1.
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
Propositional Equality and Identity Types
2.
Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
3.
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
MartinL¨of’s Intuitionistic Type Theory (Log Coll ’73, published 1975)
with the socalled Identity Type
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
4.
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 MartinL¨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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
5.
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 MartinL¨of into homotopy theory, resulting in new
examples of certain algebraic structures which are
important in topology”. (“Type Theory and Homotopy”,
preprint, 2010.)
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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 MartinL¨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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
7.
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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 omegacategories” 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 wellformed
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
9.
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
10.
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
socalled 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.
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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 truthvalues
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
12.
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 truthvalues
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
13.
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
14.
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 nonnegative 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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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
Mdirect 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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
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
MartinL¨of’s Type Theory
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
18.
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 λcalculus, 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 nonnormal 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 and Identity Types
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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 βequal or βconvertible to Q (notation P =β Q) iff Q is
obtained from P by a ﬁnite (perhaps empty) series of
βcontractions and reversed βcontractions 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.
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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 NDstyle 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 nonnormal) proof, the process of
bringing it to a normal form should be ﬁnite and conﬂuent.
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
Equality in Type Theory
MartinL¨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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
22.
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
24.
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 λcalculus;
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
25.
Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
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 metalevel 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.
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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 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
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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.
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
Propositional Equality
Idintroduction
a =s b : A
s(a, b) : IdA(a, b)
Idelimination
m : IdA(a, b)
[a =g b : A]
h(g) : C
REWR(m, ´g.h(g)) : C
Idreduction
a =s b : A
s(a, b) : IdA(a, b)
Idintr
[a =g b : A]
h(g) : C
REWR(s(a, b), ´g.h(g)) : C
Idelim
β
[a =s b : A]
h(s/g) : C
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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: 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))
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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Identity Types in Type Theory Heyting’s explanation via proofs Direct Computations The Functional Interpretation of Propositional E
The Functional Interpretation of Direct Computations
Rule of Substitution
x =g y : A f(x) : P(x)
g(x, y) · f(x) : P(y)
Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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
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) Univ. Federal de Pernambuco, Recife, Brazil
Propositional Equality and Identity Types
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