, Since Thr E (P ) is a countable set, let i be an injection from Thr E (P )/ ? to N \ {0, 1}. thread. Notice that ? asc is functional: if e 1 ? asc e 2 , we write e 2 = asc
, If (a 1 , c 1 ) ? asc (a 2 , c 2 ) then ?a 3 ? {0, 1} *, vol.2
, As in Sec. 11.1.3, we notice that, in a thread, every occurrence has the same applicative depth
, ? If e = a ? supp mut (P ), then ad(e) = ad(a)
, Asc(e) = e. As noted in Observation 11.1, p. 241, a top ascendant is either located in an ax-node or an abs-node (asc is total on app-nodes), motivating the notion of syntactic polarity (see Sec. 11.1.2 for examples), We set, for all e ? bisupp mut (P ), Asc(e) = asc i (e), where i is maximal (i.e. asc i (e) is defined, but not asc i+1 (e)). Thus, Asc(e) is the top ascendant of e. Remark that if e = (a, ?) with a ? Ax P or e = a ? supp mut (P )
, ? If thr(e) = ? and Pol
, ? If ? is left/right-consumed at e and Pol(e) = ? (resp. Pol(e) = )
, Since ? pi also defines an injective function and p 1 ? pi p 2 implies that p 1 (in a ?x) and p 2 (in an axiom) do not have ascendants, compare with Lemma, vol.13, issue.6
, Lemma 13.6 means an edge thread can have at most two connex components, THE SURJECTIVITY OF THE COLLAPSE OF SEQUENTIAL INTERSECTION TYPES Lemmas 13.5 and 13.6 may be illustrated by Fig. 11.1, p.240
, ? If the top occurrence of an ascendant thread is in an ax-rule typing a variable x s.t. x is not
, because if (a, c) ? E(P ), t(a) = ?x and (a, c) does not have an ascendant, then (1) 1 is not a prefix of c (if we had c = 1 · c 0 , then (a · 0, c 0 ) would be an ascendant of, ? Since we consider threads of mutable edges, no thread can have only negative occurrences (contrary to the blue thread in Fig. 11.1)
, Thus, c = k · c 0 for some k ? N \ {0, 1} and (a, c) has a positive polar inverse which is (pos(a · 0
, The referent of ?, denoted ref(?) is the top ascendant of the positive ascendant thread included in ?
, ? A thread whose referent is an argument referent (resp. an axiom referent resp. an inner referent) is said to be an argument thread (resp. an axiom thread, resp. an inner thread)
, The applicative depth of ?, denoted ad(?), is defined by ad(?) = ad(ref(?))
, as observed in Sec. 11.1.3. The referent of a thread ? is the unique element of the intersection ? ?ref(P ), we have, for all e ? ?, ad(e) ad(?), vol.13
, We extended the non-idempotent intersection types of the ?-calculus into non-idempotent intersection and union types for the ? µ-calculus. This allowed us to provide typetheoretic characterizations of head and strong normalization in the ? µ-calculus, along with upper bounds to the length of the head reduction strategy and of all reductions sequences in the former and in the latter case respectively, Perspectives on Part II: the ? µ-Calculus, vol.184
, Obtaining exact bounds for normalizing reduction sequence (instead of just upper bounds) à la B-L, vol.14
, Proving that inhabitation is decidable as for non-idempotent intersection
, Studying the models associated e.g., to H ?µ , investigating the quantitative to qualitative collapse as in, vol.42
, Providing quantitative inter. and union. types for other classical calculi, e.g., the ?µ?µ?µ?µ, vol.33
, Simultaneously, we gave a semantic proof that the hereditary head reduction strategy is complete for (infinitary) weak normalization in the infinitary calculus ? 001 , which is an extension of arguments that were hitherto used in the finite case. Last, we characterized the set of hereditary permutations by means of S-types, which gives a positive answer to TLCA Problem # 20. Many natural extensions of these contributions come to mind (conclusion on p. 232): Infinitary normalization and system S, beyond Klop's question ? Identifying other sets of Böhm trees (besides hereditary permutations) that can be characterized with system S. ? Characterizing strong normalization in ? 001. ? Extending the characterization of WN (and possibly SN) to the other infinitary calculi ? 111 and ? 101, Perspective on Part III: Infinitary Normalization We provided a type-theoretic characterization of the set of hereditary head normalizing ?-terms, thus answering positively to Klop's question
, Perspective on Part IV: Non-Productive Reduction
,
, ? Can we extract a tree-like semantics from R? ? Applications to observational theories (TLCA Problem # 18), possible new semantic proofs, etc. ? Is the collapse from R w to D w (irrelevant non-idempotent intersection to irrelevant idempotent intersection) surjective? ? Investigating whether the proofs of Chapters 12 and and 13 could be reformulated, ? Studying the equational theory of R. ? Does infinitary subject expansion hold
, An intersection type system is then defined as a colored operad (i.e. a symmetric multicategory with possibly more than one object) endowed with 2-arrows, that is, arrows between arrows: a multimorphism generalizes the notion of typed judgment ? t : B, where t is now thought as a multimorphism from ? to B. Intuitively, there is a 2-arrow from ? t : B to ? t : B if the latter judgment is obtained from the former by subject reduction steps. An intersection type system is built from a given restriction of the ?-calculus (e.g., the set of ?-terms endowed with head reduction) and a given subset of the simply typed polyadic calculus by a pullback construction, inspired form the Grothendieck construction. This framework enables us to almost automatically build new systems of intersection types in this way, Polyadic Approximations and Fibrations and Intersection Types: a categorical interpretation of intersection type theory We conclude by saying a few words about a work that was not presented in this thesis, that provides a categorical understanding of intersection type systems (whereas we have been more focused on the "concrete machinery" of typing throughout this document)
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, Reduction and Approximability The rigid constructions presented here ensure "trackability", contrary to multiset constructions of system R. We show here a few applications useful to prove that approximability is stable under reduction or expansion (Lemma 10.5). We consider a derivation P
, We say p 1 subjugates p 2 if, for all finite f P P , p 1 ? f P implies p 2 ? f P
, are equinecessary (written p 1 ? p 2 ) if, for all finite f P P , p 1 ? f P iff p 2 ? f P
, Moreover, let B 1 , B 2 ? P. Then B 1 and B 2 are equinecessary if every p 1 ? B 1 (resp
, There are many elementary equinecessity cases that are easy to observe. We need only a few ones and we define asc(p) and Asc(p) s.t, p ? asc(p) and p ? Asc, p.for
, ? asc(p) is defined for any p ? bisupp(P ) which is not in an axiom leaf.-Left bipositions: asc(a, x, k · c) = (a · , x, k · c)
, asc(a, 1 · c) = (a · 0, c) and asc(a, k · c) = (a · 0, x, k · c) if k 2, Right bipositions (abs): if t(a) = ?x, asc(a, ?) = (a · 0, ?)
, let h be maximal such that asc h p is defined, p) is a right biposition and is defined as the highest right biposition related to p by asc.-Right bipositions: if p = (a, c)
, let h be maximal (if it exists) such that asc h (p) exists, Left bipositions: if p = (a, x, k·c)
, Asc(p) is defined for any right biposition p. If p is quantitative, then Asc(p) is also defined for any left biposition. An examination of the app-rule shows that, if t(a) = @, for all k 2, Since t ? ? 001 (and not in ? 111 ? ? 001 )
, A very important case of equinecessity is this one (with the same notations as in Sec. 10.3.5 e.g., a = b): (a · 1, k · c) ? (a · 10 · a k , x, k · c) and (a · 1, k · c) ? (a · k, c), Assume t| b = (?x.r)s
, B * be the set obtained from Asc(B) by replacing any (a·10·a k , c) by (a·k, c)
, For the converse implication, we just have to replace 0 B by Asc( 0 B )
, This also allows us to prove that a derivation P is approximable iff P is quantitative and, for all finite set of right bipositions 0 B ? bisupp(P ), there exists f P P such that 0 B ? bisupp( f P ). The implication ? is obvious by Definition 10, 4 and Lemma, vol.10, issue.5
, Let 0 B ? bisupp( f P ) be a finite set of bipositions
, Since P is quantitative, for all p ? 0 B, Asc(b) is defined. We set 0 B * = Asc( 0 B), so that
, By equinecessity, 0 B ? bisupp( f P ). imply that, for a derivation P to be approximable, it is enough to have: "P is quantitative and, for all 0 B ? bisupp(P ) finite set of root bipositions, there exists f P P such that 0 B ? bisupp( f P ), B * is a set of right bipositions equinecessary to 0 B. By hypothesis, there is f P P such that 0 B * ? bisupp( f P )
, This allows us to define some binary relations on B, that will be denoted by an arrow. Notice that those relations are defined on B (and not on B ? B), Conditions (c1), (c2), (c3), (c4)
,
,
, For all a ? A such that t(a) = x, for all c ? N * , (a, x, k·c) ? ax (a, c)
, For all a ? A s.t. t(a) = ?x, for all c ? N * , ? Right Ascendance @ (ra@): For all a ? A s, ? Right Ascendance ? (ra?)
, For all a ? A s.t. t(a) = ?x:-(la?1) For all k 2, c ? N * , (a, k·c) ? asc (a·0, x, k·c)-(la?2) For all y ? V , k 2, ? Left Ascendance ? (la?)
, For all a ? A, x ? V , k 2 s.t. t(a) = @ and (a, x, k) ? B, for all c ? N * , (a, x, k·c) ? asc (a · , x, k · c)
, For all a ? A s.t. t(a) = @, for all k 2
, We will consider ? @ , the reflexive transitive symmetric closure of ? ax ? ? asc ? ? pi ? ?: relation ? @ helps us to track a type symbol through a derivation. For instance, in P ex
, All those bipositions also point to type variable o. An attentive observation of the relations above shows that, for P to be a derivation typing t, then B must be closed under ? t1 , ? t2 , ? @. Type formation ensures that the supports of the types (on the right-hand sides of ) are well formed e.g., closure under ? t1 ensures that the supports of types are trees and closure under ? t2 means that any non terminal node in a type has a son on track 1, We have (?, 1) ? asc (0, ?) ? asc (03, ?) ax ? (03, x, 2) asc
, Right ascendance explains how the type given in a rule is partially computed from the type given in one of its premises
, FINITE OR NOT) APPROXIMATIONS 341 at position a. Consumption is related to the app-rule: the left-hand side of the arrow type typing the left-hand side of the application (bipositions of the form (a·1, k·c) must be equal to the sequence of types given to the argument
, (c4), (c5) and closure under? t1 ,? t2 , ? @ are not enough to guarantee that P is a derivation, Conditions (c1), (c2), (c3)
, ? Non terminal nodes are arrow (lab1): For all p ? B \ Lves(B), P (p) =?
, ? Leaves are type variables (lab2): For all p ? Lves(B), P (p) ? O
, ? Matching leaves labels (lab3): for all p 1 , p 2 ? Lves(B) such that p 1 ? @ p, vol.2
, We claim and prove: Proposition A.1. The function P is a S-derivation typing t iff is satisfies (c1), (c2), (c3), (c4), (c5), (lab1), (lab2), (lab3) and is closed under ? t1
, The necessity of those conditions has been discussed
, lab2), it easy to check that, for all a ? A and x ? V , T (a) is a type and C(a)(x) is sequence type of Typ 111. Using (c5, lab1
, let k be the unique integer 2 such that (a, x, k) ? B. By (axf ), closure by ? @ and (lab3) (w.r.t. ? ax ), the function (c ? C(a)(x)(c) and (c ? T (a)(c) are equal. Moreover, by (r1) again, ? Correctness of ax-rules: let a ? A such that t(a) = x. By (r1)
, By (r2), (?, 1) ? B, so (a, ?) / ? Lves(B) and 1 ? supp(T (a)) so T (a) is an arrow type (if it were a type variable, ? Correctness of abs-rules: let a ? A such that t(a) = ?x
, (a)) comes from the definition of left ascendance (see (la?1) and (la?2)) and (lab3) (w.r.t. ? asc ). By (ra?) and (lab3) (w.r.t. ? asc ), the functions c ? T (a)(1 · c) and c ? T (a · 0)(x) are equal i
,
, The fact that C(a) = ?{1}?K C(a · ) comes from the definition of left ascendance (see (laapp)) and (lab3) (w.r.t. ? asc ), but also from (c4) and the fact that every C(a · )(x) is a correct sequence judgment. By (c1), (a, ?) ? B, so, by (raapp), ? Correctness of app-rules: let a ? A such that t(a) = x
, · c) APPENDIX A. COMPLEMENTS TO KLOP'S PROBLEM are equal i.e. T (a) = Tg(T (a · 1)). By (con) and (lab3) (w.r.t. ?), the functions (k, c) ? T (a · 1)(k · c) and (k, c) ? T (a · k, By (ra@) and (lab3) (w.r.t. ? asc ), the functions c ? T (a)(c) and c ? T (a · 1
,
, Let (P i ) i?I be a non-empty family of derivations typing the same term t, such that ?i, A.3.1 Meets and Joins of Derivations Families Lemma A.1
, We check first that if one of the P i satisfies (c1), (c2), (c3), (c4), (c5), then P also does
, This works because:-For all a ? A := out(B) such that t(a) = x, for all i ? I, ? The stability of P under ? t1 , ? t2 and ? @ comes from the stability of under those for the P i. However, ? @ depends on B := bisupp(P )
, such that t(a) = @ and (a, x, k) ? B, for all i ? I, uptr P (a, x, k) = uptr P i (a
, Thus, the relation ? @ of P is the intersection of those of the P i (i ranging over I)
, P ) iff p ? Lves(()P i ) for all i. Thus, P also satisfies (lab1) and (lab2). Finally, P satisfies
, Let (P i ) i?I be a non-empty family of derivations typing the same term, such that ?i, vol.2
, P i ) for some i, then P i (p) = o (for some o ? O), so, for all j ? I such that p ? bisupp(P j ), P j (p) = o i.e. p ? Lves(()P j ). Thus, p ? Lves(B), ? We check first that, since P i satisfies (c1), (c2), (c3), (c4)
, · S n ) ? o) ? PP o In the case above, we also say that o is the last type variable of (S, T ) (for short, · T ?(n) ) ? o)
, PPP o ) for all o ? O, o occurs at most once in S
, Let us set t = p x. Let y ? o y be an injection from B = supp(t) to O. We associate to each b ? supp(t) two indeterminates X b and Y b. The idea is that X b is a placeholder for the types of head variables and Y b is a placeholder for the types of the sub-hereditary permutations of t. We denote by B p the set of positions b of subterms of t that are y-HP for some y ? V and, for all b ? B p, The condition of properness is here to ensure that every variable occurs at a level deeper than its binder and to distinguish them from one another (see the proof of Claim A.2), p.hvp
, (b) is the longest b 0 such b 0 ? b · {0, 1} *. For all b ? B p , we abusively denote by or(b) the order or(t| b ) of t| b and by x b the head variable of hv, p.hvp
, p y?n with n = or(b) 0, y = x b and ? ? S n. We then denote by ? b the permutation ? and we set set, vol.1, p.1
, S(b) = F(b)
, T (t)) is a proper permutation pair. From there, it is not difficult to built a (quantitative) S-derivation P , such that, for all b ? B p , P (b) = x b : (2 · S(b)) t| b : T (b). Since t is a normal form, By proceeding as in Sec. A.4, we prove that, for all b ? B p , (S(b)
, P an approximable S-derivation and U a permutation type such that P t : U. Then t is a hereditary permutation
, 7, the empty sequence type ( ) does not occur in U , so that t : U is unforgetful. Since P is approximable
, Since T and S are types of order n, we must have p = q n (the abs-rule creates an arrow whereas the app-rule destroys one). Moreover, S 1, · S) ?y 1. .. y p .x t 1. .. t q : T is derivable by means of an approximable derivation P *
, o ?(p) are the respective last type variable of T ?(1)
, , vol.1
, 5 expresses the fact that, in an unforgetful derivation typing a normal form, every type nested in a subterm at applicative depth n occurs at applicative
, Let P be an approximable derivation concluding with C t : T , where t is a normal form, and a ? bisupp(P ) such that ad, p.1
, t q occur at applicative depth 1 in T 0. Moreover, the types of all subterms occurring at applicative depth 1 are nested in the APPENDIX B. RESIDUATION, THREADS AND ISOMORPHISMS IN SYSTEM S op this root interface, we will now build a hybrid derivation P concluding with C t : T with T ? T , as it was announced at the beginning of Sec. 13.2. Conventions on metavariables a and ? As in Sec. 10.3.5 and 12.4.1, the letter a will stand for a position of P that corresponds to the root of the redex (i.e. a ? supp(P ) and a = b) and the letter ? for other positions in supp(P ) or even in A. We set Ax ? (a) = Ax P a·1·0 (x) and Tr ? (a) = {tr P (? 0 ) | ? 0 ? Ax ? (a)} = Rt(T P (a · 1)). Thus, Ax ? (a) is the set of positions of the redex variable (to be substituted) above a and Tr ? (a) is the set of the axiom tracks that have been used for them. For instance, system R 0 ). Thus, the types of the arguments t 1
, After reduction, the app-rule and abs-rule at positions a and a · 0 have been destroyed and the position of this judgment ? ?. Likewise, ? represents a judgment, Thus, its position must be of the form a · 1 · 0 · ? ?
, Thus, its position must
, ? Paradigm ?: if ? = a·k R ·? 0 where a = b and k R ? Arg(a), then Res b (?) = a·a kL ·? 0 with k L = ? ?1 a (k R )
, ? Paradigm ?: if ? = a · 1 · 0 · ? 0 where a ? supp(P ), a = b and ? 0 = a k
, Outside the redex: if b ?, then Res b (?) = ? APPENDIX B. RESIDUATION, THREADS AND ISOMORPHISMS IN SYSTEM S op We define then P as the labelled tree s.t. supp(P ) = A and for all ? ? A
, We intend to prove that P is a correct hybrid derivation typing t. As hinted at Sec. 13.2, we must check that a type T(?) (where ? ? A) may only be replaced, C (? ) t | ? : T (? ) (so that C = C P and T = T P as expected)
, Quasi-Residuation in the Hybrid Setting In order to check that, it is convenient to extend residuation into quasi-residuation: namely, we define quasi-residual QRes b (?)
, ? We do not necessarily have t(?) = t (? ) or child(?) = child (? ) when ? = QRes b (?) (compare with Lemma B.1) and QRes b is usually not injective. For instance, if t = (?x.y)y, t = y, b = ? = ? = ? , then t b ? t , ? = QRes b (?) but t(?) = @ = y = t (? ) and child
, quasi-residuals will be useful to define the isomorphisms Res b|? , ResR b|? and ResL b|? below
, This lemma is also a definition: that of the quasi-residuation, vol.2
, (?) where ? = Res ?1 b (? ). Besides, if ? ? Ax or ? a · a k (for a = b, then T (? ) = T(?)
, ? More precisely, if P is endowed with an interface (? a ) a=b at position b (extending the root-interface (? a ) a=b ), then, for all ? ? A and ? ? A such that QRes b (?) = ?
, ? When Res b (?) = ? , we write Res b|? instead of QRes b|?. Moreover, Res b|? is the identity if ? ? Ax or ? a · a k for some a ? A, a = b and k ? AxTr
, We proceed by 001-induction on ? ? A := supp(P ) and split the cases as suggested in Remark B.2. ? Paradigm ?: Res b (?) = ? and ? a · a k (for some a ? A
, Subcase t(?) = y: here, t(?) = y = x and T (? ) = T(?)
, Subcase t(?) = ?y: ? · 0 ? A, Res b (? · 0) = ? · 0 and by IH, we have T (? · 0) = T(? · 0) and Res b|?·0 is the identity id T(?·0)
, Since T(?) = C(? · 0)(y) ? T(? · 0) and T (? ) = C(? · 0)(y) ? T (? · 0), we also have T(?) = T (?) and we set Res b|? = id T(?)
, Subcase t(?) = @: ? · 1 ? A, Res b (? · 1) = ? · 1 and by IH, we have T (? · 0) = T(? · 0) and Res b|?·1 is the identity id T(?·1)
, Moreover, T(?) = Tg(T(? · 1)) and T (? ) = Tg(T (? )). So T(?) = T(? ) and we set Res b|? = id T(?)
, = b):-Subcase ? = a · 1 · 0 · a kL and ? = a · a kL : ? = Res b (a · k R ) (where k R = ? a (k L )) and by IH, T (? ) = T(a · k R ). Moreover, since T(?) = L(?)| kL , we can set QRes b|? = ? a|kL
, Subcase t(?) = ?y: ?·0 ? A, Res b (?·0) = ? ·0: we set Res b|? = C(?·0)(y) ? QRes b|?·0
, Subcase t(?) = @: ?·1 ? A, Res b (?·1) = ? ·1: we set Res b|? = Tg(QRes b|?·1 )
, ? Outside the redex: ? b:-Subcase ? ? Ax : here, t(?) = y = x and T (? ) = T(?)
, a = Res b (a · 1 · 0) and by IH, we have an type isomorphism QRes b|a·1·0 : T(a · 1 · 0) ? T(a). Since T(a · 1 · 0) = T(a), we can set QRes b|a = QRes b|a·1·0
, Subcase t(?) = ?y: ? · 0 ? A, QRes b (? · 0) = ? · 0 and we set Res b|? = C(? · 0)(y) ? QRes b|?·0
, Subcase t(?) = @: ?·1 ? A, QRes b (?·1) = ? ·1: we set Res b|? = Tg(QRes b|?·1 )
, ? It is far easier to define the residual of a biposition for a derivation of S: if P is trivial, whenever ? := Res b (?) is defined, the residual biposition of p := (?, ?) ? bisupp(P ) is Res b
, bisupp(P ) such that ? = b · 1, whereas in Sec. 12.4.1, it was also defined for ? = a · 1. Actually, we will extend QRes b (?, c) in this case (see Remark B.7), but now, quasi-residuation is defined for all
, Residual Interface We notice that if ? ? supp @ (P ) and ? = b, then Res b (?) is defined. So, for ? ? A @ (Sec. B.1.1), we set L (? ) = Sc(T (? · 1)), ArgTr (? ) = {k 2 | ? ? A } and R (? ) = (k · T (? · k)) k?ArgTr (?). We write then Inter (? ) for the set of sequence type isomor
, Thanks to Lemma B.1: ? Since QRes b|?·1 is a type isomorphism from T(? · 1) to T (? · 1) and L (? ) = Sc(T (? · 1)), then T (? · 1) is an arrow type (since T(? · 1) is) and we define the sequence type isomorphism, Assume that ? ? A @. Let us write ? = Res ?1 b (? )
, We can define ResR b|? by ResR b|? (k · ?) = k · Res b|?·k (?)
, Since ? supp induces a bijection from ArgTr 1 (a 1 ) to ArgTr 2 (a 2 ) (notation ArgTr is defined on p. 294), we define a sequence type isomorphism ? R
, The isomorphisms between S-derivations defined in Appendix A.5 are isomorphisms of operable derivations
, Ax 0 , we set T 0 (a 0 ) = ? a
, {? supp (a 0 ) | al 0 ? Ax ? (x)} for all a ? A and x ? V. Since ? supp induces a bijection from Ax to Ax 0 , we can set tr 0 (?(a)) = ? tr (a) for all a ? Ax, ? Let Ax 0 (?(a))(x) =
, Since ? tr is an injection whose domain is Ax, we can define pos 0 : codom(()? tr ) ? Ax with pos 0 (k 0 ) = a 0
, can write ? a,x for the context isomorphism from C(a)(x) to C 0 (a 0 )(x) such that ? a, x (k ·?) = ? tr
, We can now define a type T 0 (?(a)) and a type isomorphism ? a from T(a) to T 0 (?(a)) for all a ? A by 001-induction
, If t(a) = ?x, we set T 0 (a 0 ) = C 0 (a 0 )(x) ? T 0 (a 0 · 0) and ? a = ? a·0,x ? ? a·0
, We set then ArgTr 0 (?(a)) = {k 0 2 | ?k ? ArgTr(a)