Propositional Equality and Identity Types

Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Propositional Equality and Identity Types
Ruy de Queiroz
(joint work with Anjolina de Oliveira)
Univ. Federal de Pernambuco, Recife, Brazil
Seminar at Logic Group, TU Darmstadt
19 Aug 2013
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil

Content
1 Identity Types in Type Theory
2 Heyting’s explanation via proofs
3 Direct Computations
4 The Functional Interpretation of Propositional Equality
5 Normal form for equality proofs

What is a proof of an equality statement?
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.”

Identity Types
Type Theory and Homotopy Theory
Indeed, a whole new research avenue has since 2005 been explored
by people like Vladimir Voevodsky and Steve Awodey in trying to
make a bridge between type theory and homotopy theory, mainly via
the groupoid structure exposed in the Hofmann–Streicher (1994)
countermodel to the principle of Uniqueness of Identity Proofs (UIP).
This has open the way to, in Awodey’s words,
“a new and surprising connection between Geometry,
Algebra, and Logic, which has recently come to light in the
form of an interpretation of the constructive type theory of
Per Martin-L¨of into homotopy theory, resulting in new
examples of certain algebraic structures which are
important in topology”. (“Type Theory and Homotopy”,
preprint, 2010.)

Identity Types
Identity Types as Topological Spaces
According to B. van den Berg and R. Garner (“Topological and
simplicial models of identity types”, ACM Transactions on
Computational Logic, Jan 2012),
“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.”.

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...”

Identity Types: Iteration
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

Propositional Equality
Proofs of equality as (rewriting) computational paths
Motivated by looking at equalities in type theory as arising from the
existence of computational paths between two formal objects, our
purpose here is to offer a different perspective on the role and the
power of the notion of propositional equality as formalised in the
so-called Curry–Howard functional interpretation. The main idea
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.

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
∀xD
.P(x) a function that turns an element a
into a proof of P(a)
∃xD
.P(x) an element a (witness)
and a proof of P(a)

Brouwer–Heyting–Kolmogorov Interpretation: Formally
Canonical proofs rather than truth-values
a proof of the proposition: has the canonical form of:
A ∧ B p, q where p is a proof of A and
q is a proof of B
A ∨ B inl(p) where p is a proof of A or
inr(q) where q is a proof of B
(‘inl’ and ‘inr’ abbreviate
‘into the left/right disjunct’)
A → B λx.b(x) where b(p) is a proof of B
provided p is a proof of A
∀xD
.P(x) Λx.f(x) where f(a) is a proof of P(a)
provided a is an arbitrary individual chosen
from the domain D
∃xD
.P(x) εx.(f(x), a) where a is a witness
from the domain D, f(a) is a proof of P(a)

Brouwer–Heyting–Kolmogorov Interpretation
What is a proof of an equality statement?
a proof of the proposition: is given by:
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?

Statman’s Direct Computations
Terms, Equations, Measure
Definition (equations and systems of equations)
Let us consider equations E between individual terms
a, b, c, . . ., possibly containing function variables, and finite sets
of equations S.
Definition (measure)
A function M from terms to non-negative integers is called a
measure if M(a) ≤ M(b) implies M(c[a/x]) ≤ M(c[b/x]), and,
whenever x occurs in c, M(a) ≤ M(c[a/x]).

Kreisel–Tait’s calculus K
Deﬁnition (calculus K)
The calculus K of Kreisel and Tait consists of the axioms a = a and
the rule of substituting equals for equals:
(1)
E[a/x] a
.
= b
E[b/x]
where a
.
= b is, ambiguously, a = b and b = a, together with the rules
(2)
sa = sb
a = b
(3)
0 = sa
b = c
(4)
a = sn
a
b = c
H will be the system consisting only of the rule (1)

Computations, Direct Computations
Deﬁnition (computation)
Computations T in K or H are binary trees of equation
occurrences built up from assumptions and axioms according
to the rules.
Deﬁnition (direct computation)
If M is a measure, we say that a computation T of E from S is
M-direct if for each term b occurring in T there is a term c
occurring in E or S with M(b) ≤ M(c).

The Functional Interpretation of Propositional Equality
Curry–Howard and Labelled Deduction
Our aims:
Using Gentzen’s methods for equality reasoning
(Statman’77)
Addressing the dichotomy: deﬁnitional vs. propositional
Giving an alternative to the ‘Equality/Identity Type’ of
Martin-L¨of’s Type Theory

The Functional Interpretation of Equality
Sequences of conversions
(λx.(λy.yx)(λw.zw))v η (λx.(λy.yx)z)v β (λy.yv)z β zv
(λx.(λy.yx)(λw.zw))v β (λx.(λw.zw)x)v η (λx.zx)v β zv
(λx.(λy.yx)(λw.zw))v β (λx.(λw.zw)x)v β (λw.zw)v η zv
There is at least one sequence of conversions 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?Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil

Propositional equality
Deﬁnition (Hindley & Seldin 2008)
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.

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.

The Functional Interpretation of Direct Computation
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
Deﬁnition (Hindley & Seldin 2008)
(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
→-introduction
[x : A]
f(x) = g(x) : B
λx.f(x) = λx.g(x) : A → B
(ξ)
→-elimination
x = y : A g : A → B
gx = gy : B
(µ)
x : A g = h : A → B
gx = hx : B
(ν)
→-reduction
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)
Role of ξ: Bishop’s constructive principles.
Role of η: “[In CL] All it says is that every term is equal to an
abstraction” [Hindley & Seldin, 1986]

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

Labelled Deduction
to ﬁnd a unifying framework factoring out meta- from
object- level features;
to keep the logic (and logical steps, for that matter) simple,
handling meta-level features via a separate, yet
harmonious calculus;
to make sure the relevant assumptions in a deduction are
uncovered, paying more attention to the explicitation and
use of resources.

Equality in 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
-
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 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
-elimination

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 computational path from a to b.

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
REWR(m, ´g.h(g)) : C
Id-reduction
a =s b : A
s(a, b) : IdA(a, b)
Id-intr
[a =g b : A]
h(g) : C
REWR(s(a, b), ´g.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
Πx : AΠy : A(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)
REWR(p,´t(σ(t))(y, x)) : IdA(y, x)
λp.REWR(p,´t(σ(t))(y, x)) : IdA(x, y) → IdA(y, x)
λy.λp.REWR(p,´t(σ(t))(y, x)) : Πy : A(IdA(x, y) → IdA(y, x))
λx.λy.λp.REWR(p,´t(σ(t))(y, x)) : Πx : AΠy : A(IdA(x, y) → IdA(y, x))

Rule of Substitution
x =g y : A f(x) : P(x)
g(x, y) · f(x) : P(y)

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.

Propositional Equality and Identity Types

More Related Content

What's hot

Similar to Propositional Equality and Identity Types

More from Ruy De Queiroz

Recently uploaded

Propositional Equality and Identity Types