Higman's Lemma in Type Theory

This thesis is about exploring the possibilities of a limited version of Martin-L of's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the rst paper, A Lambda Calculus Model of Martin-L of's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of Martin-Lof's logical framework with explicit substitution extended with some inductively de ned sets, also given in complete detail. These inductively de ned sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossibility of proving some intuitively trivial statements like, for instance, Peano's fourth axiom. We prove that this also applies to the present formalism. This is proved in the standard way by exhibiting a proof-irrelevant model which, however, is unexpectedly di cult to de ne. The solution presented in the paper consists of giving an interpretation of the logical framework in terms of the untyped lambda calculus. As particular instances of this interpretation we obtain two models: a proof-irrelevant model and a realizability model. We illustrate through examples how to extend these two models to cope with inductively de ned sets. Both the lambda calculus model and the proof-irrelevant model are given in detail. The realizability model is only outlined. The goal of the second paper, Higman's Lemma in Type Theory, is to present a proof of Higman's lemma in the limited version of type theory given in the rst paper. We present a brief account of the history of Higman's lemma which shows that in most constructive proofs the binary relation involved in the statement of the lemma is required to be decidable. This is an important assumption in constructive mathematics. Recently, Veldman showed the intuitionistic validity of the lemma without that assumption. His is the most general formulation of Higman's lemma and his proof is the one we interpreted in the limited version of type theory. The proof-irrelevant model de ned in the rst paper was very helpful during the formalization. Just as that model shows that Peano's fourth axiom is not provable, it easily shows that many other intuitively true statements are not provable either. Thus it helped avoiding hopeless attempts of proving intermediate results and it also helped nding appropriate reformulations of them. The formalization was carried out in ALF. It would have been impossible without a proof-editor. Whereas in the rst paper we prove some limitations of the formalism, in the second we show that it is nevertheless suitable for formalizing non-trivial statements, like that of Higman's lemma. Acknowledgments This thesis is certainly not the result of work made in isolation. On the contrary, this thesis and I have bene ted greatly from previous works, conversations with many people, the social and research environments, and the support of friends and relatives here and in the distance. I am deeply indebted to Thierry Coquand. He is a never-ending source of ideas, many of which contributed substantially to this thesis. He has always been patient with me, listening to my intuitions however vaguely formulated and suggesting the magic step or spotting the error. His open mind and enthusiasm in research are inspiring. I consider myself very fortunate to have had him as supervisor. I want to thank the members of my supervision committee, Thierry Coquand, Lars Hallnas and Jan Smith, for helping to lead this thesis to an end. Jan Smith has constantly supported me showing complete con dence in me and my work. I am grateful to him not only for that but also for reading and commenting previous versions of the papers. Many people have contributed to this thesis through fruitful discussions or by reading and commenting previous versions of it. Special thanks to Stefano Berardi, Gustavo Betarte, Marc Bezem, Peter Dybjer, Ver onica Gaspes, Henrik Persson and Alvaro Tasistro. I am grateful to Wim Veldman for communicating to me his not yet published proof of Higman's lemma, which is one of the main subjects of this thesis. Thanks to Bengt Nordstrom for his generosity and for always being very positive towards my work. That has been very encouraging. I was fortunate to have had the chance to come to Sweden and to this department where I had the opportunity to work in a very stimulating and friendly environment. Working here is very comfortable thanks to many people who make this the best possible place to work at. Special thanks to Christer Carlsson for all his help. My gratitude also to Lena L oov. She has been very patient with me, especially during the last time when I was so busy. I am glad that I dared to cross the invisible line which separates mathematicians and statisticians on one side and computing scientists on the other. I am very happy to have today many friends at both sides. All these people and other friends and relatives helped to the development of my work directly or indirectly, by making my life more enjoyable. To all of them, many thanks. Higman's Lemma in Type Theory: Errata Daniel Fridlender October 29, 1997 I am very grateful to Monika Seisenberger for letting me know that Diana Schmidt presented a constructive proof of Higman's lemma for decidable relations in her Habilitationsschrift Well-Partial-Orderings and Their Maximal Order Types in 1979. In the historical overview of Higman's lemma in the second paper of this thesis, I wrote that Sch utte and Simpson's proof \would be the rst constructive proof of Higman's lemma", but in fact it would have been inspired by the work by Schmidt. Schmidt in turn refers to the paper Well-Partial Orderings and Hierarchies by de Jongh and Parikh, Indagationes Mathematicae, Volume 39, 1977, where a proof of Higman's lemma for a nite alphabet is given. I have not examined the proofs in these two papers in detail as yet. I found two mistakes in the introduction to the thesis. The rst one is in the de nition of the function find' on page 2, in which the auxiliary function indexes should have rel as an argument, too. The second is in the de nition of V (w) in the third paragraph on page 9. Instead of being de ned as 8v U(wv) it should be de ned as \there exists an initial segment v of w such that U(v)". Introduction Many interesting questions arise when trying to prove Higman's lemma in a disciplined way; questions regarding the formulation of its statement, the reliability of its proofs, their computational content. Indeed, the literature o ers various di erent statements called Higman's lemma, some controversial proofs of them, many uncertainties concerning their computational content. As a logic for mathematics and as a programming language, Martin-Lof's type theory is an appropriate instrument to address some of those issues. We will refer to Martin-Lof's type theory simply as type theory from now on. This thesis is about a proof of Higman's lemma in type theory. It presents two papers. The rst one is devoted to the study of metatheoretical aspects of the particular version of type theory which is used in the second paper to prove Higman's lemma. The work presented in the rst paper was actually motivated by the proof given in the second. Proving Higman's lemma in type theory raises further questions regarding the representation of the notions involved in its formulation. The statements needed to prove the lemma, like bar induction, the fan theorem, Ramsey theorem and Higman's lemma itself, must be correctly interpreted in type theory. This is a fundamental issue, since na ve interpretations would lead to unsuccessful proof attempts. Therefore it is necessary to use more elaborated type-theoretic interpretations of those statements. With this, a statement and its interpretation become more dissimilar. It is then important to make sure that the original meaning is preserved. Higman's lemma as a programming problem This section is intended as an intuitive introduction to the problem of proving Higman's lemma in type theory, by interpreting the lemma as a programming problem. Higman's lemma will be seen as a speci cation of a program and, as it is usual in programming, a solution would consist in writing a program and giving a justi cation |in a broad sense| that it satis es the speci cation. Higman's lemma is about words and binary relations. Given a set A we denote by A the set of nite words over the alphabet A. Given a binary relation S on A and two words v and w we say that v is S-embeddable in w if there is a subword v0 of w such that v and v0 have the same length and each letter of v is S-related to the corresponding letter in v0. This relation among words is denoted S . If the alphabet is the set of natural numbers, for instance, the word 5; 7; 3; 8 is -embeddable in 8; 7; 0; 9; 9 but not in 200; 6; 500; 400; 7; 7; 7; 7. In general, given a program relS which decides S, that is, which decides for any two elements of A whether the rst of them is S-related to the second or not, it is possible to write a program which decides S . In a functional programming style we write: 1 embeddS = embedd relS embedd :: (a -> a -> Bool) -> [a] -> [a] -> Bool embedd rel [] w = True embedd rel (a:v) [] = False embedd rel (a:v) (b:w) = if rel a b then embedd rel v w else embedd rel (a:v) w The relation S is unavoidable if for all in nite sequence a0; : : : ; an; : : : there are two elements ai and aj such that ai precedes aj (that is, i < j) and ai is S-related to aj. The relation on natural numbers is unavoidable. The property of being unavoidable is called nite basis property in the second paper of this thesis. A constructive proof that a relation is unavoidable contains implicitly a program which when applied to an in nite sequence nds i and j as above. We say that such a program guarantees S. Given a program relS which decides S, then it is easy to write a program which guarantees S, if S is unavoidable. We distinguish between the type of streams {_} and the type of lists [_]. findS = find relS find :: (a -> a -> Bool) -> {a} -> (Nat,Nat) find rel as = find' rel [] as find' rel acc (a:as) = if any (\x -> rel x a) acc then indexes a acc else find' rel (a:acc) as It is a linear search program which maintains an accumulator with all the elements received so far. When a new element arrives the program tests whether any of those elements received earlier is S-related to the new one, in which case the auxiliary function indexes computes the corresponding i and j. Otherwise it stores the new element in the accumulator and continues. Since S is assumed to be unavoidable findS terminates. Higman's lemma Given a binary relation S, if S is unavoidable then S is unavoidable as well. In terms of programs, Higman's lemma states that if there is a program that guarantees S, then there is also one that guarantees S . The question is how to construct it. Again, if there is a program relS which decides S, the simple program findSstar = find embeddS 2 would be enough, provided there is a proof that it terminates. A possible proof of termination would be a classical proof of Higman's lemma, like the one below. A rst di erence with the formal proof of Higman's lemma in this thesis is precisely this last step. Appealing to a non-constructive proof of termination means to apply Markov's principle which is not intuitionistically valid. Another di erence is that, unlike other functional programming languages, in type theory it is only possible to write programs which are terminating by construction. This is not the case of find' above. But the main di erence is that in the formal proof S is not assumed to be decidable. That is, it is not assumed that there is a program relS which decides S. Hence, the problem of writing findSstar is more di cult. The question is how to nd, among in nitely many words, two such that one of them is S-embeddable in the other when it is not even possible to test whether two given letters are S-related. The answer to this question resides in exploiting the constructive content of the only assumption of Higman's lemma, that S is unavoidable. Veldman [Vel94] discovered how to do it in intuitionistic mathematics. The proof in this thesis is an interpretation in type theory of Veldman's proof. As such, it contains implicitly a program which does not use relS and solves Higman's lemma. A classical proof of Higman's lemma The program findSstar above is very simple. It is just a linear search program. Then, to prove that it terminates it is necessary to prove that the element which is being sought exists. In this case, this amounts to proving Higman's lemma itself. So the program contributes actually nothing to the proof of the lemma. This explains its simplicity. There are several di erent proofs of Higman's lemma. Many of them, if not all, use some formulation of Ramsey theorem, usually implicitly. Among those proofs, the one presented here is the shortest and the easiest to show. It is a classical and impredicative argument due to Nash-Williams [Nas63], which is adapted here to make explicit the use of the following version of Ramsey theorem. Ramsey theorem In every in nite graph, either there is an in nite subgraph in which every two vertices are adjacent, or there is an in nite subgraph in which no two vertices are adjacent. Nash-Williams' proof of Higman's lemma is by contradiction. Suppose that for some S, S is unavoidable but S is not. Then, it must exist an in nite sequence that avoids S . We will construct a minimal in nite sequence v0; : : : ; vn; : : : that avoids S . We select a word at a time, and describe this selection by induction. We take v0 to be a word v such that there is an in nite sequence of words v; w0; : : : ; wn; : : : that avoids S but there is no in nite sequence w;w0; : : : ; wn; : : : avoiding S such that w is shorter than v. 3 Once vi has been de ned, we take vi+1 to be a word v such that there is an in nite sequence v0; : : : ; vi; v; w0; : : : ; wn; : : : that avoids S but there is no in nite sequence v0; : : : ; vi; w; w0; : : : ; wn; : : : avoiding S such that w is shorter than v. The in nite sequence v0; : : : ; vn; : : : constructed avoids S . Otherwise, there would be i and j such that i < j and vi S-embeddable in vj which is impossible since when vj was chosen it existed an in nite sequence v0; : : : ; vj; w0; : : : ; wn; : : : avoiding S . Therefore, the in nite sequence v0; : : : ; vn; : : : is a sequence avoiding S , minimal by construction. The contradiction will be obtained by proving that there is an even smaller in nite sequence avoiding S . There is no n such that vn is empty. If it were such an n then vn would be S-embeddable in vn+1 and v0; : : : ; vn; : : : would not avoid S . Hence, for all n, vn is of the form anwn for a letter an and a word wn. Now we will determine an in nite graph in order to be able to apply Ramsey theorem. The vertices are pairs of the form (ai; i). Given two vertices (a; i) and (b; j) such that i < j, there is an edge between them i a is S-related to b. By Ramsey theorem either there is an in nite subgraph in which every two vertices are adjacent or there is an in nite subgraph in which no two vertices are adjacent. We will see that the latter does not hold. If it did, the vertices (ak0 ; k0); : : : ; (akn; kn); : : : of that in nite subgraph (with k0 < : : : < kn < : : :) would be such that the in nite sequence ak0 ; : : : ; akn ; : : : avoids S. This would contradict the assumption of Higman's lemma that S is unavoidable. Then, it must be an in nite subgraph in which every two vertices are adjacent. That is, an in nite sequence (ak0; k0); : : : ; (akn; kn); : : : with k0 < : : : < kn < : : : and every element akn is S-related to all of its successors. From this, and from the fact that v0; : : : ; vn; : : : avoids S it follows that v0; : : : ; vk0 1; wk0; wk1; : : : ; wkn; : : : avoids S as well, contradicting the minimality of vk0 . This nishes the proof of Higman's lemma. In this proof, classical reasoning is essential. In addition, the proof is impredicative since an in nite sequence avoiding S is constructed by referring to the totality of in nite sequences avoiding S . We also prove Ramsey theorem. Given an in nite graph we have to show that there is an in nite subgraph such that every two vertices are adjacent or an in nite subgraph such that no two vertices are adjacent. We rst try to build an in nite subgraph satisfying the rst alternative. We select a vertex at a time, and describe this selection by induction. We take rst any vertex v0 which is adjacent to in nitely many other vertices. Then, the vertex v0 and all those vertices which are not adjacent to v0 are removed from the graph. From the way v0 was selected the graph is still in nite after this removal. Once the vertex vi has been selected, we take any vertex vi+1 which is adjacent to in nitely many other vertices. Then, the vertex vi+1 and all those vertices 4 which are not adjacent to vi+1 are removed from the graph. Again, the resulting graph is still in nite after this removal. This method of selection gives an in nite subgraph of the original in which every two vertex are adjacent. But the method may certainly fail if at some stage there is no vertex v such that v is adjacent to in nitely many other vertices. In this situation we realize that we should attempt to build an in nite subgraph satisfying the second alternative of Ramsey theorem. At that stage, in spite of the removals the graph is still in nite and every vertex in it is only adjacent to nitely many other vertices. Then, a selection process analogous to the previous would build an in nite subgraph in which no two vertices are adjacent. This nishes the proof of Ramsey's theorem, also classical. But Higman's lemma can be given a constructive proof whereas Ramsey's theorem, as it stands, cannot. Interpreting Ramsey theorem Anybody familiar with the usual statements of bar induction, the fan theorem, Ramsey theorem and Higman's lemma may be surprised with the formal statements in the second paper of this thesis bearing those names. These are statements that needed to be given suitable type-theoretic interpretation. This situation is characteristic in constructive mathematics. A classical notion frequently admits di erent constructive interpretations. The justi cation for adopting a seemingly di erent formulation typically relies on the classical equivalence between the original formulation and its interpretation. Sometimes, a similar phenomenon occurs when interpreting an intuitionistic notion in type theory. The justi cation for adopting an apparently di erent formulation should rely on the intuitionistic equivalence between the intuitionistic and the type-theoretic formulations. A very evident gap occurs between Ramsey theorem as stated above and the interpretation we give of it in type theory. The reason for this is that, whereas bar induction, the fan theorem and Higman's lemma are intuitionistically valid, Ramsey theorem's original formulation is not and therefore, it is interpreted twice, rst into intuitionism and then into type theory. An intuitionistic interpretation of it is the following variation of the intuitionistic Ramsey theorem as presented in [VB93]. Intuitionistic Ramsey theorem Given two sets A and B, a relation R on A and a relation S on B, if R and S are unavoidable, so is R S. The proof that Ramsey theorem implies the intuitionistic version is as follows. Given any in nite sequence (a0; b0); : : : ; (an; bn); : : : of elements of A B we want to nd i and j such that i < j, ai is R-related to aj and bi is S-related to bj. 5 Like in the proof of Higman's lemma we can use Ramsey's theorem to build an in nite subsequence ak0 ; : : : ; akn; : : : such that k0 < : : : < kn < : : : and every akn is R-related to all of its successors. As S is unavoidable, for the in nite sequence bk0 ; : : : ; bkn; : : : there are i and j such that i < j and bki is S-related to bkj . The proof of the converse is classical. We take any in nite graph and de ne two binary relations on vertices, R and S. Given two vertices, they are R-related if they are equal or adjacent, and they are S-related if they are equal or not adjacent. Given an in nite sequence of pairwise di erent vertices v0; : : : ; vn; : : :, the sequence (v0; v0); : : : ; (vn; vn); : : : avoids R S. By the intuitionistic Ramsey theorem either R or S is avoidable. If it is R, then there is an in nite subgraph such that no two vertices are adjacent. If it is S, then there is an in nite subgraph in which every two vertices are adjacent. Inductive bars The formal statements of bar induction, the fan theorem, Ramsey theorem and Higman's lemma in the second paper of this thesis rely on the notion of inductive bar, an interpretation of the intuitionistic notion of bar in type theory. We take n to range over natural numbers, u, v and w over nite sequences of natural numbers and over in nite sequences of natural numbers. Given two such sequences, uv denotes the concatenation of u and v. We denote by (n) the (n+1)-th term of the sequence , and by (n) the nite initial segment (0); : : : ; (n 1) of . According to the de nition of bar below, a given predicate U over nite sequences of natural numbers is a bar if 8 9n U( (n)) holds. DEFINITION (Bar) Given a predicate U over the set of nite sequences of natural numbers, then U is a bar if every in nite sequence of natural numbers has an initial segment which satis es U . Given a nite sequence u, U bars u if every in nite sequence having u as initial segment, has an initial segment which satis es U . Observe that U is a bar if and only if U bars <>, where <> denotes the empty sequence. The notion of bar has no direct interpretation in type theory, since it is based on the intuitionistic notion of in nite sequence, which does not have a typetheoretic counterpart. In nite sequences and bars are of central importance in intuitionism. They are fundamental for the intuitionistic reconstruction of the continuum. Of special interest is Brouwer's analysis of the structure of a hypothetical proof that a 6 predicate is a bar. Di erent presentation of his analysis can be seen for instance in [Bro54], [Dum77] and [MG79]. There are di erent ways of formulating a statement interpreting Brouwer's analysis. One of them is expressed by the following statement which can also be found under the name of Brouwer's dogma in [MG79]. Brouwer's thesis ([MG79]) Any intuitionistic proof that U is a bar can be transformed into another proof of it where only inferences of the following three forms occur U(u) U bars u U bars u U bars un U bars u0; : : : ; U bars un; : : : U bars u Brouwer's thesis could be reinterpreted as follows. Consider U ju inductively de ned with the following introduction rules, exact copies of the inferences in Brouwer's thesis. U(u) U ju U ju U jun U ju0; : : : ; U jun; : : : U ju When U j<> we say that U is an inductive bar. Under this interpretation, Brouwer's thesis would state that every bar is an inductive bar. As the converse is easy, the meaning of Brouwer's thesis is that bar and inductive bar are notions intuitionistically equivalent. This equivalence makes sense intuitionistically, that is, it is a statement which involves two notions, namely the one of bar and the one of inductive bar, both of which are meaningful. This is not the case in type theory, because whereas the notion of inductive bar makes sense, the one of bar cannot be interpreted as it stands, since it deals with in nite sequences. We rather see the fact that they are intuitionistically equivalent as a motivation to adopt the notion of inductive bar as the type-theoretic representation of the notion of bar. This idea goes back to Martin-Lof [Mar68] where a similar inductive de nition of the notion of bar is used. In type theory, it makes sense to make this de nition generally, that is, for nite sequences of elements of any set rather than restricting it to the set of natural numbers. Given a set A and a predicate U over nite sequences of elements of A, we de ne inductive bars by means of the following inductive de nition. U(u) U ju U ju U jua 8a 2 A U jua U ju Bar induction, Ramsey theorem and Higman's lemma in the second paper of this thesis are stated in terms of the notion of inductive bar. 7 Di erent formulations of Brouwer's thesis The above formulation of Brouwer's thesis is one of several possibilities introduced in the literature to express Brouwer's arguments. Another way is through the notion of stump. A stump is a decidable subset of nite sequences of natural numbers. It is inductively de ned as follows. f<>g 2 Stumps 0; : : : ; n; : : : 2 Stumps f<>g [ Sn fnu j u 2 ng 2 Stumps With this, Brouwer's analysis is expressed by the following statement presented by Veldman [Vel81]. Brouwer's thesis ([Vel81]) If U is a bar then there is a stump such that for all v 2 maximal, there is an initial segment u of v such that U(u) holds. A third, and probably most frequent choice, is to represent Brouwer's ideas with the following statement which has been made an axiom of intuitionistic analysis. It is known as bar induction on monotone bars, or simply bar induction, and is explained for instance in [Dum77]. It is expressed by the following rule. 8u U(u)) Y (u) (U is included in Y ) 8u 8n U(u)) U(un) (U is monotone) 8u (8n Y (un))) Y (u) (Y is hereditary) 8 9n U( (n)) (U is a bar) Y (<>) In fact, the three alternatives presented above are equivalent. To avoid confusion during the proof of this equivalence we refer to the statement of Brouwer's thesis given in the previous section as Brouwer's dogma, as in [MG79]. We will see that bar induction follows from Brouwer's dogma and implies Brouwer's thesis which in turn implies Brouwer's dogma. Brouwer's dogma implies bar induction: To see this we assume Brouwer's dogma, and the premises in the rule of bar induction and prove the conclusion. As U is assumed to be monotone, inferences of the form U bars u U bars un can be eliminated. In this case then, Brouwer's dogma states that any proof that U bars <> can be transformed into another proof of it where only inferences of the following two forms occur U(u) U bars u U bars u0; : : : ; U bars un; : : : U bars u 8 It is possible to prove Y (<>) now by copying such a proof of U bars <>. Every occurrence of the rst inference would be replaced by a use of the fact that U is included in Y and every occurrence of the other inference, by a use of the fact that Y is hereditary. The result is a proof that Y (<>) holds. Bar induction implies Brouwer's thesis: We say that the stump secures the predicate U when for every v 2 maximal there is an initial segment u of v such that U(u) holds. We write it ` U . Given a nite sequence w, we write `w U when for every v 2 maximal there is an initial segment u of wv such that U(u) holds. Clearly, `<> U if and only if ` U . We want to use bar induction to prove that if U is a bar then there exists such that ` U . We de ne V (w) = 8v U(wv) and Y (w) = 9 `w V . It is easy to see that V is monotone, that V is included in Y and Y is hereditary. As U is a bar, so is V . Hence `<> V and so ` U . Brouwer's thesis implies Brouwer's dogma: Here we can prove that for all w, if `w U then there is a proof that U bars w using only the inferences of Brouwer's dogma. The proof is by induction on the stump , using the rst two forms of inference in the base case, and in the inductive case the third form of inference plus the observation that if = f<>g[Sn fnu j u 2 ng 2 Stumps and `w U , then for all n, n `wn U . References [Bro54] L. Brouwer. Points and Spaces. Canadian Journal of Mathematics, 6:1{17, 1954. [Dum77] M. Dummett. Elements of Intuitionism. Clarendon Press, Oxford, 1977. [Mar68] P. Martin-Lof. Notes on Constructive Mathematics. Almqvist & Wiksell, 1968. [MG79] E. Martino and P. Giaretta. Brouwer, Dummett and the Bar Theorem. In Atti del Convengo Nazionale di Logica. Bibliopolis, 1979. [Nas63] C. Nash-Williams. On Well-Quasi-Ordering Finite Trees. Proceedings of the Cambridge Philosophical Society, 59:833{835, 1963. [VB93] W. Veldman and M. Bezem. Ramsey's Theorem and the Pigeonhole Principle in Intuitionistic Mathematics. Journal of the London Mathematical Society, (2)47:193{211, 1993. [Vel81] W. Veldman. Investigations in Intuitionistic Hierarchy Theory. PhD thesis, Katholieke Universiteit te Nijmegen, 1981. [Vel94] W. Veldman. Intuitionistic Proof of the General Non-Decidable Case of Higman's Lemma. Personal communication, 1994. 9 A Lambda Calculus Model of Martin-L of's Theory of Types with Explicit Substitution Abstract This paper presents a proof-irrelevant model of Martin-L of's theory of types with explicit substitution; that is, a model in the style of [Smi88], in which types are interpreted as truth values and objects (or proofs) are irrelevant. The fundamental di erence here is the need to cope with a formal system which in addition to types has sets and substitutions. This di erence leads us to a whole reformulation of the model which consists in de ning an interpretation in terms of the untyped lambda calculus. From this interpretation the proof-irrelevant model is obtained as a particular instance. Finally, the paper outlines the de nition of a realizability model which is also obtained as a particular instance.