Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies

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Talk given at "Encontro Brasileiro de Lógica" (EBL), 9 Apr 2014, LNCC, Petrópolis, RJ

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Propositional Equality, Identity Types and Reversible Rewriting Sequences as Homotopies

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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