Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
1. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Propositional Equality, Identity Types
and Reversible Rewriting Sequences as
Homotopies
Ruy de Queiroz
(joint work with Anjolina de Oliveira)
Centro de Inform´atica
Universidade Federal de Pernambuco (UFPE)
Recife, Brazil
Encontro Brasileiro de L´ogica (EBL)
LNCC, Petr´opolis (RJ)
Apr 2014
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
2. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
3. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
4. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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, http://arxiv.org/abs/1010.1810.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
5. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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.”.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
6. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
7. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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...”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
8. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
9. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
10. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Identity Types
Univalent Foundations of Mathematics
From Vladimir Voevodsky (IAS, Princeton) “Univalent
Foundations: New Foundations of Mathematics”, Mar 26, 2014:
“Univalent Foundations, like ZFC-based foundations
and unlike category theory, is a complete foundational
system, but it is very different from ZFC. To provide a
format for comparison let me suppose that any
foundation for mathematics adequate both for human
reasoning and for computer verification should have
the following three components.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
11. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Identity Types
Univalent Foundations of Mathematics
From Vladimir Voevodsky (IAS, Princeton), “Univalent
Foundations: New Foundations of Mathematics”, Mar 26, 2014:
“The first component is a formal deduction system: a
language and rules of manipulating sentences in this
language that are purely formal, such that a record of
such manipulations can be verified by a computer
program.
The second component is a structure that provides a
meaning to the sentences of this language in terms of
mental objects intuitively comprehensible to humans.
The third component is a structure that enables
humans to encode mathematical ideas in terms of the
objects directly associated with the language.”
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
12. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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, i.e. proofs of equality statements as (reversible)
sequences of rewrites, 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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
13. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
14. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
15. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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?
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
16. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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]).
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
17. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Statman’s Direct Computations
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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
18. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
Statman’s Direct Computations
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).
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
19. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
20. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Equality
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.
NB: equality as the reflexive, symmetric and transitive closure
of 1-step contraction
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
21. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computation
Existential Force
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
22. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of 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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
23. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computation
Equality in Type Theory
Martin-L¨of’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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
24. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computation
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
25. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computation
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]
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
26. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
27. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
28. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
29. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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.
(This formal entity is missing in both of Martin-L¨of’s
formulations of Identity Types.)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
30. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
31. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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)
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)) : ΠyA(IdA(x, y) → IdA(y, x))
λx.λy.λp.REWR(p,´t(σ(t))(y, x)) : ΠxAΠyA(IdA(x, y) → IdA(y, x))
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
32. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Propositional Equality: A Weak Version of Function Extensionality
With the formulation of propositional equality that we have just
defined, we can actually prove a weakened version of function
extensionality, namely
A → ΠfA→B
ΠgA→B
(ΠxA
.IdB(AP(f, x), AP(g, x)) → IdA→B(f, g))
which asserts that, if the domain A is nonempty, then in case f
and g agree on all points they must be considered to be
propositionally equal elements of type A → B.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
33. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
34. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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.” (Hofmann & Streicher 1996)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
35. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
Strategy:
Analyse possibilities of redundancy
Construct a rewriting system
Prove termination and confluence
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
36. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
37. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
38. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
39. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Normal form for the rewrite reasons
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
40. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
41. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
42. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
43. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies
44. Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Reductions
Working paper:
Propositional equality, identity types, and direct
computational paths
Ruy J.G.B. de Queiroz, Anjolina G. de Oliveira
http://arxiv.org/abs/1107.1901
[v1] Sun, 10 Jul 2011 21:28:26 GMT (33kb)
[v2] Thu, 1 Mar 2012 13:27:15 GMT (34kb)
[v3] Mon, 5 Aug 2013 20:26:40 GMT (37kb)
Ruy de Queiroz (joint work with Anjolina de Oliveira)
Centro de Inform´atica Universidade Federal de Pernambuco (UFPE) Recife, Brazil
Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies