The infinitary lambda calculus of the infinite eta Böhm trees

In this paper, we introduce a strong form of eta reduction called etabang that we use to construct a confluent and normalising infinitary lambda calculus, of which the normal forms correspond to Barendregt's infinite eta Böhm trees. This new infinitary perspective on the set of infinite eta Böhm trees allows us to prove that the set of infinite eta Böhm trees is a model of the lambda calculus. The model is of interest because it has the same local structure as Scott's D ∞-models, i.e. two finite lambda terms are equal in the infinite eta Böhm model if and only if they have the same interpretation in Scott's D ∞-models.


Introduction
In the classical finitary lambda calculus (Barendregt 1984), one can express that the fixed point combinator Y (= λf. (λx.f(xx))(λx.f(xx))) can reduce to terms of the form λf.f n ((λx.f(xx))(λx.f(xx))), for any n > 0, but one cannot express that Y has an infinite reduction to the infinite term λf.f ω , where f ω is convenient shorthand for the infinite term f(f(f(. . .))).
In the infinitary lambda calculus, the set Λ of finite λ-terms is extended to explicitly include infinite terms such as λf.f ω and the notation allows for finite and infinite reductions. This makes it possible to define the concept of Böhm tree directly in the notational framework of the infinitary lambda calculus, in contrast to Barendregt (1984) where Böhm trees are defined with their own notational machinery.
Infinitary lambda calculus allows an alternative definition of the notion of tree as normal form. Figure 1 summarises the correspondences between the infinitary lambda calculi and the trees which have been studied so far. All these calculi include a notion of ⊥-reduction and they are all proved to be confluent and normalising before except for the one on the last row (Berarducci 1996;Kennaway and de Vries 2003;Kennaway et al. 1995aKennaway et al. , 1997Severi and de Vries 2002;Severi and de Vries 2011). From any infinitary lambda calculus which is confluent and normalising, we can construct a model of the finite lambda calculus by defining the interpretation of a term to be exactly the (infinite) normal form of that term (or equivalently the tree of that term).
The infinitary lambda calculi sketched in the first four rows of Figure 1 are variations of λ ∞ β⊥ = (Λ ∞ ⊥ , −→ β⊥ ). By changing the ⊥-rule, we obtain different notions of trees. If we take the terms without head normal form (HNF) as meaningless terms, then we obtain an infinitary lambda calculus which is confluent and normalising. The normal form of a term in this calculus correspond to the Böhm tree of this term. The collection of normal forms of this calculus forms a model of the lambda beta calculus, better known as Barendregt's Böhm model (Barendregt 1984). Similarly, by reducing terms without weak HNF to ⊥, we capture the notion of Lévy-Longo tree (Kennaway and de Vries 2003;Kennaway et al. 1995aKennaway et al. , 1997 and this gives rise to the model of Lévy-Longo trees. Also by reducing terms without top HNF to ⊥, we capture the notion of Berarducci tree (Berarducci 1996;Kennaway and de Vries 2003;Kennaway et al. 1995aKennaway et al. , 1997 which gives rise to the model of Berarducci trees. The infinitary lambda calculi λ ∞ β⊥ with a ⊥-rule parametric on a set of (weakly) meaningless terms encompasses the previous three cases ( Kennaway and de Vries 2003;Severi and de Vries 2011). This method to construct models of the lambda beta calculus is quite flexible as there is ample choice for the set of meaningless terms (Severi and de Vries 2005a,b;Severi and de Vries 2011). Because the collection of sets of weakly meaningless terms is uncountable, we get an uncountable class of models which are not continuous (Severi and de Vries 2005a).
The infinitary lambda calculus λ ∞ β⊥η = (Λ ∞ ⊥ , −→ β⊥η ) sketched in the last but one row incorporates the η-rule (Severi and de Vries 2002). This calculus captures the notion of η-Böhm tree, which can be described as the eta-normal form of a Böhm tree, and gives rise to an extensional model of the lambda calculus that has the same local structure as Coppo, Dezani and Zacchi's filter model D * ∞ (Coppo et al. 1987). The last row in Figure 1 represents the contribution of this paper. The infinitary lambda calculus λ ∞ β⊥η! = (Λ ∞ ⊥ , −→ β⊥η! ) is constructed with the η!-rule, a strengthening of the ηrule. The notion of η!-reduction is based on the observation that the explicit syntactic characterisation of infinite eta expansions in the definition of infinite eta Böhm trees in Barendregt (1984) can be succinctly redefined as strongly converging eta-expansions in the terminology of infinitary rewriting. The power of η!-reduction is such that it reduces the Böhm tree of J to I, see Figure 2. The main complication of this paper will be to prove that λ ∞ β⊥η! is confluent and normalising. As direct consequences of this result, we will first obtain an alternative definition of the notion of ∞η-Böhm tree of a lambda term as its normal form in λ ∞ β⊥η! which is more compact than the one in Bakel et al. (2002); Barbanera et al. (1998); Barendregt (1984). Second, we can show that the set of ∞η-Böhm trees is an extensional model of the finite lambda calculus. The model of ∞η-Böhm trees is of interest because it has the same local structure as Scott's D ∞ -models, i.e. two finite lambda terms have the same normal form in λ ∞ β⊥η! if and only if they are equal in D ∞ Hyland (1975) and Wadsworth (1976).
It may appear at first sight that extending an infinitary lambda calculus with η or η! should not be complicated. However, the two lambda calculi of Lévy-Longo and Berarducci trees do not seem to accept any variations on the ⊥-rule without losing confluence. There is a critical pair between the η-rule (η!-rule) and the ⊥-rule for terms without weak HNF: The ⊥-step follows from the fact that the term Ωx has no weak HNF. This pair can be completed only if λx.⊥ −→ ⊥ ⊥ which is true only for the ⊥-rule that equates terms without HNF. For a counterexample of confluence for β⊥η and β⊥η! where the ⊥-rule equates terms without top normal form, we use the term Ω η = λx 0 .(λx 1 .(. . .)x 1 )x 0 . Similar to Ω which β-reduces to itself in only one step, this term η-reduces to itself in only one step. The term Ω η can be obtained as the fixed point of 1 = λxy.xy. The body of the outermost abstraction in Ω η is root active (it always reduces to a β-redex) and hence Ω η −→ ⊥ λx.⊥. The span can only be joined if λx.⊥ −→ ⊥ ⊥.

Outline of this paper
Section 2 recalls some notions of infinitary lambda calculus and introduces the definition of η!-reduction. Section 3 studies properties of mainly −→ η! and −→ η −1 on their own. Section 4 proves two strip lemmas for η! and β. Section 5 proves that outermost ⊥-reduction commutes with η!. Section 6 proves confluence and normalisation of the infinitary lambda calculus λ ∞ β⊥η! . Section 7 explains in detail the connection between the infinite eta Böhm trees and the normal forms in λ ∞ β⊥η! . Section 8 shows that the set of the normal forms of λ ∞ β⊥η! is an extensional model of the finite lambda calculus.

Infinitary lambda calculus
2.1. The set Λ ∞ ⊥ of finite and infinite terms Infinitary lambda calculus provides a single framework for finite lambda terms and infinite terms. Infinite extensions of finite lambda calculus were introduced around 1994 following similar developments in first order term rewriting initiated by Dershowitz and Kaplan (Berarducci 1996;Kennaway et al. 1997). As starting point for this paper, we are interested in one particular extension λ ∞ β⊥ of the finite lambda calculus defined in Kennaway et al. (1997), namely the extension in which the normal forms correspond to the Böhm trees of Barendregt (1984). The set Λ ∞ ⊥ of finite and infinite terms of λ ∞ β⊥ can conveniently be defined as metric completion of the finite terms for a suitable chosen metric. In spirit, this construction goes back at least to Arnold and Nivat (1980). The metric context will also be used to define transfinite converging reductions.
We will now briefly recall this construction from Kennaway et al. (1997). Throughout, we assume familiarity with basic notions and notations from Barendregt (1984). Definition 2.1 (Set Λ ⊥ of finite lambda terms with ⊥). Let Λ ⊥ be the set of finite λ-terms given by the inductive grammar: where x is a variable from some fixed countable set of variables V.
We follow the usual conventions on syntax. Terms and variables will respectively be written with (super-and subscripted) letters M, N and x, y, z. Terms of the form (M 1 M 2 ) and (λxM) will respectively be called applications and abstractions. A context C[ ] is a term with a hole in it, and C[M] denotes the result of filling the hole by the term M, possibly capturing some free variables of M.

Converging reductions
In this section, we define the notion of strongly converging reduction.
Strongly converging reduction is a key concept of infinite rewriting (Kennaway et al. 1995b(Kennaway et al. , 1997) that generalises and includes finite reduction. Intuitively, an infinite reduction is strongly converging when the depth of the position of the application of the reduction rules goes to infinity along the reduction sequence. Cauchy converging reduction sequence do not behave so nicely as strongly converging reductions (Kennaway et al. 1997;Simonsen 2004). Hence, strongly converging reduction is the natural notion of reduction to study. This preference is reflected in the next notation.

Notation 2.3.
M −→ ρ N denotes a one-step reduction from M to N; M −→ → ρ N denotes a finite reduction from M to N; M −→ → → ρ N denotes a strongly converging reduction from M to N. M −→ =,ρ N denotes equality or one-step reduction from M to N.
We will sometimes write the depth of the contracted redex on top of the arrows. For example, M m −→ ρ N denotes a reduction step where the contracted redex is at depth m. Many notions of finite lambda calculus extend now more or less straightforwardly to If λ ∞ ρ is confluent and normalising, it induces a total function, denoted by nf ρ , from Λ ∞ ⊥ to Λ ∞ ⊥ such that nf ρ (M) gives the ρ-normal form of M. The set of ρ-normal forms over Λ ∞ ⊥ is denoted by nf ρ (Λ ∞ ⊥ ) and the set of ρ-normal forms over Λ is denoted by nf ρ (Λ).
2.3. The basic reductions: β, η, η −1 and ⊥-reductions We will now extend several notions of reductions on finite lambda calculus to infinite terms. The β-reduction, denoted by −→ β , is the smallest reduction on Λ ∞ ⊥ closed under the rule: The β h -reduction, denoted by −→ β h , is the β-reduction restricted to head redexes, i.e.
The η-reduction, denoted by −→ η , is the smallest reduction on Λ ∞ ⊥ closed under the η-rule: x We now define the ⊥-rule. The variant that we will use in this paper is the one that equates terms that have no HNF. The ⊥-rule is necessary because the infinitary lambda calculus with only β-reduction is not confluent. For example (Berarducci 1996), where Ω = (λx.xx)λx.xx, I = λx.x and Ω I = (λx.I(xx))(λx.I(xx)). Let M ∈ Λ ∞ ⊥ . We say that M is in HNF, if M is of the form λx 1 . . . x n .yN 1 . . . N k . We say that M has a HNF (or is head normalising), if there exists N in HNF such that M −→ → β N. The terms Ω and λx.⊥x are examples of terms without HNF.
We will also need a variation of the ⊥-reduction, called ⊥ out -reduction, that contracts only outermost ⊥-redexes and which is not closed under contexts. The ⊥ out -reduction, denoted by −→ ⊥ out , is defined as the smallest binary relation on P. Severi and F. J. de Vries 688 2.4. The new reduction: η!-reduction We will now introduce the notion of η!-rule. It is inspired by Barendregt's ∞η construction on Böhm trees (Barendregt 1984). With the current knowledge of infinite rewriting, we see that this relation 6 η on Böhm trees is nothing else but an alternative definition for strongly converging η −1 -reduction. For η-expansions, strong convergence ensures that the expanded terms remain within Λ ∞ ⊥ and are finitely branching. Thus, we define the η!-rule on Λ ∞ ⊥ as: ⊥ closed under the η!-rule. This η!-rule does not occur in the finite lambda calculus. Note that the original notion 6 η in Barendregt (1984) is defined on β⊥-normal forms (Böhm trees) only, while ηexpansion −→ → → η −1 applies to any term in Λ ∞ ⊥ . It is easy to see that 6 η and −→ → → η −1 coincide on the set of β⊥-normal forms. Hence −→ → → η −1 is an extension of 6 η to the set of all lambda terms Λ ∞ ⊥ . The strength of the new η!-reduction can be demonstrated on the Böhm tree of Wadsworth's term J mentioned above. The Böhm tree of J is represented by the term J ∞ = λxy 0 .x(λy 1 .y 0 (λy 2 .y 1 (. . .))). We see that J ∞ is of the form λxy 0 .xE y 0 where E y 0 = λy 1 .y 0 (λy 2 .y 1 (. . .)). The term E y 0 is the limit of a strongly converging η-expansion of y 0 : Therefore J ∞ reduces to I in a single η!-step, while J ∞ is not even a η-redex.

The infinitary calculus λ ∞ ⊥
The infinitary calculus λ ∞ ⊥ has some straightforward properties worthwhile to state on their own which have not been stated explicitly before. Proof. Confluence follows from Lemma 26 in Kennaway et al. (1999). Depth-first leftmost ⊥-reduction is clearly a normalising strategy. Since the depth-first left-most strategy contracts only outermost redexes, we have that M −→ → → ⊥ out nf ⊥ (M). It is not difficult to show ω-compression by adapting the proof of the compression lemma for λ ∞ β in Kennaway et al. (1997) (quite similar to our later proof of Lemma 3.4).

The infinitary lambda calculus λ ∞ β⊥
In this section, we collect some properties of the infinitary lambda calculus λ ∞ β⊥ that will be used later. Crucial is the following theorem from Kennaway et al. (1997) and Kennaway and de Vries (2003).
Since this reduction sequence is strongly convergent, for all n there exists M i such that (nf β⊥ (M)) n = (M i ) n . By Lemma 2.2, there exist m = m i > m i−1 > . . . m 0 = n such that

Properties of η!-reduction
Before we will deal with the interaction of η!-reduction with β-and ⊥-reduction in the further sections, we will study a number of useful properties of η!-reduction and η-expansion. First, we show that any η!-reduction is strongly converging. Next, we will demonstrate that λ ∞ η! and λ ∞ η −1 are dual calculi in the sense that strongly converging ηexpansion and strongly converging η!-reduction are each others inverse (cf. Lemma 3.2). This allows us to prove that strongly converging η!-reductions and strongly converging η −1 -reductions can be compressed to reductions of length at most ω. It also permits us to prove that the steps of a strongly converging η −1 -reduction can be ordered according to their depth. Finally, we will show in this section that λ ∞ η! is confluent and normalising.

Strong convergence of η!
We will prove that any η!-reduction (and hence η-reduction) starting from a term in Λ ∞ ⊥ is strongly converging. This is a direct result of our choice of depth used in the metric completion Λ ∞ ⊥ of Λ ⊥ . The infinite term Ω η is an example of a term that is not in Λ ∞ ⊥ . Clearly, Ω η η-reduces to itself by contraction of the η-redex at its root. Therefore, Ω η can perform infinitely many η-reductions at depth zero, and hence, it is not strongly converging.
Proof. Strong convergence of η! reduction follows by a counting argument. For M ∈ Λ ∞ ⊥ , let |M n | denote the number of abstractions in M n . The number |M n | decreases by one, if we contract an η!-redex in M at depth n and it remains equal if we contract an η!-redex at depth m > n. Suppose by contradiction that we have a transfinite η!-reduction sequence that is not strongly convergent, that is, suppose we have a reduction M 0 −→ η! M 1 −→ η! . . . in which infinitely many reductions occur at depth n. Then, infinitely many inequalities in the sequence . . are strict, which is impossible. Hence, the limit of the depth of the contracted redexes in any sequence M 0 −→ η! M 1 −→ η! . . . goes to infinity at each limit ordinal 6 α. This implies that all η!-reduction sequences are strongly converging.
In contrast to η!-reduction, η-expansion need not be strongly converging. For instance, the following infinite sequence of η-expansions is not Cauchy, as the distance between The infinitary lambda calculus of the infinite eta Böhm trees 691 any two terms in this sequence in this sequence is always 1.

3.2.
Relation between η −1 and η! Next, we will show that strongly converging η-expansion is the inverse of strongly converging η!-reduction: In general, η!-reduction may need less steps than its inverse. For example, while an infinite number of eta expansions is necessary to reach E x starting from x, the reverse η!-reduction can be done in only one step.
We will make frequent use of this inverse relationship. The proof of the inverse relationship (Theorem 3.1) will follow from some smaller results and ω-compression lemmas for η! and η −1 . These compression lemmas will simplify many later proofs.

Lemma 3.2 (Inverse of one-step reduction). Let
Proof. The first statement is trivial. The second statement follows directly from the definitions of η! and η −1 as illustrated in the next diagram.
If the depth of the η!-redex in M −→ η! N is n then the η −1 -redexes in N −→ → → η −1 M occur at least at depth n.

Lemma 3.3 (Inverse reductions restricted to
Proof. We only prove the first item using induction on the length α of the reduction sequence from M to N. The proof of the second item is similar.
The base case α = 0 is trivial. The successor case α = n + 1 follows easily from Lemma 3.2 and the induction hypothesis, as shown in the next diagram: Limit case α = ω. By strong convergence, the number of steps at certain depth n is finite. We can, then, always split the sequence by depth as follows.
Now consider the last step occurring at depth 0 in this sequence. The position of its redex is still present in all terms that follow M 1 , including M ω . By reversing this last η!-step at depth 0 in the limit M ω , we construct the following diagram: We repeat this process for each step at depth 0 and obtain a term N 1 such that Since all steps in the η!-reductions sequence from M to N 1 occur at depth greater than 0, the terms M and N 1 coincide at depth 0.
Repeating the above argument on the reduction sequence M >1 −→ → → η! N 1 , we find a term Moreover, M and N 2 coincide up to depth 1. Hence, we obtain an infinite η −1 -reduction sequence from N as indicated in the next diagram: Because the reduction sequence from N is strongly converging, it has a limit, say N ω . Since each term N i coincides with M up to depth i, the limit N ω of this sequence is exactly M.

Lemma 3.4 (Compression for η −1 and η!). Strongly converging reduction is
Proof. First, we consider λ ∞ η! . The proof proceeds by transfinite induction on the length of the reduction sequence. By a general argument (Kennaway and de Vries 2003;Kennaway et al. 1995b), it is sufficient to prove that a sequence of length ω + 1 can be compressed into one of length ω. Without loss of generality, we may suppose that we have a strongly The infinitary lambda calculus of the infinite eta Böhm trees 693 convergent η!-reduction sequence of length ω + 1 as follows: Contracting them, we obtain the terms of the bottom row. The reduction in the bottom row is the projection of the reduction in the top row. This way we obtain a sequence of length ω from λx.M 0 N 0 to M ω .
The proof of compression of λ ∞ η −1 is similar, but without appeal to Lemma 3.3.
Lemma 3.4 allows us to remove the conditions on length in Lemma 3.3.

Theorem 3.1 (Inverse reductions in
Thus, we have shown at the main result of this section that strongly converging η!reduction is the inverse of strongly converging η −1 -reduction.

Confluence of η!
In this section, we will show that η!-reduction is confluent. The main ingredients of the proof are Local Confluence and the Strip Lemma for η!.

Lemma 3.5 (Preservation of η!-redexes by
To prove the previous lemma, we use the next lemma, the proof of which can be found in the appendix.
Using Lemma 3.7, we can complete all the subdiagrams except for the limit case. The . . is strongly converging, say with limit N ω . Either the vertical η!-reduction M 0 −→ η! N 0 got cancelled out in one of the applications of Local Confluence or not. If it gets cancelled out, then, from that moment on, all vertical reductions are reductions of length 0, implying that M ω is equal to the limit N ω . Or the vertical η!- where all the C k [ ] have the hole at the same position at depth m, and all S ω T ω ] and the hole of C ω is also at depth m. By Lemma 3.5, λx.S ω T ω is an η!-redex. Contracting this redex in the limit M ω , we obtain C ω [S ω ] which is equal to the limit N ω of the bottom sequence.

Theorem 3.2 (η!-Confluence). The infinitary calculus
Proof. Confluence of λ ∞ η! can be shown by a simultaneous induction on the length of the two given coinitial η!-reductions. By compression (Lemma 3.4), we may assume that these reductions are at most of length ω, so here we don't need transfinite induction.

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The induction proof makes use of so called tiling diagrams (Kennaway and de Vries 2003), which can be constructed using the induction hypothesis, Lemma 3.7 (Local Confluence) and Lemma 3.8 (Strip Lemma). The important thing to note is that the depth of an η!-redex in a term does not change when we contract an η!-redex elsewhere in the term.
The double limit case is more involved. In that case, we can construct the tiling diagram shown below. The induction hypothesis allows us to construct all proper subtiling diagrams. It remains to show that the bottom row reduction and the right-most column reduction strongly converge to the same limit.
Clearly, by the fact that all subtiles are depth preserving, both the bottom row reduction and the right-most column reduction inherit the strong convergence property from respectively the top row and the left-most column reductions. Using strong convergence, we can show that for any k there exists k 1 , k 2 , such that for all i > k 1 and j > k 2 the terms M i,j have the same prefix up to depth k. Hence, the limits of the bottom row reduction and the right-most column reduction are the same.
Alternatively, one can check that the conditions of the general tiling diagram theorem in Kennaway and de Vries (2003) are satisfied to conclude that both limits are the same.
In a similar way, one can prove that strongly converging η-expansion is confluent. We skip the proof, as we don't need this result in this paper.

Normalisation of η!-reduction
We finish this section showing that λ ∞ η! -calculus is normalising in contrast to λ ∞ η −1 -calculus which is not normalising.

Theorem 3.3 (Normalisation of
. . in which each M i+1 is obtained from its predecessor M i by contracting the depthfirst left-most η!-redex in M i . By Lemma 3.1, this reduction is strongly converging. If it is finite, then the last term is an η!-normal form. If it is infinite, then by strong convergence it has a limit M ω . By a reductio ad absurdum M ω must be an η!-normal form as well: For suppose M ω contains an η!-redex λx.P X at some position p. Then, by strong convergence, there is an M n in the reduction that contains a subterm of the formλx.P X at position p, while all reduction steps after M n take place at depth greater than the depth of λx.P X . Hence X −→ → → η! X, and so X −→ → → η −1 X by Lemma 3.1. We also have that x −→ → → η −1 X, because λx.P X is an η!-redex. Therefore, x −→ → → η −1 X . Thus, λx.P X must also be an η!redex in M n . Since the later reductions steps in M n −→ → → η! M ω take place at greater depth than λx.P X . This contradicts the fact that the reduction The combination of the previous result with the confluence of η!-reduction give us uniqueness of normal forms as corollary:

Commutation properties for β and η!
In this section, we will prove various instances of commutation of β and η! to be used in the proof of confluence of λ ∞ β⊥η! . To prove local commutation of one step β and one step η! we need a preservation result which in turn is a consequence of the next lemma, the proof of which can be found in the appendix.
Proof. Suppose M 0 can do both a β-reduction and an η!-reduction at respectively depths n and m. We prove only one case. The β-redex is inside the expanded variable term of the η!-redex, that is M 0 is of the form C 1 [λx .MN] and N = C 2 [(λy.P )Q].
The infinitary lambda calculus of the infinite eta Böhm trees 697 Next, we prove the strip lemma for one step β over η!.

Lemma 4.4 (Strip Lemma for
The proof is similar to the proof of the strip lemma for η!. By Lemma 3.4 (Compression Lemma), we can assume that the sequence has length at most ω.
Using Lemma 4.3, we can complete all the subdiagrams except for the limit case. Either the vertical β-reduction got cancelled out in one of the applications of Local Confluence or not. If it gets cancelled out, then from that moment on all vertical reductions are reductions of length 0, implying that M ω is equal to the limit N ω . Or, if the vertical β-reduction did not get cancelled out, then a residual of the for all k > 0, and in all C k [ ] the hole occurs at the same position at depth m, so that all N k are of the form C k [P k ]. This holds for the limit terms as well. Contracting this residual in the limit M ω gives us the limit N ω .

Lemma 4.5 (Strip lemma for
The proof can be found in the appendix.

Commutation of η! and ⊥ out
Full commutation of η! and ⊥ does not hold. Already local commutation of η and ⊥ goes wrong (cf. Severi and de Vries (2002)) when the contracted ⊥-redex is not outermost. Take for instance Ω η! ←− λx.Ωx → ⊥ λx.⊥. However, for proving confluence of λ ∞ β⊥η! it is sufficient that η!-reduction commutes with ⊥ out -reduction. Recall that −→ ⊥ out is the reduction that replaces an outermost subterm without HNF by ⊥. The proof of this commutation property then follows the familiar pattern.
The proof can be found in the appendix.
Lemma 5.2 (Strip lemma for −→ ⊥ out over −→ → → η! ). Given a one-step reduction M −→ ⊥ out N and a strongly converging reduction M −→ → → η! P , then there exists Q such that: The proof can be found in the appendix.

Lemma 5.3 (Strip lemma for
. Given a one-step reduction M −→ η! N and a strongly converging reduction M −→ → → ⊥ out P , then there exists Q such that: The proof can be found in the appendix. Theorem 5.1 (η!⊥ out -commutation). Strongly converging η!-reduction commutes with strongly converging ⊥ out -reduction: As for the confluence of η! (Theorem 3.2), but using Lemmas 5.1-5.3 instead.

Confluence and normalisation of β⊥η!
We are now ready to prove the main results of this paper concerning confluence and normalisation of the infinite extensional lambda calculus λ ∞ β⊥η! .

Theorem 6.1 (Preservation of β⊥-normal forms by
The proof can be found in the appendix. The proof can be found in the appendix. (N) and the next diagram commutes:

Theorem 6.3 (Projecting β⊥η!-reductions
Case α = 0. This is trivial. Case α = γ + 1. By the induction hypothesis, we have that We distinguish two cases depending on the last step: 2. If the last step is an η!-reduction step, then we first normalise the term X of the η!- Then, we split the β⊥-reduction sequence from C[λx.P nf β (X)] to the normal form nf β⊥ (M γ ) into a β-reduction sequence followed by a ⊥ out -reduction sequence using Theorems 2.2 and 2.1. This is depicted in the diagram: Next, applying first the Strip Lemma 4.5 for −→ η! with −→ → → β and secondly the full Since all η!-reduction sequences are strongly convergent (Theorem 3.1), the bottom reduction sequence is strongly convergent, and hence has a limit, say N. To conclude that N is in fact nf β⊥ (M λ ), it suffices to prove that for all n, N n = (nf β⊥ (M λ )) n .
By Approximation Theorem 2.4, there exist m > n such that (Theorem 2.3). And so, from nf β⊥ (M γ 0 ) onwards, P is a prefix of all terms of the bottom reduction sequence. Hence, P is a prefix of their limit N. Therefore, N n = P n = (nf β⊥ (M λ )) n .

Infinite eta Böhm trees as normal forms
In this section, we will see that the infinite eta Böhm tree of a lambda term M denoted by ∞ηBT(M) is nothing else than the η!-normal form of BT(M), the Böhm tree of M, which in turn is nothing else than nf β⊥ (M).
We begin with the definition of Böhm tree formulated as a term in Λ ∞ ⊥ . The original notion of Böhm tree defined in Barendregt (1984) for finite terms applies to infinite terms as well. Remark 7.1. We define Böhm trees as terms in the infinitary lambda calculus and this definition is given co-recursively. In Barendregt (1984) Definition 10.1.4, a Böhm tree is defined as a function from a set of sequences or positions to a set Σ of labels. Up to a change of representation, Definition 7.1 is very similar to the informal definition of Böhm trees given in Definition 10.1.3 in Barendregt (1984).
As stated before, one purpose of the infinitary lambda calculus λ ∞ β⊥ = (Λ ∞ ⊥ , −→ β⊥ ) is to capture the notion of Böhm trees as normal forms inside the calculus (see Figure 1 Next, we redefine the ∞η-construction of Barendregt (1984) using the notation of strongly converging reduction. Barendregt's original definition in Barendregt (1984) Proposition 10.2.15 of the infinite eta expansion differs slightly from the above definition.
It uses the order 6 η on Böhm trees instead of −→ → → η −1 . To prove that our definition of infinite eta Böhm trees coincides with the definition in Barendregt (1984), it suffices to prove that the relations 6 η and −→ → → η −1 coincide on β⊥-normal forms. We leave the proof for the reader. Proof. For the first part, it is not difficult to prove that M η!-reduces to ∞η(M) and that ∞η(M) is in η!-normal form. By Corollary 3.1, the η!-normal form is unique, hence ∞η(M) = nf η! (M). The second part follows from Theorems 6.4 and 7.1 and the previous part.

Conclusion: the ∞η-Böhm trees are a model of the lambda calculus
In Barendregt (1984), the Böhm trees got a role on their own, when with help of the continuity theorem it was shown that the set of Böhm trees can be enriched to become a model of the finite lambda calculus λ β . The original definition of Böhm tree used the language of labelled trees. In that approach, lambda terms and Böhm trees live in different worlds, because lambda terms got defined with the usual syntax definition. Using infinitary lambda calculus (Kennaway et al. 1997), this separation no longer exists. Hence, it can be shown directly from the confluence and normalisation properties of λ ∞ β⊥ that the Böhm trees are a model of the finite lambda calculus λ β . Now, we have shown in this paper that λ ∞ β⊥η! is confluent and normalising, we can prove that the infinite eta Böhm trees are a model of λ ∞ βη . This is a new result that could not have been proved with a variation of continuity argument used in Barendregt (1984) to prove that the Böhm trees for a model of the finite lambda calculus. In Barendregt (1984), the proof that the set B = BT(Λ) of Böhm-like trees is a λ-model uses continuity of the context operator. However, this appeal to continuity is not possible for ∞η-Böhm trees, because neither the abstraction nor the application are continuous (Severi and de Vries 2005a). For instance, take λx.y⊥ and λx.yx. Then λx.y⊥ λx.yx, but ∞ηBT(λx.y⊥) = λx.y⊥ y = ∞ηBT(λx.yx).
From the ∞η-Böhm trees of the finite lambda terms in Λ, we will now construct an We will now show that B ∞η is a syntactic λ-model in the sense of Barendregt (1984), Definitions 5.3.1. and 5.3.2. The proof can be found in the appendix. These sections in the appendix contain all omitted proofs.
The following lemma is proved by induction on the depth of the hole in the context. Proof. This is proved by induction on the lexicographically ordered pair (n, ||P n ||) where ||P n || is the number of symbols of P n . Suppose n > 0 and P = P 1 P 2 . Then   (N n )).
We also need a variation of Lemma 2.1 with β h -reduction instead of ⊥ out -reduction. Proof. (i ⇒ ii). Kennaway et al. (1997) Suppose there exists N in HNF such that M −→ → → β N. We can assume that the length of this reduction is ω by Theorem 2.2. Since −→ → → β is strongly convergent, we have that there exists N such that It is easy to show that N is in HNF.
(ii ⇒ iii). By Theorem 2.2, M −→ → β N −→ → → β⊥ nf β⊥ (M). Since N is in HNF, so is nf β⊥ (M). We truncate the normal form of M at depth 1 and apply the well-known results on head normalisation in finite lambda calculus. By Theorem 2.4, there exists m > 1 such that nf β⊥ (M m ) (nf β⊥ (M)) 1 . Hence, nf β⊥ (M m ) is in HNF because (nf β⊥ (M)) 1 is in HNF. By Theorem 2.2 and the fact that −→ → → β⊥ is strongly convergent, we have that Since the term M m ∈ Λ ⊥ is a finite λ-term and the reduction M m >0 −→ → β P is finite, we can now apply Theorem 8.3.11 of Barendregt (1984). We have that M m −→ → β h N for some N in HNF. By Lemma A.4, there exists N such that M −→ → β h N and N N . Since N is in HNF, so is N .

Appendix B. Preservation of η-Expansions of x after η!
By postponing the η −1 (or the η!) steps at greater depth, we can re-order the steps in an η −1 (or an η!) reduction sequence by increasing order of depth.
Lemma B.1 (Postponing η −1 -steps at greater depth). Let j < i. Then, an η −1 -reduction step at depth i can be postponed over an η −1 -reduction step at depth j, that is: Since j < i, the η −1 -redex at depth j occurs either in C or in P but it cannot occur in x. We have that The infinitary lambda calculus of the infinite eta Böhm trees 705 As a consequence of the previous lemma, we obtain:

Lemma B.3 (Postponing η!-steps at greater depth).
Let j < i. There are only two possible situations that can occur when we postpone an η!-reduction step at depth i over an η!-reduction step at depth j Proof. The situation of the second tile occurs when the term M 1 contains a term λx.P Q at depth j and the η!-redex at depth i is inside Q: We know that Q is an infinite η-expansion of x and we have to show that so is Q. Since Q is obtained from Q by applying only one step of η!-reduction, by Lemma 3.2, we can reverse the reduction from Q to Q . Hence, As a consequence of the previous lemma, we obtain:

Lemma B.4 (Sorting η!-reduction sequences by order of depth). If M −→ → → η! N, then there is either a finite reduction
or an infinite reduction The first abstraction λy is not an η!-redex. In spite of this, it is possible to undo all the η −1 -steps by doing only a finite number of steps of η! at depth 0. This is proved in the following lemma which will be used to prove Commutation of β and η!.

Lemma B.5 (Inverting the expansion of a variable). If
there is a single free occurrence of x in M at depth 0.
Proof. By Theorem 3.1, x −→ → → η −1 M implies that M −→ → → η! x. By Lemma 3.4 (Compression Lemma) and Lemma B.4 (Sorting by order of depth), we have that there is either a finite reduction or an infinite reduction In both cases, the following case analysis allows us to conclude that M 1 = x.
1. If M 1 is a variable, then the rest of the reduction sequence from M 1 onwards is empty, so that M 1 = x.

Suppose M 1 = (P Q). Then, all η!-reducts from M 1 are applications (including N).
This contradicts the fact that N is a variable. 3. Suppose M 1 = λx.P . Then either all reducts from M 1 are abstractions (including N) or this abstraction disappears because it is contracted by an η!-redex. Neither case is possible. The first option contradicts the fact that N is a variable. The second option contradicts the fact that in the reduction from M 1 to N we contract only redexes at depth strictly greater than 0.
Local Confluence and the Strip Lemma for η! depend on the following lemma that says that η!-redexes are preserved by certain η!-reduction sequences.
Since η −1 does not change the depth of any subterm once the η!-redex is created, its depth remains fixed. By omitting the η-expansion step that created the abstraction of the η!-redex plus all the subsequent η-expansions from y to Q, we construct the reduction sequence at the bottom: The more general statement of the lemma follows by repeated application of the above. Proof. By compression of η!-reduction we may assume that the reduction M −→ → → η! M is at most ω steps long. By Theorem B.2, this reduction sequence can be sorted. Two possible situations can arise: Case M −→ → η! M is finite. We illustrate the proof for a sequence of length 3.
Case M −→ → → η! M has length ω. By repeated application of the previous argument, we can construct the diagonal sequence as shown in the following diagram: By construction, the diagonal sequence is strongly convergent and has a limit, say N ω . It is easy to see that the limits M ω and N ω are the same, because for all k we have N k+1 and M k+1 have the same truncation to depth k.

Appendix C. Preservation of η-Expansions of x after β
For the proof of local commutation, we need to prove preservation of η!-redexes under β-reduction. The proof follows the same pattern as the proof of preservation of η!-redexes under η!-reduction (Lemma 3.5).
Since η −1 does not change the depth of any term, once the β-redex is created, its depth remains fixed. There are only two ways in which η-expansions can create a β-redex: 1. The application of the β-redex is created before its abstraction in the η-expansion.
This happens as follows: 2. The abstraction in the β-redex gets created before its application in the η-expansion.
Proof. By strong convergence, we can assume that the β-reduction sequence is of the . .. Now, we can proceed similarly as in the proof of Theorem B.1 while exploiting Lemmas B.2 and C.1 instead.

Appendix D. Strip lemma for one step η! over infinitely many β's
The full strip lemma for β over η! is harder than the strip lemma for η! over β (see Lemma 4.4). The difficulty lies in the fact that, due to overlap, the residuals of an η! redex are not always immediately η! redexes themselves. We illustrate this with an example. Consider M = (λx.zX)Q, where x −→ → → η −1 λy 1 y 2 y 3 .xy 1 y 2 y 3 = X and Q is some arbitrary term. What are the residuals of the η!-redex λx.zX in M after contracting the β-redex (λx.zX)Q? We have that Only the first of these consecutive η!-redexes is readily present in λy 1 y 2 y 3 .Qy 1 y 2 y 3 . From the next two redexes, only their λ's are present in λy 1 y 2 y 3 .Qy 1 y 2 y 3 . These lambda's are η!-redexes 'in waiting.' The residuals in λy 1 y 2 y 3 .Qy 1 y 2 y 3 of the original η!-redex λx.zX will be the three abstractions λy 3 .Qy 1 y 2 y 3 , λy 2 y 3 .Qy 1 y 2 y 3 and λy 1 y 2 y 3 .Qy 1 y 2 y 3 in spite of the fact that not all of them are η!-redexes. We make this precise with underlining. To track the residuals of λx.zX, we will not only underline the λ of λx.zX but also all the λ's in X, i.e. (λx.zX)Q where X = λy 1 λy 2 λy 3 .xy 1 y 2 y 3 .
To simplify matters a bit, we will not do this in full generality. Instead, we will do this only with η!-redexes of the form λx. MN where N is an η-expansion of x which is in β⊥-normal form, because such expansions have a straightforward format: Proof. The reduction steps in x −→ → → η! X can be sorted by depth with Lemma B.2, so that we may assume without loss of generality that x −→ → → η! X is of the form Because X is a β⊥-normal form, X 1 must be of the form λy 1 . . . y n .xy 1 . . . y n as can be shown with a proof by induction on n: if the expansions steps at depth 0 would be executed at other positions than the sequence of positions that leads to λy 1 . . . y n .xy 1 . . . y n a β-redex would be introduced and all further terms in the sequence would contain a β-redex, contradicting the normal form of the final term X. Now, because the deeper reductions in X 1 >0 −→ → → η −1 X do not alter the left spine of X 1 , it follows that also X must be of the shape λy 1 . . . y n .xY 1 . . . Y n where y i −→ → → η −1 Y i and Y i is in β⊥-normal form for each 1 6 i 6 n.
In the proof of the restricted strip lemma for one step η! over β, we will employ the underlining technique of Barendregt (1992) to track the residuals of η!-redexes of the particular form λx.P X where X is an η-expansion of x in β⊥-normal form. To introduce this technique in the infinitary setting, we extend the set Λ ∞ ⊥ to Λ ∞ ⊥ which will contain underlined terms of the following form only: where Y i ∈ Λ ∞ ⊥ is in β⊥-normal form, Y i is an η-expansion of y i and Y i is obtained by underlining all λs in Y i . for all 1 6 i 6 n.

Definition D.1 (Family of sets E x for x ∈ V).
We define a family of sets E x on x ∈ V by simultaneous induction X ::= x | λx 1 . . . λx n .xX 1 . . . X n .
where, X i ∈ E x i for all 1 6 i 6 n.
Definition D.2 (Set Λ ⊥ of underlined finite lambda terms with ⊥). We define the set Λ ⊥ of underlined finite λ-terms by induction where, x ∈ V, x i ∈ FV (M) and X i ∈ E x i for all 1 6 i 6 n.
The metric d on Λ can be easily extended to terms in Λ ⊥ and in each of the E x .

Let x ∈ V. The set E ∞
x is the metric completion of the set E x with respect to the metric d.
2. The set Λ ∞ ⊥ is the metric completion of the set of underlined finite lambda terms Λ ⊥ with respect to the metric d. Now, we are ready to define underlined η!-and β-reduction.
The definition of −→ β is correct in the sense that Λ ∞ ⊥ is closed under underlined βreduction: one sees easily that X[x := Q] ∈ Λ ∞ ⊥ holds for any X ∈ E ∞ x and any Q ∈ Λ ∞ ⊥ . We will frequently use situations, where X i ∈ E ∞ x and x i ∈ FV (M) for all 1 6 i 6 n, in which case we have the reductions: We will denote the union of −→ β and −→ β by −→ ββ .
As in the finitary setting, we need mechanisms to remove the underlining: Definition D.5 (Removing underlinings). Let M ∈ Λ ∞ ⊥ . 1. We define |M| ∈ Λ ∞ ⊥ as the result of removing all the underlinings in M.
The infinitary lambda calculus of the infinite eta Böhm trees 711 2. We define ϕ(M) ∈ Λ ∞ ⊥ as the result of contracting all η!-redexes from M by corecursion as follows.
x where X is the result of underlining all abstractions in X. Proof.

Suppose X ∈ E ∞
x . It is not difficult to show that X = λy 1 . . . λy n .xY 1 . . . Y n and Y i ∈ E ∞ y i for all 1 6 i 6 n using Definitions D.1 and D.3. We consider the η −1 -reduction sequence: x −→ → η −1 λy 1 . . . λy n .xy 1 . . . y n . We repeat a similar argument for each Y i with 1 6 i 6 n as we did for X. This process can be repeated ad infinitum to obtain an η −1 -strongly converging reduction sequence from x to |X|. 2. Suppose x −→ → → η −1 X. We construct a Cauchy sequence M 1 , M 2 , . . . of terms in E x whose limit is X using Lemma D.1. By construction, the limit X is an element of E ∞ x . The first term M 1 in this sequence is x which belongs to E x . By Lemma D.1, we have that X = λy 1 . . . λy n .xY 1 . . . Y n and y i −→ → → η −1 Y i for each 1 6 i 6 n. The second term M 2 of the sequence is λy 1 . . . λy n .xy 1 . . . y n which belongs to E x . We repeat this process to construct all the terms in the sequence. The limit of this sequence is X and it belongs to E ∞ x by Definition D.3.
Proof. This is proved by induction on the length of the reduction sequence. We only prove it for a reduction sequence of length 1. Suppose λx.MX −→ η! M. Then, X ∈ E ∞ x and x ∈ FV (M). By Lemma D.2(i), we have that x −→ → → η −1 |X| and hence, λx.MX −→ η! M.
The next lemma is a straightforward consequence of the definition of ϕ.
Proof. Contraction of the η!-redexes using a depth-first left-most strategy gives a reduction M −→ → → η! ϕ(M) of length at most ω.  The function ϕ does not preserve truncations, i.e. ϕ(M n ) = ϕ(M) n . For example, take M = λx.y(λz.xz). We will define a notion of quasi-truncation which is preserved by ϕ. The quasi-truncation of a term at depth n truncates the term at depth n except for the η-expansions X in an η!-redex. Definition D.6 (Quasi-truncation). We define quasi-truncation of M at depth n by induction on the lexicographically ordered pair (n, m) where m is the number of abstractions and applications at depth n:  1 = λx.y⊥(λz.xz). Note that ([M] n ) n = M n , for all M ∈ Λ ∞ ⊥ . The function ϕ preserves quasi-truncations:

Lemma D.8 (Preservation of quasi-truncations). ϕ([M] n ) = ϕ(M) n .
Proof. This is proved by induction on (n, m) where n is the depth of the truncation and m is the number of abstractions and applications in M at depth n. Lemma D.9 (ϕ on many β-steps). Let M ∈ Λ ∞ ⊥ . If M −→ → → ββ N has length at most ω, then ϕ(M) −→ → → β ϕ(N).
Proof. We prove it by induction on the length of M −→ → → ββ N. The finite case follows from Lemma D.7. We prove the case when the length is ω. The following diagram can be constructed by repeated applications of Lemma D.7. Since ϕ preserves the depth of the contracted redex, we have that the bottom sequence is strongly convergent and the limit exists which is P .
It remains to prove that P = ϕ(M ω ). By strong convergence, there exists n 0 such that for all n > n 0 , .
Proof. Let X be an η-expansion of x such that X is a β⊥-normal form. Suppose M = C[λx.M 0 X]. In order to track the residuals of this η!-redex, we consider the term M = C[λx.M 0 X] where X is the result of underlining all abstractions in X. Then, X ∈ E ∞ x by Lemma D.2(ii). The reduction M −→ → → β N is lifted to M −→ → → ββ N . Using Lemmas D.3, D.5 and D.9, we obtain the following diagram:

Appendix E. Commutation properties of β and η −1
In this section, we will study some precise commutation properties of β and η −1 . We need these properties to prove that η-expansions of HNFs again have a HNF. As a consequence η!-reduction preserves ⊥ out -redexes, which plays a crucial role in the proof of the commutation property for η! and ⊥ out in Section 5.

E.1. Strip Lemmas for One
Step β 0 Over η −1 In this subsection, we concentrate on the strip lemmas for one-step β-reductions that takes place at depth 0 over η-expansion. There is a slight complication, because η-expansions can create β-redexes as shown by the next example.
In the above η −1 -reduction sequence, we have created two extra β-redexes which should be contracted to get a common reduct. These extra β-redexes are of a special nature, for which we will introduce the notion of β 0 -reduction.
Lemma E.1 (Local commutation for one step β 0 and one step η −1 ). Given M 0 −→ η −1 M 1 and M 0 −→ β 0 M 2 , there exists M 3 such that either one of the following diagrams hold: This results in an instance of Diagram (1). 2. The β 0 -redex is inside the η −1 -redex, that is M 0 is of the form C 1 [N], where N ≡ C 2 [(λy.P )Q]. This case results in an instance of Diagram (1) too. 3. The η −1 -redex is part of the body of the abstraction λy.P of the β 0 -redex, i.e. P −→ η −1 P . Since η −1 does not affect the depth of the variable y, (λy.P )Q remains a β 0 -redex and we have which is an instance of Diagram (1). 4. The η −1 -redex is part of the argument Q of the β 0 -redex. This results in the following: which corresponds to Diagram (1). Since the variable y occurs only once in P , we need only one η −1 -step from P [y := Q] to P [y := Q ]. And because the depth of this variable is 1, the depth of that η −1 -redex in P [y := Q] is m. 5. Only if m = 0, the η −1 -redex coincides with the abstraction λy.P of the β 0 -redex, that is M 0 is of the form C [NQ] where N ≡ λy.P .
The above is an instance of Diagram (2). Lemma 3.4 (Compression Lemma), we can assume that the η −1 -reduction sequence has length at most ω. If the length is finite, then the result follows by repeated application of Diagram (1) of Lemma E.1. Diagram (2) does not apply as the η-expansions are performed at depth greater than 0. When the length is ω we construct the diagram:

Lemma E.2 (Strip lemma for
Using Diagram (1) of Lemma E.1, we can complete all the subdiagrams except for the limit case. Since the η-expansions are performed at depth greater than 0, all M k with k > 0 are of the form C k [(λx.P k )Q k ], where all the C k [ ] have the hole at the same position at depth 0, and all P k have exactly one occurrence of x at depth 1. The limit term is of the form C ω [(λx.P ω )Q ω ]. The hole of C ω occurs also at depth 0 and x occurs only once in P ω and at depth 1 because η-expansions do not introduce variables and the existing variables remain at the same depth. Hence, the residual remains a β 0 -redex in the limit. Contracting this redex in the limit M ω reduces to C ω [P ω [x := Q ω ]] which is equal to the limit N ω of the bottom sequence.
instance of Diagram (3): Note that in the first step on the right vertical line, the contracted outermost β-redex that got created by the η-expansion is a β 0 -redex. Again, note the informative role of β 0 in the formulation of the lemma.
The previous local commutation lemma generalises to a finite strip lemma of Proof. By induction on the finite length of the η −1 -reduction sequence. We show the induction step.  (1) and (3) of Lemma E.3. Note that Diagram (2) does not apply because β and η −1 are performed at the same depth.
Next, we prove the strip lemma for one-step β reduction over many step η −1 .

Lemma E.5 (Strip lemma for
Proof. Similar to Lemma E.2 using Lemma E.3 Diagrams (1) and (2). Lemma E.6 (Strip lemma for β at depth 0 over η −1 ). Given a one-step reduction M 0 −→ β P and a strongly converging reduction M −→ → → η −1 N, then there exists Q such that Proof. By Lemma B.2, we can assume that the η −1 -reduction sequence is of the form
In the first example, we see that the argument of the β-redex that we have created is a variable, while in the second example the argument is ⊥.
In fact, we will define two instances of β-reduction, called respectively β v -reduction, and β V -reduction in order to deal with β-redexes created by η-expansions starting from a HNF.
Definition E.2 (β v and β V -reductions). Let C[ ] be a context with the hole at depth 0.
We defined β V because β v and η −1 do not commute if the η −1 -step is performed at depth greater than 0. The following diagram can be completed because the right vertical line is a β V -step. (λx.x⊥)y Lemma E.7 (Local commutation of β v and η −1 at depth 0). If M 0 −→ η −1 M 1 and M 0 −→ β v M 2 , then there exists an M 3 such that one of the following diagrams holds: Proof. Suppose M 0 can do both a β v -redex (λy.P )Q at depth 0 and η −1 -redex N at depth 0. The only possible situations in which this can happen are as follows: 1. The β v -redex (λy.P )Q and the η −1 -redex N are not nested, i.e. M 0 = C[(λy.P )Q, N].
This case leads to Diagram (1). 2. The β v -redex is inside the η −1 -redex, that is M 0 is of the form C 1 [N], where N ≡ C 2 [(λy.P )Q]. This case leads to Diagram (1). 3. The η −1 -redex is part of the body of the abstraction λy.P of the β v -redex. This case leads to Diagram (1). 4. The η −1 -redex cannot be part of the argument Q of the β v -redex, because the η −1 -redex occurs at depth 0. 5. The η −1 -redex coincides with the abstraction λy.P of the β v -redex, that is M 0 is of the form C [NQ], where N ≡ λy.P .
This case leads to Diagram (2). Note that C[(λx.(λy.P )x)Q] contains a β v -step and a β 0 -step. The β 0 -reduction contracts the outermost redex created by the η-expansion and the β v -reduction contracts the innermost one.
Later, we will need a sort of strip lemma of β v over η −1 where the right vertical line is a β V -step. For example For the bottom horizontal line, we will define a parallel reduction called η −1 v -reduction which replaces some of the variables and ⊥'s in a term by their η-expansions.
Proof. Suppose M 0 = C[(λx.P )Q] and (λx.P )Q is a β v -redex. Then, Q is either a variable y or ⊥. The following diagram can be completed . Note that P is obtained from P by replacing some of the variables or ⊥'s by their η-expansions. Suppose that x has been replaced in P by its η-expansion. There are now two options for Q: 3. Before one of the λy i , e.g. Lemma F.5 (Parallel η-expansions). Let N ∈ Λ ∞ ⊥ be the truncation of some term at depth Lemma F.6 (Approximation for η −1 ). If M −→ → → η −1 N, then there is a P such that M 1 −→ → → η −1 P where N 1 P N.
Proof. By Lemma 3.4, we can assume that the η −1 -reduction sequence has at most length ω and by Lemma B.2 we can assume it is sorted by increasing order of depth. Suppose the reduction sequence is finite, i.e.
We construct a reduction sequence from M 1 of the form: P k M k . Since (M k ) 1 P k and the hole in C occurs at depth 0, we have that there exist N 1 and C 1 such that P k = C 1 [N 1 ] where C 1 C 1 and N 1 N 1 . Since P k M k , we also have that C 1 C and N 1 N. By setting P k+1 = C 1 [λx.N 1 x], we have that where (M k+1 ) 1 = C 1 [λx. and P k = C 1 [⊥] for some C 1 such that C 1 C . We set P k+1 = P k = C 1 [⊥]. Note that we also have that (M k+1 ) 1 = (M k ) 1 because the position of the hole in C occurs at depth n k greater than 0. Hence, (M k+1 ) 1 = (M k ) 1 P k = P k+1 = C Finally consider the limit case. Suppose we have a strongly convergent reduction of length ω: By induction, we can construct the infinite reduction that performs the η-expansions at the same depth: The above sequence is strongly convergent and hence, it has a limit P ω .
We prove that P ω M ω by showing that (P ω ) k (M ω ) k for all k. For any k, there exists n 0 such that for all n > n 0 , (M n ) k = (M ω ) k and (P n ) k = (P ω ) k . By induction, P n M n . Hence, (P ω ) k = (P n ) k (M n ) k = (M ω ) k .
By Lemma F.7, Q has a HNF.
As a consequence of the above theorem and Theorem 3.1, we have the following: In order to prove that the η-expansions of a variable do not contain subterms without HNF, we will need the following theorem: c. Q = λx.Px. This case is similar to the previous case.
d. Q is a subterm of P in N, then so it is in M as well, and therefore head normalising.
2. Q is not an subterm of λx.Px. Then at the same position q we find a possible different subterm Q in M. By assumption Q is head normalising.
a. P is a subterm of Q , then Q −→ η −1 Q and hence by Theorem F.4, we see that Q is head normalising.
b. P is not a subterm of Q , then Q = Q , and so Q is head normalising.  Lemma 3.4, the remaining situation we have to consider is a reduction M −→ → → η −1 N of length ω. So consider a subterm Q of N at some depth n. By strong convergence, there is M −→ → η −1 C[Q ] −→ → → η −1 N so that Q occurs at depth n in C[Q ] and Q −→ → → η −1 Q. Assuming that all subterms of M are head normalising, it follows from induction hypothesis that all subterms of C [Q ] are head normalising, in particular Q . Again, by Theorem F.4, we find that Q is head normalising.