propositional equality, identity types and computational paths

What is a Proof of an Equality? Brouwer–Heyting–Kolmogorov for Equality Direct Computations The Functional Interpretation of Propositional Equality Normal form for equality proofs

Propositional Equality, Identity Typesand Computational Paths

Ruy de Queiroz(joint work with Anjolina de Oliveira)

Centro de InformaticaUniversidade Federal de Pernambuco (UFPE)

Recife, Brazil

Logic and Applications: in honor to Francisco Miragliaon the occasion of his 70th birthday

16–17 Sept 2016

Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil

Propositional Equality, Identity Types and Computational Paths


Homotopy Type TheoryUnivalent Foundations of Mathematics

Institute for Advanced Study, Princeton484–600p. Open-source book:

27 main authors. 58 contributorsAvailable on GitHub. Latest version September 3, 2016




Homotopy Type TheoryUnivalent Foundations of Mathematics

“Homotopy type theory is a new branch of mathematics thatcombines aspects of several different fields in a surprisingway. It is based on a recently discovered connectionbetween homotopy theory and type theory. Homotopytheory is an outgrowth of algebraic topology andhomological algebra, with relationships to higher categorytheory; while type theory is a branch of mathematical logicand theoretical computer science. Although theconnections between the two are currently the focus ofintense investigation, it is increasingly clear that they arejust the beginning of a subject that will take more time andmore hard work to fully understand. It touches on topics asseemingly distant as the homotopy groups of spheres, thealgorithms for type checking, and the definition of weak∞-groupoids.”




Geometry and LogicAlexander Grothendieck

Alexander Grothendieckb. 28 March 1928, Berlin, Prussia, Germany

d. 13 November 2014 (aged 86), Saint-Girons, Ariege, France




Geometry and LogicAlexander Grothendieck

. . . the study of n-truncated homotopy types (ofsemisimplicial sets, or of topological spaces) [shouldbe] essentially equivalent to the study of so-calledn-groupoids. . . . This is expected to be achieved byassociating to any space (say) X its “fundamentaln-groupoid” Πn(X ).... The obvious idea is that0-objects of Πn(X ) should be the points of X ,1-objects should be “homotopies” or paths betweenpoints, 2-objects should be homotopies between1-objects, etc.(Grothendieck 1983)




Equality in λ-Calculus: definitional vs. propositionalProofs of equality: Paths

Church’s (1936) original paper:

NB: equality as the reflexive, symmetric and transitive closure of1-step contraction: rewriting paths. An algebra of paths (with α, β, η,µ, ν, ξ, ρ, σ, τ )? E.g. σ(σ(r)) = r , τ(τ(t , r), s) = τ(t , τ(r , s)).




Equality in λ-Calculus: definitional vs. propositionalProofs of equality: Paths

Definition (Hindley & Seldin 2008)

P is βη-equal or βη-convertible to Q (notation P =βη Q) iff Q isobtained from P by a finite (perhaps empty) series ofβ-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 thatP0 ≡ P, Pn ≡ Q,(∀i ≤ n − 1) (Pi .1β Pi+1 or Pi+1 .1β Pi

or Pi .1η Pi+1 or Pi+1 .1η Pior Pi ≡α Pi+1).




The Functional Interpretation of EqualitySequences 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 tothe 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 InterpretationProofs rather than truth-values

a proof of the proposition: is given by:A ∧ B a proof of A and

a proof of BA ∨ B a proof of A or

a proof of BA→ 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)




Brouwer–Heyting–Kolmogorov InterpretationWhat is a proof of an equality statement?

a proof of the proposition: is given by:

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 betweenpaths)?




Geometry and LogicVladimir Voevodsky

“From an observation by Grothendieck:

Formalism of higher equivalences (theory of grupoids)=Homotopy theory (theory of shapes up to adeformation)

Combined with some other ideas it: leads to an encoding ofmathematics in terms of the homotopy theory. Unlike the usualencodings in terms of the set theory this one respectsequivalences.”




Type Theory and EqualityProposition 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 ) equalproofs of proposition A”.What about the judgement of “p is a proof of the statement that x andy are equal elements of type A”? This is where the so-called Identitytype comes into the picture:

p : IdA(x , y)




Type Theory and its Derivations-as-TermsInterpretationHoward on Curry-Howard

“ [de Bruijn] discovered the idea of derivations asterms, and the accompanying idea offormulae-as-types, on his own. (...)Martin-Lof suggested that the derivations-as-termsidea would work particularly well in connection withPrawitz’s theory of natural deduction.” (W.Howard,Wadler’s Blog, 2014)

Since the early 1990’s, our focus has been on insisting on thederivations-as-terms perspective, especially wrt topropositional equality.




Type Theory and Homotopy TheorySteve Awodey

“The purpose of this informal survey article is to introduce thereader to a new and surprising connection between Geometry,Algebra, and Logic, which has recently come to light in the formof an interpretation of the constructive type theory of PerMartin-Lof into homotopy theory, resulting in new examples ofcertain algebraic structures which are important in topology.This connection was discovered quite recently, and variousaspects of it are now under active investigation by severalresearchers.”(“Type Theory and Homotopy Theory”, 2010.)




Algebraic Structure: GroupoidsSteve Awodey

“A groupoid is like a group, but with a partially-definedcomposition operation. Precisely, a groupoid can be defined asa category in which every arrow has an inverse. A group is thusa groupoid with only one object. Groupoids arise in topology asgeneralized fundamental groups, not tied to a choice ofbasepoint.”




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 dichotomiesarise:

1 definitional equality versus propositional equality

2 intensional equality versus extensional equality

First step on the formalisation of proofs of equality statements: PerMartin-Lof’s Intuitionistic Type Theory (Log Coll ’73, published 1975)with the so-called Identity Type




Identity TypesIdentity Types - Topological and Categorical Structure

Workshop, Uppsala, November 13–14, 2006

“The identity type, the type of proof objects for thefundamental propositional equality, is one of the mostintriguing constructions of intensional dependent typetheory (also known as Martin-Lof type theory). Itscomplexity became apparent with the Hofmann–Streichergroupoid model of type theory. This model also hinted atsome possible connections between type theory andhomotopy theory and higher categories. Exploration of thisconnection is intended to be the main theme of theworkshop.”




Identity TypesType Theory and Homotopy Theory

Indeed, a whole new research avenue has since 2005 been exploredby people like Vladimir Voevodsky and Steve Awodey in trying tomake a bridge between type theory and homotopy theory, mainly viathe 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-Lof type theorybased on groupoids and fibrations of groupoids in whichidentity types may contain two distinct elements which arenot even propositionally equal. This shows that the principleof uniqueness of identity proofs is not derivable in thesyntax”. (“LICS ’94, pp. 208–212.)




Identity TypesIdentity Types as Topological Spaces

According to B. van den Berg and R. Garner (“Topological andsimplicial models of identity types”, ACM Transactions onComputational Logic, Jan 2012),

“All of this work can be seen as an elaboration of thefollowing basic idea: that in Martin-Lof type theory, a type Ais analogous to a topological space; elements a,b ∈ A topoints of that space; and elements of an identity typep,q ∈ IdA(a,b) to paths or homotopies p,q : a→ b in A.”.




Identity TypesIdentity Types as Topological Spaces

From the Homotopy type theory collective book (2013):

“In type theory, for every type A there is a (formerlysomewhat mysterious) type IdA of identifications of twoobjects of A; in homotopy type theory, this is just the pathspace AI of all continuous maps I → A from the unitinterval. In this way, a term p : IdA(a,b) represents a pathp : a b in A.”




Identity Types: IterationFrom Propositional to Predicate Logic and Beyond

In the same aforementioned workshop, B. van den Berg in hiscontribution “Types as weak omega-categories” draws attention to thepower 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,modulodefinitional equality; A1 consists of terms of the typesId(X ; p; q) (in context Γ) for elements p,q in A0, modulodefinitional equality; A2 consists of terms of well-formedtypes Id(Id(X ; p; q); r ; s) (in context Γ) for elements p,q inA0, r , s in A1, modulo definitional equality; etcetera...”




Identity Types: IterationThe homotopy interpretation

Here is how we can see the connections between proofs of equalityand homotopies:

a, b : Ap, q : IdA(a,b)α, β : IdIdA(a,b)(p,q)· · · : IdIdId...(· · · )

Now, consider the following interpretation:Types SpacesTerms Maps

a : A Points a : 1→ Ap : IdA(a,b) Paths p : a⇒ b

α : IdIdA(a,b)(p,q) Homotopies α : p V q




Identity TypesUnivalent Foundations of Mathematics

From Vladimir Voevodsky (IAS, Princeton) “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“There were two main problems with the existingfoundational systems which made them inadequate.Firstly, existing foundations of mathematics werebased on the languages of Predicate Logic andlanguages of this class are too limited.Secondly, existing foundations could not be used todirectly express statements about such objects as, forexample, the ones that my work on 2-theories wasabout.”





From Vladimir Voevodsky (IAS, Princeton) “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“Univalent Foundations, like ZFC-based foundationsand unlike category theory, is a complete foundationalsystem, but it is very different from ZFC. To provide aformat for comparison let me suppose that anyfoundation for mathematics adequate both for humanreasoning and for computer verification should havethe following three components.”





From Vladimir Voevodsky (IAS, Princeton), “UnivalentFoundations: New Foundations of Mathematics”, Mar 26, 2014:

“The first component is a formal deduction system: alanguage and rules of manipulating sentences in thislanguage that are purely formal, such that a record ofsuch manipulations can be verified by a computerprogram.The second component is a structure that provides ameaning to the sentences of this language in terms ofmental objects intuitively comprehensible to humans.The third component is a structure that enableshumans to encode mathematical ideas in terms of theobjects directly associated with the language.”




Propositional EqualityProofs of equality as (rewriting) computational paths

What is a proof of an equality statement? In what sense it can beseen as a homotopy? Motivated by looking at equalities in typetheory as arising from the existence of computational paths betweentwo formal objects, it may be useful to review the role and the powerof the notion of propositional equality as formalised in the so-calledCurry–Howard functional interpretation.The main idea, i.e. proofs of equality statements as (reversible)sequences of rewrites, goes back to a paper entitled “Equality inlabelled deductive systems and the functional interpretation ofpropositional equality”, , presented in Dec 1993 at the 9th AmsterdamColloquium, and published in the proceedings in 1994.




Brouwer–Heyting–Kolmogorov InterpretationProofs rather than truth-values

a proof of the proposition: is given by:A ∧ B a proof of A and

a proof of BA ∨ B a proof of A or

a proof of BA→ 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: FormallyCanonical 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 BA ∨ 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 Bprovided 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 chosenfrom the domain D

∃xD.P(x) 〈f (a),a〉 where a is a witnessfrom the domain D, f (a) is a proof of P(a)




Brouwer–Heyting–Kolmogorov InterpretationWhat is a proof of an equality statement?

a proof of the proposition: is given by:

t1 = t2 ?(Perhaps a sequence of rewritesstarting 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 ComputationsTerms, Equations, Measure

Definition (equations and systems of equations)

Let us consider equations E between individual termsa,b, c, . . ., possibly containing function variables, and finite setsof equations S.

Definition (measure)

A function M from terms to non-negative integers is called ameasure 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 ]).




Statman’s Direct ComputationsKreisel–Tait’s calculus K

Definition (calculus K )

The calculus K of Kreisel and Tait consists of the axioms a = a andthe rule of substituting equals for equals:

(1)E [a/x ] a .

= bE [b/x ]

where a .= b is, ambiguously, a = b and b = a, together with the rules

(2)sa = sba = b

(3)0 = sab = c

(4)a = snab = c

H will be the system consisting only of the rule (1)




The Functional Interpretation of EqualitySequences of contractions

(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 contractions from the initial term tothe final term. (In this case we have given three!) Thus, in the formaltheory of λ-calculus, the term (λx .(λy .yx)(λw .zw))v is declared to beequal 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 Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



The Functional Interpretation of EqualityPropositional equality


P is β-equal or β-convertible to Q (notation P =β Q) iff Q isobtained from P by a finite (perhaps empty) series ofβ-contractions and reversed β-contractions and changes ofbound variables. That is, P =β Q iff there exist P0, . . . ,Pn(n ≥ 0) such thatP0 ≡ 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 closureof 1-step contraction: arising from rewriting




The Functional Interpretation of Direct ComputationEquality: 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 iscalled =βη; that is, we define

M =βη N ⇔ λβη ` M = N.

Note again that two terms are λβη-equal if there exists a proofof their equality in the theory of λβη-equality.




The Functional Interpretation of EqualityGentzen’s ND for propositional equality

Remark

In setting up a set of Gentzen’s ND-style rules for equality weneed 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 ComputationEquality in Type Theory

Martin-Lof’s Intuitionistic Type Theory :Intensional (1975)Extensional (1982(?), 1984)

Remark (Definitional vs. Propositional Equality)

definitional, i.e. those equalities that are given as rewriterules, orelse originate from general functional principles(e.g. β, η, ξ, µ, ν, etc.);propositional, i.e. the equalities that are supported (orotherwise) by an evidence (a sequence of substitutionsand/or rewrites)




The Functional Interpretation of Direct ComputationDefinitional 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 = NN = M

(τ)M = N N = P

M = P




The Functional Interpretation of Direct ComputationIntuitionistic Type Theory

→-introduction

[x : A]

f (x) = g(x) : Bλx .f (x) = λx .g(x) : A→ B

(ξ)

→-eliminationx = y : A g : A→ B

gx = gy : B(µ)

x : A g = h : A→ Bgx = hx : B

(ν)

→-reduction a : A[x : A]

b(x) : B(λx .b(x))a = b(a/x) : B (β)

c : A→ Bλx .cx = c : A→ B

(η)




The Functional Interpretation of Direct ComputationsLessons 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.

∃xP(x)

[P(t)]

CC

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 akind 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 Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



The Functional Interpretation of Direct ComputationsEquality in Martin-Lof’s Intensional Type Theory

A type a : A b : A

Id intA (a,b) type

Id int -formation

a : A

r(a) : Id intA (a,a)

Id int -introductiona = b : A

r(a) : Id intA (a,b)

Id int -introduction

a : A b : A c : Id intA (a,b)

[x :A]d(x):C(x,x,r(x))

[x :A,y :A,z:Id intA (x,y)]

C(x,y,z) type

J(c,d) : C(a,b, c)Id int -elimination

a : A[x : A]

d(x) : C(x , x ,r(x))

[x : A, y : A, z : Id intA (x , y)]

C(x , y , z) typeJ(r(a),d(x)) = d(a/x) : C(a,a,r(a))

Id int -equality




The Functional Interpretation of Direct ComputationsEquality in Martin-Lof’s Extensional Type Theory

A type a : A b : AIdext

A (a,b) typeIdext -formation

a = b : Ar : Idext

A (a,b)Idext -introduction

c : IdextA (a,b)

a = b : AIdext -elimination

c : IdextA (a,b)

c = r : IdextA (a,b)

Idext -equality




The Functional Interpretation of Direct ComputationsThe missing entity

Considering the lessons learned from Type Theory, thejudgement of the form:

a = b : Awhich says that a and b are equal elements from domain D, letus add a function symbol:

a =s b : Awhere one is to read: a is equal to b because of ‘s’ (‘s’ beingthe rewrite reason); ‘s’ is a term denoting a sequence ofequality 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-Lof’sformulations of Identity Types.)




The Functional Interpretation of Direct ComputationsPropositional Equality

Id-introductiona =s b : A

s(a,b) : IdA(a,b)

Id-elimination

m : IdA(a,b)

[a =g b : A]

h(g) : CJ(m, λg.h(g)) : C

Id-reductiona =s b : A

s(a,b) : IdA(a,b)Id-intr

[a =g b : A]

h(g) : CJ(s(a,b), λg.h(g)) : C

Id-elim

.β[a =s b : A]

h(s/g) : CRuy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil



The Functional Interpretation of Direct ComputationsPropositional Equality: A Simple Example of a Proof

By way of example, let us prove

ΠxAΠyA(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))




The Functional Interpretation of Direct ComputationsPropositional Equality: The Groupoid Laws

With the formulation of propositional equality that we have justdefined, we can also prove that all elements of an identity typeobey the groupoid laws, namely

1 Associativity2 Existence of an identity element3 Existence of inverses

Also, the groupoid operation, i.e. composition ofpaths/sequences, is actually, partial , meaning that not allelements will be connected via a path. (The groupoidinterpretation refutes the Uniqueness of Identity Proofs.)




The Functional Interpretation of Direct ComputationsPropositional Equality: The Uniqueness of Identity Proofs

“We will call UIP (Uniqueness of Identity Proofs) the followingproperty. If a1,a2 are objects of type A then for any proofs pand q of the proposition “a1 equals a2” there is another proofestablishing equality of p and q. (...) Notice that in traditionallogical formalism a principle like UIP cannot even be sensiblyexpressed as proofs cannot be referred to by terms of theobject language and thus are not within the scope ofpropositional equality.” (Hofmann & Streicher 1996)




The Functional Interpretation of Direct ComputationsNormal form for the rewrite reasons

Strategy:

Analyse possibilities of redundancyConstruct a rewriting systemProve 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, andA 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 i th equation in S.Ruy de Queiroz (joint work with Anjolina de Oliveira)Centro de Informatica Universidade Federal de Pernambuco (UFPE) Recife, Brazil




Definition (rewrite reason)

Given a system of equations S and an equation s =r t : A, ifS ` s =r t : A, i.e. there is a deduction/computation of theequation starting from the equations in S, then the rewritereason 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 andtransitivity) are defined as follows:

x : Ax =ρ x : A

(reflexivity)

x =t y : Ay =σ(t) x : A

(symmetry)

x =t y : A y =u z : Ax =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 : AC[x ] =subR(r ,s) u : A

where C[x ] is the context in which the subterm x appears




The Functional Interpretation of Direct ComputationsReductions

Definition (reductions involving ρ and σ)

x =ρ x : Ax =σ(ρ) x : A

.sr x =ρ x : A

x =r y : Ay =σ(r) x : A

x =σ(σ(r)) y : A.ss x =r y : A

Associated rewritings:σ(ρ) .sr ρσ(σ(r)) .ss r





Definition (reductions involving τ )

x=r y :D y=σ(r)x :Dx=τ(r,σ(r))x :D

.tr x =ρ x : D

y=σ(r)x :D x=r y :Dy=τ(σ(r),r)y :D

.tsr y =ρ y : D

u=r v :D v=ρv :Du=τ(r,ρ)v :D

.rrr u =r v : D

u=ρu:D u=r v :Du=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 .





Definition

βrewr -×-reductionx =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





Definition

βrewr -×-reductionx =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 PathsComputational Paths form a Weak Category

Theorem

For each type A, computational paths induce a weakcategorical structure Arw where:

objects: terms a of the type A, i.e., a : Amorphisms: a morphism (arrow) between terms a : A andb : A are arrows s : a→ b such that s is a computationalpath between the terms, i.e., a =s b : A.

Corollary

Arw has a weak groupoidal structure.





Working papers:

1 Propositional equality, identity types, and direct computationalpathsRuy J.G.B. de Queiroz, Anjolina G. de Oliveirahttp://arxiv.org/abs/1107.1901

2 arXiv:1509.06429 [pdf, other]On Computational Paths and the Fundamental Groupoid of aTypeArthur F. Ramos, Ruy J. G. B. de Queiroz, Anjolina de Oliveira

3 arXiv:1506.02721 [pdf, other]On the Groupoid Model of Computational PathsArthur F. Ramos, Ruy J. G. B. de Queiroz, Anjolina G. de Oliveira

4 arXiv:1504.04759 [pdf, other]On the Identity Type as the Type of Computational PathsArthur F. Ramos, Ruy J.G.B. de Queiroz, Anjolina G. de Oliveira



propositional equality, identity types and computational paths

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