted to the study of continuous deformations of geometric spaces
and mappings. The basis of homotopy type theory is an interpretation
of the system of intensional type theory into abstract homotopy
theory. As a result of this interpretation, one can construct new kinds
of models of constructive logic and study that system semantically,
e.g. proving consistency and independence results. Conversely,
constructive type theory can also be used as a formal calculus to
reason about abstract homotopy.”
(A proposition is the (homotopy) type of its proofs, Jan 2016.)

Algebraic Structure: Groupoids
Steve Awodey
“A groupoid is like a group, but with a partially-deﬁned
composition operation. Precisely, a groupoid can be deﬁned as
a category in which every arrow has an inverse. A group is thus
a groupoid with only one object. Groupoids arise in topology as
generalized fundamental groups, not tied to a choice of
basepoint.”
(Type Theory and Homotopy, 2010.)
“A groupoid is a generalized group, with the multiplication being
only a partial operation – or equivalently, a category in which
every arrow has an inverse.”
(Univalence as a Principle of Logic, 2016.)

Sets in the next dimension
Vladimir Voevodsky: Groupoids vs Categories
“The successes of category theory inspired the idea that
categories are ‘sets in the next dimension’ and that the
foundation of mathematics should be based on category
theory or on its higher-dimensional analogues. (...)
The greatest roadblock for me was the idea that categories
are ‘sets in the next dimension’. I clearly recall the feeling
of a breakthrough that I experienced when I understood
that this idea is wrong. Categories are not ‘sets in the next
dimension’. They are ‘partially ordered sets in the next
dimension’ and ‘sets in the next dimension’ are
groupoids.”
(V. Voevodsky, “The Origins and Motivations of Univalent
Foundations”, The Institute Letter Summer 2014, Princeton)

The Functional Interpretation of Equality
Sequences of contractions
(λx.(λy.yx)(λw.zw))v 1η (λx.(λy.yx)z)v 1β (λy.yv)z 1β zv
(λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1η (λx.zx)v 1β zv
(λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1β (λw.zw)v 1η zv
There is at least one sequence of contractions from the initial term to
the ﬁnal term. Thus, in the formal theory of λ-calculus, the term
(λx.(λy.yx)(λw.zw))v is declared to be equal to zv.
Now, some natural questions arise:
1 Are the sequences themselves normal?
2 What are the non-normal sequences?
3 How are the latter to be identiﬁed and (possibly) normalised?

Brouwer–Heyting–Kolmogorov Interpretation
Proofs rather than truth-values
a proof of the proposition: is given by:
A ∧ B a proof of A and
a proof of B
A ∨ B a proof of A or
a proof of B
A → B a function that turns a proof of A
into a proof of B
∀x.P(x) a function that turns an element a
into a proof of P(a)
∃x.P(x) an element a (witness)
and a proof of P(a)

What is a proof of an equality statement?
t1 = t2 ?
(Perhaps a path from t1 to t2?)
What is the logical status of the symbol “=”?
What would be a canonical/direct proof of t1 = t2?
What is an equality between paths?
What is an equality between homotopies (i.e., paths between
paths)?

Type Theory and Equality
Proposition vs Judgements
In type theory, two main kinds of judgements:
1 x : A
2 x = y : A
Via the so-called Curry-Howard interpretation, “x : A” can be read as
“x is a proof of proposition A”.
Also, “x = y : A” can be read as “x and y are (deﬁnitionally) equal
proofs of proposition A”.
What about the judgement of “p is a proof of the statement that x and
y are equal elements of type A”? This is where the so-called Identity
type comes into the picture:
p : IdA(x, y)

Type Theory and Equality
Explicit Terms for Paths
Those paths are not part of the syntax of type theory. This is
clear from an answer given by Vladimir Voevodsky for the
following question in a short interview (22 Oct 2015):
- Martin Escardò: What was your first reaction when
you first saw the type of identity? Did you immediately
connect with path spaces?
- Vladimir Voevodsky: Not at all. I did not make this
connection until late 2009. All the time before it I
was hypnotized by the mantra that the only
inhabitant of the Id type is reflexivity which made
it useless from my point of view.

What is the formal counterpart of a proof of an equality?
In talking about proofs of an equality statement, two dichotomies
arise:
1 deﬁnitional equality versus propositional equality
2 intensional equality versus extensional equality
First step on the formalisation of proofs of equality statements: Per
Martin-L¨of’s Intuitionistic Type Theory (Log Coll ’73, published 1975)
with the so-called Identity Type

Identity Types
Identity Types - Topological and Categorical Structure
Workshop, Uppsala, November 13–14, 2006:
“The identity type, the type of proof objects for the fundamental
propositional equality, is one of the most intriguing constructions of
intensional dependent type theory (also known as Martin-L¨of type
theory). Its complexity became apparent with the Hofmann–Streicher
groupoid model of type theory. This model also hinted at some
possible connections between type theory and homotopy theory and
higher categories. Exploration of this connection is intended to be the
main theme of the workshop.”
Michael Shulman’s (2017) ‘Homotopy type theory: the logic of
space’: “For many years, the most mysterious part of Martin-L¨of’s
type theory was the identity types “x = y”.”

Identity Types
Type Theory and Homotopy Theory
The groupoid structure exposed in the Hofmann–Streicher (1994)
countermodel to the principle of Uniqueness of Identity Proofs (UIP).
In Hofmann & Streicher’s own words,
“We give a model of intensional Martin-L¨of type theory
based on groupoids and ﬁbrations of groupoids in which
identity types may contain two distinct elements which are
not even propositionally equal. This shows that the principle
of uniqueness of identity proofs is not derivable in the
syntax”.
(M Hofmann, T Streicher, “The groupoid model refutes uniqueness of
identity proofs”. In Logic in Computer Science, 1994 (LICS’94), pp.
208–212.)

Identity Types
Identity Types as Topological Spaces
“All of this work can be seen as an elaboration of the
following basic idea: that in Martin-L¨of type theory, a type A
is analogous to a topological space; elements a, b ∈ A to
points of that space; and elements of an identity type
p, q ∈ IdA(a, b) to paths or homotopies p, q : a → b in A.”.
(B. van den Berg and R. Garner, “Topological and simplicial models of
identity types”, ACM Transactions on Computational Logic, Jan 2012)

Identity Types
Identity Types as Topological Spaces
From the Homotopy type theory collective book (2013):
“In type theory, for every type A there is a (formerly
somewhat mysterious) type IdA of identiﬁcations of two
objects of A; in homotopy type theory, this is just the path
space AI
of all continuous maps I → A from the unit
interval. In this way, a term p : IdA(a, b) represents a path
p : a b in A.”

Identity Types: Iteration
From Propositional to Predicate Logic and Beyond
In the same aforementioned workshop, B. van den Berg in his
contribution “Types as weak omega-categories” draws attention to the
power of the identity type in the iterating types to form a globular set:
“Fix a type X in a context Γ. Define a globular set as follows:
A0 consists of the terms of type X in context Γ,modulo
definitional equality; A1 consists of terms of the types
Id(X; p; q) (in context Γ) for elements p, q in A0, modulo
definitional equality; A2 consists of terms of well-formed
types Id(Id(X; p; q); r; s) (in context Γ) for elements p, q in
A0, r, s in A1, modulo definitional equality; etcetera...”

The homotopy interpretation
Here is how we can see the connections between proofs of equality
and homotopies:
a, b : A
p, q : IdA(a, b)
α, β : IdIdA(a,b)(p, q)
· · · : IdIdId...
(· · · )
Now, consider the following interpretation:
Types Spaces
Terms Maps
a : A Points a : 1 → A
p : IdA(a, b) Paths p : a ⇒ b
α : IdIdA(a,b)(p, q) Homotopies α : p q

The homotopy interpretation (Awodey (2016))
point, path, homotopy, ...

Identity Types
Univalent Foundations of Mathematics
From Vladimir Voevodsky (IAS, Princeton) “Univalent
Foundations: New Foundations of Mathematics”, Mar 26, 2014:
“There were two main problems with the existing
foundational systems which made them inadequate.
Firstly, existing foundations of mathematics were
based on the languages of Predicate Logic and
languages of this class are too limited.
Secondly, existing foundations could not be used to
directly express statements about such objects as, for
example, the ones that my work on 2-theories was
about.”

The Homotopy Interpretation
Steve Awodey (2016): a “logic of homotopies”
“The homotopy interpretation was ﬁrst proposed by the
present author and worked out formally (with a student) in
terms of Quillen model categories – a modern, axiomatic
setting for abstract homotopy theory that encompasses not
only the classical homotopy theory of spaces and their
combinatorial models like simplicial sets, but also other,
more exotic notions of homotopy (...). These results show
that intensional type theory can in a certain sense be
regarded as a “logic of homotopy”, in that the system can
be faithfully represented homotopically, and then used to
reason formally about spaces, continuous maps,
homotopies, and so on. ”
(A proposition is the (homotopy) type of its proofs, Jan 2016.)

Propositional Equality
Proofs of equality as (rewriting) computational paths
What is a proof of an equality statement? In what sense it can be
seen as a homotopy? Motivated by looking at equalities in type
theory as arising from the existence of computational paths between
two formal objects, it may be useful to review the role and the power
of the notion of propositional equality as formalised in the so-called
Curry–Howard functional interpretation.
The main idea, namely, proofs of equality statements as (reversible)
sequences of rewrites, i.e. paths, goes back to a paper entitled
“Equality in labelled deductive systems and the functional
interpretation of propositional equality”, presented in Dec 1993 at the
9th Amsterdam Colloquium, and published in the proceedings in
1994.

BHK for Identity Types
Types and Propositions
(source: Awodey (2016))
types vs propositions:
sum/coproduct vs disjunction,
product vs conjunction,
function space vs implication
dependent sum vs existential quantiﬁer,
dependent product vs universal quantiﬁer
path space (?) vs equality symbol

t1 = t2 ?
(Perhaps a sequence of rewrites
starting from t1 and ending in t2?)
What is the logical status of the symbol “=”?
What would be a canonical/direct proof of t1 = t2?

The Functional Interpretation of Direct Computations
Propositional Equality: The Groupoid Laws
With the formulation of propositional equality that we have just
deﬁned, we can also prove that all elements of an identity type
obey the groupoid laws, namely
1 Associativity
2 Existence of an identity element
3 Existence of inverses
Also, the groupoid operation, i.e. composition of
paths/sequences, is actually, partial, meaning that not all
elements will be connected via a path. (The groupoid
interpretation refutes the Uniqueness of Identity Proofs.)

Propositional Equality: The Uniqueness of Identity Proofs
“We will call UIP (Uniqueness of Identity Proofs) the
following property. If a1, a2 are objects of type A then for
any proofs p and q of the proposition “a1 equals a2” there is
another proof establishing equality of p and q. (...) Notice
that in traditional logical formalism a principle like UIP
cannot even be sensibly expressed as proofs cannot
be referred to by terms of the object language and thus
are not within the scope of propositional equality.”
Martin Hofmann and Thomas Streicher, “The groupoid
interpretation of type theory”, Twenty-ﬁve years of constructive
type theory (Venice, 1995), Oxford Logic Guides, vol. 36,
Oxford Univ. Press, New York, 1998, pp. 83–111.

Sequences of contractions
(λx.(λy.yx)(λw.zw))v 1η (λx.(λy.yx)z)v 1β (λy.yv)z 1β zv
(λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1η (λx.zx)v 1β zv
(λx.(λy.yx)(λw.zw))v 1β (λx.(λw.zw)x)v 1β (λw.zw)v 1η zv
There is at least one sequence of contractions from the initial term to
the ﬁnal term. (In this case we have given three!) Thus, in the formal
theory of λ-calculus, the term (λx.(λy.yx)(λw.zw))v is declared to be
equal to zv.
Now, some natural questions arise:
1 Are the sequences themselves normal?
2 Are there non-normal sequences?
3 If yes, how are the latter to be identiﬁed and (possibly)
normalised?
4 What happens if general rules of equality are involved?

Propositional equality
P is β-equal or β-convertible to Q (notation P =β Q) iff Q is
obtained from P by a ﬁnite (perhaps empty) series of
β-contractions and reversed β-contractions and changes of
bound variables. That is, P =β Q iff there exist P0, . . . , Pn
(n ≥ 0) such that
P0 ≡ P, Pn ≡ Q,
(∀i ≤ n − 1)(Pi 1β Pi+1 or Pi+1 1β Pi or Pi ≡α Pi+1).
NB: equality with an existential force.
NB: equality as the reﬂexive, symmetric and transitive closure
of 1-step contraction: arising from rewriting

The Functional Interpretation of Direct Computation
Equality: Existential Force and Rewriting Path
The same happens with λβη-equality:
Deﬁnition 7.5 (λβη-equality) (Hindley & Seldin 2008)
The equality-relation determined by the theory λβη is
called =βη; that is, we deﬁne
M =βη N ⇔ λβη M = N.
Note again that two terms are λβη-equal if there exists a proof
of their equality in the theory of λβη-equality.

Gentzen’s ND for propositional equality
Remark
In setting up a set of Gentzen’s ND-style rules for equality we
need to account for:
1 definitional versus propositional equality;
2 there may be more than one normal proof of a certain
equality statement;
3 given a (possibly non-normal) proof, the process of
bringing it to a normal form should be finite and confluent.

Equality in Type Theory
Martin-Löf’s Intuitionistic Type Theory:
Intensional (1975)
Extensional (1982(?), 1984)
Remark (Definitional vs. Propositional Equality)
definitional, i.e. those equalities that are given as rewrite
rules, orelse originate from general functional principles
(e.g. β, η, ξ, µ, ν, etc.);
propositional, i.e. the equalities that are supported (or
otherwise) by an evidence (a sequence of substitutions
and/or rewrites)

Deﬁnitional Equality
(α) λx.M = λy.[y/x]M (y /∈ FV(M))
(β) (λx.M)N = [N/x]M
(η) (λx.Mx) = M (x /∈ FV(M))
(ξ)
M = M
λx.M = λx.M
(µ)
M = M
NM = NM
(ν)
M = M
MN = M N
(ρ) M = M
(σ)
M = N
N = M
(τ)
M = N N = P
M = P

Intuitionistic Type Theory
→-intro
[x : A]
f(x) = g(x) : B
λx.f(x) = λx.g(x) : A → B
(ξ)
→-elim
x = y : A g : A → B
gx = gy : B
(µ)
→-elim
x : A g = h : A → B
gx = hx : B
(ν)
→-reduc
a : A
[x : A]
b(x) : B
(λx.b(x))a = b(a/x) : B
(β)
c : A → B
λx.cx = c : A → B
(η)

Lessons from Curry–Howard and Type Theory
Harmonious combination of logic and λ-calculus;
Proof terms as ‘record of deduction steps’, i.e.
‘deductions-as-terms’
Function symbols as ﬁrst class citizens.
Cp.
∃xF(x)
[F(t)]
C
C
with
p : ∃xF(x)
[t : D, g(t) : F(t)]
h(g, t) : C
? : C
in the term ‘?’ the variable g gets abstracted from, and this enforces a
kind of generality to g, even if this is not brought to the ‘logical’ level.

Equality in Martin-L¨of’s Intensional Type Theory
A type a : A b : A
Idint
A (a, b) type
Idint
-formation
a : A
r(a) : Idint
A (a, a)
Idint
-introduction
a = b : A
r(a) : Idint
A (a, b)
Idint
-introduction
a : A b : A c : Idint
A (a, b)
[x:A]
d(x):C(x,x,r(x))
[x:A,y:A,z:Idint
A (x,y)]
C(x,y,z) type
J(c, d) : C(a, b, c)
Idint
-elimination
a : A
[x : A]
d(x) : C(x, x, r(x))
[x : A, y : A, z : Idint
A (x, y)]
C(x, y, z) type
J(r(a), d(x)) = d(a/x) : C(a, a, r(a))
Idint
-equality

Equality in Martin-L¨of’s Extensional Type Theory
A type a : A b : A
Idext
A (a, b) type
Idext
-formation
a = b : A
r : Idext
A (a, b)
Idext
-introduction
c : Idext
A (a, b)
a = b : A
Idext
-elimination
c : Idext
A (a, b)
c = r : Idext
A (a, b)
Idext
-equality

The missing entity
Considering the lessons learned from Type Theory, the
judgement of the form:
a = b : A
which says that a and b are equal elements from domain D, let
us add a function symbol:
a =s b : A
where one is to read: a is equal to b because of ‘s’ (‘s’ being the
rewrite reason); ‘s’ is a term denoting a sequence of equality
identiﬁers (β, η, ξ, etc.), i.e. a composition of rewrites. In other
words, ‘s’ is the (explicit) computational path from a to b.
(This formal entity is missing in both of Martin-L¨of’s
formulations of Identity Types.)

HoTT Book
Path terms are not in the syntax, thus: Encode-Decode Method
“To characterize a path space, the first step is to
define a comparison fibration “code” that provides a
more explicit description of the paths.”
(...)
“There are several different methods for proving that
such a comparison fibration is equivalent to the paths
(we show a few different proofs of the same result in
§8.1). The one we have used here is called the
encode-decode method.”

Propositional Equality
Id-introduction
a =s b : A
s(a, b) : IdA(a, b)
Id-elimination
m : IdA(a, b)
[a =g b : A]
h(g) : C
J(m, λg.h(g)) : C
Id-reduction
a =s b : A
s(a, b) : IdA(a, b)
Id-intr
[a =g b : A]
h(g) : C
J(s(a, b), λg.h(g)) : C
Id-elim
β
[a =s b : A]
h(s/g) : C

Propositional Equality: A Simple Example of a Proof
By way of example, let us prove
ΠxA
ΠyA
(IdA(x, y) → IdA(y, x))
[p : IdA(x, y)]
[x =t y : A]
y =σ(t) x : A
(σ(t))(y, x) : IdA(y, x)
J(p, λt(σ(t))(y, x)) : IdA(y, x)
λp.J(p, λt(σ(t))(y, x)) : IdA(x, y) → IdA(y, x)
λy.λp.J(p, λt(σ(t))(y, x)) : ΠyA(IdA(x, y) → IdA(y, x))
λx.λy.λp.J(p, λt(σ(t))(y, x)) : ΠxAΠyA(IdA(x, y) → IdA(y, x))

Normal form for the rewrite reasons
Strategy:
Analyse possibilities of redundancy
Construct a rewriting system
Prove termination and conﬂuence

Definition (equation)
An equation in our LNDEQ is of the form:
s =r t : A
where s and t are terms, r is the identifier for the rewrite reason, and
A is the type (formula).
Definition (system of equations)
A system of equations S is a set of equations:
{s1 =r1
t1 : A1, . . . , sn =rn
tn : An}
where ri is the rewrite reason identifier for the ith equation in S.

Deﬁnition (rewrite reason)
Given a system of equations S and an equation s =r t : A, if
S s =r t : A, i.e. there is a deduction/computation of the
equation starting from the equations in S, then the rewrite
reason r is built up from:
(i) the constants for rewrite reasons: { ρ, σ, τ, β, η, ν, ξ, µ };
(ii) the ri’s;
using the substitution operations:
(iii) subL;
(iv) subR;
and the operations for building new rewrite reasons:
(v) σ, τ, ξ, µ.

Definition (general rules of equality)
The general rules for equality (reflexivity, symmetry and
transitivity) are defined as follows:
x : A
x =ρ x : A
(reflexivity)
x =t y : A
y =σ(t) x : A
(symmetry)
x =t y : A y =u z : A
x =τ(t,u) z : A
(transitivity)

Deﬁnition (subterm substitution)
The rule of “subterm substitution” is split into two rules:
x =r C[y] : A y =s u : A
x =subL(r,s) C[u] : A
x =r w : A C[w] =s u : A
C[x] =subR(r,s) u : A
where C[x] is the context in which the subterm x appears

Reductions
Deﬁnition (reductions involving ρ and σ)
x =ρ x : A
x =σ(ρ) x : A
sr x =ρ x : A
x =r y : A
y =σ(r) x : A
x =σ(σ(r)) y : A
ss x =r y : A
Associated rewritings:
σ(ρ) sr ρ
σ(σ(r)) ss r

Reductions
Deﬁnition (reductions involving τ)
x=r y:D y=σ(r)x:D
x=τ(r,σ(r))x:D tr x =ρ x : D
y=σ(r)x:D x=r y:D
y=τ(σ(r),r)y:D tsr y =ρ y : D
u=r v:D v=ρv:D
u=τ(r,ρ)v:D rrr u =r v : D
u=ρu:D u=r v:D
u=τ(ρ,r)v:D lrr u =r v : D
Associated equations: τ(r, σ(r)) tr ρ, τ(σ(r), r) tsr ρ, τ(r, ρ) rrr r,
τ(ρ, r) lrr r.

Reductions
Deﬁnition
βrewr -×-reduction
x =r y : A z : B
x, z =ξ1(r) y, z : A × B
× -intr
FST( x, z ) =µ1(ξ1(r)) FST( y, z ) : A
× -elim
mx2l1 x =r y : A
Associated rewriting:
µ1(ξ1(r)) mx2l1 r

Reductions
Deﬁnition
βrewr -×-reduction
x =r x : A y =s z : B
x, y =ξ∧(r,s) x , z : A × B
× -intr
FST( x, y ) =µ1(ξ∧(r,s)) FST( x , z ) : A
× -elim
mx2l2 x =r x : A
Associated rewriting:
µ1(ξ∧(r, s)) mx2l2 r

Categorical Interpretation of Computational Paths
Computational Paths form a Weak Category
Theorem
For each type A, computational paths induce a weak
categorical structure Arw where:
objects: terms a of the type A, i.e., a : A
morphisms: a morphism (arrow) between terms a : A and
b : A are arrows s : a → b such that s is a computational
path between the terms, i.e., a =s b : A.
Corollary
Arw has a weak groupoidal structure.

Publications
Recent publications:
1 R. J. G. B. de Queiroz, A. G. de Oliveira and A. F. Ramos.
Propositional equality, identity types, and direct computational
types. Special issue of South American Journal of Formal Logic
(ISSN: 2446-6719) entitled “Logic and Applications: in honor to
Francisco Miraglia by the occasion of his 70th birthday”, M.
Coniglio & H. L. Mariano (eds.), 2(2):245–296, December 2016.
2 T. L. M. de Veras, A. F. Ramos, R. J. G. B. de Queiroz, A. G. de
Oliveira. An alternative approach to the calculation of
fundamental groups based on labeled natural deduction.
arXiv:1906.09107
Oliveira. A Topological Application of Labelled Natural
Deduction. arXiv:1906.09105

Publications
Recent publications (cont’d):
Oliveira. On the Calculation of Fundamental Groups in
Homotopy Type Theory by Means of Computational Paths.
arXiv:1804.01413
2 A. F. Ramos, R. J. G. B. de Queiroz, A. G. de Oliveira. On The
Identity Type as The Type of Computational Paths. EBL’14
special issue of Logic Journal of the IGPL, Oxford Univ Press,
Published online 26 June 2017.
3 A. F. Ramos, R. J. G. B. de Queiroz, A. G. de Oliveira. On the
Groupoid Model of Computational Paths. arXiv:1506.02721

Computations, Paths, Types and Proofs

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