Type Structures and Normalization by Evaluation for System F ω

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1 Type Structures and Normalization by Evaluation for System F ω Andreas Abel Department of Computer Science Ludwig-Maximilians-University Munich Abstract We present the first verified normalization-by-evaluation algorithm for System F ω, the simplest impredicative type theory with computation on the type level Types appear in three shapes: As syntactical types, as type values which direct the reification process, and as semantical types, ie, sets of total values The three shapes are captured by the new concept of a type structure, and the fundamental theorem now states that an induced structure is a type substructure This work is an attempt at an algebraic treatment of type theory based on typed applicative structures rather than categories 1 Introduction The Curry-Howard isomorphism, which identifies proofs with programs and propositions with types, is the basis of several type-theoretic theorem provers, such as Coq [19], Agda [20], Epigram [15] and LEGO [21] In these systems, checking the validity of proofs or typings relies on deciding equality of types Types are recognized as equal if they have the same normal form, this is why normalization plays a key role in the study of the theories underlying these theorem provers, such as the Calculus of Constructions (CC) which underlies Coq and LEGO, and Martin-Löf Type Theory which is the basis of Agda and Epigram The hardwired type equality of these systems, often referred to as definitional equality, is necessarily intensional, since fully extensional equality is undecidable The minimal equality is induced by just the computational laws (β-laws), yet, the stronger the hardwired equality, the less manual equality proofs, and the succincter proofs can get Thus, it is desirable to add bits of extensionality as long as decidability is preserved For the CC, different directions have been explored, such as rewriting [13, 16] and decision procedures [14] Our goal is to integrate η-laws which provide some extensionality for functions, into definitional equality of CC Normalization by evaluation (NbE) [12, 17] is a systematic method to perform βη-normalization In a first step, the object t of type T is evaluated The resulting value d is then reified to an η-long β-normal form v The reification process is directed by the shape of type T NbE has proven a valid tool to structure extensional normalization, especially in the case of sum types [7, 9, 11] We are convinced it is the perfect tool to account for the meta-theory of η, thus, we are researching the type systems of the lambda cube with βη-equality given by judgements The lambda cube organizes type systems in three dimensions: dependency, impredicativity, and higher-order In previous work [3], we have adapted NbE to a dependent type theory with one predicative universe and judgmental βη-equality, yielding injectivity of the function space constructor in the presence of universes What is the challenge when stepping up to impredicativity? Predicative type theories allow to define the semantics of types from below via induction-recursion, and the reification function can be defined by induction on types This fails in the presence of impredicativity, where one first has to lay out a lattice of semantic type candidates and then define impredicative quantification as the intersection over all candidates [18] Hence, the semantic type structure is not inductive, and reification cannot be defined by induction on types 1 There are at least two ways out of this dilemma: Altenkirch, Hofmann, and Streicher [8] incorporated the definition of the normalization function into their model construction for System F In previous work, I have conceived reification as a deterministic relation between value d and normal form v and their type T, and showed through a model construction that it corresponds to a total function In this work, we are moving one step closer to NbE for the CC: we are considering the simplest type system which features impredicativity and computation on the type level: the higher-order polymorphic lambda-calculus F ω It adds to the problem of impredicativity the difficulty that types are no longer fixed syntactic expressions as in System F, but 1 The reflection function, which is defined mutually inductive with reification, yields at universal quantification ( XA t)(b) = A[B/X] (t B) Clearly A[B/X] can be bigger as XA For similar reasons, the set of η-long normal forms cannot be defined by induction on the type in System F; it lacks the subformula property for normal forms 1

2 they need to be normalized as well One solution would be to keep types always in long normal form, e g, by the use of hereditary substitutions [23, 6] This is possible since the types are simply-kinded, so normalization can be defined by induction on kinds However, this approach would not scale, eg, to CC, where serious computation is happening on the type level Furthermore, we would like to use the same normalization procedure both on object and type level In the solution we have chosen here, reification of objects is directed by type values A Syntactic types T are interpreted by a pair (A, A) of a type value A and a semantic type A which is a set of objects that are reifiable at type A Furthermore, type value A reifies to a normal form V which is βη-equal to T These considerations lead us to the concept of a type structure which captures the similarities between syntactic types, type values, and semantic types Consequently, syntactic objects and their values both form an object structure over a type structure, the syntactical type structure in case of syntactic objects and the structure of type values in case of (object) values We reorganize a typical normalization proof for System F ω by model construction [5] into our framework of type and object structures Central to such a normalization proof is the fundamental theorem of typing which states that t [[T ]] for any object t of type T Herein, t interprets object t in some applicative structure, for instance in the structure of syntactical objects The semantic type [[T ]] is a collection of values, typically a set of normalizing objects, where the function type [[T 1 T 2 ]] is defined à la Tait [22], i e, as all values f such that f applied to d inhabits [[T 2 ]] for all d in [[T 1 ]] The essence of the fundamental theorem is that the (hereditarily) normalizing terms model the typing rules where these terms are a (non-proper) subset of the typable terms In our abstract setting, the fundamental theorem proves that a part of an object structure with the above function type definition is an object substructure, casting the fundamental theorem into an algebraic setting Or notions of type and object structures are very general, in essence typed versions of Barendregt s syntactical applicative structures [10, Def 531] The fundamental theorems we prove are also very general since we do not fix an interpretation of types; we only require that semantical types inhabit a candidate space By choosing different candidate spaces we can harvest different results from the same fundamental theorem, e g, soundness of NbE, completeness of NbE, or weak normalization of β- or βη-reduction [2] 11 Overview Table 1 systematizes the main structures introduced in this article Kind structures are inhabited by kinds, the types of types of System F ω The free kind structure Ki is inhabited by the syntactic kinds κ which coincide with their values, since no computation takes place on the kind level A kind κ can be interpreted as a family K of types which are drawn from a type structure T ; with Tait s interpretation of the function arrow between kinds, the collection T of subsets of T forms another kind structure Instances of type structures are Ty, which contains the syntactical types T, or T, which contains type values A of some sort, or D which contains semantical types A, sets of objects, hence, subsets of an object structure D At higher kinds κ κ, Ty is inhabited by proper type constructors, T by their values F, and D by operators F on sets of objects On the level of objects, we have syntactical objects t in structure Obj and values d in structure D All structures are sorted, so types always have a kind, and objects always have a type, syntactical objects a syntactical type, and values a type value In the following developments we omit most proofs due to lack of space They can be found in the full version of this article on the author s homepage [1] 12 Preliminaries Contexts,, Γ,, Φ, Ψ are functions from variables to some codomain We write for the totally undefined function and Φ, x : a for the function Φ with domain dom(φ) {x} such that Φ (x) = a and Φ (y) = Φ(y) for y x We say Ψ extends Ψ, written Ψ Ψ, if Ψ (x) = Ψ(x) for all x dom(ψ) Families T indexed by a context are always understood to be Kripke, i e, implies T T We use also for pointwise inclusion of families, e g, S T means S κ T κ if S, T are families indexed by a kind κ and a context The notion Kripke family is also used for maps M There it implies that M does not depend on the context parameter, i e, M (a) = M (a) for a dom(m ) and (Note that dom(m ) dom(m ) since M is Kripke) We identify a pair of functions (f, g) with the function h(x) = (f(x), g(x)), especially in the case of contexts (, Γ) or environments (σ, ρ) We will sometimes drop the parentheses and write just the comma We write (a A) B(a) for the dependent function space {f A a A B(a) f(a) B(a) for all a A} 2 Syntax In this section, we present the syntax and inference rules for System F ω The system consists of three levels: On the lowest level there live the objects, meaning polymorphic, purely functional programs On the middle level live the 2

3 syntax value semantics structures kind κ Ki family of types K T κ Ki, T type T Ty A T family of objects A D A Ty, T, D object T Ty κ κ t Obj Γ T D F :κ κ F T κ κ operator F d D A Obj, D Table 1 Systematics of kind, type, and object structures types of objects, and the type constructors, which are classified by kinds which inhabit the highest level Kinds κ Ki are given by the grammar κ ::= κ κ Kind classifies type constructors which are actually types, and kind κ κ classifies the type constructors which map type constructors of kind κ to type constructors of kind κ In the following, we will refer to all type constructors as types Assume a countably infinite set of type variables TyVar whose members are denoted by X, Y, Z, Kinding contexts, KiCxt are partial maps from the type variables into Ki The set TyCst = {, κ κ Ki} contains the type constants C Their kinds are given by the signature Σ TyCst Ki defined by Σ( ) = Σ( κ ) = (κ ) for all κ Ki Types are given by the grammar T, U, V ::= C X λx : κ T T U where X TyVar and form a simplykinded lambda calculus Objects are given by the grammar t, u, v ::= x λx : T t t u ΛX : κ t t U and form a polymorphic lambda-calculus with type abstraction and type application Herein, object variables x are drawn from a countably infinite set ObjVar which is disjoint to TyVar We write b[a/x] for caption-avoiding substitution of a for variable x in syntactic expression b, and FV for the function returning the set of all free type and object variables of a syntactic expression Kinding, typing, and equality for System F ω is given by five judgements whose inference rules are displayed in Figure 1 T : κ T = T : κ ; Γ t : T ; Γ t = t : T Herein, the auxiliary judgement Γ, read Γ is a wellformed typing context in, is defined as Γ(x) : for all x dom(γ) We set Ty κ := {T T : κ} Obj Γ T := {t ; Γ t : T } 3 Kind Structures Definition 1 (Kind structure) A kind structure is a set K with a distinguished element K and a binary operation K K K The free kind structure is Ki A kind structure can also consist of semantical kinds which are sets of types drawn from a type structure T In the following, we present this abstractly Definition 2 (Kind candidate space) Let (K κ, κ ) be a Ki-indexed family of posets with a (polymorphic) binary operation K κ K κ K κ κ A kind candidate space C consists of two families C κ, C κ K κ, written κ, κ if no ambiguity arises, such that K-BASE, K-FUN-I κ κ κ κ, K-FUN-E κ κ κ κ For K K κ we write κ C K and say κ realizes K w r t C iff κ K κ For a family K κ K κ we write C K if κ C K for all κ Lemma 3 (Realizability kind structure) In the context of the above definition: If is antitone in its first argument and monotone in its second argument, then C := {(κ, K) κ C K} constitutes a kind structure with distinguished element Proof We have, and κ K and κ K imply κ κ K K 4 Type Structures In this section, we define type structures as an abstraction over syntactic types, type values, and semantic types Type structures form a category with products Definition 4 (Type structure) An (F ω ) type structure is a Kripke family T κ of sets with the following Kripke families of maps: Cst (C dom(σ)) T Σ(C) App κ κ T κ κ T κ T κ 3

4 Kinding T : κ In context, type T has kind κ C : Σ(C) X : (X), X :κ T : κ T : κ κ U : κ λx :κ T : κ κ T U : κ Type equality T = T : κ In context, types T and T are βη-equal of kind κ, X :κ T : κ U : κ T : κ κ (λx :κ T ) U = T [U/X] : κ λx :κ T X = T : κ κ X FV(T ) C = C : Σ(C) X = X : (X), X :κ T = T : κ T : κ κ U : κ λx :κ T = λx :κ T : κ κ T U : κ Typing ; Γ t : T In contexts, Γ, object t has type T Γ ; Γ x : Γ(x) ; Γ, x:u t : T ; Γ λx:u t : U T ; Γ t : U T ; Γ u : U ; Γ t u : T, X :κ; Γ t : T X ; Γ ΛX :κ t : κ T X FV(T ) ; Γ t : κ T U : κ ; Γ t U : T U ; Γ t : T T = T : ; Γ t : T Object equality ; Γ t = t : T In contexts, Γ, objects t and t are βη-equal of type T ; Γ, x:u t : T ; Γ u : U ; Γ (λx:u t) u = t[u/x] : T, X :κ; Γ t : T U : κ ; Γ (ΛX :κ t) U = t[u/x] : T [U/X] ; Γ t : U T ; Γ λx:u t x = t : U T x FV(t) ; Γ t : κ T ; Γ ΛX :κ t X = t : κ T X FV(t) Γ ; Γ x = x : Γ(x) ; Γ, x:u t = t : T ; Γ λx:u t = λx:u T : U T ; Γ t = t : U T ; Γ u = u : U ; Γ t u = t u : T, X :κ; Γ t = t : T X ; Γ ΛX :κ t = ΛX :κ t : κ T X FV(T ) ; Γ t = t : κ T U = U : κ ; Γ t U = t U : T U ; Γ t = t : T T = T : ; Γ t = t : T Figure 1 F ω : kinding, type equality, typing, object equality 4

5 Usually, we will just write F G for App κ κ (F, G) Furthermore, we stipulate an interpretation function [[ ]] : Ty κ T T κ where ρ T (X) iff ρ(x) T for all X dom() The interpretation function must satisfy the following equations [[C]] ρ = Cst (C) [[X]] ρ = ρ(x) [[λx :κ T ]] ρ G = [[T ]] ρ[x G] [[T U]] ρ = [[T ]] ρ [[U]] ρ We use interpretation synonymous to evaluation We may suppress the index in Cst or even drop Cst altogether, i e, we just write T instead of Cst( ) T To avoid ambiguities when different type structures are in scope, we may write T, κ T, T and T [[ ]] to emphasize that we mean the type structure operations of T Simple examples of type structures are Ty and Ty modulo β, βη, or judgmental equality In these instances, the interpretation function is parallel substitution Our notion of type structure essentially coincides with Barendregt s syntactical applicative structure [10, 531], except that we are working in a kinded setting and Barendregt in untyped lambda-calculus Definition 5 (Type structure morphism) Given two type structures S and T, a type structure morphism M : S T is a Kripke family of maps M κ Sκ T κ that commute with the operations of S, i e, M κ(c S) = C T (C :κ) Σ M κ (F S G) = M κ κ (F ) T M κ (G) F Sκ κ, G S κ M κ(s[[t ]] ρ ) = T [[T ]] M ρ T Ty κ, ρ S forms a new type structure with C S T := (C S, C T ) for all C TyCst (F, F ) (G, G ) := (F G, F G ) [[F ]] ρ,ρ := ([[F ]] ρ, [[F ]] ρ ) Proof One easily validates the laws [[X]] ρ,ρ = (ρ, ρ )(X) [[λx :κ T ]] ρ,ρ (G, G ) = [[T ]] (ρ,ρ )[X (G,G )] [[T U]] ρ,ρ = [[T ]] ρ,ρ [[U]] ρ,ρ using the definition of application and evaluation The two projections π 1 : S T S and π 2 : S T T are trivially type structure morphisms, and is a product in the category of type structures and their morphisms 41 Type Substructures and the Fundamental Theorem for Kinding Definition 8 (Type substructure) Given a type structure T, the Kripke family S κ T κ is a type substructure of T if C T S κ for all (C :κ) Σ T S κ κ S κ Sκ T [[ ]] Ty κ S Sκ Basically, a type substructure is a subfamily which is still a type structure with the original operations One could also say that S is a type substructure of T iff the identity on S is a type structure morphism from S to T However, then the notion of type structure morphism must be set up more liberally, not requiring S to be a type structure a priori Lemma 9 (Projection type substructure) If S T 1 T 2 is a type substructure, so are π 1 (S) T 1 and π 2 (S) T 2 where (M (X) ρ)(x) := M (ρ(x)) Definition 6 (Environment pairing) Given two type structures S, T and environments ρ S and ρ T, we define (ρ, ρ ) (X dom()) S (X) pointwise by (ρ, ρ )(X) = (ρ(x), ρ (X)) T (X) For the environment update operation then holds (ρ, ρ )[X (G, G )] = (ρ[x G], ρ [X G ]) Def and Lem 7 (Product type structure) Given type structures S, T, the pointwise product (S T ) κ = Sκ T κ two Definition 10 (Function space) We write K T κ if K is a Kripke family of subsets K T κ Given K T κ and K T κ we define the Kripke function space (K bt K ) = {F T κ κ F G K for all and G K } If no ambiguities arise, we write for bt Definition 11 (Induced type structure) Let T be a type structure and S T be Kripke If C T S κ for all (C :κ) Σ S κ κ = (S κ bt S κ ) then S is called induced or an induced type substructure of T (see Thm 12) 5

6 Such an S is called induced since it is already determined by the choice of S Theorem 12 (Fundamental theorem of kinding) Let T be a type structure If S T is induced, then S is a type substructure of T Proof We mainly need to show that evaluation is welldefined This is shown by induction on the kinding derivation, as usual Let ρ S We show T ρ S κ by induction on T : κ The cases for variables and constants are trivial, X :κ T : κ λx :κ T : κ κ Since S is induced, it is sufficient to show λx : κ T ρ G S κ for arbitrary and G S κ This follows from the induction hypothesis T ρ S κ with ρ = ρ[x G] S,X:κ T : κ κ U : κ T U : κ Since S κ κ S κ bt S κ, the goal follows by the induction hypotheses and T U ρ = T ρ U ρ 42 NbE for Types and Its Soundness We are ready to define the reification relation for type values and show that NbE, i e, the composition of evaluation of a syntactic type T and reification to a normal form V, is sound, i e, T and V are judgmentally equal As a byproduct, we show totality of NbE on well-kinded types The structure T of type values is left abstract, a concrete definition will be given in Sect 8 However, not every T permits reification of its inhabitants It needs to include the variables which need to be distinguishable from each other and other type values Neutral types, i e, of the shape X G, need to be analyzable into head X and spine G We call a suitable T term-like; on such a T we can define contextual reification [4, 2] Definition 13 (Type structure with variables) A type structure T has variables if there exists a Kripke family of maps Var (X dom()) T (X) such that Var (X) T κ iff (X) = κ Usually, we simply write X for Var (X) 1 Types X G are called neutral 2 Types C G are called constructed If clear from the context of discourse, we drop the index to Var To disambiguate, we sometimes write Var T to refer to the variable embedding of type structure T Usually, we write T [[T ]] instead of T [[T ]] VarT for the interpretation in the identity environment Definition 14 (Term-like type structure) A type structure T is term-like if it has variables and there exists a Kripke family of partial maps View κ T κ {(C, G) TyCst T κ Σ(C) = κ κ} + {(X, G) TyVar T κ (X) = κ κ} such that: View(F ) = (C, G) iff F = Cst(C) G View(F ) = (X, G) iff F = Var(X) G Lemma 15 (Injectivity in term-like type structures) In a term-like type structure T, Cst and Var are families of injective maps and neutral and constructed application is injective, i e, H G = H G implies H = H and G = G for neutral or constructed H, H Proof Assume, for instance H = X G, H = X G neutral We have (X; G, G) = View(H G) = View(H G ) = (X ; G, G ), thus X = X, G = G, and G = G Immediately, H = H follows Trivially, Ty is a term-like type structure with Var Ty (X) = X The product of term-like type structures S and T is again term-like with Var S T = (Var S, Var T ) Ty/= β is term-like due to confluence of β-reduction Ty modulo judgmental equality is not trivially term-like, since first injectivity of the type constructors needs to be proven Definition 16 (Reifiable type structure) A type structure T with variables is reifiable, if there are relations F V κ in, F reifies to V at kind κ, H U κ in, H reifies to U, inferring kind κ, (where F, H T κ with H neutral or constructed, and V, U Ty κ ) which obey the following rules: C C Σ(C) X X (X) H U κ κ G V κ ================================= H G U V κ H U ============ H U, X :κ F X V κ ====================== F λx :κ V κ κ 6

7 Further, these reification relations must be deterministic in the following sense: For all, κ, F (inputs) and neutral or constructed H (input) there is at most one V (output) such that F V κ and at most one U and κ (outputs) such that H U κ Seen as logic programs with inputs and outputs as indicated above, these relations denote partial functions, where is defined by cases on the kind κ and and by cases on the neutral value H Lemma 17 (Reification returns long normal form) If F V κ, then V is η-long β-normal Lemma 18 (Term-like type structure is reifiable) Any term-like type structure T is reifiable Proof In the presence of View, the two relations can simply defined inductively by the above rules We continue by constructing a model of the kinding rules which proves soundness of NbE for types Kinds κ are interpreted as sets G κ of pairs (F, T ) glued together by reification, i e, the type value F reifies to syntactic type T up to βη-equality Function kinds are interpreted via Tait s function space, thus, the fundamental theorem of kinding yields that G is indeed a type structure Definition 19 (Glueing candidate) Fix a reifiable type structure T We define the families Gl, Gl T Ty by Gl κ = {(F, T ) T κ Tyκ F V κ and T = V : κ}, Gl κ = {(H, T ) T κ Tyκ H U κ and T = U : κ} A family S with Gl κ S κ Gl κ is called a glueing candidate Lemma 20 Any glueing candidate S contains the constants Proof Since C C Σ(C) we have Cst S (C) = (C, C) Gl Σ(C) S Σ(C) Lemma 21 (Glueing candidate space) Gl κ, Gl κ form a kind candidate space according to Def 2 Proof Let us write κ, κ for Gl κ, Gl κ We show K-FUN-I Assume (F, T ) (κ κ ) and show (F, T ) κ κ We have (X, X) κ,x:κ, hence, (F X, T X) κ,x:κ That is,, X :κ F X V κ and, X :κ T X = V : κ It follows F λx : κ V κ κ and λx : κ T X = λx : κ V : κ Since T Ty κ κ by assumption, we have X FV(T ), hence, T = λx : κ V : κ Thus, (F, T ) κ κ Definition 22 (Glueing type structure) Given a type structure T, we define G T Ty by G :=, G κ κ := (G κ T Ty Gκ ) Lemma 23 (G κ is a glueing candidate) Gl κ G κ Gl κ Proof By induction on κ By definition, Gl G, and κ Gl G κ and κ Gl G κ imply κ κ κ κ G κ G κ κ κ κ κ Corollary 24 G is induced Proof C G G Σ(C) since any glueing candidate contains the constants, and G κ κ = G κ G κ by definition Since G is induced, by the fundamental theorem of kinding it is a type substructure of T Ty Theorem 25 (Soundness of NbE for types) Let T be an reifiable type structure If T : κ then there is a V Ty κ such that T [[T ]] VarT V κ and T = V : κ Proof Since (X, X) (X) G (X) for all X dom(), for the identity valuation Var T Ty (X) = (X, X) we have Var G = Var T Ty G By the fundamental theorem, G[[T ]] VarG = (T [[T ]] VarT, T ) G κ Since Gκ Gl κ, we have that T [[T ]] VarT V κ with T = V : κ The V returned by reification is the long normal form of T 5 Type Groupoids Completeness of NbE means that it models judgmental type equality, i e, if T = T : κ then [[T ]] V κ and [[T ]] V κ Completeness will be shown by a fundamental theorem of type equality Judgmental equality is usually modelled by partial equivalence relations, which can be seen as groupoids Hence, we introduce the notion of a groupoidal type structure, or type groupoid Definition 26 (From Wikipedia, :) A groupoid is a set G with a function 1 : G G and a partial function : G G G that have the following properties: 1 Associativity: For all a, b, c G, if a b and b c are defined, then (a b) c and a (b c) are defined and equal Conversely, if either of these last two expressions is defined, then so is the other (and again they are equal) 7

8 2 Inverse: For all a G, a 1 a and a a 1 are defined 3 Identity: For all a, b G, if a b is defined, then a b b 1 = a, and a 1 a b = b (The previous axioms already show that these expressions are defined and unambiguous) Example 27 (PER as groupoid) A partial equivalence relation R over set S is a groupoid with (s, t) 1 = (t, s) and (r, s) (s, t) = (r, t) Example 28 (Discrete groupoid) Any set S gives rise to a trivial groupoid with s 1 = s and is defined exactly on the diagonal of S S, and there s s = s Lemma 29 (b 1 ) 1 = b Proof (Uli Schoepp) First, note that b 1 = b 1 b b 1 Hence (b 1 ) 1 = (b 1 ) 1 b 1 (b 1 ) 1 = (b 1 ) 1 b 1 b b 1 (b 1 ) 1 = b b 1 (b 1 ) 1 = b Lemma 30 If a b is defined then (a b) 1 = b 1 a 1 Proof First a = a b b 1 Since (a b) 1 (a b) is defined, (a b) 1 = (a b) 1 a a 1 = (a b) 1 a b b 1 a 1 = b 1 a 1 Definition 31 (Subgroupoid) Given a groupoid G, a set H G is a subgroupoid if it is closed under inversion and composition, ie, if 1 a H implies a 1 H, and 2 a, b H implies a b H if a b is defined 51 Type Groupoids and the Fundamental Theorem of Type Equality Definition 32 (Type groupoid) A type structure is groupoidal if each T κ is a groupoid, and the following laws are fulfilled: C 1 = C for all C TyCst, (F G) 1 = F 1 G 1, (F F ) (G G ) = (F G) (F G ) Example 33 Each type structure is groupoidal for the trivial choice of inversion (identity) and composition (idempotent, defined on the diagonal) Def and Lem 34 (Square type groupoid) Given a type structure T we define the square type groupoid 2 T as the product type structure T T equipped with (F, G) 1 = (G, F ), (F, G) (G, H) = (F, H) Proof The laws of a type groupoid are satisfied, eg, for the last law we have ((F 1, F 2 ) (F 2, F 3 )) ((G 1, G 2 ) (G 2, G 3 )) = (F 1, F 3 ) (G 1, G 3 ) = (F 1 G 1, F 3 G 3 ) = (F 1 G 1, F 2 G 2 ) (F 2 G 2, F 3 G 3 ) = ((F 1, F 2 ) (G 1, G 2 )) ((F 2, F 3 ) (G 2, G 3 )) Lemma 35 (Function space is groupoid) If K T κ and K T κ are groupoids, so is K K T κ κ Proof Analogous to the proof of Lemma 81 Definition 36 (Induced type groupoid) Let T be a type structure and E 2 T We say E is induced if E is an induced type structure and E is a family of groupoids Since type equality refers to kinding, we will have to refer to the fundamental theorem of kinding in the proof of the fundamental theorem of type equality Lemma 37 (Fund thm of kinding for type groupoids) Let T be a type structure and E 2 T be induced Then, 1 E is a type subgroupoid of 2 T, and 2 if T : κ and (ρ, ρ ) E then (T [[T ]] ρ, T [[T ]] ρ ) E κ Proof 1 By induction on κ and Lemma 35 we show that E κ is groupoidal 2 This is just an instance of Theorem 12, since E[[T ]] (ρ,ρ ) = (T [[T ]] ρ, T [[T ]] ρ ) We are now ready to show the fundamental theorem of type equality Since the β-law is formulated in terms of ordinary substitution and not explicit substitution, we need to ask from interpreting type structures that substitution commutes with evaluation Since this is a characterizing property of combinatory algebras (separating them from mere applicative structures), we use the term combinatory type structure Definition 38 (Combinatory type structure) A structure T is combinatory if [[U]] ρ = [[U]] ρ if (ρ = ρ ) FV(U) [[T [U/X]]] ρ = [[T ]] ρ[x [U ]ρ ] for all U Ty κ, T Ty κ,x:κ and ρ, ρ T type 8

9 The first condition, namely that evaluation of an expression depends only on the values of its free variables, is needed for the η-law Definition 39 (Modelling type equality) Let T a type structure We say that E 2 T models type equality if T = T : κ and (ρ, ρ ) E imply (T [[T ]] ρ, T [[T ]] ρ ) E κ Theorem 40 (Fundamental theorem of type equality) Let T be an combinatory type structure and E 2 T an induced type structure Then E models type equality Proof By induction on T = T : κ We give a few representative cases T : κ κ λx :κ T X = T : κ κ X FV(T ) Since E is induced, it is sufficient to assume arbitrary and (G, G ) E κ and show ([[λx : κ T X]] ρ, [[T ]] ρ ) (G, G ) E κ By Lemma 37, ([[T ]] ρ, [[T ]] ρ ) E κ κ, hence, ([[T ]] ρ G, [[T ]] ρ G ) E κ We are done, since [[λx : κ T X]] ρ G = [[T X]] ρ[x G] = [[T ]] ρ G, for X FV(T ) and T is combinatory, X :κ T : κ U : κ (λx :κ T ) U = T [U/X] : κ Here we use the law [[T [U/X]]] ρ = [[T ]] ρ[x [U ]ρ ] of combinatory type structures T 1 = T 2 : κ T 2 = T 3 : κ T 1 = T 3 : κ Let F i = [[T i ]] ρ for i = 1, 2, 3 By induction hypothesis, (F 1, F 2 ), (F 2, F 3 ) E κ Since E κ is a subgroupoid of 2 T κ and (F 1, F 2 ) (F 2, F 3 ) is defined, we have (F 1, F 3 ) E κ Definition 41 (Respect type equality) A type structure T respects type equality if T = T : κ implies [[T ]] ρ = [[T ]] ρ for all ρ T Lemma 42 (Type structure modulo) Let T and E 2 T be type structures such that E models type equality Then T /E is a type structure respecting type equality Proof Application and evaluation are well-defined in T /E since E is a type structure T /E respects type equality since E models type equality 52 Completeness of NbE for Types Def and Lem 43 (Kind candidate space for completeness) Let T be an reifiable type structure Per κ = {(F, F ) 2 T κ F V κ and F V κ for some V Ty κ } Per κ = {(F, F ) 2 T κ F V κ and F V κ for some V Ty κ } Per κ and Per κ are Kripke families of subgroupoids, and form a kind candidate space Proof Composition (F 1, F 2 ) (F 2, F 3 ) is well-defined since reification is deterministic Def and Lem 44 (Type groupoid for completeness) Let T be a type structure We define P κ 2 T κ by recursion on κ: P := Per, P κ κ := P κ c2 T Pκ P is an induced type groupoid Theorem 45 (Completeness of NbE for Types) Let T be an reifiable type structure If T = T : κ then [[T ]] Var V κ and [[T ]] Var V κ for some V Proof Since (Var, Var) P, by the fundamental theorem of type equality we have ([[T ]] Var, [[T ]] Var ) P κ Per κ which entails the goal 53 Type structure modulo reification Lemma 46 (Strengthening of reification) Let T be a term-like type structure, F T κ, and {, } If F T κ for some, then F T κ Proof By induction on F T κ In the case of a variable X T κ, we know (X) = κ, hence, X X κ Lemma 47 (Reification of neutral and constructed types) Let T be a reifiable type structure Let H T κ be neutral or constructed If H V κ then H U κ and U = V : κ Proof By induction on κ Trivial for κ = In case κ = κ 1 κ 2, we have Xi, X :κ 1 H X V κ 2 ======================== H λx :κ 1 V κ 1 κ 2 Analyzing the induction hypothesis, we get, X : κ 1 H U κ 1 κ 2 and, X :κ 1 U X = V : κ 2 Since H T κ, we get by Lemma 46, H U κ 2, hence, X FV(U) This entails U = λx : κ 1 V : κ 1 κ 2 9

10 Corollary 48 If (H, T ) Per κ and H is neutral or constructed, then (H, T ) Per κ Theorem 49 (Type structure modulo reification) Let T be a reifiable type structure and P defined as above Then T /P is reifiable and respects type equality Proof Since P models type equality, we only need to show that reification is well-defined on T /P First, if F V κ and F V κ and (F, F ) P κ Perκ then V = V since reification is deterministic Secondly, if H U κ and H U κ for neutral or constructed (H, H ) P κ, then by Cor 48 (H, H ) Per κ, hence, U = U 6 Object Structures In this section, we introduce object structures which model both the syntactic object structure Obj indexed by syntactic types in Ty as structures of values D indexed by type values from a structure T The following development, leading up the fundamental theorem of typing and the soundness of NbE for objects, parallels the preceding one on the type level However, while we could define the glueing type structure G κ by induction on kind κ, we cannot define a similar glueing objects structure gl by induction on types, due to impredicativity Hence, we will define gl as a structure of candidates for semantical types Definition 50 (Typing context) Given a type structure T, a T -context T is a partial map from the term variables into T If Γ Ty and ρ T then [[Γ]] ρ T is defined by [[Γ]] ρ (x) = [[Γ(x)]] ρ Definition 51 (Object structure) Let T be a type structure An object structure over T is a family D A (A T ) of Kripke sets indexed by T -contexts such that implies D A = D A It respects type equality, i e, T = T : implies D [T ] ρ = D [T ] ρ for any ρ T, and there are operations: app A B D A B D A TyApp κ F D κ F (G T κ D B, F G ) D We write for both of these operations For, Ψ T, let η D Ψ iff η(x) D Ψ(x) for all x dom(ψ) We stipulate a family of evaluation functions ρ Obj T Γ indexed by ρ T D [Γ] ρ D [T ] ρ which satisfy the following equations: x ρ η = η(x) r s ρ η = r ρ η s ρ η t U ρ η = t ρ η [[U]] ρ λx:u t ρ η d = t ρ η[x d] if d D [U ] ρ ΛX :κ t ρ η G = t η ρ[x G] if G T κ With parallel substitution, Obj (modulo β, βη, or judgmental equality) forms an object structure over Ty (modulo the same equality) Definition 52 (Object substructure) Let S, T be type structures with S T and let D be an object structure over T Let E A D A be a Kripke family of subsets indexed by S for all A S Then E is an object substructure of D over S if application and evaluation are well defined on E app A B E A B E A TyApp κ F E κ F (G S κ ρ Obj Γ T E S [Γ] ρ E B ) E F G E S [T ] ρ for all S and ρ S Definition 53 (Reindexed object structure) Let M : S T a type structure morphism and D an object structure over T The type structure E A := D M(A) over S with is called D reindexed by M f E d := f D d d E G := d D (M(G)) E ρ := D M ρ Definition 54 (Object structure morphism) Let M : S T a type structure morphism and D an object structure over S and E one over T An object structure morphism m : D E is a Kripke family of maps m A D A E M (A) M which commute with application and evaluation, i e, m(f D d) = m(f) E m(d) for all f D A B, d D A m(d D G) = m(d) E M(G) for all d D κ F, G T κ m(d t ρ η) = E t M ρ m η for all T Ty, t ObjΓ T, ρ S S [Γ], η D Definition 55 (Product object structure) Given object structures D 1 over T 1 and D 2 over T 1 we define the product object structure D 1 D 2 over T 1 T 2 by (D 1 D 2 ) (A1,A2) A ( 1, 2) := D 1 A 1 1 D with (f 1, f 2 ) D1 D 2 (d 1, d 2 ) := (f 1 D1 d 1, f 2 D2 d 2 ) (d 1, d 2 ) D1 D 2 (G 1, G 2 ) := (d 1 D1 G 1, d 2 D2 G 2 ) (D 1 D 2 ) t (ρ1,ρ2) η 1,η 2 := (D 1 t ρ1 η 1, D 2 t ρ2 η 2 ) 10

11 61 Realizability Type Structure and the Fundamental Theorem of Typing Fix some term-like type structure T and an object structure D over T Let A D A if A D A and A is Kripke DA forms a complete lattice for all, A Lemma 56 If then D A = D A Proof D A = D A Definition 57 (Function space and type abstraction on D) bd DA D B A B D (A B) := {f D A B for all d,, d A implies f d B } ( ) κ F (G T κ F G ) D D κ F (GA) κ F := {d D κ F d G A } Constructors of higher kind are interpreted as operators on Kripke sets Definition 58 (Kripke operators of higher kind) D A: := D A D F :κ κ := (G Tκ ) D G:κ D F G:κ Definition 59 (Type candidate space) A type candidate space C for D over T consists of two Kripke sets C A, C A D A, (written A, A if no ambiguities arise) for each type A T such that the following conditions hold K-NE H H D H if H neutral K-FUN-E A B A bd B K-FUN-I A bd B A B A B D A B D K-ALL-E κ F GF G D κ F K-ALL-I XF X κ F D κ F if G T κ if X dom() Definition 60 (Realizable semantic types) If F T κ and F :κ F D then F κ C F (pronounced F realizes F) is defined by induction on κ as follows: Definition 61 (Neutral realizable semantic types) Given a type candidate space, for a neutral H T κ we define H:κ H, H D by induction on κ For base kind, it is already defined, for higher kinds we set H(G, G) := H G, H(G, G) := H G Lemma 62 For neutral H T κ, H κ H and H κ H Definition 63 (Type constants in D) D : (A, A)(B, B) := A B κ D κ :(κ ) κ (F, F) := G κ G GF(G, G) Definition 64 (Interpretation into D) For T Ty κ and (σ, ρ) T D we define D[[T ]] σ,ρ D T [T ] σ :κ as follows: D[[C]] σ,ρ = C bd D[[X]] σ,ρ = ρ(x) D[[λX :κ T ]] σ,ρ = ((G, G) T D κ ) D[[T ]] (σ,ρ)[x (G,G)] D[[T U]] σ,ρ = D[[T ]] σ,ρ (T [[U]] σ, D[[U]] σ,ρ ) Lemma 65 (Well-definedness of interpretation into D) If T Ty κ and (σ, ρ) T D then D[[T ]] σ,ρ D T [T ] σ :κ Proof D[[C]] σ,ρ D[[C]] σ,ρ D T [C ] σ :Σ(C) = D C T :Σ(C) = C bd D[[X]] σ,ρ D T [X ] σ :κ = D[[X]] σ,ρ D σ(x):κ = ρ(x) D[[λX :κ T ]] σ,ρ D T [λx:κ T ] σ = ((G, G) T D κ ) D T [λx:κ T ] σ G = ((G, G) T D κ ) D T [T ] σ[x G] D[[λX :κ T ]] σ,ρ = ((G, G) T D κ ) D[[T ]] (σ,ρ)[x (G,G)] A C A : A A A F κ κ C F : F G κ C F(G, G) for all G κ C G We define the Kripke families T D and C T D by T D κ = {(F, F) T κ F :κ D } C κ = {(F, F) T κ F :κ D F κ C F} For the remainder of this section, we fix a type candidate space C D[[T U]] σ,ρ D T [T U ] σ = D T [T ] σ T [U ] σ D[[T U]] σ,ρ = D[[T ]] σ,ρ (T [[U]] σ, D[[U]] σ,ρ ) where, in the last line, D[[T ]] σ,ρ D T [T ] σ :κ κ = ((G, G) T D κ ) D T [T ] σ G:κ 11

12 Lemma 66 (Pre-realizability type structure) T D is a type structure with application (F, F) (G, G) = (F G, F(G, G)) and evaluation T D[[T ]] σ,ρ = (T [[T ]] σ, D[[T ]] σ,ρ ) Since the kind function space is the full set-theoretic one, D is combinatory and respects type equality Lemma 67 ( D is combinatory) Let U Ty κ, T Ty κ,x:κ and (σ, ρ), (σ, ρ ) T D 1 D[[U]]σ,ρ = D[[U]] σ,ρ if ρ(x) = ρ (X) for all X FV(U) 2 D[[T [U/X]]]σ,ρ = D[[T ]] (σ,ρ)[x T b D [U ]σ,ρ ] Proof 1 By induction on U 2 By induction on T, using 1 κ = κ 1 κ 2 and T = λy : κ 1 T Using the induction hypothesis and 1, we have for all κ1 (G, G) T D, since Y FV(U), D[[T [U/X]]] (σ,ρ)[y (G,G)] = D[[T ]] (σ,ρ)[y (G,G)][X T b D [U ](σ,ρ)[y (G,G)] ] = D[[T ]] (σ,ρ)[y (G,G)][X T b D [U ]σ,ρ ] = D[[T ]] (σ,ρ)[x T b D [U ]σ,ρ ][Y (G,G)] hence, D[[λY : κ1 T [U/X]]] σ,ρ = D[[λY : κ 1 T ]] (σ,ρ)[x T b D [U ]σ,ρ ] Lemma 68 ( D respects type equality) If T = T : κ and (σ, ρ) T D then D[[T ]] σ,ρ = D[[T ]] σ,ρ Proof By induction on T = T : κ, using Lemma 67 Lemma 69 (Realizability of type constructors) and κ (κ ) κ Proof Assume A D A and B D B with A A and B B We show A B A B A B A B A B A B A B D F :κ Assume F with F κ F We show κ F κ X:κ (F, F) Let X dom() Then X D,X:κ and X κ X κ F G T GF G κ G κ G GF G G κ G GF(G, G) = κ (F, F) XF(X, X) XF X κ F Theorem 70 (Realizability) C is a type substructure of T D Proof By Def 60 and Lemma 69, C is induced, hence, by Theorem 12, it is a type structure Corollary 71 If T Ty and σ ρ then T [[T ]] σ T [[T ]] σ Proof By the theorem, T [[T ]] σ D[[T ]]σ,ρ Theorem 72 (Fundamental theorem of typing) Let D be an object structure over T and C, C D a type candidate space Let S C a type substructure of the associated realizability type structure C Then the family E (A,A) (,Φ) := A is an object substructure of D reindexed by π 1 : S T Proof Note that E (A,A) (,Φ) D A 1 E respects type equality For T = T : and (σ, ρ) S we have E S [T ] σ,ρ,φ = ( D[[T ]] σ,ρ ) = ( D[[T ]] σ,ρ ) = E S [T ] σ,ρ,φ since D respects type equality 2 Object application is well-defined Let (A, A), (B, B) S (A,A) (B,B) and f E,Φ = (A bd B) and d E (A,A),Φ = A By definition of bd, f d B = E (B,B),Φ 3 Type application is well-defined Let (F, F) S κ, (G, G) S κ, and d E κ (F,F),Φ = ( κ F) = ( G κ G GF(G, G)) = G κ G (GF(G, G)) Since G G, d (GF(G, G)), hence, d E (G, G) = d D G F(G, G) = E (F,F) (G,G),Φ 4 Evaluation is well-defined We show E t σ,ρ η E S [T ] σ,ρ,φ E S [Γ] σ,ρ,φ for (, Φ) S, (σ, ρ) S, and η by induction on ; Γ t : T By definition of E, it is sufficient to show D t σ η ( D[[T ]] σ,ρ ) for η ( D[[Γ]] σ,ρ ) Γ ; Γ x : Γ(x) We have D x σ η = η(x) ( D[[Γ(x)]] σ,ρ ), X :κ; Γ t : T X ; Γ ΛX :κ t : κ T X FV(T ) To show D ΛX : κ t σ η ( D[[ κ T ]] σ,ρ ), assume arbitrary (G, G) S κ, let σ := σ[x 12

13 G] and ρ := ρ[x G], and show D ΛX : κ t σ η E (G, G) ( D[[T ]] σ,ρ (G, G)) Let d := D ΛX : κ t σ η E (G, G) = D t σ η and note that S[[X]] σ,ρ = (G, G) Since (σ, ρ ) S,X:κ, by induction hypothesis d ( D[[T X]] σ,ρ ) = ( D[[T ]] σ,ρ (G, G)), because X F V (T ) and D is combinatory ; Γ t : κ T U : κ ; Γ t U : T U Let (F, F) := S[[T ]] σ,ρ and (G, G) := S[[U]] σ,ρ By induction hypothesis d := D t σ η ( D[[ κ T ]] σ,ρ ) = ( κ b D (F, F)) Since G G, d E (G, G) = d D G F(G, G) = ( D[[T U]] σ,ρ ) ; Γ t : T T = T : ; Γ t : T If view(f) = (e, G) then f = e G and e is neutral If view(f) = then f is not neutral If D is a term-like structure over T, we may write D t ρ for D t ρ var D If T is also term-like we may further abbreviate D t Var T to D t Lemma 74 (Injectivity in a term-like object structure) In a term-like object structure, var and neutral application are injective Definition 75 (Object reification) Given a term-like type structure T and a term-like object structure D over T, we define the relations ; d v A d reifies to v at type A, ; e u A e reifies to u, inferring type A, (where d, e D A and v, u are syntactical objects) inductively by the following rules: ; x x (x) Clear, since D respects type equality ; e u A B ; d v A ; e d u v B 62 Soundness of Nbe for Objects Definition 73 (Term-like object structure) An structure D over T is term-like if: object 1 There exists a Kripke family of maps var (x dom()) D (x) indexed by T In the presence of var, we can define the neutral objects inductively as follows: var (x) D (x) If e D A B f d D B If e D κ F F G e G D is neutral is neutral and d D A is neutral then is neutral and G T κ, then is neutral 2 There exists a Kripke family of computable maps view B such that: D B ObjVar + A T D A B D A + F T κ + { } Definition 76 (Glueing type candidate space) Let S T Ty a glueing candidate, Gl S For (A, T ) {G T κ F G = B} S we define the Kripke families gl (A,T ), gl (A,T ) D Obj (A,T ) by D κ F If view(f) = x then f = var(x) If view(f) = (e, d) then f = e d and e is neutral ; e u κ F G V κ ; e G u V F G ; e u H ; e u H H neutral ;, x:a f x v B A U ; f λx:u v A B, X :κ; f X v F X ; f ΛX :κ v κ F As for types, object reification is deterministic Note that we cannot say now in which Obj Γ T the objects u and v live The conjecture is those Γ, T with Γ and A T However, this does not follow directly from the definition (Not even if we suppose (, Γ) S and (A, T ) S, because of the -elimination rule) The typings of u and v will be a byproduct of Thm 78 gl (A,T ) (,Γ) := {(d, t) ; d v A and ; Γ t = v : T for some v}, gl (A,T ) (,Γ) := {(e, t) ; e u A and ; Γ t = u : T for some u} 13

14 Lemma 77 gl is a type candidate space Proof We show gl A,T gl B,U gl A B,T U (f, v) (gl A,T gl B,U ),Γ =, x:a and Γ = Γ, x:t x x A and Γ x = x : T (x, x) gl A,T,Γ (f x, v x) gl B,U,Γ f x w B Γ v x = w : U A T Γ T = T : f λx:t w A B Γ v = λx:t w : T U (A,T ) (B,U) (f, v) gl,γ assumption by def by def from assumption and for some w and since (A, T ) S and since x FV(w) by def Theorem 78 (Soundness of NbE for objects) Let D be a term-like object structure over a term-like type structure T If ; Γ t : T then there is a long normal form v such that ; T [[Γ]] D t v T [[T ]] and ; Γ t = v : T Proof Consider the type candidate space gl, gl D Obj for the glueing type structure G Recall that gl κ = {((F, T ), F) G κ D Obj (F,T ):κ (F, T ) κ gl F} ((A,T ),A) By the fundamental theorem of typing, E((,Γ),Φ) := A (,Γ) is an object substructure of D Obj reindexed by π 1 : gl G Since (X, X) G (X) and (X, X) (X) gl gl (X,X) we have a variable embedding Var gl (X) = ((X, X), gl (X,X) ) gl (X) into type structure gl Also, since ; x x (x) and ; Γ x = x : Γ(x) we have (x, x) gl ((x),γ(x)) The setting of such (,Γ) that G[[Γ]] = (, Γ) implies (x, x) gl G[Γ(x)] G[Γ] ( D Obj[[Γ(x)]]) G[Γ] = E gl[γ(x)] gl[γ] Hence, setting var gl[γ] Vargl E (x) = (x, x) we have var [Γ] E E gl[γ] gl[γ] It follows that E t = E t Var gl var E E gl[t ] gl[γ] = ( D Obj[[T ]]) G[Γ] gl G[T ] With d := D t we have (d, t) = (D Obj) t = E t, thus, ; d v T [[T ]] and ; Γ t = v : T for some v 7 Object Groupoids Definition 79 (Object groupoid) An object structure D over a type structure T is an object groupoid if each D A is a groupoid and 1 app(f, d) 1 = app(f 1, d 1 ), 2 TyApp(f, G) 1 = TyApp(f 1, G), and 3 if f f and d d are defined, then app(f, d) app(f, d ) = app(f f, d d ) and TyApp(f, G) TyApp(f, G) = TyApp(f f, G) Definition 80 (Groupoidal at higher kinds) Let D be a object groupoid over T Then A D A: is ( -)groupoidal if A is a subgroupoid for each F D F :κ κ is (κ κ -)groupoidal if F(G, G) is (κ -)groupoidal for all G T κ G:κ and (κ-)groupoidal G D Let D F :κ D F :κ := {F F is κ-groupoidal }, T D κ := {(F, F) T κ F :κ T D } D A Lemma 81 (Function space is groupoidal) If A and B D B are groupoidal, so is A bd B D A B Proof First, we show that (A B) is closed under inversion Assume f (A B) To show f 1 (A B), assume arbitrary and d A and show f 1 d B Since d 1 A we have f d 1 B, hence, B (f d 1 ) 1 = f 1 d Then, we show that (A B) is closed under composition Assume f, f (A B) and f f defined Further, assume and d A arbitrary and show (f f ) d B Note that d = d d 1 d, hence, it suffices to show (f d) (f (d 1 d)) B Yet this holds, since f d B and f (d 1 d) B (because d 1 d A ) Lemma 82 (Type abstraction is groupoidal) If A D F G is groupoidal, so is GA D κf Proof Assume f, f (GA) and f f defined Since f G A, also (f G) 1 = f 1 G A Further, (f G) (f G) = (f f ) G A, hence f f (GA) Corollary 83 (Impredicative quantification is groupoidal) κ D κ :(κ ) is (κ ) -groupoidal Theorem 84 (Fund thm of kinding for type groupoids) Let D be an object groupoid over T Then T D is a type substructure of T D 14

15 D Proof By Lemma 81 and Cor 83, T D b T and κ T D b (κ ) ) T D By definition of groupoidal, κ (F, F) T D b (G, G) = (F G, F(G, G)) T D for all D κ κ (F, F) T and (G, G) T D κ, thus, application is well-defined It remains to show that interpretation is well-defined Let T : κ and (σ, ρ) T D One easily shows T D[[T ]] ρ T D κ by induction on T : κ Definition 85 (Square object groupoid) Let D be an object structure over T Then 2 D A = D A D A with (d, d ) 1 = (d, d) (d 1, d 2 ) (d 2, d 3 ) = (d 1, d 3 ) is a type groupoid over T 71 Fundamental theorem of object equality Definition 86 (Combinatory object structure) Let D be an object structure over some type structure T We say D is combinatory if u σ η = u σ η t[u/x] σ η = t σ[x [U ] σ ] η t[u/x] σ η = t σ η[x u σ η ] if σ = σ FV(u) and η = η FV(u) Theorem 87 (Fundamental theorem of object equality) Assume a type structure T that respects type equality and a combinatory object structure D over T Let C, C 2 D be a type candidate space and C its associated realizability type structure If ; Γ t = t : T and (σ, ρ) C and (η, η ) 2 D[[Γ]] σ,ρ then (D t σ η, D t σ η ) 2 D[[T ]] σ,ρ Proof By induction on ; Γ t = t : T We show a few representative cases, X :κ; Γ t : T U : κ ; Γ (ΛX :κ t) U = t[u/x] : T Let (G, G) := T 2 D[[U]] σ,ρ C κ and (σ, ρ ) := (σ, ρ)[x (G, G)] C,X:κ By induction hypothesis, (d, d ) := ( t σ η, t σ η ) 2 D[[T ]] σ,ρ This entails the goal, since 2 D[[T [U/X]]] σ,ρ = 2 D[[T ]] σ,ρ and ( (ΛX : κ t) U σ η, t[u/x] σ η ) = (d, d ) by the combinatory laws ; Γ t : κ T ; Γ ΛX :κ t X = t : κ T X FV(t) Let F := 2 D[[T ]] σ,ρ, d := t σ η and d := t σ η Let further ˆd := ΛX : κ t X σ η We have to show ( ˆd, d ) κ F Assume arbitrary (G, G) C κ It is sufficient to show ( ˆd, d ) G F(G, G) Note that ˆd G = t X η σ[x G] = t σ[x G] η G = d G, since X FV(t) and D is combinatory The desired (d G, d G) F(G, G) now follows from the induction hypothesis (d, d ) κ F ; Γ t = t : κ T U = U : κ ; Γ t U = t U : T U Let d := t σ η and d := t σ η Let further (G, G) := (T 2 D[[U]] σ,ρ ) = (T 2 D[[U ]] σ,ρ ), since T (hence also T 2 D) respects type equality First, observe that t U σ η = d G t U σ η = d G 2 D[[T U]] σ,ρ = 2 D[[T ]] σ,ρ (G, G) By induction hypothesis, (d, d ) 2 D[[ κ T ]] σ,ρ = (G,G) C κ G( 2 D[[T ]] σ,ρ (G, G)) By the fundamental theorem of kinding, (G, G) C κ, hence (d, d ) G = (d G, d G) 2 D[[T ]] σ,ρ (G, G), as expected ; Γ t = t : T T = T : ; Γ t = t : T Trivial, since 2 D respects type equality 72 Completeness of Nbe for objects Definition 88 (Type candidate space for completeness) Let D be a term-like object structure over term-like T We define the Kripke families per A, per A 2 D A: by per A per A := {(d, d ) 2 D A ; d v A and ; d v A for some v} := {(e, e ) 2 D A ; e u A and ; e u A for some u} Note that composition (d 1, d 2 ) (d 2, d 3 ) = (d 1, d 3 ) is welldefined in per A and per A since reification is deterministic Lemma 89 per, per form a type candidate space for 2 D 15

16 Theorem 90 (Completeness of NbE for objects) Let D 0 be a term-like object structure over reifyable T 0 If ; Γ t = t : T then there is a long normal form v such that ; t v A and ; t v A where := T [[Γ]] and A := T [[A]] Proof The term structure T := T 0 /P is reifyable and respects type equality Let D be defined as D 0 reindexed by the representation function T T 0 Consider the type structure per κ = {(F, F) T κ 2 D F :κ F κ per F} The variable embedding for per is given by Var = (X (X, per X )) per E (A,A) (,Φ) := A is an object subgroupoid of 2 D reindexed by π 1 : per T Its variable embedding is given by var E = (var π1 D, var π1 D ) = (x (x, x)) E (,Φ) (,Φ) per By the fundamental theorem of object equality we obtain (D t, D t ) 2 D[[T ]] per A, thus, t v A and t v A for some v 8 An Nbe Algorithm Definition 91 (Value domain) Define D X O as the family of recursive datatypes indexed by X TyVar and O ObjVar with the strict constructors: vard O D X O tyvard X D X O absd [D X O DX O ] DX O appd D X O DX O DX O arrowd D X O foralld Ki D X O Application D [D X O [DX O DX O ]] is defined by (absd f) d = f(d) e d = appd e d if e not an absd Lemma 92 (Monotonicity) D X O is monotone in X and O can be constructed by solving the domain equa- Proof tion D X O D X O = O X [D X O D X O] D X O D X O 1 Ki Since the continuous function space is covariant, O and X appear only positively in this equation Definition 93 (Value type structure) We define Val κ := with D dom() Val Val Val := arrowd κ Val Val (κ ) κ Val := foralld κ Val Val κ κ [Val κ Val κ ] F Val G := F D G Val[[ ]] Ty κ [Val Val κ ] Val[[C]] ρ := C Val Val[[X]] ρ := ρ(x) Val[[T U]] ρ := Val[[T ]] ρ Val Val[[U]] ρ Val[[λX :κ T ]] ρ := (G Val κ ) Val[[T ]] ρ[x G] Lemma 94 Val a term-like and combinatory type structure with computable application and evaluation Hence, it is reifyable Definition 95 (NbE for types) By Thm 25, the following is a well-defined total and computable function: Nbe ( (KiCxt)) (κ Ki) Ty κ Ty κ Nbe κ (F ) := the V with Val[[F ]] V κ Theorem 96 (Nbe decides type equality) Let T, T : κ Then T = T : κ iff Nbe κ (T ) = Nbe κ (T ) Proof Nbe is computable, total, sound, and complete by theorems 25 and 45 Definition 97 (Value object structure) We define val A := D dom dom indexed by A Val and Val with val f val d val f val G val val x ρ η val r s ρ η val t U ρ η val A B := f D d [val A val B ] val κ F (G Val κ ) val := f D G Obj Γ T [val [Γ] ρ = η(x) [(ρ Val ) = val r ρ η val val s ρ η = val t ρ η val [[U]] ρ val [T ] ρ ]] val λx:u t ρ η = (d val [U ] ρ ) val t ρ η[x d] val ΛX :κ t ρ η = (G Val κ ) val t ρ[x G] η F G Lemma 98 val is a term-like and combinatory object structure over Val with computable application and evaluation 16