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Seminar given at the Logic Group, Technische Universität Darmstadt, 19 August 2013

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- 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 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) 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 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) 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 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.) 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 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) 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
- 8. 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) 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 so-called Curry–Howard functional interpretation. The main idea goes back to a paper entitled “Equality in labelled deductive systems and the functional interpretation of propositional equality”, , presented in Dec 1993 at the 9th Amsterdam Colloquium, and published in the proceedings in 1994. Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil Propositional Equality and Identity Types
- 11. 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) 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 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) 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 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) Univ. Federal de Pernambuco, Recife, Brazil Propositional Equality and Identity Types
- 15. 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
- 16. 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) Univ. Federal de Pernambuco, Recife, Brazil Propositional Equality and Identity Types
- 17. 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 Martin-L¨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 λ-calculus, the term (λx.(λy.yx)(λw.zw))v is declared to be equal to zv. Now, some natural questions arise: 1 Are the sequences themselves normal? 2 Are there non-normal sequences? 3 If yes, how are the latter to be identiﬁed and (possibly) normalised? 4 What happens if general rules of equality are involved?Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil Propositional Equality and Identity Types
- 19. 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. Ruy de Queiroz (joint work with Anjolina de Oliveira) Univ. Federal de Pernambuco, Recife, Brazil Propositional Equality and Identity Types
- 20. 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) 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 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) 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
- 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 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 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) 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 meta-level features via a separate, yet harmonious calculus; to make sure the relevant assumptions in a deduction are uncovered, paying more attention to the explicitation and use of resources. 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 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) 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
- 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 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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