Normalization in the simply typed 
${\lambda \mu \mu '}\rho \theta \varepsilon$
 -calculus

Abstract In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the 
$\lambda \mu$
 -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual 
$\beta$
 -, 
$\mu$
 -, and 
$\mu '$
 -reductions, we consider the 
$\lambda \mu$
 -calculus augmented with the reduction rules 
$\rho$
 , 
$\theta$
 and 
$\varepsilon$
 . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules 
$\rho$
 , 
$\theta$
 , and 
$\epsilon$
 , while the 
$\lambda \mu$
 -calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the 
$\beta \mu \mu '\rho \theta \varepsilon$
 -reduction in the de Groote-style calculus.


Introduction
The works of Murthy (1991) and Griffin (1990) revealed in the nineties that the Curry-Howard correspondence can be extended to classical logic. Since then, many different calculi have appeared with the aim of representing natural deduction or Gentzen-style derivations in classical logic (Barbanera and Berardi 1994;Curien and Herbelin 2000;Munch-Maccagnoni 2013;Parigot 1992;Rehof and Sørensen 1994;Saurin 2008;Wadler 2003Wadler , 2005. The systems based on the sequent calculus, like the λμμ-calculus of Curien-Herbelin (Curien and Herbelin 2000), have the advantage of possessing symmetric reduction rules, which makes their study more pleasant. This is also the case for the system of Barbanera-Berardi (Barbanera and Berardi 1994), which is exclusively based on two connectives ∧ and ∨. Parigot's λμ-calculus (Parigot 1992) was originally defined as a calculus for establishing connections between logical calculi and natural deductionstyle proofs in second-order predicate logic. The λμ-calculus can be considered as a simple extension of the λ-calculus: one obtains the λμ-calculus by adding classical variables (so-called μ-variables) to the λ-calculus together with their term formation rules and reduction rules.
The λμ-calculus has two distinct presentations. One of them is the original calculus of Parigot, and the other one was introduced by de Groote by relaxing some of the term formation rules. The investigation of the difference between the two presentations in terms of normalization properties can be considered as the main aim of the paper. It turns out that, despite their great resemblance, the two calculi differ in important respects: if one adds to the β-, μ-, and μ -reductions some structural rules, e.g., the ρor ε-rules, which are necessary to ensure the uniqueness of the representation of data, one can find that the Parigot-style calculus retains its strong normalization property, whereas the de Groote-style calculus does not. We will return to this question in several parts of the paper, and we will give the precise definitions for the different calculi in Sections 2 and 4.
Regarding the Parigot-style λμ-calculus, originally, his system contains only two reductions (the βand μ-rules). They stem from cut eliminations with respect to the implicational fragment of natural deduction both in the intuitionistic and in the classical case. He adds two more rules (ρ and θ), which he calls simplification rules, and which correspond to cut elimination regarding negation. From a logical point of view, the λμ-calculus makes it possible to easily encode the mathematical proofs. However, the calculus is not symmetric, which makes the examination of its properties more difficult. We note that we are concerned with the simply typed version of the λμ-calculus throughout the paper.
The simply typed λμ-calculus enjoys many useful proof-theoretical properties, like confluence (Parigot 1993;Py 1998), subject reduction, and strong normalization (Parigot 1993). The calculus has, from the computer science point of view, a severe drawback: it does not meet the requirement of the unique representation of data. In the second-order typed λ-calculus, any term having the integer type N is β-equal to a Church numeral. This property enables the automatic generation of programs for the integers. In fact, a λ-term of type N → N is a program that represents a function for the integers: it takes a Church-integer and returns, after reduction, another one. This is no longer true in the λμ-calculus: we can find normal terms of integer type that are not Church numerals. This should not come as a surprise, since we are able to prove that a type is N by indirect reasoning. Hence, when we consider a type even as simple as N , it is only the intuitionistic proof of the type that we can recover by utilizing exclusively the existing reduction rules.
There are several possible solutions to this problem. One can define an algorithm that, to every classical integer, assigns its value, which is represented by a Church numeral (Nour 1997;Parigot 1993). The other possibility is to apply output operators in order to transform a classical integer into an intuitionistic one, namely, into a Church numeral (Nour 1997;Parigot 1993). We choose here another solution which was also proposed by Parigot (1993). If we add the reduction rule μ to the λμ-calculus, which is the symmetric counterpart of the μ-rule and can be justified by the underlying logical rules, we get closer to reducing the Church integers to normal forms by simply applying the rules of the calculus. We discuss this question in more detail in Section 3. Similar assertions hold for the other algebraic data types defined formally in Krivine (1994).
The calculus augmented with the μ -rule preserves the subject reduction property (at least for the simply typed version), but confluence is lost. The μ -rule, however, plays an important role in the presentation of the call-by-value λμ-calculus. It permits a calculated value, as the functional part, to find its argument. De Groote (1998) managed to integrate this rule with the λμ-calculus while preserving confluence. Py studied the call-by-value μ -reduction in his thesis (Py 1998). In particular, he proved confluence and strong normalization properties for μand μreductions. He also enriched the syntax with new operators so that the subject reduction property was preserved and computational contents were given to the quantifiers. Parigot (1997) proved by an application of the Tait-Girard reducibility argument that the strong normalization property holds for the λμ-calculus with second-order type assignments. It was a long standing open problem of Parigot whether the μμ -rules in the untyped calculus enjoy the strong normalization property. This was solved by David and the second author Nour 2005, 2007). Subsequently, we have found in Battyányi and Nour (2020) another proof for this result. Until now, no simple weak normalization algorithm has been known for the μμ -reduction. In this paper, we also provide one.
As was mentioned, there has mainly been two different presentations of the λμ-calculus. Originally, Parigot defined his calculus with severe restrictions on the term formation rules. His syntax, however, does not make it possible to encode with typed λμ-terms any proof of the classical tautology ¬¬A → A without free variables of type ¬⊥ (similarly for some other classical tautologies). Furthermore, in this syntax with many reduction rules Böhm's theorem does not hold. In fact, Py (1998) gave two distinct closed normal terms M and N for which there is no term L, such that (L)M reduces to the first projection λx.λy.x and (L)N reduces to the second projection λx.λy.y. De Groote proposed a syntax that is less restrictive than that of Parigot's by splitting the rule concerning negation into two: one for the introduction and one for the elimination of negation and he even omitted certain restrictions that applied to the original calculus of Parigot. In this calculus, new rules can be considered like the ε-rule, which allows to handle two consecutive proofs both leading to absurdity. The version proposed by de Groote (1998) allowed him to construct an abstract machine for the λμ-calculus. Additionally, Saurin (2012) proved that the de Groote version enjoys the separation property. Taking all these into account, we have adopted the de Groote-style calculus for the majority of our work. Our results also show that the two calculi are quite different: one of them has the strong normalization property while the other one only possesses the weak normalization property. The paper is structured in the following way.
• In the second section, we present the definitions necessary for our discussion.
• The third section demonstrates the role of the structural rules: by making use of the additional rules we can prove that every term of integer type that weakly normalises behaves like a Church numeral in the sense that when it occurs in applications formed with terms of appropriate types it reduces to the well-known form of Church numerals. • The fourth section contains a proof that the Parigot-style λμ-calculus enjoys the strong normalization property. The proof generalizes Py's method (Py 1998): we prove that the simplification rules can be strongly postponed with respect to the β-, μand μ -rules and then we resort to the already mentioned result of David and the second author, which states that the de Groote-style λμμ -calculus enjoys the strong normalization property Nour 2005, 2007). The Parigot-style λμμ -calculus is a subsystem of the de Groote-style λμμ -calculus, hence, its strong normalization property follows. • In the subsequent sections, we discuss the normalization properties of the de Groote version of the symmetric calculi μμ and λμμ when augmented with the structural rules ρ, θ, and ε. We show by an example that μμ ρε-reduction is not strongly normalizing. Then we construct explicit algorithms in order to demonstrate that the weak normalization property for μμ ρε-reduction is still valid. We point out that our algorithm allows finding several normal forms for the same term. • In the last section, we extend our result to the case of β-reduction in the setting of typed calculus. More precisely, we show that, if we alternate our weak normalization algorithm for μμ ρε-reduction together with an arbitrary one for β-reduction, the process necessarily terminates. In fact, performing these reductions strictly reduces the maximal rank of β-redexes. The proofs of this section are rather technical and are based on a fine analysis of the origin of the redexes in the reduction sequences. We observe that θ-reduction does not give rise to any particular problem and we can postpone it. Our method allows us to give a weak normalization algorithm for the complete calculus.
Some of the results of this paper, especially the results in Sections 4, 5, and 6, have already been presented in the thesis of the first author (Battyányi 2007). We would like to clarify, however, that these results have not been published before and some of the proofs of the thesis were even incorrect. We found that it was very difficult to correct them using the same methods. The method applied here for proving the different variants of weak normalization properties are totally different. The proof of weak normalization of μμ ρθε-reduction is direct and gives an easy algorithm. The demonstration of weak normalization of βμμ ρθε-reduction does not utilize residuals (which are quite complicated to properly formulate), rather we proceed, in most of the cases, by a simple induction on the structure of terms.

The λμ-Calculus
The λμ-calculus was introduced by Parigot (1992) with the intention of providing a term encoding for classical natural deduction. In our paper, we restrict ourselves to the simply typed calculus. The λμ-calculus possesses two kinds of variables: one of them is the original set of λ-variables, the so called intuitionistic variables, and the other kind of variables is the set of μ-variables, which are termed classical variables. Parigot's calculus has certain restrictions imposed on the term formation rules: namely, a μ-abstraction must be followed by an application with a μ-variable and vice versa. That is, a term μα.M can only be of the form μα. [β]N for some N.
De Groote (1998) has proposed a new version of the λμ-calculus by modifying this syntax. Namely, the term formation rules became more flexible: a μ-abstraction can now be followed by an arbitrary term (in the untyped version); hence, it does not need to be of the form μα. [β]N. Since in the greater part of the paper we are concerned with the de Groote version, we introduce the de Groote-style calculus in what follows. In Section 4, we deal with the Parigot-style calculus. We postpone defining the Parigot-style syntax until then.
We decided to adopt Krivine's notation for the applications, i.e., we write (M)N instead of (M N) if we apply M to N. 2. In a λμ-term, the λ and μ operators bind the variables in their scope. We therefore consider terms modulo equivalence, which allows to rename the variables bound by a λor a μ-abstraction. The notion of subterm plays a very important role in our paper. Many of our subsequent results require a fine analysis of the propagation and creation of certain subterms. A subterm N of a term M is a term that appears in the formation tree for M. Obviously, M can contain several identical subterms. However, when we consider a subterm N of M, we intend to take only one particular occurrence of N in M.

Definition 2.2 (Subterms
In the definition below denotes a (possibly empty) context, that is, a finite set of declarations of the form x : A (resp. α : ¬A) for a λ-variable x (resp. a μ-variable α) and type A such that a λ-variable x (resp. a μ-variable α) occurs at most once in an expression x : A (resp. α : ¬A) of . The typing rules are as follows.  In this paper, we work with several kinds of substitutions. All of them are implicitly understood with the necessary renaming of bound variables to avoid the capturing of the free variables coming from the substitution. We present μ-substitution in detail. We need it in Definition 2.6. Definition 2.4 (μ-substitution). We adopt the convention that substitution has higher precedence than application and abstraction.

A μ-substitution
In order to define ε-reduction, we need the following notion.

Definition 2.5 (α-translation). Let M be a λμ-term and α a μ-variable.
We define the α-translation M α of M by induction on M.
In what follows, we define the redexes examined in this paper and the reductions induced by them. Next, we recall some results about the λμ-calculus. Regarding the necessary notions and definitions and the λ-calculus analogues of the results listed below, the reader is referred to the standard textbooks, for example Girard et al. (1989), Krivine (1993). Definition 2.6 (Redex). The six reductions presented in Definition 2.6 can be justified by the cut elimination rules in natural deduction. This is why we will have type preservation property in the calculus (Theorem 2.10), which is also known as subject reduction property.   . When talking about reductions, it is customary to define residuals, that is, the results of performing reductions in a term. However, in Section 6, when we examine the outcome of a reduction starting from a term N and leading to N , we are not interested in knowing about the residuals of a given subterm of N occurring in N (which is a one-to-many correspondence). Rather, we consider redexes of N and we are interested in their possible origin in N, that is, we trace back the reduction path, and this is a one-to-one correspondence. From the technical point of view, we do not need to formulate the notions of origin or residual to state the results of that section.

A β-redex
We list here the most important results that we are going to use in our paper. For the proofs of the next theorems, the interested reader is referred to Battyányi and Nour (2020), David and Nour (2005), Py (1998). Theorem 2.11. The μμ -reduction is strongly normalizing for the untyped λμ-terms.
Theorem 2.12. The βμμ -reduction is strongly normalizing for the typed λμ-terms.
In the sequel, let IH be an abbreviation for the phrase "the induction hypothesis".

Natural Numbers
The aim of this section is to justify our choices regarding the reduction rules. The major benefit of the "proof as program" paradigm (also known as Curry-Howard isomorphism) is that the proof itself ensures that the program extracted is correct (Krivine 1993;Leivant 1983). For example, in intuitionistic second-order λ-calculus, to program functions on integers, it suffices to encode the proof of the totality of the function to extract the correct program. Since the only λ-terms up to β-reduction that have an integer type are the Church numerals, to run the program one just has to apply it to a Church numeral and then reduce it to find the exact result. In classical logic the situation is different. Without μ -reduction we can have closed λμ-terms of integer type that are not Church numerals themselves (see Example 3.8). For data types, like integers, Parigot (1993) observed that the classical integers (the terms that encode proofs of integer type) may contain subterms which can be discarded by simple reduction rules without changing the type. This phenomenon comes from the fact that one does not need double negation elimination in order to prove that these terms are well typed. Hence, the idea is to add rules to tidy up the terms for classical integers. We must keep in mind, however, that we are not able to do the same thing in higher order logics, since, in that case, the μ -rule does not preserve the types. We emphasize that this method does not mean that every classical proof can be transformed into an intuitionistic one but, at least for many of the data types in the simply typed lambda calculus, it is possible. In higher order logics there exists another solution: one can give computational content to the quantifiers in such a way that the μ -rule preserves the types (Py 1998). Henceforth, we adopt the strategy of adding certain rules to the calculus, most notably the μrule. In this section, we prove that in the presence of all of our mentioned rules, the only λμ-terms, regarding the de Groote syntax, that can be typed with integer type are the λμ-terms that behave as Church numerals. In order to obtain the result, all the rules are necessary, that is, we consider βμμ ρθε-reduction. The results of this section can be generalized to the various algebraic data types defined by Krivine in (1994). We omit their treatment in our paper. We remark that the successor function really behaves as the arithmetical operator on natural numbers: we can verify that, for all n ∈ IN, (s)n β n + 1.

Let X be a variable type and
The following results are valid for λ-terms only. Later, we shall see examples of λμ-terms of type N (see Example 3.3).
With the introduction of the μ -rule, we have lost confluence. It is easy to construct terms leading to different normal forms. More interestingly, we are able to define a λμ-term of integer type that can evaluate to different integers. This permits us the coding of algorithms that can yield multiple possible results. The following example shows how a typed λμ-term can be reduced to two different Church numerals. This phenomenon is not a shortcoming of the calculus. On the contrary, it even allows us the construction of programs that generate a set of solutions for a given problem.
Lemmas 3.4 and 3.5 are technical and indispensable for obtaining the characterization of λμterms of type N . Proof.

By induction on
Since M is normal and, by inspecting the typing rules, Proof. By induction on M.
• Because of the typing rules and by our assumption, Since M 1 is normal and, if we take into account the typing rules, we find that We are now in a position to prove the main result in this section. We deduce that M = μα.M , and we apply Lemma 3.5.
Hence, we obtain the following characterization of λμ-terms of type N .
In the light of our subsequent results, the assumption ((M)x)f ∈ WN βμμ ρθε can in fact be omitted in the above corollary. Namely, we will show in the subsequent sections that βμμ ρθεreduction is weakly normalizing. Hence, if we wish to obtain the corresponding Church numeral, it is enough to apply M to the numeral 0 and to the successor s, thus one arrives at the result (s) n 0, which reduces to n.
The following example presents a λμ-term of integer type for which we need all reduction rules to obtain its value.
• Let x : X, f : X → X, g : X → X, α : ¬((X → X) → X), β : ¬X and γ : ¬⊥, then we have the typing relation M : N . • In the rest of the paper, in order to make the reductions more readable, at each step we underline the successive redexes. We have Remark 3.9. We recover the Church numerals directly from M if we make use of the ν-rule, which was introduced by Parigot (1993). The rule can be presented as follows: If we consider again Example 3.8, we obtain the following reduction sequence.
We prefer not to use the ν-rule, however, since it contains a constraint on the type of a bound variable. If we considered the untyped calculus together with the ν-rule, then the calculus would diverge for all λμ-terms containing a μ. We prefer a calculus where reduction does not depend on the types of terms, hence we omit the ν-rule. In addition, the ν-rule is not strictly necessary for our main goal in this section, which is the demonstration that the introduction of certain rules suffices for finding the value of a term of integer type. We need βμμ ρθεreduction for this purpose, this is the statement of Corollary 3.7.

The Parigot-Style λμ-Calculus
Prior to turning to our main task and detailing the normalization results for βμμ ρθε-reduction, we make a little detour and examine the original version of the λμ-calculus due to Parigot (1992). We will refer to the calculus discussed in this section as the Parigot-style λμ-calculus as opposed to the one appearing in Definition 2.1, which was developed by de Groote (1998). The term formation rules of Parigot's calculus differ from the ones discussed so far in the following manner: in Parigot's calculus the forms of λμ-terms are restricted in such a way that μ-abstractions and applications with μ-variables are tied together. This is achieved by distinguishing named and unnamed λμ-terms. Named λμ-terms are formulated from unnamed terms by an application of a μ-variable. Conversely, μ-abstracting over named terms returns unnamed terms. This is formalized in the definition below.
The typing rules are the same as in Definition 2.3, except that the last two rules are replaced with the following single rule: Concerning the reduction rules, due to the syntax of λμ-terms, the ε-rule cannot be defined here, hence it is omitted.
To better understand the difference between the Parigot and de Groote-style versions, we present an example that will be discussed in detail in the rest of the paragraph. We can check that the λμ-term T = λx.μα.x is of type ⊥ → X. This term can be formulated in the de Groote syntax only, since, following the μα, there is no [β]. The λμ-term T behaves quite interestingly. We can verify that, when we apply T to an arbitrarily long finite sequence of λμ-terms, after some steps of μ-reduction we obtain μα.P, provided P was the first element of the list. Roughly speaking, T takes the first element of a list and erases the others. The second author has proved with Saber (Nour and Saber 2009) that every λμ-term of type ⊥ → X enjoys the same property. The simplest proof of the formula ⊥ → X in the Parigot-style calculus turns out to be T = λx.μα. [ϕ]x. The free variable ϕ is of type ¬⊥, and it is quite embarrassing to acknowledge that we need to resort to a free variable of type ¬⊥ in order to construct a term of type ⊥ → X, which is a classical tautology. Anyway, the λμ-term T behaves in the same way as T.
In what follows, we prove that Parigot's λμ-calculus with ρθ-reduction possesses the strong normalization property. First of all, we show that ρand θ-reduction can be strongly postponed with respect to β-, μ-, and μ -reduction.
Definition 4.1 (Strong postponement). Let → 1 and → 2 be two reduction relations. The reduction → 2 can be strongly postponed with respect to Our first aim is to demonstrate the following theorem.
Theorem 4.2. In the Parigot-style calculus, ρθ-reduction can be strongly postponed with respect to μμ β.
We prove the theorem in two steps. First, we establish that θ-reduction can be strongly postponed with respect to all the other rules, namely, β, μ, μ , ρ. We then show that the ρ-rule can be strongly postponed with respect to μ, μ , and β.
First of all, we define two reductions that are special cases of μand μ -reductions. These new rules appear in the permutation lemmas that we are going to prove and are easy to handle.

The case of θ -reduction
We show that, in the Parigot-style λμ-calculus, θ-reduction can be strongly postponed with respect to βμμ ρ-reduction. We prove for β-, μ-, μ -, and ρ-reductions, one by one, that the execution of a θ-reduction followed by either of the mentioned reductions can be replaced by a reduction sequence where the respective reductions precede θ-reduction (Lemmas 4.4,4.5,4.6,4.7). Additionally, new μ 0 -or μ 0 -redexes may be created, but these redexes can be executed before the θ-redex. We can generalize the results on swapping θ-reduction with βμμ ρ-reductions to reduction sequences by reasoning by induction on the length of the reduction sequences. This is stated in Theorem 4.9.
The following lemma shows how θ-reduction relates to the β-rule. In the next lemma, we observe the usefulness of the new μ 0 -reduction rule. - The next lemma shows how θ-reduction commutes with μ-reduction. We notice the appearance of μ 0 -reduction.

By induction on M.
• - 3. The proof follows the pattern of point 1 of this lemma with the necessary changes implemented. 4. Applying case 2 of this lemma and proceeding by induction on n.

Let
Proof. Similar to Lemma 4.5.
The lemma below demonstrates that θ-reduction can be strongly postponed with respect to ρ-reduction.
Proof. An obvious induction on M.
We are now in a position to prove the next theorem, which describes how θ-reduction commutes with all the other rules.
Theorem 4.8. Let M θ The proof proceeds by induction on n and by applying the previous lemmas. Let us suppose first that M → n θ P → μ N.
• n = 0: the result is immediate.
In the former case, the IH applies to M → m θ K → μ R. In the latter case, by Lemma 4.5, there exists an S such that M μ 0 S → m θ R, and we can apply the IH to S → m θ R → μ L.
The cases P → μ N and P → β N are analogous. Finally, the case P → ρ N is straightforward by Lemma 4.7.
We have obtained the strong postponement of → θ with respect to → βμμ ρ . This is stated in the next theorem. Proof. By induction on the reduction sequence P → + βμμ ρ N using Theorem 4.8.
Remark 4.10. We notice that the above proofs, hence Theorem 4.9, are also valid in the de Groote-style calculus. By contrast, the ρ-rule exhibits a different behavior in de Groote's calculus. In what follows, we show that ρ-reduction can be strongly postponed with respect to βμμreduction in the simply typed Parigot calculus, and we demonstrate by giving counterexamples that this is not valid in the de Groote-style syntax.

The case of ρ-reduction
Now we show that, in the simply typed Parigot-style λμ-calculus, ρ-reduction can be strongly postponed with respect to βμμ -reduction. The proofs of this subsection can also be found in Py (1998). The presentation there, however, may differ in some places from ours. For the sake of completeness, we outline the proofs below. We prove that ρ-reduction commutes with β-, μand μ -reduction. This results in a reduction sequence starting with β-reduction (or with a μor μ -reduction, respectively) and ending in ρ-reduction (Lemmas 4.12 and 4.14). The subsection culminates in Theorem 4.16, which can be obtained from the above-mentioned results by induction on the length of the reduction sequences.
The following lemma will be used in the proof of Lemma 4.12. Proof. By induction on M.
The next lemma proves postponement of ρ-reduction with respect to β-reduction.

By induction on M.
• The following lemma is useful for proving Lemma 4.14. The following lemma proves postponement of ρ-reduction with respect to μand μ -reductions. Lemma 4.14.

Let
Proof. We only detail some of the more interesting cases of the first point.

By induction on M.
• The cases M = μα.M 1 and M = λx.M 1 are straightforward. This phenomenon gives rise to the counterexample falsifying the strong normalization property of the de Groote-style calculus. We will discuss this in Section 5.
Now we can prove the theorem below, which asserts that the ρ-rule can be strongly postponed with respect to βμμ -reduction.

Strong normalization
We are now in a position to state the main assertion of this section, which is Theorem 4.18.
Proof. Assume M → ρθ M . Then it is easy to observe that M , as a word, consists of fewer symbols than M, which proves our assertion.
We are now ready to prove our strong normalization result.
Proof. It is known that de Groote's simply typed λμ-calculus with βμμ -reduction is strongly normalizing (David and Nour 2005). Since de Groote's calculus is an extension of the original Parigotstyle calculus, we immediately have the strong normalization property for the simply typed Parigot calculus with βμμ -reduction. Let η(M) denote the longest βμμ -reduction sequence starting from M. Let M be such that there is an infinite βμμ ρθ-reduction sequence starting from M and η(M) is minimal. By Theorem 4.17, the infinite reduction sequence starting from M must contain a βμμ -reduction, hence there is an M and M such that M ρθ M → βμμ M and M has an infinite βμμ ρθ-reduction sequence. If we apply Theorem 4.9 successively to the reduction sequence M ρθ M → βμμ M , then we obtain an N and P such that M ρ N → + βμμ P θ M . Obviously, P has an infinite βμμ ρθ-reduction sequence, since the reduction sequence P θ M can be continued with the infinite reduction sequence starting from M . We distinguish the following cases: • M ρ N is empty: since η(P) < η (M), this contradicts the assumption on M.
• M ρ N is not empty: by Theorem 4.16, there is a Q such that M → + βμμ Q ρ P, then η(Q) < η (M). This is a contradiction again.

The μ-Calculus
For the rest of the paper, we return to the de Groote-style syntax. With the creation of the λμcalculus, Parigot introduced a formal method to encode proofs in classical logic via terms of an appropriate calculus. The addition of the rules in connection with classical variables made it possible to represent the proof-theoretical machinery called reasoning by absurdity, which is very characteristic of classical reasoning. Parigot posed the question whether the rules corresponding to classical variables, that is, the rules without β, are normalizing. In this spirit, we devote this section to the calculus with rules other than β. The rationale behind our choice is the expectation that examining the proof-theoretical properties can be simpler without the β-rule, since we know that the β-rule in the untyped case is not strongly normalizing and we know, for example, from Battyányi and Nour (2020) and David and Nour (2005) that the μμ -calculus has the strong normalization property even in the untyped case. It looks quite tempting to try and prove the strong normalization property for the calculi obtained by adding more simplification rules like ρ-, ε-, and θ-rules to the μμ -rules. The θ-rule actually does not cause any difficulty, since it is strongly normalizing and, moreover, we can always move a θ-redex to the end of a reduction sequence. However, contradicting our first guess, μμ -reduction fails to be strongly normalizing with the simplification rules ρ or ε, however, innocent these rules may look. We demonstrate by examples that μμ ρand μμ ε-reductions are not strongly normalizing. We are able to prove, however, that weak normalization is preserved: in the second half of this section we show that μμ ρε-reduction is weakly normalizing. We call the above calculus without β-rule the μ-calculus, that is, we consider μμ ρε-reduction. In this respect, we talk about μ-normalization and μ-normal form if the reduction is clear from the context. Otherwise, we indicate explicitly the reductions we are dealing with. We could retain the structure of λμ-terms by encoding a λμ-term with a μ-term by inductively replacing the subterms of the form λx.M with ((l)x)M, where l is a constant. This way any μμ ρε-reduction of a λμ-term would be simulated by an identical μμ ρε-reduction of the appropriate μ-term. For the sake of simplicity, we abandon this option. Since only the typed β-reduction is strongly normalizing, in the last section, when we consider β-rule, we exclusively remain in the typed setting. Several results of this section, however, work in the untyped case as well. This is the reason why most of the present section will be concerned with the untyped calculus. Let us first define the μ-terms, where we have deliberately omitted λ-abstractions, since they do not play any role concerning the normalization properties of the calculus without β-rule.

The reductions μμ ρ and μμ ε
It is already known that μμ -reduction is strongly normalizing (Battyányi 2007;Battyányi and Nour 2020;David and Nour 2005). We demonstrate in this section that this property is lost when we add either ρor ε-reduction to these rules. This phenomenon is actually a peculiarity of the  we present examples of μ-terms that are either not strongly or even not weakly normalizing. Both results appeared first in the thesis of the first author (Battyányi 2007).  1. Note that the μ-term M cannot be written in Parigot syntax because we will not be able to write two successive [α]s. 2. We remark that, when θ-reduction is allowed, the example above returns the initial μ-term M. 3. We can normalize the μ-term M as follows : M = (Q)P→ μ Q. 4. We can use this example to find a λμ-term of integer type which is not strongly normalizable. Just consider the λμ-term W = λy.λf .μγ .M[x := [γ ]y].
To convince ourselves that this reduction can be continued without reaching a normal form, it is enough to observe that there is a certain analogy between lines 5 and 10. The idea of the proof of non-termination is roughly as follows. We have two occurrences of the subterm K in the μ-term μγ .((z)(y)K)K, from which we can start an infinite reduction sequence. In the first stage of each iteration, we reduce exclusively the μand μ -redexes lying around the second occurrence of K until we arrive at the first occurrence of K wrapped in a subterm (z)(y)K. Then we distribute that copy of (z)(y)K among all the subterms of the emerging μredex which are named with α, and afterwards we finish the actual iteration round with an ε-reduction. Intuitively, none of the performed reduction steps decreases the total number of μand μ -redexes in the term, which anticipates that the reduction sequence cannot be terminating.
1. The variable z is used only to type the μ-term N. 2. We can show that the μ-term N is not μμ ε-normalizable. We omit the rather technical proof of this result, since it is not needed in the sequel.
We may guess that one of the reasons for the failure of strong normalization for the above terms could be the presence of two successive μ-applications [.] (M and N contain [α][α]). To represent these terms in a typed environment, we need variables of type ¬⊥. We have already seen in the beginning of the previous section that terms of this type appear in the first works of Parigot with the aim of encoding classical tautologies with typed terms. Let us give another example to better understand this phenomenon. In Parigot's syntax, it is not possible to write a λμ-term with type ¬¬X → X without using a free variable of type ¬⊥. The simplest λμ-term he proposed is: λy.μα. [ϕ](y)λx.μδ.[α]x. In this term, the variable ϕ is free and has type ¬⊥. In de Groote's syntax, we can write a term with type ¬¬X → X without free variables and without a variable of type ¬⊥. Here is an example: λy.μα.(y)λx. [α]x. Many researchers have been interested in this kind of variables. For example, Ariola and Herbelin have made use of variables of type ¬⊥ in Ariola and Herbelin (2003) (they called them variables for continuation) in order to establish translations between calculi based on classical logic. It is clear that due to the type of these variables, we can pile up two μ-applications [α] [α], which causes the loss of the strong normalization property. The first author studied a calculus in his thesis (Battyányi 2007) where variables are not allowed to have type ¬⊥. He showed that the obtained calculus enjoys strong normalization.

Weak normalization of the μ-calculus
In this section we show that μμ ρε-reduction is weakly normalizing. This result can be surprising in view of the results that we have discussed before, in particular, given that μμ ε-reduction is not even normalizing. We will see that the normalization property of μμ ρε-reduction essentially originates from the structure of normal form terms with respect to all four rules. For the proof, we generalize our method for the weak normalization of μμ -calculus (Battyányi and Nour 2020). We obtain a very simple normalization algorithm that is discussed in detail at the end of the section. The algorithm will be able to provide us with different normal forms of a given μ-term. The main idea of our approach is to decide what to do when we face an application (M)N. We start with normalizing the μ-terms M and N separately to obtain the normal forms M and N . The calculation can only continue if either M or N starts with μ. This is where the necessity of studying the substitutions P[α := s Q], where s ∈ {r, l} and P, Q are in normal form, emerges.
Lemma 5.7 characterizes the redexes that can be created in such substitutions, and Lemma 5.9 demonstrates how to handle them properly.
We present the proofs in more detail in order to help the reader to easier understand the algorithm.
The following lemma depicts the μ-terms obtained by erasing μ-variables from a normal form.
Lemma 5.5. Let M be a μ-term and α a μ-variable. If M ∈ N F μμ ρε , then M α ∈ N F μμ ρε and, if M = μβ.M for some β, then there is no γ such that M α = μγ .M .
Proof. By induction on M using Definition 2.5.
The next lemma is intuitive and has the consequence that we are not able to create a head μ-redex by a μor μ -substitution. then Mσ ∈ N F μρε . • If M = (P)Q, then Mσ = (Pσ )Qσ and, since M is neither a μand nor a μ -redex, we have P = μβ.P and Q = μβ.Q . Using IH, Pσ , Qσ ∈ N F μρε and, by Lemma 5.6, Pσ = μβ.P , Qσ = μβ.Q . Hence, Mσ ∈ N F μρε and the stipulation for the μ -redexes is valid for Mσ .
The requirement for the μ -redex can be checked in all the above cases.

A μ-term
M is said to be μ-good if M ∈ N F μρε and its μ -redexes are named with μ-variables.

A μ-term M is said to be μ -good if M ∈ N F μ ρε and its μ-redexes are named with μ-variables.
Let us now show how a μ-term that is μ or μ -good can be normalized.
Proof. We only prove the first point by induction on M.
• The result is obvious if M = x.
We can now prove our weak normalization result.
Proof. Let M be any μ-term. We prove by induction on M that M ∈ WN μμ ρε .
• The result is obvious if M = x.
P)Q, then, by IH, ∃ P , Q ∈ N F μμ ρε such that P μμ ρε P and Q μμ ρε Q . We have M μμ ρε (P )Q . We distinguish four cases. Remark 5.11. Let us summarize the steps that we took to present a normalization algorithm. We proceed by induction on μ-terms.
1. If M = μα.N, we normalize N and, if we arrive at a normal form that begins with a μβ, then we reduce the ε-redex. It is now sufficient to reduce the remaining μ -redexes and then the created ρand ε-redexes starting with the innermost ones. ii. Or one can reduce the μ -redex so that one obtains a μ -good term μβ.N 2 [α := l N 1 ]. It is now sufficient to reduce the remaining μ-redexes and then the created ρand ε-redexes starting with the innermost ones. Observe that in case (d) we can obtain two different normal forms for M.

The Case of β-Reduction
In this section, we examine how the weak normalization property is preserved when we extend the calculus with β-reduction. In what follows, since we include β-reduction in our treatment again, instead of the abbreviations μ-reduction and μ-normal form for μμ ρε-reduction and μμ ρεnormal form, we explicitly indicate the particular reductions we are talking about. Although the algorithms of the previous section also work in the untyped setting, we now must consider a typed framework, since β-reduction is not weakly normalizing in the untyped form. Hence, all results we state in this section are for typed λμ-terms. Our algorithm depends on the weak normalization algorithm presented in the previous section, and we also make use of the fact that the typed λ-calculus has the weak normalization property. The idea of the proof is as follows: • We start from a λμ-term M 1 in μμ ρε-normal form, i.e., from a λμ-term in which there are no μ-, μ -, ρor ε-redexes. • We eliminate all β-redexes from M 1 by applying an arbitrary weak β-normalization algorithm. Having done this, we arrive at a λμ-term M 2 in β-normal form. • Next, we find a μμ ρε-normal form M 3 of M 2 by our weak normalization algorithm. The λμ-term M 3 may contain β-redexes. • We prove, however, that the maximum rank of β-redexes in M 1 is strictly greater than that of M 3 . We will give the necessary definitions in the section.
By this, the process must terminate. Finally, θ-reduction causes no problem since it is strongly normalizing and does not create new redexes when applied to μμ ρε-normal forms.
We have to consider the evolution of types throughout the reduction. In order to simplify our presentation, we apply the typability relation of Church (Sørensen and Urzyczyn 2006): we explicitly write the types of the variables in the abstractions together with the types of some of the subterms. To begin with, we present the reduction rules in the typed case. Recall that this evolution of types corresponds exactly to the notion of cut elimination in natural deduction proofs. We will not indicate the types of all subterms of a particular λμ-term, but only the ones that are relevant for our purposes.
Definition 6.1. The length of a type A, which is denoted by lh(A), is defined as the number of arrows in A.

The origin of a μ-redex in a β-reduction
The aim of this section is to prove Corollary 6.7, which asserts that the creation of a μ-redex during β-reduction of a μμ ρε-normal term M forces M to contain a β-redex of a particular type. We start from M and M , where M ∈ N F μμ ρε and M β M . We suppose that M contains a μ-redex (μα ¬(A→B) .P ⊥ 1 )Q A 1 . We intend to say something about the term M, where M is such that the sequence of β-reductions leading to M has created a new μ-redex in M . We show that M must contain a β-redex (λx C .P D 2 )Q C 2 with lh(C → D) > lh(A → B). We consider this β-redex the origin of the μ-redex. (In fact, a β-redex like that can be uniquely assigned to every μ-redex of the above form, but we do not need this statement.) To establish this claim, we have to start from a more general assumption where the property M ∈ N F μμ ρε is no longer valid.
The assertion of the following lemma explicitly shows the origins of βand μ-redexes in a β-substitution. Proof. By induction on M.
Lemmas 6.3 and 6.4 determine in some sense the origin of a βor μ-redex in a β-reduction while also stating important relations between the type of the redexes in the original term and in the resulting term, respectively. We can conclude that, during a β-reduction step, a μ-redex can stem from either a βor a μ-redex. When the origin is a β-redex, then it must be a redex of longer type. On the contrary, if we consider the origin of a β-redex, then it must be a β-redex again with type not smaller than that of the descendant. This is the statement of Lemma 6.4. Hence, if we examine a sequence of β-reductions starting from a term in μ-normal form, the origin of a μ-redex in the final term must be a β-redex in the original term of longer type (Corollary 6.7). We obtain the corollaries below about β-reduction sequences. Intuitively, the origin of a μredex in a β-reduction sequence is either a μ-redex of the same type or a β-redex of a type of greater length. • or (μα.P )Q M, U [β := s N] = U and Q [β := s N] = Q for some terms P , Q , • or s = r, [β]μα.P M, N = Q and P [β := r N] = P for some term P , • or s = l, [β]Q M, N = μα.P and Q [β := l N] = Q for some term Q .

Proof. By induction on M.
In what follows, we show that the origin of a μ-redex through a μμ ρε-reduction sequence is always a μ-redex of the same type (Lemma 6.14). However, we need to add an auxiliary condition on the μ-redex to ensure this property during the search for the origin of a redex (Lemma 6.16). Hence, we give Definitions 6.9 and 6.12. Definition 6.9. Let M be a λμ-term, A, B types and α ¬(A→B) ∈ fv (M) The next lemma helps understanding the notion of α-good with respect to a substitution and will be used in subsequent proofs. (a) If M is α-good, then either P is α-good or Q is α-good.
Proof. By induction on M.
The following lemma intuitively means that we will not be able to create an α-good term by a μμ ρε-reduction. -If M → μμ ρε [α](N 1 )N 2 = M where N 1 → μμ ρε N 1 , then N 1 or N 2 is α-good, thus, by IH, In the next definition, we give the condition that should be imposed on the μ-redex to ensure that our typability assertions hold throughout the reduction sequence. For a redex (μα.P)Q, we demand that there exists a chain, in the sense of Definition 6.9, that starts with [α] and ends with a λ-abstraction. We will see in the proof of Lemma 6.13 how this condition propagates and how it prevents the creation of certain μ-redexes. The lemma below will be applied in the proof of Lemma 6.16 in the case of ε-reduction.

Proof. By induction on M.
The next lemma shows that the origin of a β-redex of type A → B through a μμ ρε-reduction sequence is either a β-redex or a μ-redex verifying the condition (A, B) of the same type. Proof. By Lemma 6.14, it is enough to examine the case of one-step reduction. We proceed by induction on M. The cases M = λy.N, M = μβ.N, and M = [β]N are treated by induction and using Lemma 6.15. We obtain the following corollaries about μμ ρε-reduction sequences. Intuitively, the origin of a β-redex through a μμ ρε-reduction sequence is a μ-redex of the same type.

Weak normalization
In this subsection, we state our main result (Theorem 6.24), which asserts that λμμ ρεθ-reduction in the typed λμ-calculus is weakly normalizing.
We define the rank of a β-redex first, and then the rank of a λμ-term. We could also define the rank of all the other redexes (and, especially, that of a μ-redex), but we do not need this for our subsequent proofs. At this point, we can combine all of our previous results on the creation of redexes to prove that βμμ ρε-reduction is weakly normalizing (Theorem 6.21). In the statements below, we implicitly make use of the facts that typed β-reduction is strongly normalizing (Krivine 1993) (in fact, weak normalization suffices here) and that μμ ρε-reduction is weakly normalizing (Theorem 5.10).
Proof. Let (λx A .P B 1 )Q A 1 be a β-redex of M 3 having maximal rank. By Corollary 6.17, (μα ¬(A→B) .P ⊥ 2 )Q A 2 M 2 and, by Corollary 6.7, (λy C .P  We are now in a position to prove our weak normalization result.