Proof theoretical strength of Martin-Löf Type Theory

with W-type and one universe

Dissertation

zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften

der Fakultät für Mathematik der Ludwig-Maximilians-Universität

München

vorgelegt von Anton Setzer

eingereicht am 10. September 1993

1. Berichterstatter: Prof. Dr. W. Buchholz

2. Berichterstatter: Prof. Dr. H. Schwichtenberg

Tag der mündlichen Prüfung:

Contents

1 Introduction 3

I Definition of the Theories 7

2 Definition of Martin-Löf’s type theory 8 3 Comparison of Russel and Tarski-version 21 4 Definition of KP i 27

II Upperbound 30

5 Interpretation of terms and types 31 6 Correctness of the Interpretation 41 7 Arithmetical Formulas 56

III Lower bound 62

8 Well-ordering proof 63 9 Ordinals 89 10 Comparison of ordinals 121

Chapter 1

Introduction

In this thesis, we bring together two different fields of Mathematical Logic, Martin-Löf’s type theory and proof theory. Since we hope, that some type theorists will read it as well, which do not know proof theory very well, we will take the chance to try to motivate the work of proof theory.

Proof theoretical strength is a measure for theories. First results go back to Gentzen ([Gen36], [Gen38] and [Gen43]) how showed the consistency of Peano Arithmetic P A by means of transfinite induction up to ϵ_{0}. He used finitistic arguments. Schütte has given a clearer version of it by using the ω-rule. We want to sketch this argument very briefly.

This was done by interpreting the proof of P A in a semi-formal, non finitistic, system, where we have the ω-rule for ∀-introduction:

\frac{A(n) for all n ∈ IN}{∀x.A(x)}

If we have interpreted the base case and induction step of an induction axiom in this system, with conclusion ∀x.A(x), we can prove A(n) by using the base case and n-times the induction step, and conclude ∀x.A(x) in our non finitistic system. Note, that the height of the nth premise is the sum of the height of the base case and n-times the height of the induction step, so the height of the new derivation is infinite. For the interpretation, we needed the cut-rule

\frac{Γ, A Γ, ¬A}{Γ}

(here Γ is a sequence of formulas, which can be read as the disjunction of the formulas). This rule corresponds fo a lemma: If we have proven A and a lemma, that proves, that from A follows B, (classically this is ¬A ∨ B, written as a sequence B, ¬A), we can conclude B. In our interpretation of the induction, in the n-th premise we have used n cuts, to conclude, that from A(0) and A(Sm), ¬A(m) follows A(n). We can eliminate these cuts, by induction over the derivation. A cut-free proof is a truth-definition, so, there is no cut-free proof of 0 = 1 and we have shown that P A ⊬ 0 = 1.

To do this, we needed induction over trees. We can measure the size of the trees by ordinals and replace the induction over trees by transfinite induction. A careful analysis shows, that induction up to ϵ_{0} is sufficient. We can even formalize this analysis in P A plus transfinite induction up to ϵ_{0}. So P A cannot prove transfinite induction up to ϵ_{O}, since by Gödel, P A does not prove its own consistency. For each α < ϵ_{0} we can prove transfinite induction up to α in P A. So we have proven, that ϵ_{0} is the supremum of ordinals, up to which we can prove transfinite induction in P A, and we say, the proof theoretical strength of P A is ϵ_{0}, |P A| = ϵ_{0}.

Now we have found a measure for the strength of a theory. The theorem, that P A proves transfinite induction up to ω_{n}, where ω_{0} = 0, ω_{n+1} = ω^{ω_{n}}, ϵ_{0} = sup{ω_{n}|n ∈ IN}, are in a logical sense the most complicated sentences, which we can prove in P A – the diagonalization of it can not be proven any more in P A.

This sort of analysis or related work could be carried out for many other theories as well (see for instance the books [BFPS81] [BS88], [Gir87], [Poh89], [Sch77a], [Tak87], the overviews [Poh91], [Poh92],[Rat91b], [WW92]), it has turned out to be a good measure, and gives an ordering of theories. The landscape below |Δ_{1}^{2} − CA + BI| is now very well known (see for instance Rathjen’s dissertation [Rat89]), knowing the proof theoretical strength gives a very good intuition, what can essentially be done in a theory. For instance, Γ_{0} is the bound for predicative theories – for the analysis of stronger theories we need totally new methods, where we can prove complete cut elimination with ordinal analysis only for a restricted set of formulas (in subsystems of analysis for Π_{2}^{0}-sentences). |Π_{2}^{1}−CA|, which is not calculated yet, and we do not know if it can ever be calculated, seems to be the bound, after which some sort of hyper-impredicativity starts – when we come closer to Π_{2}^{1} − CA, the ordinals explode.

We think, that this analysis is important, For instance, if we are playing around with Martin-Löf’s type theory, we can find out, how different concepts change the strength of it.

Let us therefore have a look on the landscape of Martin-Löf’s type theory, seeing, what is known about it. We follow the introduction of [Pal92]. Aczel has shown in [Acz77] that Martin-Löf’s type theory with one universe but no W-type has strength |ÎD_{1}| = ϕ_{ϵ_{0}}0. Martin-Löf’s type theory with n Universes has strength |ÎD_{n}| = α_{n} with α_{0} := ϵ_{0}, α_{n+1} := ϕ_{α_{n}}0, (see in [Fef82b]), and if we have infinitely many universes, but still no W-type, the theory (ML_{<ω}) has strength |ÎD_{<ω}| = Γ_{0}, the last result due to Beeson and Feferman (see remark II.6(iv) in [Fef82b] or [Bee85]). The extensional version, (ML_{<ω}^{e}) has the same strength (see Beeson, [Bee85]). This is astonishing, since these are all predicative bounds. As soon as we add the W-type, the proof theoretical ordinals become impredicative. Palmgren has shown in [Pal92] (part of his thesis [Pal91]) the following results: If we have type theory with one universe and an induction principle on it, we have |ML_{1}^{i}| ≥ |ID_{1}|. Adding extensionality gives the same strength, and if we have infinitely many universes with induction principles, we have |ML_{<ω}^{e}V | ≥ |ML_{<ω}V | ≥ |ID_{<ω}|. For intensional type theory with one universe and W-type, he has shown |ML_{1}^{e}W | ≥ |ML_{1}W | ≥ |(Δ_{2}^{1} − CA)|, but conjectured, that the strength will certainly far exceed this bound.

The strength we prove, is in fact far bigger, it is slightly bigger than the strength of |KP i| = |(Δ_{2}^{1} − CA) + (BI)|. Until recently (work of Rathjen, [Rat90], [Rat91a], [Rat92], see also [Buc93], [Sch91a], [Sch91b], [Sch92b]), KP i has been essentially the strongest theory, for which proof theoretical analysis could be carried out. For the author, |KP i| is an ordinal which seems to be an ordinal of significance similar to that of Γ_{0}. It is quite difficult, really to exceed this strength, and most of mathematics can be carried out within theories of this strength. Feferman stated in remark I, 13.2 and 4 of his paper [Fef75], that Martin-Löf’s type theory with one Universe and W -type, as well as his theory T_{0}, which is of strength KP i, are in accordance with Bishop’s approach (see [BB85]) — all the constructive mathematics can be carried out in these theories. So the result proven here underlines this arguments, and shows, that Martin-Löf’s type theory is important for the foundations of Mathematics. (For more discussions on the constructivism as an approach towards better foundations of mathematics, see [Bee85], [DT88], [Fef79], [Fef82a]and for discussions on intuitionism [Tro73]) It also justifies the use of it as basic theory for proof development systems.

Precisely we calculate the proof theoretical strength of intensional Russel-, extensional Tarski- and extensional Russel-version of Martin-Löf’s type theory with one Universe and the W -type, ML_{1}^{i}W_{R}, ML_{1}^{e}W_{T} and ML_{1}^{e}W_{R}, namely |ML_{1}^{i}W_{R}| = |ML_{1}^{e}W_{T} | = |ML_{1}^{e}W_{R}| = | ⋃_{n∈IN} KP i_{n}^{+}| = ψ_{Ω_{1}}Ω_{I+ω}. Here Russel-version stands for the formulation à là Russel of the universe, Tarski-version for the formulation à là Tarski, described in [ML84].

The thesis is organized as follows: In the first part, we introduce Martin-Löf’s type theory and Kripke-Platek set theory. In chapter 2 we give an introduction on Martin-Löf’s type theory. To make a precise definition of the substitution, we introduce sets of r-objects, g-terms, g-types, r-terms and r-types, which should contain all the terms and types occurring in Martin-Löf’s type theory (g-terms and g-types correspond to the Tarski-formalization, r-terms and r-types to the Russel-formalization). These concepts will be needed afterwards for the interpretation of ML_{1}^{e}W_{T} in KP i_{n}^{+}. We define then the theories ML_{1}^{i}W_{T}, ML_{1}^{e}W_{T}, ML_{1}^{i}W_{R}, ML_{1}^{e}W_{R}. In chapter 3 we compare the Tarski and the Russel-versions and in chapter 4 we introduce the Kripke-Platek set theories, we need.

In the following second big part, we prove |ML_{1}^{e}W_{T} | ≤ ψ_{Ω_{1}}Ω_{I+ω} by interpreting ML_{1}^{e}W_{T} in set theoretical systems of Kripke-Platek style, called KP i^{+} and KP i_{n}^{+}. This embedding is a quite general and flexible method, which can be adopted to variations of Martin-Löf’s type theory. In chapter 5, we develop, how to interpret terms and types in KP i^{+}. Types will be introduced as sets of pairs of terms, which are considered to be equal in that type. In this chapter, we really see, how W -type and the Universe extend the proof theoretical strength of Martin-Löf’s type theory.

In chapter 6 we first prove essentially, that the interpretation is correct, that is

if ML_{1}^{e}W_{T} ⊢ a : A then KP i^{+} ⊢ pair(a, a) ∈ A^{∗}

Further we conclude, that the extended version |ML_{1}^{e}W_{T,U}| can be interpreted as well.

In chapter 7 we interpret sentences of Arithmetic A as Â in ML_{1}^{e}W_{T} and prove, that for arithmetical sentences A from ML_{1}^{e}W_{T} ⊢ a ∈ Â follows KP i^{+} ⊢ A. It follows, |ML_{1}^{e}W_{T} | ≤ |KP i^{+}|. Since every proof can be interpreted actually in KP i_{n}^{+} for some n, and in the essential lemmas KP i^{+} can be replaced by KP i_{n}^{+}, we conclude, that

ML_{1}^{e}W_{T} ⊢ r ∈ Â ⇒ ∃n.KP i_{n}^{+} ⊢ A.

In the third part we show, that ML_{1}^{i}W_{R} proves the well-ordering up to ψ_{Ω_{1}}Ω_{I+n} for each n < ω, again using very flexible methods. Since ML_{1}^{i}W_{R} is a subsystem of ML_{1}^{e}W_{R} follows that the upper bound is a lower bound as well.

In chapter 8 we actually carry out the well-ordering proof. The hard work is, having introduced a concept for the power set of IN, to define W (X), some sort of very strong inductive definition on ordinals. Having introduced these two concepts, we can use well known techniques for carrying out such strong proofs in systems of analysis. We will refer to a version, ([Buc90]) where we need only some properties of an ordinal denotation system, such as the Bachmann property, which makes relatively clear, what is going on, and avoids, having to deal too much in this chapter with ordinals.

The properties of this denotation system, which can all be proven by using only induction on N, are verified in the very technical chapter 9. This chapter is only important to check, that the properties are valid, the proofs are not necessary for understanding, what is going on in Martin-Löf’s type theory.

In chapter 10 we compare the system introduced in chapter 9 with the system [Buc92b]. The first one we used for the well ordering proof, in the latter we expressed the upper bound. We show, that lower bound and upper bound are the same. This chapter rewards the hard work of chapter 9, to do all in a system containing the Veblen-function. Without this function, the work would have been far easier, but it is quite complicated (see for instance my Diplomarbeit [Set90] or now the new work [Sch92a] ) to express the Veblenfunctions within a system, that does not contain them.

Future research: We did not prove, but hope to do it soon, that all arithmetical Π_{2}^{0}-sentences provable in KP i_{n}^{+} can be proven in ML_{1}^{i}W_{R} as well. There are some ideas, using the well-ordering proof to do this. From there might follow, too, some bounds for the fast or slow growing hierarchy (see for instance proof theoretical work on this in [Sch77b], [Sch92c], [BW87], [Wai89], [Ara91], [Buc91b], [Buc91a])

Remark: While the author was finishing his thesis, M. Rathjen has told him, that in collaboration with E. Griffor independently and in parallel he has found similar results, although he has not submitted it for publication, yet, or told anything to the author.

Acknowledgement: The main amount of the research for this thesis was done as part of the EC-Twinning-Project Leeds-Munich-Oslo on “Proof theory and computation”, where most of part one was written during a visit of the author at the University of Leeds. It benefited very much from the collaboration of the three universities, since it is an example for the application of proof theoretical techniques, which have a long tradition in Munich, on type theoretical systems, on which semantical investigations are carried out in Oslo, by using the experience in the relationship between proof theory and recursion theory of Leeds. The author wants to thank D. Norman, H. Schwichtenberg for help and inspiration, S. Wainer for motivating me very much and especially W. Buchholz, who gave me an excellent insight in proof theory.

Part I

Definition of the Theories Definition of the formal system of Martin-Löf’s type theories

In this chapter we define the Martin-Löf’s type theories, we are going to analyze afterwards. After an introduction to it, we define the formal language (definition 2.1). Although this contradicts a little bit the spirit of Martin-Löf’s type theory, we introduce a set of b-objects, containing all terms and types that will actually occur in Martin-Löf’s type theory. This approach makes it easier to define substitution, allows to make definitions for all b-objects and will later be used for the interpretation of Martin-Löf’s type theory in KP i_{n}^{+}. The b-objects a very general class of objects containing all terms and types, b-substitutions, and the concept of substitution (we introduced b-objects for making the definition of substitution easy). Afterwards we define (definition 2.3) g-terms, g-types, gcontext-pieces, g-contexts, g-substitutions, g-judgements and g-statements, which contain all the terms, types etc. which occur in the Tarski-version of Martin-Löf’s type theory, and the corresponding r-terms, r-types (definition 2.4), terms and types occurring in the Russel-version. Next we define, what derivability in the four theories ML_{1}^{i}W_{T}, ML_{1}^{i}W_{R}, ML_{1}^{e}W_{T} and ML_{1}^{e}W_{R} (definition 2.5) and give a few remarks.

Martin-Löf’s type theory is a system used to formalize constructive mathematics, as used informally in [BB85]. One of the key ideas was, that, constructively seen, proofs and programs are the same, as are propositions and program specifications. In Martin-Löf’s type theory we have only one concept for propositions, program specifications and sets, the types. The advantage of this concept is, that it makes clear, why the Curry-Howard terms for realization of a proposition A → B and a proposition ∀x ∈ N.B are both λ-terms: a realization of A → B maps any element of the proposition A to an element of the proposition B, and realization of ∀x ∈ N.B is a function mapping elements n of the set N to elements of the proposition B(n) — they can both be represented in Martin-Löf’s type theory by the same concept Πx ∈ A.B: elements of this type are functions, mapping elements r of the type A to elements of the type B(r).

A(x) depends on the choice of x, therefore we need in Martin-Löf’s type theory types, depending on variables. This concept makes obvious, why, if we apply a Curry-Howard term for the formula ∀x ∈ A to an element r of type IN, we get a Curry-Howard term of the formula A[x/r], a formula, which depends on the actual choice of r. The rules are designed in such a way, that this is naturally explained.

Martin-Löf’s type theory differs from other mathematical theories in the sense that it has no fixed set of terms and types, but they are introduced during the derivation process. Therefore we have additional judgements of the form A type and in the rules we need premises which guarantee, that the type-part of the conclusion is a type. For instance to conclude that i(a) : A + B (where A + B is the disjunctive union of the types A and B or, considered as a proposition, A + B stands for A ∨ B) the premise a : A is not sufficient, we need additionally a premise B : type.

Terms, considered as programs, can be evaluated. If we reduce a term, we want to get an equivalent term, equivalent within the type, therefore we need an equality on the types.

In the formalization of [DT88] statements of Martin-Löf’s type theory are of the form Γ ⇒ Θ where Γ is a context and Θ is a judgement. A context is a sequence of assumptions x_{0} : A_{0}, …, x_{n} : A_{n}, where x_{i} are free variables, A_{i} are types with free variables among {x_{0}, …, x_{i−1}}. Judgements are of the form A type, t : A, t = s : A (t are equal elements of the type A) and A = B (A and B are equal types).

We have four sorts of rules: the first sort is the sort of type introduction rules for introducing types (in the extensional case we conclude here that certain types are equal, in the intensional version we conclude statements “A : type”). The second sort is the sort of introduction rules for introducing the canonical elements of a type, the elements formed by a introductory constructor of the type. Next sort is the sort of elimination rules, rules for deriving from some elements of one type elements of another type,(for instance the induction or primitive recursion for the N type, the induction or recursion over trees for the W -type, application for the Π-type). Last sort is the sort of equality rules, which are rules for conversion of a term, we get by introducing and immediately eliminating an element of a type. For propositions, this last sort corresponds to cut elimination, for sets it corresponds to the evaluation of a term. Additionally we have some general rules, some sort of structural rules.

The types, we have in this version are the following:

The type N, corresponding to the natural numbers.

the type N_{k}, corresponding to a set with k elements, where N_{0} is the empty type or the falsum for the propositions, having as elimination rule the ex-falsum-quodlibet, N_{1} corresponds to the always true formula, N_{2} is the type of booleans. The types N_{k}, k > 0 have case distinction as elimination rule.

the type Πx ∈ A.B, which corresponds to the formula ∀x ∈ A.B(x) or considered as a set to the set of functions mapping x : A to an element of B(x).

the type Σx ∈ A.B, corresponding to the formula ∃x ∈ A.B(x), if x ∉ F V (B) to the formula A ∧ B or as a set to the set of pairs {pair(x, y)|x : A, y : B(x)}.

the type A + B, which corresponds to the formula A ∨ B, and as a set to the disjoint union of A and B.

the type I(A, r, s) which corresponds to the formula r = s for r, s : A.

the type W x ∈ A.B, which corresponds to a set of trees with branching degrees B(x) for x : A.

Universe U, which is a type, closed under all other operations that create types.

For a more general introduction to Martin-Löf’s type theory the reader might refer to Martin-Löf’s fascinating monograph [ML84], or the books [Bee85] and [DT88].

There are two versions of Martin-Löf’s type theory, an extensional and an intensional version. The extensional version has additional rules for deriving equalities between types and between terms. For the formulation of the universe, there are two versions: the Russel-version (we write ML_{1}^{e}W_{R} for the extensional and ML_{1}^{i}W_{R} for the intensional Russel-version; in the Russel-version the elements of the Universe are types) and the Tarski-version (we write ML_{1}^{e}W_{T} for the extensional and ML_{1}^{i}W_{T} for the intensional Tarksi-version; here the elements of the Universe are indices for types, and, if we have Γ ⇒ a : U, we can conclude Γ ⇒ T (a) type). For some remarks on these two versions, see [ML84].

For the formulation of the version à là Russel, we follow essentially [DT88] (an earlier version of this is [Tro87]) in the version à là Tarski, we remain in the spirit of [DT88], taking rules as defined in [Smi84]. From [DT88], we only differ a little bit in the exact notation, and the following aspects:

We add the type N_{k}, as in [ML84], which can be derived from the other types, but does not cause any problems, see [DT88].

We make α-conversion explicit (see the rule (ALP HA), note, that from this rule follows full α-conversion).

We add some rules to (REF L), which are shown in [DT88] to be derived rules, but only for a weaker system. We just do not want to bother about deriving again these rules, since they do not at all cause any difficulties.

At the end of this introduction, having praised a lot Martin-Löf’s type theory, the author wants to say a few critical remarks to it. One problem is, that the set of rules is immense, and this makes it difficult to handle it proof theoretically. It seems to be a good system from the foundational point of few, but it is hard to be analyzed and might be difficult for the practical use, although various proof assistants are based on it. Another problem is, that substitution is not very well solved. For this proof theoretical analysis it was necessary, to introduce the concept of b-objects, a set theoretical concept, which is not in the spirit of Martin-Löf’s type theory. Additionally, this caused a lot of problems in the interpretation, which just reflects, the unsatisfactory solution of this detail.

Definition 2.1 We define the formal language of Martin-Löf’s type theory L_{ML}.

We assume some infinite set of variables V ar_{ML} = {z_{i}^{ML}|i ∈ IN}, where i ≠ j → z_{i}^{ML} ≠ z_{j}^{ML}. Further we have the basic symbols type, ⇒, :, ,, (, ), =, ∈.

Constructor symbols are either term constructors or type constructors, , defined as follows:

We have the 0-ary term constructors 0, \underline{r}, \underline{n}, n_{k} (for each n < k, n, k ∈ IN a new symbol n_{k}), and \underline{n}_{k} (for each k ∈ IN a new symbol \underline{n}_{k}); the 1-ary term constructors: S, i, j, p_{0}, p_{1}; the 2-ary term constructors: p, sup, R, Ap, ˜+, π, σ, w; the 3-ary term constructors: D, P, ĩ; for every n ∈ IN an n + 1-ary term constructor C_{n}. the 0-ary type constructors N_{k} (for each k ∈ IN), N, U; the 1-ary type constructor T; the 2-ary type constructors: +, Π, Σ, W; the 3-ary type constructor: I;

Definition 2.2 (a) To shorten the definition of free Variables and substitution we define inductively a set of b-objects (basic objects), which will contain all terms and types occurring in Martin-Löf’s type theory, together with the set of free Variables F V (t) for every b-object t .

(BO1) If x ∈ V ar_{ML}, then x is a b-object, F V (x) := ∅.
(BO2) If C is a n-ary constructor symbol, t_{1}, …, t_{n} are

b-objects, then C(t_{1} ⋯ t_{n}) is a b-object,

F V (C(t_{1} ⋯ t_{n})) := F V (t_{1}) ∪ ⋯ ∪ F V (t_{n}).
(BO3) If x ∈ V ar_{ML}, t a b-object, then
(λx.t) is a b-object, F V ((λx.t)) := F V (t) \ {x}.

The b-terms are defined as the b-objects, except that rule (BO2) is only applied for C being a n-ary term-constructor.

We write +, ˜+ infix (that is (a + b) for +(a, b)) (x)t for λx.t, (x, y)t for λx.λy.t, (x, y, z)t for λx.λy.λz.t. Further, if S ∈ {Σ, Π, W, σ, π, w}, Sx ∈ s.t := S(s, (x)t). We omit brackets, as long as there is no confusion, using the convention, that the scope of λx. is as long as possible, for instance λx.st should be read as λx.(st), and s+t+v := s+(t+v) similarly for longer sums and for ˜+ (and later for other defined operators like →, ∧).

(b) A b-substitution is a sequence [x_{1}/t_{1}, …, x_{n}/t_{n}], , where x_{i} ∈ V ar_{ML}, t_{i} are b-objects. (x_{i} = x_{j} is allowed, and in this situation, only the first occurrence of a variable will be relevant) If x⃗ = x_{1}, …, x_{n}, t⃗ = t_{1}, …, t_{n}, then [x⃗/t⃗] := [x_{1}/t_{1}, …, x_{n}/t_{n}].

If X ⊂ V ar_{ML}, {i_{j}|j ≤ m} = {i|x_{i} ∉ X}, i_{0} < i_{1} <, …, i_{m}, then [x_{1}/t_{1}, …, x_{n}/t_{n}] \ X := [x_{i_{1}}/t_{i_{1}}, …, x_{i_{n}}/t_{i_{n}}]. Similar, we define [x⃗/t⃗] ∩ X := [x⃗/t⃗] \ (V ar_{ML} \ X).

(c) We define for a b-substitution [x_{1}/t_{1}, …, x_{n}/t_{n}] and a b-term s the application of the substitution to the term, written s[x_{1}/t_{1}, …, x_{n}/t_{n}], having as result a b-term, (we define s[x⃗/t⃗] := s[x_{1}/t_{1}, …, x_{n}/t_{n}]), and the relation “s[x_{1}/t_{1}, …, x_{n}/t_{n}] is an allowed substitution” (or s[x⃗/t⃗] is an allowed substitution).

The definition is by induction on the definition of the b-term s.

Case (B1): If x ∈ V ar_{ML}, then

x[x_{1}/t_{1}, …, x_{n}/t_{n}] := \{ \begin{matrix}x & if x_{i} ≠ x for all i = 1, …, n, \\ t_{i} & if x_{i} = x, (i minimal) \end{matrix} ,

and x[x_{1}/t_{1}, …, x_{n}/t_{n}] is always an allowed substitution.

Case (B2): If C is a m-ary constructor symbol, s_{1}, …, s_{m} are b-objects, then C(s_{1} ⋯ s_{m})[x_{1}/t_{1}, …, x_{n}/t_{n}] := C(s_{1}[x_{1}/t_{1}, …, x_{n}/t_{n}], …, s_{m}[x_{1}/t_{1}, …, x_{n}/t_{n}]), C(s_{1}, …, s_{m})[x_{1}/t_{1}, …, x_{n}/t_{n}] is an allowed substitution iff for all i = 1, …, m,

s_{i}[x_{1}/t_{1}, …, x_{n}/t_{n}] is an allowed substitution.

Case (B3): If x ∈ V ar_{ML}, t is a b-object, [x′_{1}/t′_{1}, …, x′_{n′}/t′_{n′}] := [x_{1}/t_{1}, …, x_{n}/t_{n}] \ {x}, then (λx.t)[x_{1}/t_{1}, …, x_{n}/t_{n}] := λx.(t[x′_{1}/t′_{1}, …, x′_{n′}/t′_{n′}]), and (λx.t)[x_{1}/t_{1}, …, x_{n}/t_{n}] is an allowed substitution, iff t[x′_{1}/t′_{1}, …, x′_{n′}/t′_{n′}]) is an allowed substitution, and for all i = 1 ⋯ n′, whenever x′_{i} ∈ F V (t), then x ∉ F V (t′_{i}).

(d) We define α-equality as the least relation between b-objects, such that for t, t′, t′′, t_{i}, t′_{i} bobjects, x, x′ ∈ V ar_{ML} :

t=_{α}t.

If t=_{α}t′, t′=_{α}t′′, then t=_{α}t′′.

If t=_{α}t′, then λx.t=_{α}λx.t′.

If x′ ∉ F V (λx.t), t[x/x′] is an allowed substitution, λx.t=_{α}λx′.(t[x/x′]).

If t_{i}=_{α}t′_{i}, C a n-ary constructor, then C(t_{1}, …, t_{n})=_{α}C(t′_{1}, …, t′_{n}).

(e) We will write rs for Ap(r, s) (if r, s are b-objects) (we will need then parenthesis, using the usual conventions, especially, that the binding of λx. and Πx ∈ A. etc. is as long as possible).

A × B := A ∧ B := Σz_{i}^{ML} ∈ A.B, A → B := Πz_{i}^{ML} ∈ A.B, where i is minimal, such that z_{i}^{ML} ∉ F V (B).

Definition 2.3 (a) We define inductively a set of generalized terms, called g-terms, which will contain all terms occurring in Martin-Löf’s type theory, and is a subset of the b-objects:

If x ∈ V ar_{ML}, then x is a g-term. If n < k, n, k ∈ IN, then n_{k} is a g-term and if k ∈ IN, then \underline{n}_{k} is a g-term. If r, s, t are g-terms, x, y, z, x′ ∈ V ar_{ML}, x ≠ y ≠ z ≠ x, then 0, \underline{r}, \underline{n}, Sr, λx.r, p(r, s), sup(r, s), i(r), j(r), P (r, s, (x, y)t), Ap(r, s), p_{0}(r), p_{1}(r), R(r, (x, y, z)s), D(r, (x)s, (x′)t), πx ∈ r.s, σx ∈ r.s, wx ∈ r.s, r ˜+s, ĩ(r, s, t) are g-terms If n ∈ IN and r, s_{1}, …, s_{n} are g-terms, then C_{n}(r, s_{1}, …, s_{n}) is a g-term.

(b) We define inductively a set of generalized types, called g-types, which will contain all types occurring in Martin-Löf’s type theory:

If k ∈ IN, then N_{k} is a g-type. N, U are g-types. If A, B are g-types, x ∈ V ar_{ML}, r, s g-terms, then Πx ∈ A.B, Σx ∈ A.B, W x ∈ A.B, A+B, I(A, r, s), T (r) are g-types.

(c) The generalized context pieces (g-context-pieces) are (possibly empty) sequences x_{1} : A_{1}, …, x_{n} : A_{n}, where x_{i} ∈ V ar_{ML} are distinct, and A_{i} are g-types. V ar(x_{1} : A_{1}, …, x_{n} : A_{n}) := {x_{1}, …, x_{n}}. Note that, if Γ, Γ′ are g-context-pieces, V ar(Γ′)∩V ar(Γ) = ∅, then Γ, Γ′ (the concatenation of the strings Γ, ”,” and Γ′) is a g-context-piece.

Further we define the set of generalized contexts (g-contexts), which is a subset of the gcontext-pieces:

The empty string Γ is a g-context. If Γ is a g-context, x ∈ V ar_{ML} \ V ar(Γ), A a g-type, F V (A) ⊂ V ar(Γ), then Γ, x : A is a g-context.

We define for a g-context-piece Γ, x ∈ V ar_{ML} and a g-term r the relation “t is substitutable for x in Γ” and Γ[x/t] as follows: If Γ is empty, then t is substitutable for x in Γ and Γ[x/t] := Γ. t is substitutable for x in Γ, x′ : A if t is substitutable for x in Γ and in A and (Γ, x′ : A)[x/t] := Γ[x/t], x′ : (A[x/t]).

(d) A g-substitution is a b-substitution [x_{1}/t_{1}, …, x_{n}/t_{n}], where t_{i} are g-terms.

(e) A generalized judgement, short g-judgement, is a string A : type or A = B or r : A or r = s : A, where A, B are g-types, and r, s are g-terms. The set of free variables is F V (A : type) := F V (A), F V (A = B) := F V (A) ∪ F V (B), F V (r : A) := F V (r) ∪ F V (A) F V (r = s : A) := F V (r) ∪ F V (s) ∪ F V (A). We define (A : type)[x/r] := A[x/r] : type, (A = B)[x/r] := A[x/r] = B[x/r], (r : A)[x/t] := r[x/t] : A[x/t], (r = s : A)[x/t] := r[x/t] = s[x/t] : A[x/t]. A generalized statement, short g-statement, is a string Γ ⇒ Θ, where Γ is a g-context, Θ is a g-judgement.

Definition 2.4 We define inductively the set of Russel-terms, short r-terms, and the Russel-types, short r-types, which will contain all terms and types occurring in the Russelversion of Martin-Löf’s type theory, and are subsets of the b-objects:

If n < k, n, k ∈ IN, then n_{k} is a r-term,

if k ∈ IN, then N_{k} is a r-term, if r_{i} are r-terms, n ∈ IN ,then C_{n}(r_{0}, r_{1}, …, r_{n}) is a r-term, if r, s, t, are r-terms, x, y, z, x′ ∈ V ar_{ML}, x ≠ y ≠ z ≠ x, then x, 0, \underline{r}, N, Sr, λx.r, p(r, s), sup(r, s), i(r), j(r), P (r, s, (x, y)t), Ap(r, s), p_{0}(r), p_{1}(r), R(r, (x, y, z)s), D(r, (x)s, (x′)t), Πx ∈ r.s, Σx ∈ r.s, W x ∈ r.s, r + s and I(r, s, t) are r-terms.

U is a r-type. If s, t are r-terms, A, B are r-types, x ∈ V ar_{ML}, then s, Πx ∈ A.B, Σx ∈ A.B, W x ∈ A.B, A + B and I(A, s, t) are r-types.

R-context-pieces, r-contexts, r-judgements, r-statements are defined as the corresponding g-definitions, only replacing g-terms by r-terms, g-types by r-types. Definition 2.5 Definition of intensional and extensional Martin-Löf’s type theory with W -type and one Universe. We will in the following define the rules of intensional

(ML_{1}^{i}W_{T} is the Tarski- and ML_{1}^{i}W_{R} the Russel-version) and extensional Martin-Löf’s type theory (ML_{1}^{e}W_{T} is the Tarski- and ML_{1}^{e}W_{R} the Russel-version), which are of the form

(Rule) \frac{Γ_{1}⇒Θ_{1}···Γ_{n}⇒Θ_{n}}{Γ⇒Θ} ,

where Γ_{1}, …, Γ_{n}, Γ are g-context-pieces, Θ_{1}, …, Θ_{n}, Θ are g-judgements (n = 0 is allowed).

Then ML_{1}^{i}W_{T} ⊢ Γ ⇒ Θ is defined inductively by:

If (Rule) is a rule of ML_{1}^{i}W_{T} as above, Δ is a g-context-piece such that Δ, Γ_{1}, , …, Δ, Γ_{n}, Δ, Γ are g − contexts, and if ML_{1}^{i}W_{T} ⊢ Δ, Γ_{i} ⇒ Θ_{i} for i = 1, …, n, then ML_{1}^{i}W_{T} ⊢ Δ, Γ ⇒ Θ.

Analogously we define ML_{1}^{e}W_{T} ⊢ Γ ⇒ Θ. ML_{1}^{e}W_{R} ⊢ Γ ⇒ Θ and ML_{1}^{i}W_{R} ⊢ Γ ⇒ Θ is again defined analogously, but we refer to r-judgements, -contexts etc. instead of gjudgements, -contexts etc.

The rules for ML_{1}^{i}W_{T} are listed as the general rules, the rules for intensional Martin-Löf’s type theory and the intensional Tarski-rules for the universe.

The rules for ML_{1}^{i}W_{R} are the general rules, the rules for intensional Martin-Löf’s type theory and the intensional Russel-rules for the universe.

The rules for ML_{1}^{e}W_{T} are the general rules, the rules for extensional Martin-Löf’s type theory and the extensional Tarski-rules for the universe.

The rules for ML_{1}^{e}W_{R} are the general rules, the rules for extensional Martin-Löf’s type theory and the extensional Russel-rules for the universe.

We will write Θ for ⇒ Θ as a premise of a rule.

Rules for Martin-Löf’s type theory

In the following, let A, B be g-types, a, b, r, s, t, r_{i}, s_{i}, t_{i} be g-terms, θ be g-judgements, Γ′ be a g-context-piece in the Tarski-versions, A, B be r-types, a, b, r, s, t, r_{i}, s_{i}, t_{i} be r-terms, θ be r-judgements, Γ′ be a r-context-piece in the Russel-versions. (So although we state that Tarski- and Russel-version have many rules in common, in fact the rules are different in the sense that for the Tarski-version we refer to g-objects, for the Russel-version we refer to r-objects). Further let x, y, z, u ∈ V ar_{ML}. Additionally assume for all rules, that all substitution mentioned explicitly are allowed. For instance in the rule (N_{S}^{=}), assume that s_{1}[x/t, y/P (t, s_{0}, (x, y)s_{1})] and A[z/St] are allowed substitutions.

CHAPTER 2. DEFINITION OF MARTIN-LÖF’S TYPE THEORY 14

General rules

(Common for all 4 versions of Martin-Löf’s type theory)

(ASS) \frac{A type}{x:A⇒x:A}
(T HIN) \frac{A type  Γ′⇒Θ}{x:A,Γ′⇒Θ}
(REF L) \frac{t:A}{t=t:A} \frac{t=t:A}{t:A} \frac{A=A}{A type} \frac{A type}{A=A} \frac{t:A}{A type}
(SY M) \frac{t=t′:A}{t′=t:A}  \frac{A=B}{B=A}
(T RANS) \frac{t=t′:A t′=t′′:A}{t=t′′:A}  \frac{A=B B=C}{A=C}
(SUB) \frac{x:A,Γ′⇒Θ ⇒t:A}{Γ′[x/t]⇒Θ[x/t]}
(REP L1) \frac{x:A,Γ′⇒B type ⇒t=t′:A}{Γ′[x/t]⇒B[x/t]=B[x/t′]}
(REP L2) \frac{x:A,Γ′⇒s:B ⇒t=t′:A}{Γ′[x/t]⇒s[x/t]=s[x/t′]:B[x/t]}
(REP L3) \frac{t:A A=B}{t:B} \frac{t=t′:A A=B}{t=t′:B}
(ALP HA) \frac{x:A,Γ′⇒θ}{x:A′,Γ′⇒θ} \frac{A type}{A=A′} \frac{t:A}{t=t′:A} (if A=_{α}A′, t=_{α}t′)

Rules for intensional Martin-Löf’s type theory

(rules are common for Russel- and Tarski-version)

Type introduction rules

(N_{k}^{T}) N_{k} type (k ∈ IN) (N^{T}) N type

(Π^{T}) \frac{x:A⇒B type}{Πx∈A.B type}
(Σ^{T}) \frac{x:A⇒B type}{Σx∈A.B type}
(W^{T}) \frac{x:A⇒B type}{W x∈A.B type}

(+^{T}) \frac{A type B type}{A+B type}

(I^{T}) \frac{t:A s:A A type}{I(A,t,s) type}

Introduction rules

(N_{k}^{I}) n_{k} : N_{k} (n < k, n, k ∈ IN)

(N^{I}) 0 : N \frac{t:N}{St:N}

(Π^{I}) \frac{x:A⇒t:B x:A⇒B type}{λx.t:Πx∈A.B}

(Σ^{I}) \frac{s:A t:B[x/s] x:A⇒B type}{p(s,t):Σx∈A.B}

(W^{I}) \frac{s:A t:B[x/s]→W x∈A.B x:A⇒B type}{sup(s,t):W x∈A.B}

(+^{I}) \frac{s:A A type B type}{i(s):A+B} \frac{s:B A type B type}{j(s):A+B}

(I^{I}) \frac{t=t′:A}{\underline{r}:I(A,t,t′)}

Elimination rules

(N_{k}^{E}) \frac{t:N_{k} s_{i}:A[x/i_{k}](i=0...k−1) x:N_{k}⇒A type}{C_{k}(t,s_{0},...,s_{k−1}):A[x/t]} (k ∈ IN)

(N^{E}) \frac{t:N s_{0}:A[z/0] x:N,y:A[z/x]⇒s_{1}:A[z/Sx] x:N⇒A[z/x] type}{P (t,s_{0},(x,y)s_{1}):A[z/t]}

(Π^{E}) \frac{t_{0}:Πx∈A.B t_{1}:A x:A⇒B type}{Ap(t_{0},t_{1}):B[x/t_{1}]}

(Σ^{E}) \frac{r:Σx∈A.B x:A⇒B type}{p_{0}(r):A} \frac{r:Σx∈A.B x:A⇒B type}{p_{1}(r):B[x/p_{0}(r)]}

(W^{E}) ^{\frac{{t_{0}:W x∈A.B  x:A,y:(B→W x∈A.B),z:Πv∈B.C[u/Ap(y,v)]⇒t_{2}:C[u/sup(x,y)] \atop u:W x∈A.B⇒C type}}{R(t_{0},(x,y,z)t_{2}):C[u/t_{0}]}}

(+^{E}) \frac{t_{0}:A+B x:A⇒t_{1}:C[z/i(x)] y:B⇒t_{2}:C[z/j(y)] z:A+B⇒C type}{D(t_{0},(x)t_{1},(y)t_{2}):C[z/t_{0}]}

(I^{E}) \frac{t_{0}:I(A,t_{1},t_{2}) t_{1}:A t_{2}:A}{t_{1}=t_{2}:A}

Equality rules

(N_{k}^{=}) \frac{s_{i}:A[x/i_{k}](i=0...k−1) x:N_{k}⇒A type}{C_{k}(n_{k},s_{0},...,s_{k−1})=s_{n}:A[x/n_{k}]} (n < k, n, k ∈ IN)

(N_{0}^{=}) \frac{s_{0}:A[z/0] x:N,y:A[z/x]⇒s_{1}:A[z/Sx] x:N⇒A[z/x] type}{P (0,s_{0},(x,y)s_{1})=s_{0}:A[z/0]}
(N_{S}^{=}) \frac{t:N s_{0}:A[z/0] x:N,y:A[z/x]⇒s_{1}:A[z/Sx] x:N⇒A[z/x] type}{P (St,s_{0},(x,y)s_{1})=s_{1}[x/t,y/P (t,s_{0},(x,y)s_{1})]:A[z/St]}
(Π^{=}) \frac{λx.t_{0}:Πx∈A.B t_{1}:A x:A⇒B type}{Ap(λx.t_{0},t_{1})=t_{0}[x/t_{1}]:B[x/t_{1}]}
(Σ_{0}^{=}) \frac{p(r,s):Σx∈A.B A type}{p_{0}(p(r,s))=r:A}
(Σ_{1}^{=}) \frac{p(r,s):Σx∈A.B x∈A⇒B type}{p_{1}(p(r,s))=s:B[x/r]}
(Σ_{2}^{=}) \frac{t:Σx∈A.B}{t=p(p_{0}(t),p_{1}(t)):Σx∈A.B}

(W^{=}) ^{\frac{r:A  s:(B[x/t_{0}]→W x∈A.B)  x:A,y:B→W x∈A.B,z:(Πv∈B.C[u/Ap(y,v)])⇒t:C[u/sup(x,y)] u:W x∈A.B⇒C typex:A⇒B type}{R(sup(r,s),(x,y,z)t)=t[x/r,y/s,z/λv.R(Ap(s,v),(x,y,z)t)]:C[u/sup(r,s)]}}
(If v ∉ F V (s) ∪ F V ((x, y, z)t))
(+_{0}^{=}) \frac{t_{0}:A x:A⇒t_{1}:C[z/i(x)] y:B⇒t_{2}:C[z/j(y)] z∈A+B⇒C type}{D(i(t_{0}),(x)t_{1},(y)t_{2})=t_{1}[x/t_{0}]:C[z/i(t_{0})]}
(+_{1}^{=}) \frac{t_{0}:B x:A⇒t_{1}:C[z/i(x)] y:B⇒t_{2}:C[z/j(y)] z∈A+B⇒C type}{D(j(t_{0}),(x)t_{1},(y)t_{2})=t_{2}[y/t_{0}]:C[z/j(t_{0})]}
(I^{=}) \frac{t_{0}:I(A,t_{1},t_{2}) A type}{t_{0}=\underline{r}:I(A,t_{1},t_{2})}

Intensional Tarski rules for the universe Tarski type introduction rules for the universe

(U^{I}) U type
(T^{I}) \frac{a:U}{T (a) type} \frac{a=a′:U}{T (a)=T (a′)}

Tarski introduction rules for the universe CHAPTER 2. DEFINITION OF MARTIN-LÖF’S TYPE THEORY 17 (\underline{n}_{k}^{I}) \underline{n}_{k} : U (k ∈ IN) (\underline{n}^{I}) \underline{n} : U

(π^{I}) \frac{a:U x:T (a)⇒b:U}{πx∈a.b:U}
(σ^{I}) \frac{a:U x:T (a)⇒b:U}{σx∈a.b:U}
(w^{I}) \frac{a:U x:T (a)⇒b:U}{wx∈a.b:U}
( ˜+^{I}) \frac{a:U b:U}{a_{+}b:U}
(i^{I}) \frac{a:U t:T (a) s:T (a)}{_{i}(a,t,s):U}

Tarski equality rules for the universe

(\underline{n}_{k}^{=}) T (\underline{n}_{k}) = N_{k} (k ∈ IN) (\underline{n}^{=}) T (\underline{n}) = N

(π^{=}) \frac{a:U x:T (a)⇒b:U}{T (πx∈a.b)=Πx∈T (a).T (b)}
(σ^{=}) \frac{a:U x:T (a)⇒b:U}{T (σx∈a.b)=Σx∈T (a).T (b)}
(w^{=}) \frac{a:U x:T (a)⇒b:U}{T (wx∈a.b)=W x∈T (a).T (b)}
( ˜+^{=}) \frac{a:U b:U}{T (a_{+}b)=T (a)+T (b)}
(i^{=}) \frac{a:U t:T (a) s:T (a)}{T (_{i}(a,t,s))=I(T (a),t,s)}

Russel rules for the universe Russel type introduction rules for the universe

(U_{R}^{I}) U type \frac{A:U}{A type} \frac{A=A′:U}{A=A′}

Russel introduction rules for the universe CHAPTER 2. DEFINITION OF MARTIN-LÖF’S TYPE THEORY 18 ((N_{k})_{R}^{U,I}) N_{k} : U (k < ω) (N_{R}^{U,I}) N : U

(Π_{R}^{U,I}) \frac{A:U x:A⇒B:U}{Πx∈A.B:U}
(Σ_{R}^{U,I}) \frac{A:U x:A⇒B:U}{Σx∈A.B:U}
(W_{R}^{U,I}) \frac{A:U x:A⇒B:U}{W x∈A.B:U}
(+_{R}^{U,I}) \frac{A:U B:U}{A+B:U}
(i_{R}^{U,I}) \frac{A:U t:A s:A}{i(A,t,s):U}

CHAPTER 2. DEFINITION OF MARTIN-LÖF’S TYPE THEORY 19

Rules for extensional Martin-Löf’s type theory

(Common for both ML_{1}^{e}W_{R} and ML_{1}^{e}W_{T})

Extensional type introduction rules

(N_{k}^{T}) and (N^{T}) as before. Additional

(Π^{T,=}) \frac{A=A′ x:A⇒B=B′}{Πx∈A.B=Πx∈A′.B′}
(Σ^{T,=}) \frac{A=A′ x:A⇒B=B′}{Σx∈A.B=Σx∈A′.B′}
(W^{T,=}) \frac{A=A′ x:A⇒B=B′}{W x∈A.B=W x∈A′.B′}
(+^{T,=}) \frac{A=A′ B=B′}{A+B=A′+B′}
(I^{T,=}) \frac{A=A′ t=t′:A s=s′:A}{I(A,t,s)=I(A′,t′,s′)}

Extensional Introduction rules

(N_{K}^{I}) as before.

(N^{I,=}) 0 = 0 : N \frac{t=t′:N}{St=St′:N}
(Π^{I,=})  \frac{x:A⇒t=t′:B x:A⇒B type}{λx.t=λx.t′:Πx∈A.B}
(Σ^{I,=}) \frac{s=s′:A t=t′:B[x/s] x:A⇒B type}{p(s,t)=p(s′,t′):Σx∈A.B}
(W^{I,=}) \frac{s=s′:A t=t′:B[x/s]→W x∈A.B x:A⇒B type}{sup(s,t)=sup(s′,t′):W x∈A.B}
(+^{I,=}) \frac{s=s′:A A type B type}{i(s)=i(s′):A+B} \frac{s=s′:B A type B type}{j(s)=j(s′):A+B}

Extensional elimination rules

The rule intensional elimination rule (I^{E}) and in addition:

(N_{k}^{E,=}) \frac{t=t′:N_{k} s_{i}=s′_{i}:A[x/i_{k}](i=0...k−1) x:N_{k}⇒A type}{C_{k}(t,s_{0},...,s_{k−1})=C_{k}(t′,s′_{0},...,s′_{k−1}):A[x/t]} (k ∈ IN)

(N^{E,=}) \frac{t=t′:N s_{0}=s′_{0}:A[z/0] x:N,y:A[z/x]⇒s_{1}=s′_{1}:A[z/Sx] x:N⇒A[z/x] type}{P (t,s_{0},(x,y)s_{1})=P (t′,s′_{0},(x,y)s′_{1}):A[z/t]}
(Π^{E,=}) \frac{t_{0}=t′_{0}:Πx∈A.B t_{1}=t′_{1}:A x:A⇒B type}{Ap(t_{0},t_{1})=Ap(t′_{0},t′_{1}):B[x/t_{1}]}

(Σ^{E,=}) \frac{r=r′:Σx∈A.B x:A⇒B type}{p_{0}(r)=p_{0}(r′):A} \frac{r=r′:Σx∈A.B x:A⇒B type}{p_{1}(r)=p_{1}(r′):B[x/p_{0}(r)]}

\frac{(W^{E,=}) {t0=t′0:W x∈A.B x:A,y:B→W x∈A.B,z:Πv∈B.C[u/Ap(y,v)])⇒t2=t′2:C[u/sup(x,y)] \atop u:W x∈A.B⇒C type}}{R(t_{0},(x,y,z)t_{2})=R(t′_{0},(x,y,z)t′_{2}):C[u/t_{0}]}

(+^{E,=}) \frac{t_{0}=t′_{0}:A+B x:A⇒t_{1}=t′_{1}:C[z/i(x)] y:B⇒t_{2}=t′_{2}:C[z/j(y)] z:A+B⇒C type}{D(t_{0},(x)t_{1},(y)t_{2})=D(t′_{0},(x)t′_{1},(y)t′_{2}):C[z/t_{0}]}

Extensional equality rules

All the intensional Equality rules, in addition

(Π^{η}) \frac{t:Πx∈A.B}{λx.Ap(t,x)=t:Πx∈A.B}  If x ∉ F V (t)

Extensional Tarski rules for the universe Extensional Tarski type introduction rules for the universe

All the Tarski introduction rules for the universe.

Extensional Tarski introduction rules for the universe

The rules (\underline{n}_{k}^{I}) and (\underline{n}^{I}) of intensional Tarski version ML_{1}^{i}W_{T}, in addition:

(π^{I,=}) \frac{a=a′:U x:T (a)⇒b=b′:U}{πx∈a.b=πx∈a′.b′:U}
(σ^{I,=}) \frac{a=a′:U x:T (a)⇒b=b′:U}{σx∈a.b=σx∈a′.b′:U}
(w^{I,=}) \frac{a=a′:U x:T (a)⇒b=b′:U}{wx∈a.b=wx∈a′.b′:U}
( ˜+^{I,=}) \frac{a=a′:U b=b′:U}{a_{+}b=a′_{+}b′:U}
(i^{I,=}) \frac{a=a′:U t=t′:T (a) s=s′:T (a)}{_{i}(a,t,s)=_{i}(a′,t′,s′):U}

Extensional Tarski equality rules for the universe

The equality rules of the intensional Tarski version ML_{1}^{i}W_{T}

Extensional Russel rules for the universe Extensional Russel type introduction rules for the universe

All the Russel type introduction rules for the universe.

Extensional Russel introduction rules for the universe

((N_{k})_{R}^{U,I}), (N_{R}^{U,I}) as before

(Π_{R}^{U,I,=}) \frac{A=A′:U x:A⇒B=B′:U}{Πx∈A.B=Πx∈A′.B′:U}
(Σ_{R}^{U,I,=}) \frac{A=A′:U x:A⇒B=B′:U}{Σx∈A.B=Σx∈A′.B′:U}
(W_{R}^{U,I,=}) \frac{A=A′:U x:A⇒B=B′:U}{W x∈A.B=W x∈A′.B′:U}
(+_{R}^{U,I,=}) \frac{A=A′:U B=B′:U}{A+B=A′+B′:U}
(i_{R}^{U,I,=})  \frac{A=A′:U t=t′:A s=s′:A}{I(A,t,s)=I(A′,t′,s′):U}

Remark 2.6 All the rules of ML_{1}^{i}W_{T} are derived rules in ML_{1}^{e}W_{T}.

All the rules of ML_{1}^{i}W_{R} are derived rules in ML_{1}^{e}W_{R}.

Comparison of the formulation à là Russel and the formulation à là Tarski

In this chapter we first prove, that the Russel-version contains the Tarski-version (lemma 3.3). We prove the converse (lemma 3.11) for a little bit extended versions of extensional Tarski- and Russel-Martin-Löf’s type theory, which we define in 3.4. For the proof theoretical strength, this is sufficient, since we can embed the extended Tarski-version in KP i^{+}. Definition 3.1 (a) Define for C constructors, ϕ(\underline{n}) := N, ϕ(\underline{n}_{k}) := N_{k}, ϕ(π) := Π, ϕ(σ) := Σ,

ϕ(w) := W, ϕ(ĩ) := I, ϕ( ˜+) := +, ϕ(C) := C otherwise.

(b) Define ϕ : b-object → b-object by recursion on the b-object:

ϕ(x) := x (x ∈ V ar_{ML}), ϕ(C(t_{1}, …, t_{n})) := ϕ(C)(ϕ(t_{1}), …, ϕ(t_{n})), (C ≠ T), ϕ(T (t)) := ϕ(t), ϕ(λx.t) := λx.ϕ(t).

(c) If Γ = x_{1} : A_{1}, …, x_{n} : A_{n} is a g-context-piece, then ϕ(Γ) := x : ϕ(A_{1}), …, x_{n} : ϕ(A_{n}).

(d) If r, s are g-terms, A is a g-type, then ϕ(A : type) := (ϕ(A) : type), ϕ(r : A) := (ϕ(r) : ϕ(A)), ϕ(r = s : A) := (ϕ(r) = ϕ(s) : ϕ(A)), ϕ(A = B) := (ϕ(A) = ϕ(B)).

Lemma 3.2 Assume r, s, t, s_{i} b-objects, x_{i} ∈ V ar_{ML}.

(a) F V (t) = F V (ϕ(t)).

(b) If t[x_{1}/s_{1}, …, x_{n}/s_{n}] is allowed, then ϕ(t)[x_{1}/ϕ(s_{1}), …, x_{n}/ϕ(s_{n})] allowed.

(c) ϕ(t[x_{1}/s_{1}, …, x_{n}/s_{n}]) = ϕ(t)[x_{1}/ϕ(s_{1}), …, x_{n}/ϕ(s_{n})].

(d) If t is a g-term, -type, -judgement, -statement, -context, -context-piece, then ϕ(t) is a r-term, -type, -judgement, -statement, -context, -context-piece.

(e) r=_{α}s → ϕ(r)=_{α}ϕ(s)

(f) If r is a g-term and a r-term, then ϕ(r) = r. If r is a g-type and a r-type, then ϕ(r) = r.

Proof: (a) by induction on definition of b-objects, (b), (c) and (e) by (a) and the same induction, (d), (f) by induction on definition of g-objects. Lemma 3.3 If ML_{T} is the Tarski-version of extensional or intensional Martin-Löf’s type theory, and ML_{R} is the corresponding Russel-version, then

If ML_{T} ⊢ Γ ⇒ θ then ML_{R} ⊢ ϕ(Γ) ⇒ ϕ(θ).

Especially, if the statement is a statement both in ML_{T} and ML_{R}, then, if ML_{T} ⊢ Γ ⇒ θ then ML_{R} ⊢ Γ ⇒ θ. Proof:

Induction on the derivation. The general rules follow using lemma 3.2, all the rules, not dealing with the universe are immediate.

(U^{I}) is trivial, (T^{I}) is obvious. Introduction rules for the universe, e.g. (σ^{I}): By IH ML_{R} ⊢ ϕ(Γ) ⇒ ϕ(a) : U, ML_{R} ⊢ ϕ(Γ), x : ϕ(b) ⇒ ϕ(b) : U, therefore ML_{R} ⊢ ϕ(Γ) ⇒ ϕ(σx ∈ a.b) : U, similarly follow the other rules or (σ^{I,=}). In the equality rules for the Universe, both sides of the conclusion are identical, and if the statement is Γ ⇒ A = B, we can derive ML_{R} ⊢ ϕ(Γ) ⇒ ϕ(A) : type, and by (REF L) follows the assertion. Definition 3.4 (a) Let ML_{1}^{e}W_{T,U} be defined as ML_{1}^{e}W_{T}, only with the additional rules

(σ^{E}) \frac{σx∈a.b:U}{a:U} \frac{σx∈a.b:U}{x:T (a)⇒b:U}

(π^{E}) \frac{πx∈a.b:U}{a:U} \frac{πx∈a.b:U}{x:T (a)⇒b:U}

(w^{E}) \frac{wx∈a.b:U}{a:U} \frac{wx∈a.b:U}{x:T (a)⇒b:U}

( ˜+^{E}) \frac{a_{+}b:U}{a:U} \frac{a_{+}b:U}{b:U}

(i^{E}) \frac{_{i}(a,s,t):U}{a:U} \frac{_{i}(a,s,t):U}{s:T (a)}  \frac{_{i}(a,s,t):U}{t:T (a)}

(b) Let ML_{1}^{e}W_{R,U} be defined as ML_{1}^{e}W_{R}, only with the additional rules

(Σ_{U}^{E}) \frac{Σx∈A.B:U}{A:U} \frac{Σx∈A.B:U}{x:A⇒B:U}

(Π_{U}^{E}) \frac{Πx∈A.B:U}{A:U} \frac{Πx∈A.B:U}{x:A⇒B:U}

(W_{U}^{E}) \frac{W x∈A.B:U}{A:U} \frac{W x∈A.B:U}{x:A⇒B:U}

(+_{U}^{E}) \frac{A+B:U}{A:U} \frac{A+B:U}{B:U}

(I_{U}^{E}) \frac{I(A,s,t):U}{A:U} \frac{I(A,s,t):U}{s:A}  \frac{I(A,s,t):U}{t:A}

Remark 3.5 If ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ θ, then ML_{1}^{e}W_{R,U} ⊢ ϕ(Γ) ⇒ ϕ(θ). Proof:

As 3.3.

Lemma 3.6 In ML_{1}^{i}W_{T}, ML_{1}^{e}W_{T}, ML_{1}^{e}W_{T,U}, ML_{1}^{i}W_{R}, ML_{1}^{e}W_{R} and ML_{1}^{e}W_{R,U}

\frac{Γ, x : A, Γ′ ⇒ θ Γ ⇒ A = A′}{Γ, x : A′, Γ′ ⇒ θ}

is a rule, derived from the General rules only. Proof:

Let y be a fresh variable. Then by (T HIN) Γ, y : A′, x : A, Γ′ ⇒ θ, by (SY M), (T RANS) (REF L) Γ ⇒ A′ type, Γ, y : A′ ⇒ y : A′ and Γ, y : A′ ⇒ y : A, By (SUB) Γ, y : A′, Γ′[x/y] ⇒ θ[x/y], by (T HIN) and (SUB) Γ, x : A′, Γ′ ⇒ θ. (eventually we need some (ALP HA), to make θ[x/y][y/x] is allowed and θ[x/y][y/x] = θ, the same for Γ′. Definition 3.7 (a) Define for C constructors, ψ(N) := \underline{n}, ψ(N_{k}) := \underline{n}_{k}, ψ(Π) := π, ψ(Σ) := σ,

ψ(W) := w, ψ(I) := ĩ, ψ(+) := ˜+, ψ(C) := C otherwise.

(b) Define ψ : b-object → b-object by recursion on the b-objects:

ψ(x) := x (x ∈ V ar_{ML}), ψ(C(t_{1}, …, t_{n})) := ψ(C)(ϕ(t_{1}), …, ϕ(t_{n})), ψ(λx.t) := λx.ψ(t).

(c) Define the function ρ : b-object → b-object by recursion on the b-objects:

ρ(Sx ∈ r.s) := Sx ∈ ρ(r).ρ(s) (S ∈ Σ, Π, W) ρ(r + s) := ρ(r) + ρ(s), ρ(I(r, s, t)) := I(ρ(r), ψ(s), ψ(t)), ρ(C) := C for C ∈ {N, N_{k}, U}, ρ(t) := T (ψ(t)), otherwise.

(d) If Γ = x_{1} : A_{1}, …, x_{n} : A_{n} is a g-context-piece, then ρ(Γ) := x_{1} : ρ(A_{1}), …, x_{n} : ρ(A_{n})

(e) If r, s are g-terms, A is a g-type, then ρ(A : type) := (ρ(A) : type), ϕ(r : A) := (ψ(r) : ρ(A)), ρ(r = s : A) := (ψ(r) = ψ(s) : ρ(A)), ρ(A = B) := (ρ(A) = ρ(B)).

(f) Define μ : b-object → b-object by recursion on the b-objects:

μ(T (sx ∈ r.s)) := ϕ(s)x ∈ μ(T (r)).μ(T (s)) (s ∈ {σ, π, w}), μ(T (r ˜+s)) := μ(T (r)) + μ(T (s)), μ(ĩ(r, s, t))) := I(μ(T (r)), s, t), μ(T (C)) := ψ(C) for C ∈ \underline{n}, \underline{n}_{k}, μ(Sx ∈ r.s) := Sx ∈ μ(r).μ(s) (S ∈ {Σ, Π, W }) μ(r + s) := μ(r) + μ(s), μ(I(r, s, t)) := I(μ(r), s, t), μ(C) := C for C ∈ {N, N_{k}, U}, μ(t) := t, otherwise.

Lemma 3.8 Assume r, s, t, s_{i} b-objects, x_{i} ∈ V ar_{ML}.

(a) F V (t) = F V (ψ(t)) = F V (ρ(t)) = F V (μ(t)).

(b) If t[x_{1}/s_{1}, …, x_{n}/s_{n}] allowed, then ψ(t)[x_{1}/ψ(s_{1}), …, x_{n}/ψ(s_{n})], μ(t)[x_{1}/ψ(s_{1}), …, x_{n}/ψ(s_{n})], ρ(t)[x_{1}/ψ(s_{1}), …, x_{n}/ψ(s_{n})] are allowed.

CHAPTER 3. COMPARISON OF RUSSEL AND TARSKI-VERSION 24

(c) If t is a r-term, then ψ(t) is a g-term.

If t is a r-type, then ρ(t) is a g-type. If t is a g-type, then μ(t) is a g-type. If t is a r-term, then ρ(t) = μ(T (ψ(t))).

(d) If t, s_{i} are b-objects, then ψ(t[x_{1}/s_{1}, …, x_{n}/s_{n}]) = ψ(t)[x_{1}/ψ(s_{1}), …, x_{n}/ψ(s_{n})]. If t is a g-type, s_{i} are g-terms, then μ(μ(t)) = μ(t), μ(t[x_{1}/s_{1}, …, x_{n}/s_{n}]) = μ(μ(t)[x_{1}/s_{1}, …, x_{n}/s_{n}]), If t is a r-type, s_{i} are r-terms, then ρ(t[x_{1}/s_{1}, …, x_{n}/s_{n}]) = μ(ρ(t)[x_{1}/ψ(s_{1}), …, x_{n}/ψ(s_{n})]).

(e) If t is a g -judgement, -statement, -context, -context-piece, then ρ(t) is a r-judgement, - statement, -context, -context-piece.

(f) r=_{α}s → ψ(r)=_{α}ψ(s), ρ(r)=_{α}ρ(s), μ(r)=_{α}μ(s).

(g) If r is a g-term and b-term, then ψ(r) = r.

If r is a g-type and b-type, then ρ(r) = r. Proof: (a) by induction on definition of b-objects, (b) by (a) and the same induction.

(c) follows by induction of b-objects, g-types, by the first two ones, and by induction on rterms.

(d): The first assertion is easy by induction on the definition of b-objects. The second is shown by induction on the definition of μ(t). The third follows again by induction, the fourth one follows using the third and first one.

(e), (f), (g) by induction on definition of the different objects. Lemma 3.9 Let ML_{T} be ML_{1}^{i}W_{T}, ML_{1}^{e}W_{T} or ML_{1}^{e}W_{T,U} and Γ, Γ′ be g-context-pieces, a, b, t g-terms, A, B g-types, x a variable. The following applies:

(a) If ML_{T} ⊢ Γ ⇒ s : A, or ML_{T} ⊢ Γ ⇒ s = t : A then ML_{T} ⊢ Γ ⇒ A type.

(b) If ML_{T} ⊢ Γ ⇒ Sx ∈ A.B type (S ∈ {Σ, Π, W }), then ML_{T} ⊢ Γ ⇒ A, ML_{T} ⊢ Γ, y : A ⇒ B[x/y] type, for all y ∈ V ar_{ML} \ X for some finite set X.

(c) If ML_{T} ⊢ Γ ⇒ A + B type then ML_{T} ⊢ Γ ⇒ A type, ML_{T} ⊢ Γ ⇒ B type.

(d) If ML_{T} ⊢ Γ ⇒ I(A, b, c) type, then ML_{T} ⊢ Γ ⇒ b : A, ML_{T} ⊢ Γ ⇒ c : A.

(e) If ML_{T} ⊢ Γ ⇒ T (b) : type then ML_{T} ⊢ Γ ⇒ b : U.

(f) If ML_{T} ⊢ Γ ⇒ A = B, then ML_{T} ⊢ Γ ⇒ A type, ML_{T} ⊢ Γ ⇒ B type.

(g) If ML_{T} ⊢ Γ, x : A, Γ′ ⇒ θ then ML_{T} ⊢ Γ ⇒ A type. Proof: We first add the rules

(REP L1a) \frac{x:A,Γ′,Γ′′⇒B type ⇒t=t′:A}{Γ′[x/t],Γ′′[x/t′]⇒B[x/t′] type}

and

(ALP HA1) \frac{A type}{A′ type} if A=_{α}A′

and in the extensional case and in ML_{1}^{e}W_{T,U} additionally the intensional rules, and remove the (REF L)-rules

\frac{A = A}{A type}  \frac{t : A}{A type}

If for this calculus the theorem is provable, then this calculus is equivalent to the original: If we have a proof in the original calculus, then embed it into the calculus, by applying, whenever we need the removed rules the weak inferences. If we have a proof in the new calculus, the result is a proof in the original calculus, since we only added derived rules.

Now induction on the length of the derivation:

In the cases (ASS), (T HIN), (REF L), (SY M) and (T RANS) follow trivially using the IH.

In (SUB) the difficulty are the the second conclusion in the cases (b): let the conclusion be for instance Γ, Γ′[x/t] ⇒ (Σx′ ∈ A.B)[x/t]. By IH Γ, Γ′, y : A′ ⇒ B′[x′/y] : type for y ∉ X, therefore Γ, Γ′[x/t], y : A′[x/t] ⇒ B′[x′/y][x/t] for x ≠ y, x ∉ X. (the substitution is allowed) Then for y ∉ X∪{x}, if x = x′ or x ∉ F V (B) follows (Σx′ ∈ A.B)[x/t] = Σx′ ∈ A[x/t].B and we have the assertion, otherwise x′ ∉ F V (t) and B[x′/y][x/t] = B[x/t][x′/y].

In (REP L1) the assertion follows since by IH we have Γ ⇒ t : A and (SUB) or using (REP L1a).

In (REP L1a), we argue as in (SUB) .

In (REP L2), the assumption follows by (SUB) and IH.

(REP L3), (ALP HA) follow by IH.

The type introduction rules and the introduction rules are immediate, in the elimination rules we need (SUB), and for (Σ_{2}^{E}) we need one application of (Σ_{1}^{E}), and in the equality rules we need the introduction rules and (SUB).

In the type introduction rules introduction rules and the equality rules for the universe nothing is to prove or the assertion is easy by IH.

In the extensional type introduction rules, for (Π^{T,=}), (Σ^{T,=}) and (W^{T,=}), the first part is easy by the intensional type introduction and IH for the second part we have by IH x : A ⇒ B′ type, and now use 3.6, (where we replace the argument using one of the removed rules by the IH), the extensional introduction rules are immediate, the extensional elimination rules follow as the introduction rules, (Π^{η}) follows by IH and in the extensional introduction rules there is nothing to prove. In the rules of 3.4 (a), there is nothing to prove. Lemma 3.10 (a) ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ r : U , then ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ T (r) = μ(T (r))

(b) ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ A type , then ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ A = μ(A) Proof:

(a): Induction on the definition of r b-object. If for instance ML_{1}^{e}W_{T} ⊢ Γ ⇒ σx ∈ a.b : U, then by the rules of 3.4 Γ ⇒ a : U, Γ, x : T (a) ⇒ b : U, by IH Γ ⇒ T (a) = μ(T (a)), Γ, x : T (a) ⇒ T (b) = μ(T (b)), by extensional type introduction follows the assertion, similarly for the other terms, for which μ(T (t)) does something.

(b) Induction on length of the type. Consider for instance the case ML_{1}^{e}W_{T} ⊢ Γ ⇒ Πx ∈ A.B type, then by 3.9 Γ ⇒ A type, Γ, y : A ⇒ B[x/y] type, by IH Γ ⇒ A = μ(A), Γ, y : A ⇒ B[x/y] = μ(B[x/y]), by extensional type introduction and (ALP HA) follows the assertion, similarly for the other types, but the T (t)-type.

If Γ ⇒ T (t) : type by 3.9 we have Γ ⇒ t : U, and by the (a) follows the assertion. Lemma 3.11 If ML_{1}^{e}W_{R,U} ⊢ Γ ⇒ θ then ML_{1}^{e}W_{T,U} ⊢ ρ(Γ) ⇒ ρ(θ).

Especially, if Γ ⇒ θ is a statement of both ML_{1}^{e}W_{R,U} and ML_{1}^{e}W_{T,U}, then we have: If ML_{1}^{e}W_{R,U} ⊢ Γ ⇒ θ then ML_{1}^{e}W_{T,U} ⊢ Γ ⇒ θ.

Proof: Induction on the derivation.

In most rules, the assertion follows by the same rules. Difficult rules: (SUB), (REP L): Use 3.8 (d), 3.10 and 3.6. Equality rules and extensional equality rules: use for the substitution part the same

argument.

Second and third rule in (U^{I}): we conclude T (ψ(A)), and using 3.10 and an easy argument follows the assertion.

Universe introduction rules (possibly extensional): easy.

Definition of KP i

In this chapter we motivate Kripke-Platek set theory,and define the language and theory.

Kripke-Platek set theory is the result of omitting in axiomatic set theory the power set axiom and restricting the separation and collection axioms. In the ordinary separation axiom ∃b.b = {x ∈ a|ϕ(x)}, if ϕ(x) = ∃y.ψ(x, y), the quantifier refers to all sets, even b itself — we have self-reference. If we restrict separation to Δ_{0}-formulas, e.g. ϕ = ∃y ∈ c.ψ(x, y)} with ψ quantifier-free, the quantifiers refer only to subterms of ϕ (here c). Then we can assign an ordinal level(b) to b in such a way, that the validity of ϕ(x) for x ∈ a is determined by all sets of level less than b (in our case level(b) = max{level(a), level(c)}). We have no self-reference. The collection axiom need to be restricted to Δ_{0}-formulas as well. We can use the method called ramified set theory, to analyze the theory proof theoretically.

The resulting theory is KP ω which is examined in [Bar75](more precisely, Barwise examines KP U, KP with urelements, KP ω is KP U with no urelements) In order to have stronger theories, Jäger added in [Jäg79] a predicate Ad(u) and axioms, expressing, that Ad(u) means u is admissible, a transitive inner model of KP ω, and further axioms claiming the existence of certain admissibles. The step to the next admissible corresponds to the step to the next inductive definition and will correspond in Martin-Löf’s type theory to the building of one further nested W -type. In some of his theories, Jäger restricted the Δ_{0} collection axiom and the foundation axiom, to have lower proof theoretical bounds.

We interpret ML_{1}^{e}W_{T} in KP i^{+}, KP i_{n}^{+}, theories which are slight extensions of the Jäger’s theory KP i, developed in [Jäg79] and analyzed proof theoretically in [Jäg83] and [JP82].

For further proof theoretical investigations of Kripke-Platek set theories, the reader might refer to [Jäg86]or [Poh82].

Definition 4.1 Definition of Kripke-Platek set theory:

CHAPTER 4. DEFINITION OF KP I 28

(a) Let L_{KP} be the classical first-order language, with terms being variables, atomic formulas being u ∈ v, ¬(u ∈ v), Ad(u), ¬Ad(u). The set of Variables should be V ar_{KP} = {u_{i}^{KP} |i ∈ N} (a meta-set), u_{i}^{KP} ≠ u_{j}^{KP} for i ≠ j. The formulas are built from atomic formulas by ∧, ∨, ∀, ∃. We define ¬A by the deMorgan’s laws. The quantifier in ∀x.ϕ (∃x.ϕ) is bounded, if ϕ of the form x ∈ v → B (x ∈ v ∧ B) with x ≠ v. A Δ_{0}-formula is a formula with no unbounded quantifier. We abbreviate A → B := ((¬A) ∨ B), ∀x ∈ v.B := ∀x.x ∈ v → B, ∃x ∈ v.B := ∃x.(x ∈ v ∧ B), (u = v) := ((∀x ∈ u.x ∈ v) ∧ (∀x ∈ v.x ∈ u)), u ∉ v := ¬(u ∈ v), tran(u) := ∀x ∈ u.∀y ∈ x.y ∈ u, infinite(u) := ∃x ∈ u.(x = x) ∧ ∀x ∈ u.∃y ∈ u.x ∈ y. ψ a formula, then ψ^{u} means the replacing of every unbounded quantifier ∀v by ∀v ∈ u and

∃v by ∃v ∈ u.

(b) Definition of axiom schemes:

(Ext) ∀x.∀y.∀z.x = y → (x ∈ z → y ∈ z) ∧ (Ad(x) → Ad(y))

(F ound) ∀z⃗.[∀x.(∀y ∈ x.ϕ(y, z⃗) → ϕ(x, z⃗)) → ∀x.ϕ(x, z⃗)] (ϕ an arbitrary formula )

(P air) ∀x.∀y.∃z.x ∈ z ∧ y ∈ z.

(Union) ∀x.∃z.∀y ∈ x.∀u ∈ y.u ∈ z.

(Δ_{0} − sep) ∀z⃗.∀w.∃y.[∀x ∈ y.(x ∈ w ∧ ϕ(x, z⃗)) ∧ ∀x ∈ w.ϕ(x, z⃗) → x ∈ y]

(ϕ a Δ_{0}-formula).

(Δ_{0} − coll) ∀z⃗.∀w.[∀x ∈ w.∃y.ϕ(x, y, z⃗) → ∃w′.∀x ∈ w.∃y ∈ w′.ϕ(x, y, z⃗)]

(ϕ a Δ_{0}-formula ). (Ad.1) ∀x.Ad(x) → tran(x) ∧ ∃w ∈ x.infinite(w). (Ad.2) ∀x.∀y.Ad(x) ∧ Ad(y) → x ∈ y ∨ x = y ∨ y ∈ x.

(Ad.3) ∀x.Ad(x) → ψ^{x},

(ψ an instance of (Pair), (Union), (Δ_{0}-sep), (Δ_{0}-coll) ).

(Lim) ∀x.∃y.Ad(y) ∧ x ∈ y. (inf) ∃x.infinite(x). (+) ∃z.Ad(z) ∧ ∀x ∈ z.∃y ∈ z.Ad(y) ∧ x ∈ y. (+_{n}) ∃x_{1}, …, x_{n−1}, z.Ad(z)∧

(∀x ∈ z.∃y ∈ z.Ad(y) ∧ x ∈ y)∧.

Ad(x_{1}) ∧ ⋯ ∧ Ad(x_{n−1})∧.

z ∈ x_{1} ∧ x_{1} ∈ x_{2} ∧ ⋯ ∧ x_{n−2} ∈ x_{n−1}.

(c) ES is the theory (Ext) + (Found) + (Pair) + (Union) + (Δ_{0}-sep) + (inf) KP is the theory (Ext) + (Found) + (Pair) + (Union) + (Δ_{0}-sep) + (Δ_{0}-coll) KP ω is the theory ES + (Δ_{0}-coll) = KP + (inf) KP l is the theory ES + (Ad.1 - 3) + (Lim) . KP i is the theory ES + (Ad.1 - 3) + Lim + (Δ_{0}-coll)

(= KP l + KP ω). KP i^{+} is the theory ES + (Ad.1 - 3) + (Lim) + (+). KP i_{n}^{+} is the theory ES + (Ad.1 - 3) + (Δ_{0}-coll) + (+_{n}).

ES is a weak basic theory, KP is Kripke-Platek set theory without infinity axiom, KP ω is the usual Kripke-Platek set theory (without urelements but with infinity) KP l is a theory claiming the existence of infinitely many admissibles, but the model itself need not be an admissible. A model of KP l is is L_{Ω_{ω}^{rec}}. KP i formalizes the existence of infinitely many admissible, and the model itself need to be admissible. Its model is L_{I}, where I is an admissible fixed point of the enumeration of the admissible, the first recursive inaccessible. KP i^{+} is a theory which formalizes the existence of I and and infinitely many admissibles above I and is modeled by L_{Ω_{I+ω}^{rec}}, and KP i_{n}^{+} expresses the existence of n admissibles above I, modeled by L_{Ω_{I+n}^{rec}}.

Definition 4.2 (a) pair(x, y) := {{x}, {x, y}}, tripel(x, y) := pair(x, pair(x, y)). (We use this notation, to distinguish pair from the coding of pairs in the natural numbers).

(b) On should be the class or ordinals. (a ∈ On is a Δ_{1}-predicate).

Part II

An upperbound for the proof theoretical strength of Martin-Löf’s

type theory Interpretation of terms and types

In this chapter, after an introduction, how to interpret Martin-Löf’s type theory in Kripke-Platek set theory KP i^{+}, we formalize g-terms in it and introduce the reduction relation →_{red}. (5.1), and prove some lemmata. Then we introduce the concept, how to interpret types as Σ functions. (5.7), and define the interpretation of the types, first for all but the Universe (5.8), and after that we give the definition of Û, which is needed for the interpretation of U and T (a) (5.9).

But let us start with an introduction in how to interpret Martin-Löf’s type theory in KP i^{+}. The basic idea is to interpret a type as the set of those terms, for which we can prove in Martin-Löf’s type theory, that they belong to the type: (T erm_{Cl} should be the set of closed g-terms.)

A ↦ A^{∗} := {t ∈ T erm_{Cl}|ML_{1}^{e}W_{T} ⊢ t : A}

The problem is, that in this definition t ∈ A^{∗} does not give any information about the validity of the formula represented by A. We replace this definition by a set of terms, which are introduced by an introductory rule. An example might be (A×B = Σx ∈ A.B, where x ∉ B, we take this example to avoid treating dependent types now)

A × B ↦ (A × B)^{∗} := {p(a, b) ∈ T erm_{Cl}|a ∈ A ∧ b ∈ B},

but since we have reduction of terms, we have to replace this definition by

(A × B)^{∗} := Compl({p(a, b) ∈ T erm_{Cl}|a ∈ A ∧ b ∈ B})

where

Compl(u) := {r ∈ T erm_{Cl}|∃s ∈ u.s in normal-form ∧ r→_{red}s}

The next task is to treat equality. We could treat this by saying, t and t′ are equal if they have the same normal-form. But to make sure, that we do not get any nonsense like t = t′, where t→_{red}0, t′→_{red}S0, we had to prove the existence of a unique normal-form, which is a technically very difficult area (note that we had to carry these proofs out in a restricted set theoretical framework), where the reduction rules have to be modified. We better try to avoid this problematic area, and use a far easier approach: We interpret types as sets of pairs of terms — the terms, which are considered to be equal.

A type is represented by a set of pairs of closed terms

An example would be

(A × B)^{∗} := Compl({pair(p(a, b), p(a′, b′)) ∈ T erm_{Cl} × T erm_{Cl}|
pair(a, a′) ∈ A^{∗} ∧ pair(b, b′) ∈ B^{∗}}

where now

Compl(u) := {pair(r, r′) ∈ T erm_{Cl} × T erm_{Cl}|∃s, s′ in normal-form .
r→_{red}s ∧ s→_{red}s′ ∧ pair(s, s′) ∈ u}

Another example is

(A + B)^{∗} := Compl(  {pair(i(r), i(r′))|pair(r, r′) ∈ A^{∗}}
∪ {pair(j(r), j(r′))|pair(r, r′) ∈ B^{∗}})

or the easiest example

N^{∗} := Compl({pair(S^{n}0, S^{n}0)|n ∈ IN})

Now we want to interpret the type A → B (= Πx ∈ A.B where x is a new variable). The problem is, that the introduction rules deduces from an open term a closed term: from x : A ⇒ t : B we conclude λx.t : A → B. But the intended meaning of x : A ⇒ t : B is, that, (as long as x ∉ F V (B)) for every s ∈ A^{∗}, t[x/s] ∈ B^{∗}, and this is independent from the choice of equal elements of A^{∗}: If pair(s, s′) ∈ A^{∗}, pair(t[x/s], t[x/s′]) ∈ B^{∗}. Putting these two definitions together in one, we can define

(A → B)^{∗} := Compl({pair(λx.t, λx′.t′)|∀pair(r, r) ∈ A^{∗}.pair(t[x/r], t′[x′/r′]) ∈ B^{∗}})

Now, the problem of confluence, we still have, is solved: The main reason for it is the use of open terms, which are part of the closed term λx.t. But we have in our whole definition avoided the use of closed terms, and in the definition of (A → B)^{∗}, we see, that, if pair(λx.t, λx.t) ∈ (A → B)^{∗}, and if t → t′ in a general sense for open terms, t[x/r]→_{red}t′[x/r] for closed terms r. But now, if B^{∗} is closed under →_{red}, we conclude pair(λx.t, λx.t′) ∈ (A → B)^{∗}.

So we can define, that all terms λx.t are in normal form. To make the technicalities even more easier, we choose a deterministic reduction strategy, some kind of call-by-value strategy: we allow Ap(λx.r, s) →_{red} r[x/s] only if s is in normal-form (note that λx.r is in normal-form), the same for all other constructors.

Now there is a new problem: Since we have dependent types, we have types containing free variables. Let us consider the Π-type:

(Πx ∈ A.B)^{∗} :=
Compl({pair(λx.t, λx′.t′)|∀pair(r, r′) ∈ A^{∗}.pair(t[x/r], t′[x′/r′]) ∈ B^{∗}[x/r]})

(We will see later, that t[x/r] is always allowed, as long as r is closed.)

So in fact, for a type with free variable x we could treat B^{∗}[x/r] := B[x/r]^{∗}. But we can not introduce the interpretation of all types, but only for finitely many types, since we are working in a restricted set theoretical system. The solution is, that we can interpret open types as Σ functions (see [Bar75]), and give some operation A[x_{1}/r_{1}, …, x_{n}/r_{n}] for Σ functions A and x_{i} ∈ V ar_{ML}, r_{i} g-terms, in such a way, that the equality B^{∗}[x/r] = B[x/r]^{∗} holds. Since we want to prove, that relation defined by the equality relation on the interpretation of a type is symmetric and transitive (we could prove this as part of the Main Lemma for all deduced types, but this would make this lemma even heavier), we add the additional condition B^{∗}[x/r] = B^{′∗}[x/r′] which will be fulfilled, whenever we have derived x : A ⇒ B type:

(Πx ∈ A.B)^{∗} :=
Compl({pair(λx.t, λx′.t′)|∀pair(r, r′) ∈ A^{∗}.pair(t[x/r], t′[x′/r′]) ∈ B^{∗}[x/r]
∧B^{∗}[x/r] = B^{∗}[x/r′]})

Now original Martin-Löf’s type theory has no fixed set of terms and types: they are introduced during the derivation of formulas. But we have extracted in chapter 2 a set of general terms, the g-terms, such that every term occurring in Martin-Löf’s type theory is a g-term, but not vice versa, and a set of g-types. Now we can interpret every g-type as a Σ function. Up to this stage we do not know any correctness properties. This begins, in the last step, where we prove, that this interpretation is correct for derivations in Martin-Löf’s type theory, in the sense that

ML_{1}^{e}W_{T} ⊢ r ∈ A ⇒ KP i^{+} ⊢ pair(r, r) ∈ A^{∗},

or, considering a more complicated case,

(ML_{1}^{e}W_{T} ⊢ x ∈ B ⇒ s = t ∈ A) ⇒
KP i^{+} ⊢ ∀pair(r, r′) ∈ B^{∗}.pair(s[x/r], t[x/r′]) ∈ A^{∗}[x/r].

The reader might have at this stage a first look at the following definitions, omitting all concepts dealing with the W -type and the universe, e.g. W, U, T, (πx ∈ s.t), (σx ∈ s.t)

(wx ∈ s.t), ĩ(r, s, t), r ˜+s, \underline{n}, \underline{n}_{k}, which will be explained next.

Let us now consider the W -type. From the rules follows, that an interpretation of it must be the least set closed under the applications of the W^{I}-rule. This set must be the least fixed point of the operator F, where, if we for simplicity switch back to the state, where types are interpreted as sets of terms rather than pairs of terms, F is defined as

F (u) := {sup(r, λx.s)|r ∈ A^{∗} ∧ ∀t ∈ B^{∗}[x/r].s[x/t] ∈ u}

To get the least fixed point we have to iterate this operator up to the least admissible such that A^{∗} ∈ L_{α} and ∀pair(r, r′) ∈ A^{∗}.B^{∗}[x/r] ∈ L_{α} (in fact we will iterate it up to an ordinal such that even ∀s ∈ T erm_{Cl}.B^{∗}[x/s] ∈ L_{α}).

The most powerful type of Martin-Löf’s type theory is the Universe. When we consider a rule like

(π^{I})  \frac{a∈U x∈T (a)⇒b∈U}{πx∈a.b:U}

we see, that we have to define simultaneously the elements of U and its interpretations as types T (a). We will therefore first define a set Û, which will be a set of tripels tripel(a, A, b). The intended meaning of tripel(a, A, b) ∈ Û is, that a and b are equal elements in U and T (a)^{∗} = A. This will be done in such a way, that, if we define

∼:= {pair(a, b) ∈ T erm_{Cl} × T erm_{Cl}|∃A.tripel(a, A, b) ∈ Û}

and

f := {pair(a, A)|∃b ∈ T erm_{Cl}.tripel(a, A, b) ∈ Û}

then ∼ is a symmetric and transitive relation, f is a function and ∀a, b.a ∼ b → f(a) = f(b).

Having defined Û, we can define U^{∗} :=∼ and T (a) := f(a)

As for the W -type, Û will be the least fixed point of an operator Ũ. Since U is closed under the W -type, in the definition of Ũ we have to go to the next admissible. So, to get Û, we have to iterate Ũ up to an admissible, which is closed under the formation of the next admissible, e.g. the first recursive inaccessible. This is the point, where we need the power of KP i^{+}. We need the existence of a recursive inaccessible, and in order to form W -types above U, we need the existence of infinitely many admissibles above this recursive inaccessible.

In fact, to interpret a particular proof of ML_{1}^{e}W_{T}, the nesting of W -types is limited, so we can interpret the types of this proof in KP i_{n}^{+} for some n.

To prove in chapter 7, that all sentences provable in ML_{1}^{e}W_{T} are provable in KP i_{n}^{+} for some n we will add some constructors, having not necessary recursive reduction rules.

Definition 5.1 (a) We add to the set of constructors a finite number of additional extended term constructors (A_{i})_{i∈I} with arities arity(A_{i}). (I = {1, …, n} for some n ∈ IN).

The extended b-terms are defined as the b-terms, only using additionally the extended term constructors.

We extend substitution and α-conversion for this extended system. Let (⌈C⌉)_{C constructor} Gödel-numbers for each constructor, such that we have a primitive recursive set T ermConstr_{(A_{i})_{i∈I}} of these Gödel-numbers (we will usually omit the index (A_{i})_{i∈I}), a function

arity : T ermConstr → N,

and a primitive recursive function

n ↦ ⌈C_{n}⌉

(C_{n} being the eliminator for the N_{n}-type).

Further let ⌈λ⌉, ⌈V ariable⌉, λ be Gödelnumbers. (Gödelnumbers always means, that we have chosen different natural numbers).

We interpret terms t as follows:

⌈z_{i}^{ML}⌉ :=< ⌈V ariable⌉, i >, ⌈C(t_{1}, …, t_{n})⌉ :=< ⌈C⌉, ⌈t_{1}⌉, …, ⌈t_{n}⌉ >, ⌈λz_{i}^{ML}.t⌉ :=< ⌈λ⌉, i, ⌈t⌉ >.

We have a primitive recursive set of all codes of extended b-terms T erm_{(A_{i})_{i∈I}} (again we omit usually the index), can break it primitive recursively into its part, can define a primitive recursive function sub : N → N → N such that

sub(⌈t⌉, << i_{1}, ⌈t_{1}⌉ >, …, < i_{n}, ⌈t_{n}⌉ >>) = ⌈(t[z_{i_{1}}^{ML}/t_{1}, …, z_{i_{n}}^{ML}/t_{n}])⌉,

(< · > being the coding for sequences of natural numbers in natural numbers), a primitive recursive function F V′ : N → N such that, if F V (t) = {z_{i_{1}}^{ML}, …, z_{i_{n}}^{ML}} with i_{1} < ⋯ < i_{n}, F V′(⌈t⌉) =< i_{1}, …, i_{n} >, a primitive recursive definable subset T erm_{Cl} of indices of closed terms. In the following we will omit usually the Gödel-parenthesis ⌈ ⌉, if it is clear that we are speaking of natural numbers.

(b) The introductory term constructors are the term constructors 0, \underline{r}, \underline{n}, S, i, j, p, sup, ˜+, π, σ, w, ĩ.

(c) We assume, that for each A_{i}, by which we extended our constructors, we have a function A_{i}^{∗} : N^{arity(A_{i})} → N, defined in KP i^{+}.

Let →_{red,imm}_{(A_{i},A_{i}^{∗})_{i∈I}} (again we will omit the index (A_{i}, A_{i}^{∗})_{i∈I}) be the set of pairs of closed terms, written t→_{red,imm}_{(A_{i},A_{i}^{∗})}t′ or short t→_{red,imm}t′ for pair(t, t′) ∈ →_{red,imm}, defined as follows:

If r, s, t, t′, r_{i} are extended b-terms (and the left side of →_{red,imm} are closed terms), then p_{0}(p(r, s))→_{red,imm}r, p_{1}(p(r, s))→_{red,imm}s, Ap(λx.r, s)→_{red,imm}r[x/s], C_{n}(i_{n}, r_{1}, …, r_{n})→_{red,imm}r_{i}, D(i(r), s, t)→_{red,imm}sr, D(j(r), s, t)→_{red,imm}tr, P (0, s, t)→_{red,imm}s, P (Sr, s, t)→_{red,imm}(trP (r, s, t)), (note that we write rs for Ap(r, s)), R(sup(r, s), t)→_{red,imm}(trs(λz_{i}^{ML}.R(sz_{i}^{ML}, t))), where i is minimal such that z_{i}^{ML} ∉ F V (s)∪ F V (t) A_{i}(S^{n_{1}}0, …, S^{n_{k}}0)→_{red,imm}S^{A_{i}^{∗}(n_{1},...,n_{k})}0, the pairs mentioned above are all pairs. Note that it is primitive recursive decidable, if there exists a t′ such that t→_{red,imm}t′, and for each term t there is at most one term t′ such that t→_{red,imm}t′, so →_{red,imm} represents are partial function.

(d) We define inductively a set of (indices for) terms in normal-form T erm_{nf} ⊂ T erm_{Cl}:

If C is an introductory n-ary term constructor, t_{1}, …, t_{n} ∈ T erm_{nf}, then C(t_{1}, …, t_{n}) ∈ T erm_{nf}. If C is a n-ary term constructor (possibly an extended term constructor) that is not introductory, t_{1}, …, t_{n} ∈ T erm_{nf}, and there exists no t such that C(t_{1}, …, t_{n})→_{red,imm}t, then C(t_{1}, …, t_{n}) ∈ T erm_{nf}. If t ∈ T erm, x ∈ V ar_{ML}, F V (t) ⊂ {x}, then λx.t ∈ T erm_{nf}.

We easily see, that T erm_{nf} is a primitive recursively decidable relation.

(e) We define terms t ∈ T erm_{Cl}, the next reduced term t^{red}. For t ∈ T erm_{nf}. t^{red} := t. If C is a n-ary (possibly extended) term constructor, r_{i} ∈ T erm_{Cl}, ∃i.r_{i} ∉ T erm_{nf}, then C(r_{1}, …, r_{n})^{red} := C(r_{1}^{red}, …, r_{n}^{red}). If t := C(r_{1}, …, r_{n}) ∉ T erm_{nf}, r_{i} ∈ T erm_{nf}, then t→_{red,imm}t′ for some t′, t^{red} := t′.

We define r→_{red}s :↔ ∃n ∈ ω, s ∈ ω.sequence(s) ∧ lh(s) = n ∧ (s)_{0} = r ∧ (s)_{n} = s ∧ ∀i < n.(s)_{i}^{red} = (s)_{i+1}.

Lemma 5.2 Assume x_{i}, y_{i} ∈ V ar_{ML}.

(a) KP i^{+} ⊢ ∀r, s′ ∈ T erm.r=_{α}r′ → F V (r) = F V (r′).

(b) KP i^{+} ⊢ ∀r, r_{i} ∈ T erm.F V (r[x⃗/r⃗]) ⊂ F V (r) ∪ ⋃_{i=1}^{n} F V (r_{i}).

(c) In KP i^{+} we prove that for all s, t_{i} ∈ T erm, all r_{i} ∈ T erm_{Cl}, s[x⃗/r⃗] is allowed substitution ∧ (s[y⃗/t⃗] is allowed ) → s[x⃗/r⃗][y⃗/t⃗] is allowed )∧ (s[x⃗/r⃗][y⃗/t⃗] = s[x⃗/r⃗, y⃗/t⃗]).

(d) KP i^{+} ⊢ ∀t, t′ ∈ T erm.∀r_{i}, r′_{i} ∈ T erm_{Cl}.(t=_{α}t′ ∧ r_{i}=_{α}r′_{i}) → t[x⃗/r⃗]=_{α}t′[x⃗/r′].

(e) KP i^{+} ⊢ ∀x, x′ ∈ V ar_{ML}.∀t, t′ ∈ T erm.∀r, r′ ∈ T erm_{Cl}. (λx.t=_{α}λx′.t′ ∧ r=_{α}r′) → t[x/r]=_{α}t′[x′/r′] Proof:

(a), (b): easy.

(c): If x ∈ V ar_{ML} this is trivial, and if s = C(t_{1}, …, t_{n}) it follows by IH. If s = λx.t, [x′/r′] := [x⃗/r⃗] \ {x}, [y′/t⃗′] := [y⃗/t⃗] \ {x}, s[x⃗/r⃗] = λx.(t[x′/r′]) allowed, since x ∉ F V (r_{i}) and the IH. If s[y⃗/t⃗] is allowed follows t[y′/t⃗′] is allowed and x ≠ y ∈ F V (t) → x ∉ F V (t_{i}) therefore t[x′/r′][y′/t⃗′] is allowed and x ≠ y ∈ F V (t[x′/r′]) = F V (t) → x ∉ F V (t_{i}), s[x⃗/r⃗][y⃗/t⃗] is allowed. Further follows s[x⃗/r⃗][y⃗/t⃗] = λx.(s[x′/r′][y′/t⃗′]) = λx.(s[x′/r′, y′/t⃗′]) = t[x⃗/r⃗, y⃗/t⃗].

(d) We prove first, that if t=_{α}t′, r_{i} ∈ T erm_{Cl}, t[x⃗/r⃗]=_{α}t′[x⃗/r⃗].

Induction on t=_{α}t′. The case t = t′ is trivial, the cases of t=_{α}t′′=_{α}t′, t = C(t_{1}, …, t_{n}) ∧ t′ = C(t′_{1}, …, t′_{n}) ∧ t_{i}=_{α}t′_{i} and t =_{N} λx.s ∧ t′ =_{N} λx.s′ ∧ s=_{α}s′ follows by IH.

Case t = λx.s, t′ = λy.s[x/y], y ∉ F V (s), t[x/y] is allowed, [x′/r′] := [x⃗/r⃗]\{x, y}. Then y ∉ F V (t), t[x⃗/r⃗] = λx.(s[x′/r′]), by (c) therefore s[x′/r′][x/y] is allowed, y ∉ F V (t[x⃗/t⃗], s[x′/r′][x/y] = s[r′/r′, x/y] = s[x/y][s′/r′], t′[x⃗/r⃗] = λy.(s[x′/r′][x/y])=_{α}t[x⃗/r⃗].

Next we prove r_{i}=_{α}r′_{i} → t′[x⃗/r⃗]=_{α}t′[x⃗/r′], which follows by a trivial induction on the construction of t′.

(e) Induction on definition of λx.t=_{α}λx′.t′: The case λx.t = λx′.t′ is trivial, if λx.t=_{α}s=_{α}λx′.t′, then s = λx′′.t′′ and the assertion follows by IH, and if x = x′, t=_{α}t′ the assertion follows by (d). If t′ = t[x/x′], x′ ∉ F V (λx.t), t[x/x′] substitutable, t′[x′/r′]=_{α}t′[x′/r] = t[x/x′][x′/r] = t[x/r, x′/r] = t[x/r], since x′ ∉ F V (t) ∨ x′ = x. Lemma 5.3 (a) KP i^{+} ⊢ ∀r, s, s′ ∈ T erm_{Cl}.(r→_{red}s ∧ r→_{red}s′) → (s→_{red}s′ ∨ s′→_{red}s).

(b) KP i^{+} ⊢ ∀r, s, s′ ∈ T erm_{Cl}.(r→_{red}s ∧ r→_{red}s′ ∧ s, s′ ∈ T erm_{nf}) → s = s′.

(c) If C is a n-ary constructor, then

KP i^{+} ⊢ ∀r_{1}, …, r_{n}, r′_{1}, …, r′_{n}.(r_{1}→_{red}r′_{1} ∈ T erm_{nf} ∧ ⋯ ∧ r_{n}→_{red}r′_{n} ∈ T erm_{nf})
→ C(r_{1}, …, r_{n})→_{red}C(r′_{1}, …, r′_{n})

(d) KP i^{+} ⊢ ∀t, t′, s ∈ T erm_{Cl}.(t→_{red}s ∧ t=_{α}t′) → ∃s′ ∈ T erm_{Cl}.t′→_{red}s′.

(e) KP i^{+} ⊢ ∀t, t′ ∈ T erm_{Cl}.t=_{α}t′ → (t ∈ T erm_{nf} ↔ t′ ∈ T erm_{nf}). Proof: (a), (b): Immediate, since we have chosen t ↦ t^{′red} as a function.

(c): Induction on maximum of the reduction sequences of r_{i}→_{red}r′_{i}.

(d), (e): We show t=_{α}t′ → ((t ∈ T erm_{nf} ↔ t′ ∈ T erm_{nf}) ∧ t^{red}→_{red}t′^{red}), by induction on the definition of t=_{α}t′, side induction on construction on t:

If t = t′ the assertion is trivial, if t=_{α}t′′=_{α}t′ it follows by IH.

Case t = C(t_{1}, …, t_{n}). Then t′ = C(t′_{1}, …, t′_{n}), t_{i}=_{α}t′_{i}: If t_{i} ∉ T erm_{nf}, follows t′_{i} ∉ T erm_{nf}, t_{i}^{red}=_{α}t′_{i}^{red}, t^{red} = C(t_{1}^{red}, …, t_{n}^{red})=_{α}t′ by IH. If t_{i} ∈ T erm_{nf}, t→_{red,imm}s, t′→_{red,imm}s′, where s′ is just the same arrangement of the terms as in s, except that in the case of C = R we might choose another variable z_{i}^{ML}, and in the case C = Ap where we have to conclude using lemma 5.2 (d), and we have in all cases s→_{red}s′. If t_{i} ∈ T erm_{nf}, ¬(t→_{red,imm}s) we have t, t′ ∈ T erm_{nf}, t^{red} = t=_{α}t′ = t′^{red}.

If t = λx.s, then t′ = λy.s′, t, t′ ∈ T erm_{nf}. Definition 5.4 (a) α_{I} := min{γ ∈ On|Ad(L_{γ}) ∧ ∀x ∈ L_{γ}.∃y ∈ L_{γ}.Ad(y) ∧ x ∈ y} (definable in

KP i^{+}),

(b)

α_{I,n} := min{γ ∈ On|∃α_{1}, …, α_{n−1} ∈ On.Ad(L_{α_{1}}) ∧ ⋯ Ad(L_{α_{n−1}}) ∧
α_{I} ∈ α_{1} ∧ α_{1} ∈ α_{2} ∧ ⋯ ∧ α_{n−2} ∈ α_{n−1} ∧ γ = α_{n−1}}

(definable in KP i^{+} or KP i_{m}^{+} for n < m)

(c) If α is an ordinal, then α^{+} := min{γ ∈ On|Ad(L_{γ}) ∧ α ∈ L_{γ}} (definable in KP i^{+}), α^{+,n} := min{γ ∈ On|Ad(L_{γ}) ∧ α ∈ L_{γ}} ∪ {α_{I,n}} (definable in KP i_{n}^{+})

(d) a(u) := min{c|Ad(c) ∧ u ∈ c} (definable in KP i^{+}). α(u) := ⋃{γ ∈ a(u)|γ ∈ On}, definable in KP i^{+}. a(u)^{n} := min{c|Ad(c) ∧ u ∈ c} ∪ {L_{α_{I,n}}}, definable in KP i_{n}^{+}. α(u)^{n} := ⋃{γ ∈ a(u)^{n}|γ ∈ On}, definable in KP i_{n}^{+}.

We need for the interpretation of W and U the transfinite iteration of an operator, defined as follows: Definition 5.5 (a) If F is an (n + 1)-ary Σ function, let F (n⃗, ·) be the Σ function, such that

F (n⃗, ·)(u) := F (n⃗, u).

(b) If F is a Σ function, we define by recursion on α

F^{α}(v) := \{ \begin{matrix}v & if α = 0, \\ F (F^{β}(v)) & if α = β + 1, \\ ⋃_{β<α,β∈On} F^{β}(v) & if α ∈Lim. \end{matrix}

Definition 5.6 We define the Σ function Compl

Compl(a) := {pair(r, s) ∈ T erm_{Cl} × T erm_{Cl}|∃r′, s′ ∈ T erm_{nf}.
r→_{red}r′ ∧ s→_{red}s′ ∧ pair(r′, s′) ∈ a}

Compl(a) ⊂ T erm_{Cl} × T erm_{Cl}, and will be used only, if a ⊂ T erm_{Cl} × T erm_{Cl}.

Note that in fact this definition is relative to the extension of our set of constructors, so actually we have to write Compl_{(A_{i},A_{i}^{∗})_{i∈I}}.

We will interpret each g-type occurring in a proof of Martin-Löf’s type theory (note that there are only finitely many — we cannot give a general interpretation of all types) as Σ functions, with arguments represented by the free variables of the type. More precisely, if F V (A) = {z_{1}^{ML}, …, z_{n}^{ML}}, (z_{i}^{ML} as in the definition 2.1 of V ar_{ML}) the arguments of the interpretation A^{∗} will have arguments given by the variables {u_{1}^{KP} , …, u_{n}^{KP} } (u_{i}^{KP} as in definition 4.1 (a) of V ar_{KP}). We introduce the following abbreviation: Definition 5.7 If A is a Σ function in KP i^{+} with arguments represented by the free variables {u_{i_{1}}^{KP} , …, u_{i_{n}}^{KP} }, i_{1} <, …, i_{n}, u_{i}^{KP} as in the definition 4.1 (a) of V ar_{KP} z_{i}^{ML} as in the definition 2.1 of V ar_{ML}, r_{1}, …, r_{m} extended b-objects, j_{k} := min{l|x_{l} = z_{j_{k}}^{ML}} (k = 1, …, m), then

A[x_{1}/r_{1}, …, x_{m}/r_{m}] := A[u_{i_{1}}^{KP} /r_{j_{1}}, …, u_{i_{n}}^{KP} /r_{j_{n}}],

where on the right-hand side we have the real substitution or (having introduced a Σ function symbol) the application of the function symbol to the arguments.

We will write A[x⃗/n⃗] for A[x_{1}/n_{1}, …, x_{n}/n_{n}]. Note that, if a variable occurs more than once in the sequence x_{1}, …, x_{n}, only the first one is relevant. Definition 5.8 Definition of the interpretation of g-types A, namely A^{∗}, which will be a Σ function, that has, if F V (A) = {z_{1}^{ML}, …, z_{n}^{ML}}, (z_{i}^{ML} as in definition 2.1 of V ar_{ML}), arguments given by the variables {u_{1}^{KP} , …, u_{n}^{KP} } (u_{i}^{KP} as in defition 4.1 (a) of V ar_{KP}). We will define it by giving the values A^{∗}[x⃗/s⃗].

Additionally we define lev(A) ∈ IN. The interpretation is relative to a choice of term constructors (A_{i})_{i∈I}, extending the given term constructors, together with some interpretation A_{i}^{∗}, as stated in 5.1 (c), so in fact we have to write (we do not for simplicity) A_{(A_{i},A_{i}^{∗})_{i∈I}}^{∗}.

We define in all, but the W -, U- and T (t)-cases

A^{∗}[x⃗/s⃗] := Compl(A^{basis}[x⃗/s⃗]),

where A^{basis} is defined as follows:

For k ∈ IN,

N_{k}^{basis}[ ] := {pair(n_{k}, n_{k})|n < k},

(note that we have a constant Σ function) lev(N_{k}) := 0

N^{basis}[ ] := {pair(S^{n}0, S^{n}0)|n < ω},

lev(N) := 0 Let A, B be g-types, m := max{lev(A), lev(B)}. Then lev(Πx ∈ A.B) := lev(Σx ∈ A.B) := m, lev(W x ∈ A.B) := m+1, and we define:

(Πx ∈ A.B)^{basis}[x⃗/s⃗] := {pair(λx.r, λx′.r′) ∈ T erm_{nf} × T erm_{nf}|

∀pair(s, s′) ∈ A^{∗}[x⃗/s⃗].pair(r[x/s], r[x′/s′]) ∈ B^{∗}[x/s, x⃗/s⃗]

∧B^{∗}[x/s, x⃗/s⃗] = B^{∗}[x/s′, x⃗/s⃗]},

(more precisely we have to write:

(Πx ∈ A.B)^{basis}[x⃗/s⃗] := {pair(t, t′) ∈ T erm_{nf} × T erm_{nf}|

∃x, x′ ∈ V ar_{ML}, r, r′ ∈ T erm.t = λx.r ∧ t′ = λx′.r′ ∧

∀s, s′ ∈ T erm.pair(s, s′) ∈ A^{∗}[x⃗/s⃗] →
pair(r[x/s], r[x′/s′]) ∈ B^{∗}[x/s, x⃗/s⃗]
∧B^{∗}[x/s, x⃗/s⃗] = B^{∗}[x/s′, x⃗/s⃗]},

similarly in the following definitions)

(Σx ∈ A.B)^{basis}[x⃗/s⃗] := {pair(p(r, s), p(r′, s′)) ∈ T erm_{nf} × T erm_{nf}|
pair(r, r′) ∈ A^{∗}[x⃗/s⃗] ∧ pair(s, s′) ∈ B^{∗}[x/r, x⃗/s⃗] ∧
B^{∗}[x/r, x⃗/s⃗] = B^{∗}[x/r′, x⃗/s⃗]},
W x ∈ A.B^{∗}[x⃗/s⃗] := F^{α_{I,m}}(s⃗, ·)(∅)

(note that m = max{lev(A), lev(B)})

where F (s⃗, u) := Compl(F^{basis}(s⃗, u)), and

F^{basis}(s⃗, u) :=
{pair(sup(r, λx.s), sup(r′, λx′.s′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ A^{∗}[x⃗/s⃗] ∧
B[x/r, x⃗/s⃗] = B[x/r′, s⃗]
∧∀pair(t, t′) ∈ B^{∗}[x/r, x⃗/s⃗].pair(s[x/t], s′[x′/t′]) ∈ u}

If A, B are g-types, then lev(A + B) := max{lev(A), lev(B)},

(A + B)^{basis}[x⃗/s⃗] := {pair(i(r), i(r′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ A^{∗}[x⃗/s⃗]}

∪{pair(j(r), j(r′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ B^{∗}[x⃗/s⃗]}

If A is a g-type, s, t are g-terms, then lev(I(A, s, t)) = lev(A) and

(I(A, r, s))^{basis}[x⃗/s⃗] := \{pair(\underline{r}, \underline{r})|pair(r[x⃗/s⃗], s[x⃗/s⃗]) ∈ A^{∗}[x⃗/s⃗]\}.

lev(U) := 1 and

U^{∗}[ ] := {pair(r, r′) ∈ T erm_{Cl} × T erm_{Cl}|∃b ∈ T C(Û)tripel(r, b, r′) ∈ Û}.

(Û will be defined in the next definition)

If t is a g-term, then lev(T (t)) := 1

T (t)^{∗}[x⃗/s⃗] := ⋃{b ∈ T C(Û)|tripel(t[x⃗/s⃗], b, t[x⃗/s⃗]) ∈ Û}.

Definition 5.9 Definition of Û:

Û := Ũ^{α_{I}}(∅), where Ũ is a Σ function symbol, which can be defined in KP i^{+} in the following way:

Let Compl_{U}(a) := {tripel(r, b, s) ∈ T erm_{Cl} × T C(a) × T erm_{Cl}|∃r′, s′ ∈ T erm_{nf}.

r→_{red}r′ ∧ s→_{red}s′ ∧ tripel(r′, b, s′) ∈ a}.

Ũ(u) := Compl_{U}(Ũ^{basis}(u)),

where

Ũ^{basis}(u) :=  \{tripel(\underline{n}_{k}, u_{fin}(k), \underline{n}_{k}) ∈ a(u)|k ∈ IN\}
∪ \{tripel(\underline{n}, u_{nat}, \underline{n})\}
∪ {tripel(πx ∈ r.s, u_{π}(b, f), πx′ ∈ r′.s′) ∈ a(u)|ϕ(r, x, s, r′, x′, s′, b, f, u)}
∪ {tripel(σx ∈ r.s, u_{σ}(b, f), σx′ ∈ r′.s′) ∈ a(u)|ϕ(r, x, s, r′, x′, s′, b, f, u)}
∪ {tripel(wx ∈ r.s, u_{w}^{α(u)}(b, f, ·)(∅), wx′ ∈ r′.s′) ∈ a(u)|
ϕ(r, x, s, r′, x′, s′, b, f, u)}
∪ {tripel(r ˜+s, u_{+}(b, c), r′ ˜+s′) ∈ a(u)|ψ_{+}(r, s, r′, s′, b, c, u)}
∪ {tripel(ĩ(r, s, t), u_{i}(b, s, t), ĩ(r′, s′, t′) ∈ a(u)|ψ_{i}(r, s, t, r′, s′, t′, b, u)}

and

ϕ(r, x, s, r′, x′, s′, b, f, u) := r, r′ ∈ T erm_{nf} ∧ s, s′ ∈ T erm ∧ f ∈ a(u)
∧F V (s) ⊂ {x} ∧ F V (s′) ⊂ {x′} ∧ tripel(r, b, r′) ∈ u ∧
(∀pair(t, t′) ∈ b.tripel(s[x/t], f(t), s′[x′/t′]) ∈ u)

(note that f(t) = ⋃{c ∈ T C(f)|pair(t, c) ∈ f}, so f need not be a function)

ψ_{+}(r, s, r′, s′, b, c, u) := r, s, r′, s′ ∈ T erm_{nf} ∧ tripel(r, b, r′) ∈ u ∧ tripel(s, c, s′) ∈ u,

ψ_{i}(r, s, t, r′, s′, t′, b, u) := r, s, t, r′, s′, t′ ∈ T erm_{nf} ∧ tripel(r, b, r′) ∈ u ∧
pair(s, s′) ∈ b ∧ pair(t, t′) ∈ b,

CHAPTER 5. INTERPRETATION OF TERMS AND TYPES 40 u_{fin}(k) : Compl(u_{fin}^{basis}(k), where

u_{fin}^{basis}(k) := {pair(n_{k}, n_{k})|n < k}

u_{nat} := Compl(u_{nat}^{basis}), where

u_{nat}^{basis} := {pair(S^{n}0, S^{n}0)|n ∈ IN}

u_{π}(b, f) := Compl(u_{π}^{basis}(b, f)), where

u_{π}^{basis}(b, f) := {pair(λx.t, λx′.t′) ∈ T erm_{nf} × T erm_{nf}|
∀pair(r, r′) ∈ b.pair(t[x/r], t′[x′/r′]) ∈ f(r)}

u_{σ}(b, f) := Compl(u_{σ}^{basis}(b, f)), where

u_{σ}^{basis}(b, f) := {pair(p(r, s), p(r′, s′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ b ∧
pair(s, s′) ∈ f(r)},

u_{w}(b, f, v) := Compl(u_{w}^{basis}(b, f, v)), where

u_{w}^{basis}(b, f, v) := { pair(sup(r, λx.s), sup(r′, λx′.s′)) ∈ T erm_{nf} × T erm_{nf}|

r, r′ ∈ T erm_{Cl}, s, s′ ∈ T erm, F V (s) ⊂ {x}, F V (s′) ⊂ {x′}, pair(r, r′) ∈ b, ∀pair(t, t′) ∈ f(r).pair(s[x/t], s′[x′/t′]) ∈ v)}

u_{+}(b, c) := Compl(u_{+}^{basis}(b, c)), where

u_{+}(b, c) := {pair(i(r), i(r′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ b}
∪ {pair(j(r), j(r′)) ∈ T erm_{nf} × T erm_{nf}|pair(r, r′) ∈ c}

u_{i}(b, r, s) := Compl(u_{i}^{basis}(b, r, s)), where u_{i}^{basis}(b, r, s) := \{pair(\underline{r}, \underline{r})|pair(r, s) ∈ b\}

Correctness of the Interpretation

In this chapter, we show, that the interpretation is correct, in the sense, that, if ML_{1}^{e}W_{T} ⊢ r : A, then KP i^{+} ⊢ pair(r, r) ∈ A^{∗}. In chapter 7 we will then conclude, that this shows, that all arithmetical sentences provable in ML_{1}^{e}W_{T}, can be proven in KP i^{+}.

We first prove some correctness property: The relation defined by the type is a symmetric and transitive relation (lemma 6.10), we examine the relationship to substitution (lemma 6.12) and α-conversion (lemma 6.15). Further we introduce abbreviations, to state easily our Main Lemma (definitions 6.11 and 6.16). The heart of this chapter is the Main Lemma 6.18, which proves, that, if ML_{1}^{e}W_{T} ⊢ r : A, then KP i^{+} ⊢ pair(r, r,) ∈ A^{∗}. In fact we prove something more general, to have a stronger induction hypothesis. The proof is tedious, since we have to check the correctness of each rule, but not very complicated — the intuition for the upper bound is concentrated in. At the end we see, that we can even interpret an a little bit extended version ML_{1}^{e}W_{T,U} in this way. This version is equivalent to ML_{1}^{e}W_{R,U}, an extension of |ML_{1}^{e}W_{R}| and therefore we can interpret ML_{1}^{e}W_{R} as well (lemma 6.19).

The next lemma shows, that by iterating u_{w}(b, f, ·) up to the next admissible, we get a fixed point. Lemma 6.1 (a) ∀γ < δ.u_{w}^{γ}(b, f, ·)(∅) ⊂ u_{w}^{δ}(b, f, ·)(∅).

(b) (b ∈ a ∧ f ∈ a ∧ Ad(a) ∧ α = ⋃_{δ∈a∩On} δ) → ∀γ > α.u_{w}^{γ}(b, f, ·)(∅) = u_{w}^{α}(b, f, ·)(∅).

(c) The proof can be done in KP i^{+}. Proof: (a) follows by induction on δ.

(b) It is sufficient to show, with u′ := u_{w}^{α}(b, f, ·)(∅), that u_{w}(b, f, u′) ⊂ u′. (Then the assertion follows immediately by induction on γ)

Since Compl(u′) ⊂ u′ it is sufficient to prove u_{w}^{basis}(b, f, u′) ⊂ u′. Now, if

pair(sup(r, λx.s), sup(r, λx′.s′)) ∈ u_{w}^{basis}(b, f, u′)

follows

∀t, t′ ∈ T erm_{Cl}.pair(t, t′) ∈ f(r) → ∃δ < α.pair((s[x/t], s′[x′/t′]) ∈ u_{w}^{δ}(b, f, u))

Since Ad(a) (here we need (Δ_{0}-coll), there exists ρ < α, such that

∀t, t′ ∈ T erm_{Cl}.pair(t, t′) ∈ f(r) → ∃δ < ρ.pair((s[x/t], s′[x′/t′]) ∈ u_{w}^{δ}(b, f, u))

Now follows

pair(sup(r, λx.s), sup(r′, λx′.s′)) ∈ u_{w}^{ρ+1}(b, f, u) ⊂ u′

and the assertion. Definition 6.2 (a) equiv(u) :⇔

∀r, s, t, r′, s′ ∈ T erm_{Cl}.(pair(r, s) ∈ u → pair(s, r) ∈ u)∧

((pair(r, s) ∈ u ∧ pair(s, t) ∈ u) → pair(r, t) ∈ u). (note that we do not claim reflexivity)

(b) Cor(u) :⇔

∀r, r′, r′′ ∈ T erm_{Cl}.∀b, b′.(tripel(r, b, r′) ∈ u → [tripel(r′, b, r) ∈ u ∧ equiv(b) ∧[tripel(r′, b′, r′′) ∈ u → (tripel(r, b, r′′) ∈ u ∧ b = b′)]])

Remark 6.3 (a) (Cor(u) ∧ tripel(r, b, r′) ∈ u ∧ tripel(r, b′, r′′) ∈ u) → (b = b′ ∧ tripel(r, b, r) ∈ u).

(b) If Cor(u) and

∼:= {pair(a, b) ∈ T erm_{Cl} × T erm_{Cl}|∃A ∈ T C(u).tripel(a, A, b) ∈ u}

and

f := {pair(a, A) ∈ T erm_{Cl} × T C(u)|∃b ∈ T erm_{Cl}.tripel(a, A, b) ∈ u}

then ∼ is a symmetric and transitive relation, f is a function and ∀a, b.a ∼ b → f(a) = f(b). Proof: Obvious. Lemma 6.4 Assume r, s, t, r′, s′, t′ ∈ T erm, x, x′ ∈ V ar_{ML}, b, f, u sets.

(a) (ϕ(r, x, s, r′, x′, s′, b, f, u) ∧ Cor(u)) → (b ∈ a(u) ∧ ∃f ∈ a(u).∀pair(t, t′) ∈ b.f(t) = f(t′) = f′(t) = f′(t′)).

(b) ψ_{+}(r, s, r′, s′, b, c, u) → b, c ∈ a(u)

(c) ψ_{i}(r, s, t, r′, s′, t′, b, u) → b ∈ a(u)

Proof:

(a) b ∈ T C(u) ∈ a(u).

Let f′ := {pair(t, c) ∈ T erm_{Cl} × T C(u)|pair(t, t) ∈ b ∧ tripel(s[x/t], c, s[x/t]) ∈ u}.

f′ ∈ a(u). Further, if pair(t, t′) ∈ b follows tripel(s[x/t], f(t), s′[x/t′]) ∈ u, by Cor(u) tripel(s[x/t], f(t), s[x/t]) ∈ u, further ∀c.tripel(s[x/t], c, s′[x/t]) ∈ u → c = f(t), f′ is a function, f(t) = f′(t) and by

tripel(s′[x/t], f(t), s[x/t]) ∈ u ∧ tripel(s′[x/t], f(t′), s[x/t′]) ∈ u

follows f(t) = f(t′).

(b), (c) follow by b, c ∈ T C(u). Lemma 6.5 Assume r, s, t, r′, s′, t′ ∈ T erm, x, x′ ∈ V ar_{ML}, b, f, u sets.

(a) ∀k ∈ IN.tripel(\underline{n}_{k}, u_{fin}(k), \underline{n}_{k}) ∈ a(u).

(b) tripel(\underline{n}, u_{nat}, \underline{n}) ∈ a(u).

(c) (Cor(u) ∧ ϕ(r, x, s, r′, x′, s′, b, f, u)) →

(tripel(πx ∈ r.s, u_{π}(b, f), πx′ ∈ r′.s′), tripel(σx ∈ r.s, u_{σ}(b, f), σx′ ∈ r′.s′) ∈ a(u)∧ tripel(wx ∈ r.s, u_{w}^{α(u)}(b, f, ·)(∅), wx′ ∈ r′.s′) ∈ a(a(u)).

CHAPTER 6. CORRECTNESS OF THE INTERPRETATION 43

(d) (Cor(u) ∧ ψ_{+}(r, s, r′, s′, b, c, u)) → tripel(r ˜+s, u_{+}(b, c), r ˜+s) ∈ a(u).

(e) (Cor(u) ∧ ψ_{i}(r, s, t, r′, s′, t′, b, u)) → tripel(ĩ(r, s, t), u_{i}(b, s, t), ĩ(r′, s′, t′)) ∈ a(u). Proof: By 6.4 Lemma 6.6 Assume r, s, t, r′, s′, t′, r′′, s′′, t′′ ∈ T erm, x, x′, x′′ ∈ V ar_{ML}, b, b′, f, f′, u, u′ sets.

(a) ϕ(r, x, s, r′, x′, s′, b, f, u) → Cor(u) → ϕ(r′, x′, s′, r, x, s, b, f, u).

(b) (ϕ(r, x, s, r′, x′, s′, b, f, u) ∧ ϕ(r′, x′, s′, r′′, x′′, s′′, b′, f′, u′) ∧ Cor(u ∪ u′)) → ϕ(r, x, s, r′′, x′′, s′′, b, f, u ∪ u′) ∧ b = b′∧ ∀pair(t, t′) ∈ b.f(t) = f′(t) = f(t′) = f′(t′)

(c) ψ_{+}(r, s, r′, s′, b, c, u) → Cor(u) → ψ_{+}(r′, s′, r, s, b, c, u).

(d) (ψ_{+}(r, s, r′, s′, b, c, u) ∧ ψ_{+}(r′, s′, r′′, s′′, b′, c′, u′) ∧ Cor(u ∪ u′) → (ψ_{+}(r, s, r′′, s′′, b, c, u ∪ u′) ∧ b = b′ ∧ c = c′))

(e) ψ_{i}(r, s, t, r′, s′, t′, b, u) → Cor(u) → ψ_{i}(r′, s′, t′, r, s, t, b, u).

(f) (ψ_{i}(r, s, t, r′, s′, t′, b, u) ∧ ψ_{i}(r′, s′, t′, r′′, s′′, t′′, b′, u′) ∧ Cor(u ∪ u′) → (ψ_{i}(r, s, t, r′′, s′′, t′′, b, u ∪ u′) ∧ b = b′))

Proof:

(a) tripel(r′, b, r) ∈ u. If pair(t, t′) ∈ b, then

tripel(s[x/t′], f(t′), s′[x′/t]) ∈ u, tripel(s′[x′/t], f(t′), s[x/t′]) ∈ u,

f(t) = f(t′) by 6.4 (a).

(b) u′′ := u ∪ u′. b = b′ by Cor(u′′). tripel(r, b, r′′) ∈ u′′. If pair(t, t′) ∈ b, follows

tripel(s[x/t], f(t), s′[x′/t]) ∈ u′′, tripel(s′[x′/t], f(t), s′′[x′′/t′]) ∈ u′′,

therefore tripel(s[x/t], f(t), s′′[x′′/t′]) ∈ u′′. Further

tripel(s′[x′/t], f(t), s[x/t]) ∈ u′′, tripel(s′[x′/t], f′(t), s′′[x′′/t]) ∈ u′′,

therefore f(t) = f(t′)

(c) - (f): Easy. Lemma 6.7 Assume r, s, t, r′, s′, t′ ∈ T erm, x, x′ ∈ V ar_{ML}, b, f, u sets.

(a) ∀k ∈ IN.equiv(u_{fin}^{basis}(k)).

(b) equiv(u_{nat}^{basis}).

(c) (Cor(u) ∧ ϕ(r, x, s, r′, x′, s′, b, f, u)) → (equiv(u_{π}^{basis}(b, f))equiv(u_{σ}^{basis}(b, f))∧ ∀v.equiv(v) → equiv(u_{w}^{basis}(b, f, v))).

(d) (Cor(u) ∧ ψ_{+}(r, s, r′, s′, b, c, u)) → equiv(u_{+}^{basis}(b, c))

(e) (Cor(u) ∧ ψ_{i}(r, s, t, r′, s′, t′, b, u)) → equiv(u_{i}^{basis}(b, s, t))

Proof:

(a), (b) are trivial.

(c): Assume pair(λx.t, λx′.t′) ∈ u_{π}^{basis}(b, f), pair(r̃, r̃′) ∈ b. Then pair(r̃′, r̃) ∈ b, pair(t[x/r̃′], t′[x/r̃]) ∈ f(r̃′), pair(t′[x/r̃], t[x/r̃′]) ∈ f(r̃′) = f(r̃) so we have symmetry of u_{π}^{basis}(b, f), similarly for transitivity and u_{σ}^{basis}, u_{w}^{basis}.

(d)- (e) are easy. Lemma 6.8 (a) (Cor(u) ∧ (∀tripel(a, b, a′) ∈ u.a, a′ ∈ T erm_{nf})) → Cor(Compl_{U}(u)).

(b) (equiv(u) ∧ u ⊂ T erm_{nf} × T erm_{nf}) → equiv(Compl(u)). Proof:

By lemma 5.3 (b). Lemma 6.9 (a) Cor(u) → Cor(Ũ(u)),

(b) u ⊂ u′ ∧ Cor(u′) → Ũ(u) ⊂ Ũ(u′),

(c) Cor(Û). Proof: (a): by 6.6, 6.7, 6.8.

(b) The only difficulty is, to show that

(ϕ(r, x, s, r′, x′, s′, b, f, u) ∧ ϕ(r, x, s, r′, x′, s′, b, f, u′)) → u_{w}^{α(u)}(b, f, ·)(∅) = u_{w}^{α(u′)}(b, f, ·)(∅).

This follows by 6.1.

(c) Follows by (a), (b).

We conclude, that the interpretation of each g-types gives a symmetric and transitive relation: Lemma 6.10 If A g-type, then KP i^{+} ⊢ ∀s_{1}, …, s_{n} ∈ T erm_{Cl}.equiv(A^{∗}[x⃗/s⃗]) Proof: Induction on the definition of types.

In the cases (Πx ∈ A.B)^{∗} we need the additional condition in the definition of the interpretation B^{∗}[x/s, x⃗/s⃗] = B^{∗}[x/s′, x⃗/s⃗], similarly for Σ, W.

The difficult cases are equiv(U^{∗}), which follows from Cor_{U}(u), and equiv(T (t))^{∗}[r⃗], which is trivial if ¬∃b.tripel(t^{∗}[r⃗], b, t^{∗}[r⃗]) ∈ Û, and if tripel(t^{∗}[r⃗], b, t^{∗}[r⃗]) ∈ Û for some b, follows T (t)^{∗}[r⃗] = b, equiv(b).

Now we define the interpretation of the judgements in the L_{KP}. Definition 6.11 Let A, B g-types, s, t g-terms, F V (A), F V (B), F V (s), F V (t) ⊂ {x_{1}, …, x_{n}}, r_{1}, …, r_{n}, s_{1}, , …, s_{n} be extended g-terms.

(a) (A type)^{∗}[x⃗/r⃗; s⃗] :⇔ (A type)^{∗}[x_{1}/r_{1}; s_{1}, …, x_{n}/r_{n}; s_{n}] :⇔ (A^{∗}[x⃗/r⃗] = A^{∗}[x⃗/s⃗]).

(b) (A = B)^{∗}[x⃗/r⃗; s⃗] :⇔ (A = B)^{∗}[x_{1}/r_{1}; s_{1}, …, x_{n}/r_{n}; s_{n}] :⇔ (A^{∗}[x⃗/r⃗] = B^{∗}[x⃗/s⃗]).

(c) (t : A)^{∗}[x⃗/r⃗; s⃗] :⇔ (t : A)^{∗}[x_{1}/r_{1}; s_{1}, …, x_{n}/r_{n}; s_{n}] :⇔ pair(t[x⃗/r⃗], t[x⃗/s⃗]) ∈ A^{∗}[x⃗/r⃗].

(d) (t = t′ : A)^{∗}[x⃗/r⃗; s⃗] :⇔ (t = t′ : A)^{∗}[x_{1}/r_{1}; s_{1}, …, x_{n}/r_{n}; s_{n}] :⇔ pair(t[x⃗/r⃗], t′[x⃗/s⃗]) ∈ A^{∗}[x⃗/r⃗].

We will not mention the variables x_{1}, …, x_{n} explicitly, if they are the variables, mentioned in the context, writing (A = B)^{∗}[r⃗, s⃗], (t : A)^{∗}[r⃗; s⃗], (t = t′ : A)^{∗}[r⃗; s⃗]. Lemma 6.12 (Substitution lemma).

Let C, D be g-types, r, s, t_{i}, t′_{i} g-terms, x_{i}, y_{i} ∈ V ar_{ML}. Then:

(a) If r[x⃗/t⃗] is an allowed substitution, F V (r[x⃗/t⃗]) ⊂ {y_{1}, …, y_{n}}, then

KP i^{+} ⊢ ∀r⃗ ∈ T erm_{Cl}.r[x⃗/t⃗][y⃗/r⃗] = r[x_{1}/t_{1}[y⃗/r⃗], …, x_{n}/t_{n}[y⃗/r⃗], y⃗/r⃗].

(Note that, if variables occur more than once in [y⃗/r⃗], only the first substitution is relevant.)

(b) If C[x⃗/t⃗] is an allowed substitution, F V (C[x⃗/t⃗]) ⊂ {y_{1}, …, y_{n}}, then

KP i^{+} ⊢ ∀r⃗, r′ ∈ T erm_{Cl}.C[x⃗/t⃗]^{∗}[y⃗/r⃗] = C^{∗}[x_{1}/t_{1}[y⃗/r⃗], …, x_{n}/t_{n}[y⃗/r⃗], y⃗/r⃗].

(c) If A[x⃗/t⃗], B[x⃗/t⃗′] are allowed substitutions, F V (A[x⃗/t⃗]), F V (B[x⃗/t⃗′]) ⊂ {y_{1}, …, y_{n}}, then

KP i^{+} ⊢ ∀r⃗, s⃗ ∈ T erm_{Cl}.(A = B)^{∗}[x⃗/(t⃗[x⃗/r⃗]); (t⃗′[x⃗/s⃗]), x⃗/r⃗; s⃗] ↔ (A[x⃗/t⃗] = B[x⃗/t⃗′])^{∗}[x⃗/r⃗; s⃗].

(d) If A[x⃗/t⃗], r[x⃗/t⃗] are allowed substitutions, F V (A[x⃗/t⃗]), F V (r[x⃗/t⃗]) ⊂ {y_{1}, …, y_{n}}, then

KP i^{+} ⊢ ∀r⃗, s⃗ ∈ T erm_{Cl}.(r : A)^{∗}[x⃗/(t⃗[x⃗/r⃗]); (t⃗[x⃗/s⃗]), x⃗/r⃗; s⃗] ↔ (r[x/t] : A[x/t])^{∗}[x⃗/r⃗; s⃗].

(e) If A[x⃗/t⃗], r[x⃗/t⃗], s[x⃗/t⃗′] are allowed substitutions, F V (A[x⃗/t⃗]), F V (r[x⃗/t⃗]), F V (s[x⃗/t⃗′]) ⊂ {y_{1}, …, y_{n}}, then

KP i^{+} ⊢ ∀r⃗, s⃗ ∈ T erm_{Cl}.(r = s : A)^{∗}[x⃗/(t⃗[x⃗/r⃗]); (t⃗′[x⃗/s⃗]), x⃗/r⃗; s⃗] ↔ (r[x⃗/t⃗] = s[x/t⃗′] : A[x/t⃗])^{∗}[x⃗/r⃗; s⃗].

Proof: (a): Induction on the definition of r g-term:

If r is a variable, the assertion is trivial, if r = C(r_{1}, …, r_{n}) it follows by IH. Let r = λy.s. W.l.o.g. x_{i} ≠ y, x_{i} ∈ F V (r). Then, since the substitution is al-

lowed, follows y ∉ F V (t_{i}). W.l.o.g. y_{i} ≠ y. Then r[x⃗/t⃗][y⃗/s⃗] = (λy.(s[x⃗/t⃗]))[y⃗/s⃗] = λy.(s[x⃗/t⃗][y⃗/s⃗]) = λy.(s[x⃗/(t⃗[y⃗/s⃗]), y⃗/s⃗]) = r[x⃗/(t⃗[y⃗/s⃗]), y⃗/s⃗].

(b) Induction on the definition of A g-type: C = N, N_{k}, U is trivial. C = Πx ∈ A.B. Let [x′/t⃗′] := [x⃗/t⃗] \ {x}, [y′/r′] := [y⃗/r⃗] \ {x}, Then

C[x⃗/t⃗]^{basis}[y⃗/r⃗]
{pair(λx.t, λx′.t′) ∈ T erm_{Cl} × T erm_{Cl}|∀pair(r, r′) ∈ A[x⃗/t⃗]^{∗}[y⃗/r⃗].
pair(t[x/r], t′[x′/r′]) ∈ B[x′/t⃗′]^{∗}[x/r, y⃗/r⃗]
∧B[x′/t⃗′]^{∗}[x/r, y⃗/r⃗]B[x′/t⃗′]^{∗}[x/r′, y⃗/r⃗]}

Now x′_{i} ∈ F V (B) → x ∉ F V (t_{i}). Therefore

C[x⃗/t⃗]^{basis}[y⃗/r⃗]

= {pair(λx.t, λx′.t′) ∈ T erm_{Cl} × T erm_{Cl}|∀pair(r, r′) ∈ A^{∗}[x⃗/t⃗[y⃗/r⃗], y⃗/r⃗]^{∗}.

pair(t[x/r], t′[x′/r′]) ∈ B^{∗}[x′/(t⃗′[x/r, y⃗/r⃗]), x/r, y⃗/r⃗]
∧B^{∗}[x′/(t⃗′[x/r, y⃗/r⃗]), x/r, y⃗/r⃗] = B^{∗}[x′/(t⃗′[x/r, y⃗/r⃗]), x/r′, y⃗/r⃗]}
= {pair(λx.t, λx′.t′) ∈ T erm_{Cl} × T erm_{Cl}|∀pair(r, r′) ∈ A^{∗}[x⃗/t⃗[y⃗/r⃗], y⃗/r⃗]^{∗}.
pair(t[x/r], t′[x′/r′]) ∈ B^{∗}[x/r, x⃗/(t⃗[y⃗/r⃗]), y⃗/r⃗] ∧
B^{∗}[x/r, x⃗/(t⃗[y⃗/r⃗]), y⃗/r⃗] = B^{∗}[x/r′, x⃗/(t⃗[y⃗/r⃗]), y⃗/r⃗]}
= C^{basis}[x⃗/(t⃗[y⃗/r⃗]), y⃗/r⃗].

The cases C = Σx ∈ A.B, W x ∈ A.B, A + B follow in a similar way.

Case C = I(A, r, s): (a) and IH for A. Case C = T (t): (a).

(c) - (e) are immediate consequences of (a), (b).

Lemma 6.13 For every g-type A F V (A) ⊂ {x_{1}, …, x_{n}}, follows

(a) ∀r⃗, r, r′, s, s′ ∈ T erm_{Cl}.(r→_{red}r′) → (s→_{red}s′) → pair(r, s) ∈ A^{∗}[x⃗/r⃗] → pair(r′, s′) ∈ A^{∗}[x⃗/r⃗].

(b) ∀r⃗, r, r′ ∈ T erm_{Cl}.pair(r, r′) ∈ A^{∗}[x⃗/r⃗] → ∃s, s′ ∈ T erm_{nf}.r→_{red}s ∧ r′→_{red}s′.

Proof: easy, since for each type, Compl was applied to some set. Definition 6.14 (a) Define

Stable(a) := ∀r, s, r′, s′ ∈ T erm_{Cl}.pair(r, s) ∈ a → r=_{α}r′ → s=_{α}s′ → pair(r′, s′) ∈ a

(b) For every g-type A with F V (A) = {x_{1}, …, x_{n}} we define

F lex(A) := ∀r_{1}, …, r_{n}, s_{1}, …, s_{n} ∈ T erm_{Cl}.(r_{1}=_{α}s_{1} ∧ ⋯ ∧ r_{n}=_{α}s_{n}) → A^{∗}[x⃗/r⃗] = A^{∗}[x⃗/s′]

Lemma 6.15 For every g-type C, D with F V (C) = {x_{1}, …, x_{n}} and C=_{α}D we have

(a) KP i^{+} ⊢ F lex(C)

(b) KP i^{+} ⊢ ∀r_{1}, …, r_{n}.Stable(C^{∗}[x⃗/r⃗])

(c) KP i^{+} ⊢ ∀r_{1}, …, r_{n}.C^{∗}[x⃗/r⃗] = D^{∗}[x⃗/r⃗].

Proof of (a) - (c) by induction on definition of g-types, for (c) by side induction on C=_{α}D. In (c) the case C = D is trivial, and if C=_{α}C′=_{α}D, the assertion follows by IH.

We write r⃗=_{α}s⃗ for (r_{1}=_{α}s_{1} ∧ ⋯ ∧ r_{n}=_{α}s_{n}).

First note, that Stable(a) → Stable(Compl(a)) by lemma 5.3 (d).

Case N, N_{k}: Easy.

Case C = Πx ∈ A.B: Since

∀r⃗, s⃗.r⃗=_{α}s⃗ → A[x⃗/r⃗] = A[x⃗/s⃗] ∧ ∀r ∈ T erm_{Cl}.B[x/r, x⃗/r⃗] = B[x/r, x⃗/s⃗]

follows F lex(Πx ∈ A.B), and if λx.t=_{α}λx′′.t′′, λx′.t′=_{α}λx′′′.t′′′,

pair(λx.t, λx′.t′) ∈ C^{basis}[x⃗/r⃗],

then for pair(r, r′) ∈ A^{∗}[x⃗/r⃗] follows by 5.2 (e) t[x/r]=_{α}t′′[x′′/r], t′[x′/r′]=_{α}t′′′[x′′′/r′], B^{∗}[x/r, x⃗/r⃗] = B^{∗}[x/r′, x⃗/r⃗], and by Stable(B^{∗}[x/r, x⃗/r⃗], follows

pair(t′′[x′′/r], t′′′[x′′′/r′]) ∈ B^{∗}[x/r, x⃗/r⃗]

and

pair(λx′′.t′′, λx′′′.t′′′) ∈ C^{basis}[x⃗/r⃗]

For (c), if D = Πx ∈ A′.B′ with A=_{α}A′ B=_{α}B′, the assertion follows by IH, if D = Πy ∈ A.B[x/y], y ∉ F V (B), B[x/y] allowed, follows assuming under the assumption r⃗=_{α}s⃗

∀r ∈ T erm_{Cl}.B[x/y]^{∗}[y/r, r⃗] = B^{∗}[x/r, y/r, r⃗] = B^{∗}[x/r, r⃗],

and therefore follows C^{∗}[r⃗] = D^{∗}[r⃗].

Case C = Σx ∈ A.B: Stability follows immediately by IH, and if r=_{α}r′′, s=_{α}s′′, r′=_{α}r′′′, s′=_{α}s′′′, pair(p(r, s), p(r′, s′)) ∈ C^{basis}[x⃗/r⃗], pair(r′′, r′′′) ∈ A^{∗}[x⃗/r⃗], follows by IH B[x/r, x⃗/r⃗] = B[x/r′′, x⃗/r⃗], and B[x/r′, x⃗/r⃗] = B[x/r′′′, x⃗/r⃗]. therefore by stability

pair(s′, s′′) ∈ B[x/r, x⃗/r⃗] = B[x/r′′, x⃗/r⃗]

(c) follows as for Πx ∈ A.B.

Case C = W x ∈ A.B: As in the Σ-case follows using the IH, F (r⃗, u) = F (s⃗, u). Further we see, that Stable(u) → Stable(F^{basis}(r⃗, u)).

(c) follows as for Πx ∈ A.B.

Case C = (A + B): as before, (c) is easy.

Case C = I(A, r, s): Stable(I^{basis}(A, r, s)) is trivial. F lex(I(A, r, s)): If r⃗=_{α}s⃗, then s[x⃗/r⃗]=_{α}s[x⃗/s⃗], t[x⃗/r⃗]=_{α}t[x⃗/s⃗], A^{∗}[x⃗/r⃗] = A^{∗}[x⃗/s⃗], and therefore pair(s[x⃗/r⃗], t[x⃗/r⃗]) ∈ A[x⃗/r⃗] ↔ pair(s[x⃗/s⃗], t[x⃗/s⃗]) ∈ A[x⃗/s⃗].

For (c), D = I(B, r′, s′), with r=_{α}r′, s=_{α}s′, A=_{α}B, A^{∗}[r⃗] = B^{∗}[r⃗], by 5.2 (d) r[r⃗]=_{α}r′[r⃗], s[r⃗]=_{α}s′[r⃗] and, using Stable(A^{∗}[r⃗]), follows pair(r[r⃗], s[r⃗]) ∈ A^{∗}[r⃗] ↔ pair(r′[r⃗], s′[r⃗]) ∈ A^{∗}[r⃗] ↔ pair(r′[r⃗], s′[r⃗]) ∈ B^{∗}[r⃗].

Cases C = U, T (t): We define

Stable_{U}(u) := ∀s, s′, t, t′ ∈ T erm_{Cl}.∀b ∈ T C(u).

s=_{α}s′ → t=_{α}t′ → pair(s, b, t) ∈ u → (pair(s′, b, t′) ∈ u ∧ Stable(b))

We have Stable_{U}(u) → Stable_{U}(Compl_{U}(u)). We have Stable(u_{nat}^{basis}), Stable(u_{fin}^{basis}(k)),

(∀a, a′, a′′, a′′′.(pair(a, a′) ∈ b ∧ a=_{α}a′′ ∧ a′=_{α}a′′′) → (f(a) = f(a′′) ∧ f(a′) = f(a′′′) ∧ Stable(f(a)))) → Stable(b) → (Stable(u_{π}^{basis}(b, f)) ∧ Stable(u_{σ}^{basis}(b, f)) ∧ (Stable(v) → Stable(u_{w}^{basis}(b, f, v)))),

(Stable(b) ∧ Stable(c)) → Stable(u_{+}^{basis}(b, c)),

Stable(u_{i}(b, r, s)).

Further Stable(b) → r=_{α}r′ → s=_{α}s′ → u_{i}^{basis}(b, r, s) = u_{i}^{basis}(b, r′, s′).

(Stable_{U}(u) ∧ r=_{α}r′ ∧ s=_{α}s′ ∧ λx.t=_{α}λx′.t′ ∧ λy.v=_{α}λy′.v′)) → ϕ(r, x, t, s, y, v, b, f, u) → ϕ(r′, x′, t′, s′, y′, v′, b, f, u).

(Stable_{U}(u) ∧ r=_{α}r′ ∧ s=_{α}s ∧ t=_{α}t′ ∧ v=_{α}v′) → ψ_{+}(r, s, t, v, b, c, u) → ψ(r′, s′, t′, v′, u).

(Stable_{U}(u) ∧ r=_{α}r′ ∧ s=_{α}s′ ∧ t=_{α}t′r̃=_{α}r̃′ ∧ s̃=_{α}s̃′ ∧ t̃=_{α}t̃′) → ψ_{i}(r, s, t, r̃, s̃, t̃, b, c, u) → ψ_{i}(r′, s′, t′, r̃′, s̃′, t̃′, b, c, u)

From all this together we conclude Stable_{U}(u) → Stable_{U}(Ũ(u)) and therefore Stable_{U}(Û). Now immediately follows Stable(U^{∗}), F lex(U) is trivial, Stable(T (t)^{∗}[x⃗/r⃗]), and since for r⃗=_{α}s⃗, t[x⃗/r⃗]=_{α}t[x⃗/s⃗], follows

pair(t[x⃗/r⃗], b, t[x⃗/r⃗]) ∈ Û ↔ pair(t[x⃗/s⃗], b, t[x⃗/s⃗]) ∈ Û,
(T (t))^{∗}[x⃗/r⃗]=_{α}(T (t))^{∗}[x⃗/s⃗]

Now, if T (t)=_{α}D, D = T (t′) with t=_{α}t′, T (t)^{∗}[r⃗] = T (x)^{∗}[x/t[r⃗]] = T (x)^{∗}[x/t′[r′]] = T (t′)^{∗}[r⃗].

To state our Main Lemma, we need to express, that, if we assume elements of the types of the context, the interpretation of the conclusion Θ of a statement of Martin-Löf is valid. Since we need, that this is independent of the choice of equal elements of A_{i}, we will introduce the following abbreviation: Definition 6.16 Let Γ ≡ x_{1} : A_{1}, …, x_{k} : A_{k} be a g-context.

∀Γ^{=}(r⃗; s⃗).ϕ :≡ ∀r_{1}, …, r_{k}, s_{1}, …, s_{k} ∈ T erm_{Cl}.(pair(r_{1}, s_{1}) ∈ A_{1}^{∗}[ ] ∧

pair(r_{2}, s_{2}) ∈ A_{2}^{∗}[x_{1}/r_{1}] ∧ ⋯ ∧ pair(r_{k}, s_{k}) ∈ A_{k}^{∗}[x_{1}/r_{1}, …, x_{k−1}/r_{k−1}]) → ϕ

“Assume Γ^{=}(r⃗; s⃗)” means: “Assume r_{1}, …, r_{k}, s_{1}, …, s_{k} ∈ T erm_{Cl} such that pair(r_{1}, s_{1}) ∈ A_{1}^{∗}[ ]∧

pair(r_{2}, s_{2}) ∈ A_{2}^{∗}[x_{1}/r_{1}] ∧ ⋯ ∧ pair(r_{k}, s_{k}) ∈ A_{k}^{∗}[x_{1}/r_{1}, …, x_{k−1}/r_{k−1}].

Remark 6.17 All the proofs, carried out before, can be carried out in KP i^{+}.

Now we can state our Main Lemma. We need to prove that

ML_{1}^{e}W_{T} ⊢ t : A ⇒ KP i^{+} ⊢ (t : A)^{∗}.

But to carry out the proof, we need an assertion for each judgement of ML_{1}^{e}W_{T}, and further, that it respects equality: Lemma 6.18 (Main lemma)

Let Γ, Δ be g-context-pieces, x, x_{i} ∈ V ar_{ML}, A_{i}, A, B g-types, t, t′ g-terms, θ a gjudgement. Assume Γ = x_{1} : A_{1}, …, x_{n} : A_{n}.

(a) If ML_{1}^{e}W_{T} ⊢ Γ ⇒ t : A, then

(i) KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(t : A)^{∗}[x⃗/r⃗; s⃗].

(ii) KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(A type)^{∗}[x⃗/r⃗; s⃗].

(b) If ML_{1}^{e}W_{T} ⊢ Γ ⇒ t = t′ : A, then

(i) KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(t = t′ : A)^{∗}[x⃗/r⃗; s⃗].

(ii) KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(A type)^{∗}[x⃗/r⃗; s⃗].

(c) If ML_{1}^{e}W_{T} ⊢ Γ ⇒ A = A′, then

KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(A = A′)^{∗}[x⃗/r⃗; s⃗].

(d) If ML_{1}^{e}W_{T} ⊢ Γ ⇒ A type, then

KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(A type)^{∗}[x⃗/r⃗; s⃗].

(e) If ML_{1}^{e}W_{T} ⊢ Γ, x : A, Δ ⇒ θ, then

KP i^{+} ⊢ ∀Γ^{=}(r⃗; s⃗).(A type)^{∗}[x⃗/r⃗; s⃗].

Proof of the Main Lemma: We proof simultaneously (a) - (e) by induction on the derivation. We write IH 3 for the Induction-hypothesis for the 3rd premise, etc. IH 3(d) for the Induction-hypothesis (d) for the 3rd premise of the rule etc. If there is more than one rule of one category (as in the case of (REF L) ), we refer to them by (REF L)_{1}, (REF L)_{2}, etc. Let Γ = x_{1} : A_{1}, …, x_{n} : A_{n}, Γ′ = y_{1} : B_{1}, …, y_{m} : B_{m}. If r⃗ = r_{1}, …, r_{n}, i ≤ n, then r̂_{i} := r_{1}, …, r_{i}. If θ = t : A or t = t′ : A or A type or A = B, let θ′ = A type (the judgement treated in the cases (i) of (a) - (c), or which follows from the assertion in (d), (e). Distinction by the last rule applied. Case (ASS): Assume Γ^{=}(r⃗; s⃗), pair(r, r′) ∈ A^{∗}[x⃗/r⃗]. Then x[x⃗/r⃗, x/r] = r, x[x⃗/s⃗, x/r′] = r′, (x : A)^{∗}[x⃗/r⃗; s⃗, x/r; r′]

ad (a,ii): Direct by IH.

ad (e): if ,,y : B” part of Γ, by IH (e). if y : B ≡ x : A by IH (a,ii).

Case (T HIN): Assume Γ^{=}(r⃗; s⃗), pair(r, r′) ∈ A^{∗}[n⃗], pair(r′_{i}, s′_{i}) ∈ B_{i}^{∗}[x⃗/r⃗, x/r, ŷ_{i}/r̂_{i}].

Then pair(r′_{i}, s′_{i}) ∈ B_{i}^{∗}[x⃗/r⃗, ŷ_{i}/r̂_{i}], by IH θ^{∗}[x⃗/r⃗; s⃗, y⃗/r′; s′], therefore θ^{∗}[x⃗/r⃗; s⃗, x/r; r′, y⃗/r′; s′]. and (θ′)^{∗}[x⃗/r⃗; s⃗, x/r; r′, y⃗/r′; s′]. (e): If ,,y : B” ≡ x : A, follows (e) from IH 1(d), otherwise as before by IH. Case (REF L)_{1} − (REF L)_{5}: Direct by IH. Case (SY M)_{1} Assume Γ^{=}(r⃗; s⃗). From pair(r_{i}, s_{i}) ∈ A_{i}^{∗}[r̂_{i}] follows pair(s_{i}, r_{i}) ∈ A_{i}^{∗}[r̂_{i}] and by IH (b,ii) pair(s_{i}, r_{i}) ∈ A_{i}^{∗}[ŝ_{i}]. By IH (b,i) follows pair(t[s⃗], t′[r⃗]) ∈ A^{∗}[s⃗], and by IH (b,ii) A^{∗}[r⃗] = A^{∗}[s⃗], and by 6.10 follows (t′ = t : A)^{∗}[r⃗; s⃗].

(b,ii) follows from IH (b,ii). Case (SY M)_{2} Assume Γ^{=}(r⃗; s⃗). As for (SY M)_{1} we have pair(s_{i}, r_{i}) ∈ A_{i}^{∗}[ŝ_{i}], by IH A^{∗}[s⃗] = B^{∗}[r⃗] and therefore the assertion. Case (T RANS)_{1} Assume Γ^{=}(r⃗; s⃗). From pair(r_{i}, s_{i}) ∈ A_{i}^{∗}[r̂_{i}] follows pair(r_{i}, r_{i}) ∈ A_{i}^{∗}[r̂_{i}], pair(t[r⃗], t′[r⃗]) ∈ A^{∗}[r⃗], pair(t′[r⃗], t′′[s⃗]) ∈ A^{∗}[r⃗] and therefore from equiv(A^{∗}[n⃗]) follows (t = t′′ : A)^{∗}[r⃗; s⃗]. (b,ii) follows by IH. Case (T RANS)_{2} Assume Γ^{=}(r⃗; s⃗). From pair(r_{i}, s_{i}) ∈ A_{i}^{∗}[r̂_{i}] follows pair(r_{i}, r_{i}) ∈ A_{i}^{∗}[r̂_{i}], A^{∗}[r⃗] = B^{∗}[r⃗], B^{∗}[r⃗] = C^{∗}[s⃗] and therefore the assertion. Case (SUB) Assume Γ^{=}(r⃗; r′), pair(s_{i}, s′_{i}) ∈ B_{i}[x/t]^{∗}[r⃗, ŝ_{i}]. Now by lemma 6.12

B_{i}[x/t]^{∗}[r⃗, ŝ_{i}] = B_{i}^{∗}[x/t[r⃗, ŝ_{i}], r⃗, ŝ_{i}] = B_{i}^{∗}[x/t[r⃗], r⃗, ŝ_{i}]

By IH 2 (a,i) pair(t[r⃗], t[s⃗]) ∈ A^{∗}[r⃗], therefore

θ^{∗}[x⃗/r⃗; r′, x/t[r⃗]; t′[r′], y⃗/s⃗; s′],

and by lemma 6.12 θ[x/t]^{∗}[x⃗/r⃗; r′, s⃗], similarly for θ′. Proof for (e): If ,,y : B” in Γ, follows assertion by IH. If ,,y : B” in Γ′[x/t], follows by IH B^{∗}[r⃗, x/t⃗[r⃗], ŝ_{i}] = B^{∗}[r′, x/t[r′], ŝ′_{i}], and by 6.12 the assertion. Case (REP L1) Assume Γ^{=}(r⃗; r′), pair(s_{i}, s′_{i}) ∈ B_{i}[x/t]^{∗}[r⃗, ŝ_{i}]. pair(t[r⃗], t′[r′]) ∈ A^{∗}[r⃗]. By 6.12 follows B_{i}^{∗}[x⃗/r⃗, x/t[r⃗], ŷ_{i}/ŝ_{i}] = B_{i}[x/t]^{∗}[r⃗, ŝ_{i}]. Therefore we have pair(s_{i}, s′_{i}) ∈ B_{i}^{∗}[x⃗/r⃗, x/t[r⃗], ŷ_{i}/ŝ_{i}] Then by IH 1 B^{∗}[x⃗/r⃗, x/t[r⃗], y⃗/s⃗] = B^{∗}[x⃗/r′, x/t′[r′], y⃗/s′], and by 6.12 follows the assertion. Proof for (e): From IH 2 follows as in (REF L) the assertion for Γ ⇒ t′ = t : A and further as in (T RANS) the assertion for Γ ⇒ t = t : A, which is the same as for Γ ⇒ t : A and now the proof follows as in (SUB). Case (REP L2) Assume Γ^{=}(r⃗, r′), pair(s_{i}, s′_{i}) ∈ B_{i}[x/t]^{∗}[r⃗, ŝ_{i}]. Then pair(t[r⃗], t′[r′]) ∈ A^{∗}[r⃗], and by IH 1(a,i)

(s = s : B)^{∗}[x⃗/r⃗; r′, x/t[r⃗]; t′[r′], y⃗/s⃗; s′]

and by the 6.12 follows the assertion for (b,i). (b,ii) follows as in (REP L1), using that we have the assertion for Γ ⇒ t = t : A, and (e) follows exactly as in (REP L1). Case (REP L3) Assume Γ^{=}(r⃗; s⃗). We have (t = t : A)[r⃗; s⃗], and, since we have pair(r_{i}, r_{i}) ∈ A_{i}^{∗}[r̂_{i}], A^{∗}[r⃗] = B^{∗}[r⃗], therefore the assertion for the first rule. The assertion for the second rule is similar. Using the proofs of (REF L) and (T RANS) follows the assertion for Γ ⇒ B = B and therefore the assertion for (b,ii). Case (ALP HA): Immediate by the IH since if A=_{α}A′, t=_{α}t′, A[s⃗] = A′[s⃗], t[r⃗]=_{α}t′[r⃗] and pair(t[r⃗], t[r⃗]) ∈ A^{∗}[r⃗] ↔ pair(t[r⃗], t′[r⃗]) ∈ A^{∗}[r⃗]. Case (N_{k}^{T}), (N^{T}) Nothing to prove.

Case (Π^{T,=}) Assume Γ^{=}(r⃗; s⃗). By IH (e) A^{∗}[r⃗] = A^{′∗}[s⃗], and, if pair(r, s) ∈ A^{∗}[r⃗], follows pair(r, r), pair(s, s) ∈ A^{∗}[r⃗], therefore by IH B^{∗}[r⃗, x/r] = B^{′∗}[r′, x/r], B^{∗}[s⃗, x/s] = B^{′∗}[r′, x/s], Πx ∈ A.B^{∗}[r⃗] = (Πx ∈ A′.B′)^{∗}[s⃗]. Cases (Σ^{T,=}), (W^{T,=}), (+^{T,=}): similarly. Case (I^{T,=}): Assume Γ^{=}(r⃗; s⃗). By IH (1) we have

pair(t[r⃗], t′[s⃗]) ∈ A^{∗}[r⃗], pair(s[r⃗], s′[s⃗]) ∈ A^{∗}[r⃗])

and by IH 1,(b,ii) A^{∗}[r⃗] = A^{′∗}[s⃗]. Therefore we have pair(t[r⃗], s[r⃗]) ∈ A^{∗}[r⃗] ⇔ (t′[s⃗], s′[s⃗]) ∈ A^{′∗}[s⃗]. Cases (N_{k}^{I}), (N^{I,=})_{1} Nothing to prove. Case (N^{I,=})_{2}: Assume Γ^{=}(r⃗; s⃗). By IH we have for some k ∈ IN t[r⃗]→_{red}S^{k}0 ∧ t′[s⃗]→_{red}S^{k}0, therefore St[r⃗]→_{red}S^{k+1}0, St′[s⃗]→_{red}S^{k+1}0, and we have the assertion. Case (Π^{I,=}): Assume Γ^{=}(r⃗; s⃗), pair(r, s) ∈ A^{∗}[r⃗]. Then by IH (b,i) pair(t[x/r, r⃗], t′[x/s, s⃗]) ∈ B^{∗}[x/r, r⃗] = B^{∗}[r⃗, x/r], pair((λx.t)[x⃗/r⃗], (λx.t′)[x⃗/s⃗]) ∈ (Πx ∈ A.B)^{∗}[r⃗]. (b,ii) follows as in (Π_{1}^{T,=}), since from IH (b,ii) follows (d) for x : A ⇒ B type. Case (Σ^{I,=}): Assume Γ^{=}(r⃗; s⃗). By IH pair(s[r⃗], s′[s⃗]) ∈ A^{∗}[r⃗], further pair(t[r⃗], t′[s⃗]) ∈ (B[x/s])^{∗}[r⃗] = B^{∗}[x/s[r⃗], r⃗] = B^{∗}[x/s′[s⃗], s⃗]. Therefore pair(p(s, t)[r⃗], p(s′, t′)[s⃗]) ∈ (Σx ∈ A.B)^{∗}[n⃗] . (b,ii) follows as in (Σ^{T,=}). Case (W^{I,=}): Let α_{I,n}, F be as in the definition of W x ∈ A.B^{∗}, α := α_{I,n}. Assume Γ^{=}(r⃗; s⃗). Then by IH pair(r[r⃗], r′[s⃗]) ∈ A^{∗}[r⃗], s[r⃗]→_{red}λx.t, s′[s⃗]→_{red}λx′.t′, B^{∗}[x/r[r⃗], r⃗] = B^{∗}[x/r′[s⃗], s⃗], and

∀pair(u, u′) ∈ B[x/t]^{∗}[x⃗/r⃗](= B^{∗}[x/t[r⃗], x⃗/r⃗])∃γ < α.pair(t[x/u], t′[x/u′]) ∈ F (r⃗, γ)

By (Δ_{0} − coll) and Ad(L_{α}) there exist a δ < α such that the γ can be chosen to be < δ. Then pair(sup(r, s)[r⃗], sup(r, s)[s⃗] ∈ F (r⃗, γ + 1) ⊂ W x ∈ A.B^{∗}[r⃗]. (b,ii) follows as in (W^{T,=}). Case (+^{I,=}): easy. Case (I^{I,=}): obvious. Case (N_{k}^{E,=}): Assume Γ^{=}(r⃗; s⃗). By I.H. 1 follows t[r⃗]→_{red}n_{k}, t′[s⃗]→_{red}n_{k}, for some n < k, s_{i}[r⃗]→_{red}s̃_{i} ∈ T erm_{nf}, s′_{i}[s⃗]→_{red}s̃′_{i} ∈ T erm_{nf} for some pair(s̃_{i}, s̃′_{i}) ∈ A[x/i_{k}]^{∗}[r⃗] = A^{∗}[x/i_{k}, r⃗]. Therefore by lemma 5.3 (c)

C_{k}(t, s_{0}, …, s_{k−1})[r⃗]→_{red}C_{k}(n_{k}, s̃_{0}, …, s̃_{k−1})→_{red}s̃_{n},
C_{k}(t′, s_{0}, …, s_{k−1})[s⃗]→_{red}s̃′.

t[r⃗]→_{red}n_{k}, therefore pair(t[r⃗], n_{k}) ∈ N_{k}. By pair(r_{i}, r_{i}) ∈ A_{i}[r̂_{i}] and IH for the last premise follows A[x/t]^{∗}[r⃗] = A^{∗}[x/t^{∗}[r⃗], r⃗] = A^{∗}[x/n_{k}, r⃗] and we are done. (b,ii) follows, since from IH 1 follows as in (SY M), (T RANS) the assertion for t = t : N_{k}, as in (SUB). (b,ii) follows as in (SUB). Case(N^{E,=}): Assume Γ^{=}(r⃗; s⃗). Then by IH 1 pair(t, t′)[r⃗] ∈ N, therefore t[r⃗]→_{red}S^{n}0, t′[r⃗]→_{red}S^{n}0 for some n < ω. Further by IH 2 and 6.13 (b) exist s̃_{0}, s̃′_{0} ∈ T erm_{nf} such that s_{0}[r⃗]→_{red}s̃_{0}, s_{1}[r⃗]→_{red}s̃′_{0}, pair(s̃_{0}, s̃′_{0}) ∈ A[x/0]^{∗}[r⃗] = A^{∗}[x/u, r⃗].

Let [x′/r′] := [x⃗/r⃗] \ {x, y}, [x′/s′] := [x⃗/s⃗] \ {x, y}.

P (t, s_{0}, (x, y)s_{1})[r⃗]→_{red}P (S^{n}0, s̃_{0}, λx.λy.(s_{1}[x′/r′])),
P (t′, s′_{0}, (x, y)s′_{1})[s⃗]→_{red}P (S^{n}0, s̃′_{0}, λx.λy.(s′_{1}[x′/r′])).

Let P_{0}(r) := P (r, s̃_{0}, λx.λy.(s_{1}[x′/r′])), P_{1}(r) := P (r, s̃′_{0}, λx.λy.(s′_{1}[x′/s′])). We show: ∀m ∈ IN.pair(P_{0}(S^{m}0), P_{1}(S^{m}0)) ∈ A^{∗}[z/S^{m}0, r⃗]. We will now conclude as in the case N_{k}^{E,=}, A[z/t]^{∗}[r⃗] = A^{∗}[z/t[r⃗], r⃗] = A^{∗}[z/S^{n}0, r⃗], and have now assertion (b,i). If m = 0, P_{0}(S^{m}0)→_{red}s̃_{0}, P_{1}(S^{m}0)→_{red}s̃′_{0}, pair(s̃_{0}, s̃′_{0}) ∈ A^{∗}[z/0, r⃗]. If m = k + 1, follows by IH P_{0}(S^{k}0)→_{red}s̃, P_{1}(S^{k}0)→_{red}s̃′, s̃, s̃′ ∈ T erm_{nf}, pair(s̃, s̃′) ∈ A^{∗}[z/S^{k}0, r⃗] = A[z/x]^{∗}[x/S^{k}0, r⃗].

P_{0}(S^{m}0) →_{red} (λx.λy.(s_{1}[r′]))S^{k}0P_{0}(S^{k}0)→_{red}(λy.s_{1}[x/S^{k}0, r′])s̃
→_{red} s_{1}[x′/r′, x/S^{k}0, y/s̃]
P_{1}(S^{m}0) →_{red} s′_{1}[x′/s′, x/S^{k}0, y/s̃′].

Now pair(S^{k}0, S^{k}0) ∈ N^{∗}, therefore by IH 3 follows

pair(s_{1}[x′/r′, x/S^{k}0, y/s̃], s′_{1}[x′/s′, x/S^{k}0, y/s̃′]) ∈ A[z/Sx]^{∗}[r⃗] = A^{∗}[z/S^{m}0, r⃗],

and the side induction is finished. (b,ii) follows as in the case N_{k}^{E,=}. Case(Π^{E,=}): Assume Γ^{=}(r⃗; s⃗). By IH 1,2 there exist t̃_{1}, t̃′_{1} ∈ T erm_{nf} such that t_{1}[r⃗]→_{red}t̃_{1}, t′_{1}[s⃗]→_{red}t̃′_{1} pair(t̃_{1}, t̃′) ∈ A^{∗}[r⃗], and there are r, r′ ∈ T erm and Variables x, x′ ∈ V ar_{ML} such that

t_{0}[r⃗]→_{red}λx.r, t′_{0}[r′]→_{red}λx′.r′, pair(λx.r, λx′.r′) ∈ (Πx ∈ A.B)^{basis}[r⃗].

Therefore

Ap(t_{0}, t_{1})[r⃗]→_{red}Ap(λx.r, t̃_{1})→_{red}r[x/t̃_{1}, r⃗],
Ap(t′_{0}, t′_{1})[s⃗]→_{red}r′[x′/t̃_{1}, s⃗]
pair(r[x/t̃_{1}, r⃗], r′[x′/t̃′_{1}, s⃗]) ∈ B^{∗}[x/t̃_{1}, r⃗].

As before we conclude

pair(t_{1}[r⃗], t_{1}[r⃗]) ∈ A^{∗}[r⃗]
pair(t̃_{1}, t_{1}[r⃗]) ∈ A^{∗}[r⃗]
B^{∗}[x/t̃_{1}, r⃗] = B^{∗}[x/t_{1}[r⃗], r⃗] = B[x/t_{1}]^{∗}[r⃗],

and we have IH (b,i). (b,ii) follows as in the case (N_{k}^{E,=}). Case(Σ^{E,=}): Assume Γ^{=}(r⃗; s⃗). By IH 1 exist s, s′, t, t′ ∈ T erm_{nf} such that

r[r⃗]→_{red}p(s, t), r′[s⃗]→_{red}p(s′, t′), pair(s, s′) ∈ A^{∗}[r⃗], pair(t, t′) ∈ B^{∗}[x/s, r⃗].

Then p_{0}(r[r⃗])→_{red}s, p_{0}(r′[s⃗])→_{red}s′, and we are done for the first rule, and p_{1}(r[r⃗])→_{red}t, p_{1}(r′[s⃗])→_{red}t′, and since from pair(s, s′) ∈ A[r⃗], follows

pair(s, s′) ∈ A[r⃗], pair(p_{0}(r)[r⃗], s) ∈ A[r⃗],

therefore by IH 2

B^{∗}[x/s, r⃗] = B^{∗}[x/p_{0}(r)[r⃗], r⃗] = B[x/p_{0}(r)]^{∗}[r⃗]

follows (b,i) for the second rule. (b,ii) is in (Σ^{E,=})_{1} trivial, in (Σ^{E,=})_{2} we use the proof of (Σ^{E,=})_{1} and argue as before. Case(W^{E,=}): Assume Γ^{=}(r⃗; s⃗), α := α_{I,n} as in the definition of (W x ∈ A.B)^{∗}. By IH t_{0}[r⃗]→_{red}t̃_{0}, t′_{0}[s⃗]→_{red}t̃′_{0}, pair(t̃_{0}, t̃′_{0}) ∈ F^{δ}(r⃗, ·)(∅). for some δ < α. Let

[x′/r′] := [x⃗/r⃗] \ {x, y, z},
[x′/s′] := [x⃗/s⃗] \ {x, y, z},
R_{0}(r) := R(r, (x, y, z)t_{2})[r⃗](= R(r, λx.λy.λz.(t_{2}[r′])))
R_{1}(r) := R(r, (x, y, z)t′_{2})[s⃗].

We show by induction on γ,

∀γ < α.∀pair(s̃, s̃′) ∈ F^{γ}(r⃗, ·)(∅).pair(R_{0}(s̃), R_{1}(s̃′)) ∈ C^{∗}[u/s̃, r⃗] (+)

Since C[u/t_{0}]^{∗}[r⃗] = C^{∗}[u/t_{0}[r⃗], r⃗] = C^{∗}[u/t̃_{0}, r⃗] = C^{∗}[u/t̃′_{0}, s⃗] (using arguments as before), follows the assertion. The case γ = 0 is trivial, and if γ ∈ Lim follows the assertion by IH Let now γ = γ′ + 1, u′ := F^{γ′}(r⃗, ·)(∅), pair(s̃, s̃′) ∈ F (r⃗, u′). If s̃→_{red}s, s̃′→_{red}s′, pair(s, s′) ∈ F^{basis}(s⃗, ·)(∅), pair(R_{0}(s), R_{1}(s′)) ∈ C^{∗}[u/s, r⃗], follows pair(R_{0}(s̃), R_{1}(s̃′)) ∈ C^{∗}[u/s, r⃗], further, like similar arguments before, C^{∗}[u/s, r⃗] = C^{∗}[u/s̃, r⃗] = C^{∗}[u/s̃′]. We therefore assume pair(s̃, s̃′) ∈ F^{basis}(r⃗, u′). Let pair(s̃, s̃′) = pair(sup(r, λx.s), sup(r′, λx′.s′)), pair(r, r′) ∈ A^{∗}[r⃗]. Let pair(r′′, r′′′) ∈ B^{∗}[x/r, r⃗]. Then r′′→_{red}r̃, r′′′→_{red}r̃′ for pair(r̃, r̃′) ∈ B^{∗}[x/r, r⃗], r̃, r̃′ ∈ T erm_{nf}, and we have pair(s[x/r′′], s′[x′/r′′′]) ∈ u′ and

pair(s[x/r̃], s′[x′/r̃′]) ∈ u′ (∗)

Since u′ ⊂ (W x ∈ A.B)^{∗}[r⃗] follows from the first of these assertions

pair(λx.s, λx′.s′) ∈ (B → W x ∈ A.B)^{∗}[r⃗]

Further, for pair(r̃, r̃′) ∈ B^{∗}[x/r, r⃗],

(R_{0}((λx.s)v))[v/r′′]→_{red}R_{0}(s[x/r̃])(v ∉ F V (λx.s))
(R_{1}((λx.s′)v′))[v′/r′′′]→_{red}R_{1}(s′[x/r̃′])(v′ ∉ F V (λx.s))

and by side IH, follows

pair((R_{0}((λx.s)v))[v/r′′], (R_{1}((λx′.s′)v))[v′/r′′′]) ∈ C^{∗}[u/(s[x/r̃], r⃗])
= C^{∗}[u/(s[x/r′′]), r⃗])

Now we have pair(r_{i}, r_{i}) ∈ A_{i}[r̂_{i}], Ap(λx.s, r′)→_{red}s[x/r̃], and by (∗), u′ ⊂ (W x ∈ A.B)^{∗}[r⃗], equiv((W x ∈ A.B)^{∗}[r⃗]) and 6.13 follows

pair(s[x/r̃], Ap(λx.s, r̃)) ∈ (W x ∈ A.B)^{∗}[r⃗]

therefore

C[u/Ap(y, v)]^{∗}[v/r′′, y/λx.s, x⃗/r⃗] = C^{∗}[u/Ap(λx.s, r′′), r⃗] = C^{∗}[u/(s[x/r̃]), r⃗]

further

C[u/Ap(y, v)]^{∗}[v/r′′, y/λx.s, x⃗/r⃗] = C[u/Ap(y, v)]^{∗}[v/r′′′, y/λx.s, x⃗/r⃗],

and we have

pair(λv.R_{0}((λx.s)v), λv′.R_{1}((λx′.s′)v′)) ∈ (Πv ∈ B.C[u/Ap(y, v)])^{∗}[y/λx.s, x⃗/r⃗]

Now by IH 2 follows

pair(t_{2}[x/r, y/λx.s, z/λv.R_{0}((λx.s)v), r⃗], t′_{2}[x/r′, y/λx′.s′, z/λv′.R_{1}((λx.s′)v′), s⃗])
∈ C[u/sup(x, y)]^{∗}[x/r, y/λx.s, r⃗]

Since

C[u/sup(x, y)]^{∗}[x/r, y/λx.s, r⃗] = C^{∗}[u/sup(r, λx.s), r⃗] = C^{∗}[u/s, s],

and

R_{0}(s)→_{red}(λx.λy.λz.t_{2}[x′/r′])r(λx.s)(λv.R_{0}((λx.s)v))
→_{red}t_{2}[x/r, y/λx.s, z/(λv.R_{0}((λx.s)v))]
R_{1}(s′)→_{red}t′_{2}[x/r′, y/λx.s′, z/(λv′.R_{1}((λx′.s′)v′))]

follows (+), and we are done. (b,ii) follows as in the case (N_{k}^{E,=}). Case (+^{E,=}): Assume Γ^{=}(r⃗; s⃗). By IH t_{0}[r⃗]→_{red}i(r) ∈ T erm_{nf}, t′_{0}[s⃗]→_{red}i(r′) ∈ T erm_{nf} and pair(r, r′) ∈ A^{∗}[r⃗] or t_{0}[r⃗]→_{red}j(r) ∈ T erm_{nf}, t′_{0}[s⃗]→_{red}j(r′) ∈ T erm_{nf} and pair(r, r′) ∈ B^{∗}[r⃗]. Let [x′/r′] := [x⃗/r⃗] \ {x}. In the first case we have

D(t_{0}, (x)t_{1}, (y)t_{2})[r⃗]→_{red}(λx.(t_{1}[r′]))r→_{red}t_{1}[x/r, x⃗/r⃗],
D(t′_{0}, (x)t′_{1}, (y)t′_{2})[r⃗]→_{red}t′_{1}[x/r′, x⃗/s⃗],
pair(t_{1}[x/r, r⃗], t′_{1}[x/r′, s⃗]) ∈ C[z/i(x)]^{∗}[x/r, r⃗] = C^{∗}[z/i(r), r⃗]

and using arguments as before

C^{∗}[z/i(r), r⃗] = C^{∗}[z/t_{0}[r⃗], r⃗] = C[z/t_{0}]^{∗}[r⃗]

and we are done. The second assertion follows in the same way. (b,ii) follows as before. Case (I^{E}): Assume Γ^{=}(r⃗; s⃗). By IH 1 follows (I(A, t_{1}, t_{2}))^{∗}[r⃗] ≠ ∅, pair(t_{1}[r⃗], t_{2}[r⃗]) ∈ A^{∗}[r⃗]. Further by IH 3 pair(t_{2}[r⃗], t_{2}[s⃗]) ∈ A^{∗}[r⃗], and by equiv(A^{∗}[r⃗]) follows (b,i). (b,ii) is trivial. Case (Π^{=}), (Σ_{0}^{=}), (Σ_{1}^{=}): By using the proof for the elimination rules we see, that if the conclusion is r = s : C, we conclude assuming Γ^{=}(r⃗; s⃗), that (r = r : C)[r⃗; s⃗], further (r[s⃗]→_{red}t ∈ T erm_{nf}) → (s[s⃗]→_{red}t), therefore follows (r = s : C)[r⃗; s⃗]. Case (Π^{η}):Assume Γ^{=} By IH we have

pair(t[r⃗], t[s⃗]) ∈ (Πx ∈ A.B)^{∗}[r⃗],

therefore t[r⃗]→_{red}λx.s, t[s⃗]→_{red}λx′.s′,

pair(λx.s, λx′.s′) ∈ (Πx ∈ A.B)^{basis}[r⃗],

Assume pair(r, r′) ∈ A^{∗}[r⃗]. Then r→_{red}r̃, r′→_{red}r̃′, pair(r̃, r̃′) ∈ A^{∗}[r⃗], r̃, r̃′ ∈ T erm_{nf}.

Ap(t, x))[r⃗][x/r] = Ap(t[r⃗], r)→_{red}Ap(λx.s, r̃)→_{red}s[x/r̃],

and since

pair(s[x/r̃], s′[x′/r̃′]) ∈ B^{∗}[x/r̃, r⃗] = B^{∗}[x/r, r⃗],

follows

pair(Ap(t, x)[r⃗][x/r], s′[x′/r̃′]) ∈ B^{∗}[x/r, r⃗],

therefore

pair(Ap(t, x)[r⃗], λx′.s′) ∈ (Πx ∈ A.B)^{∗}[r⃗],
pair(Ap(t, x)[r⃗], t[r⃗]) ∈ (Πx ∈ A.B)^{∗}[r⃗].

Case (Σ_{3}^{=}): Assume Γ^{=}(r⃗; s⃗). By IH t[r⃗]→_{red}r, t[s⃗]→_{red}r′ for some pair(r, r′) ∈ Σx ∈ A.B^{∗}[r⃗] ∩ (T erm_{nf} × T erm_{nf}), p(p_{0}(t), p_{1}(t))[r⃗]→_{red}r and we are done. Case (I^{=}): Assume Γ^{=}(r⃗; s⃗). By IH we conclude pair(t_{0}[r⃗], t_{0}[s⃗]) ∈ I(A, t_{1}, t_{2})^{∗}[r⃗], therefore t_{0}[s⃗]→_{red}\underline{r}, pair(\underline{r}, \underline{r}) ∈ I(A, t_{1}, t_{2})^{∗}[r⃗], pair(t_{0}[r⃗], \underline{r}) ∈ I(A, t_{1}, t_{2})^{∗}[r⃗]. (b,ii) is trivial. Case other equality rules: Let r = s : A be the conclusion of the rules. By using several times the rules general rules, elimination rules and in case W^{=} the introduction rules we can conclude r = r : A, and s = s : A. (For (W^{=}) we argue that Γ, v : B[x/t_{0}] ⇒ Ap(t_{1}, v) : W x ∈ A.B, by (W^{E,=}) Γ, v : B[x/t_{0}] ⇒ R(Ap(s′, v), (x, y, z)t′) : C[u/Ap(s′, v)], by Π^{I,=} Γ ⇒ λv.R(Ap(s′, v), (x, y, z)t′) : Πv ∈ B.C[u/Ap(s′, v)], by (ALP HA) for the z_{i}^{ML}, that we need, and it follows Γ ⇒ λz_{i}^{ML}.R(Ap(s′, z_{i}^{ML}), (x, y, z)t′) : Πv ∈ B.C[u/Ap(s′, v)], and now by (SUB) follows the assertion). Now, assuming Γ^{=}(r⃗, s⃗), and using the proofs above we can conclude pair(r[r⃗], r[s⃗]) ∈ A^{∗}[r⃗] and A^{∗}[r⃗] = A^{∗}[s⃗], so (b,i). In all the cases, we have, if the right side is written as t[x_{1}/r_{1}, …, x_{n}/t_{n}], if x_{i} corresponds to the type B_{i} (read off from the rule) follows easily by IH and using the proofs of several rules handled before the assertion for Γ ⇒ r_{i} : B_{i}, therefore r_{i}[r⃗]→_{red}r̃_{i} ∈ T erm_{nf} for some r̃_{i}, pair(r̃_{i}, r_{i}[r⃗]) ∈ B_{i}[r⃗], further r[r⃗]→_{red}t[x_{1}/r̃_{1}, ⋯ x_{n}r̃_{n}, r⃗]. We conclude

pair(t[x_{1}/r̃_{1}, …, x_{n}/r̃_{n}, r⃗], t[x_{1}/r_{1}[r⃗], …, x_{n}/r_{n}[r⃗], r⃗]) ∈ A^{∗}[r⃗].

Now using equiv(A^{∗}[r⃗]) and lemma 6.13 we conclude

pair(r[r⃗], s[r⃗]) ∈ A^{∗}[r⃗], pair(s[r⃗], s[s⃗]) ∈ A^{∗}[r⃗],

and have (b,i). Case (U^{I}): trivial. (T^{I,=}) we have by IH, assuming Γ^{=}(r⃗; s⃗),

pair(a[r⃗], a′[s⃗]) ∈ U^{∗}

therefore,

tripel(a[r⃗], b, a′[s⃗]) ∈ Û

for some b, by Cor_{U}(Û),

tripel(a[r⃗], b, a[r⃗]) ∈ Û, tripel(a[s⃗], b, a[s⃗]) ∈ Û,

and

T (a)^{∗}[r⃗] = b = T (a′)^{∗}[s⃗]

Case (\underline{n}_{k}^{I}), (\underline{n}^{I}): trivial. Case (π^{I,=}): Assume Γ^{=}(r⃗; s⃗). By IH a[r⃗]→_{red}ã, a′[s⃗]→_{red}ã′,

∃γ < α_{I}∃b′ ∈ T C(Ũ^{γ}(·)(∅))(tripel(ã, b′, ã), tripel(ã′, b′, ã) ∈ Ũ^{γ}(·)(∅)),

and

∀pair(t, t′) ∈ b′ → ∃δ < α_{I}.∃c ∈ T C(Ũ^{δ}(·)(∅)).
(tripel(b[x/t, r⃗], c, b′[x/t′, s⃗]) ∈ Ũ^{δ}(·)(∅)).

Since Ad(L_{α_{I}}) (here is the central point where we need (Δ_{0}-coll) and and admissible a which is closed under the step to the next admissible), and T C(Ũ^{β}(·)(∅)) ∈ L_{α_{I}} (β < α_{I}), there is a ρ < α_{I}, such that γ < ρ and δ can be chosen < ρ. There are now b, f such that ([x′/r′] := [x⃗/r⃗] \ {x}, [x′/s′] := [x⃗/s⃗] \ {x}) ϕ(ã, x, b[r′], ã′, x, b[s′], b, f, Util^{ρ}(·)(∅)), (note that the c we used above is correct by Cor(Û)) and by 6.5 follows

pair(πx ∈ a.b, u_{π}(b, f), πx ∈ a′.b′) ∈ Ũ^{ρ+1}(·)(∅).

The cases (σ^{I,=}), (w^{I}, =), ( ˜+^{I,=}), (i^{I,=}) follow in a similar way.

Cases (\underline{n}_{k}^{=}), (\underline{n}^{=}) are immediate. Case (π^{=}): Assume Γ^{=}(r⃗; s⃗) and chose b′, f, ρ as in (π^{I,=}). Then T (a)^{∗}[s⃗] = b′, and if pair(t, t′) ∈ b′, T (b)^{∗}[x/t, r⃗] = f(t) = f(t′) = T (b)^{∗}[x/t′, s⃗]. Since we have Cor(Û) (by lemma 6.9) follows

T ((πx ∈ a.b))^{∗}[r⃗] = u_{π}(b′, f) = (Πx ∈ T (a).T (b))^{∗}[r⃗].

Cases (σ^{=}), ( ˜+^{=}) are treated in a similar way. In the case of (w^{=}) we conclude as before, that u_{w}^{γ}(b, f, ·)(∅) = F^{γ}(s⃗, ·)(∅), and, since α(Ũ^{ρ}(·)(∅)) < α_{I}, (ρ chosen as in (π^{I,=})) follows by 6.1

T (wx ∈ a.b)^{∗}[r⃗] = u_{w}^{α(_{U^{ρ}}(·)(∅))}(b, f, ·)(∅) = F^{α_{I}}(s⃗, ·)(∅)c = (W x ∈ T (a).T (b))^{∗}[s⃗].

Lemma 6.19 The Main Lemma 6.18 is valid, if we replace ML_{1}^{e}W_{T} by ML_{1}^{e}W_{T,U}. Proof: As before, we only hae to check the rules, defined in 3.4 (a): Case (σ^{E}): Assume Γ^{=}(r⃗; s⃗). By IH exists c, α ≺ α_{I} such that

tripel((σx ∈ a.b)[r⃗], c, (σx ∈ a.b)[s⃗]) ∈ Ũ^{α}(∅).

Let α be chosen minimal. Then, α = α′ +1, and with u := Ũ^{α′} there exist r, r′ ∈ T erm_{nf}, c, c′, f, f′ such that a[r⃗]→_{red}r, a[s⃗]→_{red}r′, tripel(r, c, r′) ∈ u and (with [x′/r′] := [x⃗/r⃗]\{x}, [x′/s′] := [x⃗/s⃗] \ {x}) ∀pair(t, t′) ∈ c.tripel(s[r′][x/t], f(t), s[s′][x/t′]) ∈ u. Therefore T (a[r⃗])^{∗} = c = T (a[s⃗])^{∗}, and for pair(t, t′) ∈ T (a[r⃗])^{∗}, pair(s[x/t, r⃗], s[x/t′, s′]) ∈ U^{∗}[ ]. (π^{E}), (w^{E}), ( ˜+^{E}) are checked in the same way. For (ĩ^{E}) we observe, that a[r⃗]→_{red}ã, a[s⃗]→_{red}ã′, s[r⃗]→_{red}s̃, s[s⃗]→_{red}s̃′, t[r⃗]→_{red}t̃, t[s⃗]→_{red}t̃′, and tripel(ã, c, ã′) ∈ u, for some u as before, T (a)^{∗}[r⃗] = c, pair(s̃, s̃′) ∈ c, pair(t̃, t̃′) ∈ c, and since c is closed under →_{red} follows the assertion.

Arithmetical formulas in ML_{1}^{e}W_{T} and

KP i+

In this chapter we want to evaluate the results we have found out to get the proof theoretical strength of Martin-Löf’s type theory. We will interpret the language of Peano Arithmetic (L_{P A}, introduced in 7.1) in L_{ML} and L_{KP} (definition7.2) and prove that it permutes with the interpretation of Martin-Löf’s type theory in KP i^{+} (lemmata 7.4 and 7.6). Next we observe, that we could interpret every proof in some theory KP i_{n}^{+}, and have a stronger bound (lemma 7.7). At the end we analyze the proof theoretical strength of KP i_{n}^{+} and have the desired upper bound (theorem 7.8).

Definition 7.1 Definition of the language of Peano Arithmetic L_{P A}: Variables should be a set V ar_{P A} := {v_{i}^{P A}|i ∈ IN}, v_{i}^{P A} ≠ v_{j}^{P A} for i ≠ j. Further we have symbols for each primitive recursive function, =, ∧, ∨, →, ∀, ∃, ⊥, and ., , , (,). Terms are Variables and f(t_{1}, …, t_{n}) if t_{i} are terms and f is a symbol for a n-ary primitive recursive function. Prime formulas are ⊥ and equations r = s for r, s terms. Formulas are prime formulas and A → B, A ∧ B, A ∨ B, ∀x.A, ∃x.A, if A, B formulas, x ∈ V ar_{P A}. Definition 7.2 (a) For each primitive recursive g : IN^{k} → IN we define a closed g-term

int_{P A,ML}(g), (we abbreviate this as ĝ := int_{P A,ML}(g)) such that

ML_{1}^{e}W_{T} ⊢ ĝ : \underbrace{N → ⋯ N}_{k times} → N,

and we define a set int_{P A,KP} (g) short g̃ in L_{KP}
such that KP i^{+} ⊢ fun(g̃) ∧ dom(g̃) = IN^{k} ∧ ∀x ∈ IN^{k}.g̃(x) ∈ IN.
Case g = S: ĝ := λx.Sx, g̃ := {pair(x, x + 1)|x ∈ IN}.
Case g = P roj_{i}^{n}:
ĝ := λx_{1}, …, x_{n}.x_{i}, g̃ := {pair(tupel(x_{1}, …, x_{n}), x_{i})|x_{1}, …, x_{n} ∈ IN}.
Case g = Cons_{c}^{n}:
ĝ := λx_{1}, …, x_{n}.S^{c}0, g̃ := {pair(tupel(x_{1}, …, x_{n}), c)|x_{1}, …, x_{n} ∈ IN}.
Case g(x_{1}, …, x_{n}) = h(g_{1}(x_{1}, …, x_{n}), …, g_{m}(x_{1}, …, x_{n})):
ĝ := λx_{1}, …, x_{n}.ĥ(ĝ_{1}x_{1} ⋯ x_{n}) ⋯ (ĝ_{m}x_{1} ⋯ x_{n}),
g̃ := {pair(tupel(x_{1}, …, x_{n}), h(g_{1}(x_{1}, …, x_{n}), …, g_{m}(x_{1}, …, x_{n})))
|x_{1}, …, x_{n} ∈ IN}.
Case g(x_{1}, …, x_{n}, 0) = h(x_{1}, …, x_{n}),
g(x_{1}, …, x_{n}, y + 1) = k(x_{1}, …, x_{n}, y, g(x_{1}, …, x_{n}, y)):

ĝ := λx_{1}, …, x_{n}, y.P (y, ĥx_{1} ⋯ x_{n}, (u, v)k̂x_{1} ⋯ x_{n}uv),

define a(x_{1}, …, x_{n}, 0) := h̃(x_{1}, …, x_{n}),

a(x_{1}, …, x_{n}, Sy) := k̃(x_{1}, …, x_{n}, y, a(x_{1} ⋯ x_{n}, y)),

then

g̃ := {pair(tupel(x_{1}, …, x_{n}, y), a(x_{1}, …, x_{n}, y))|x_{1}, …, x_{n}, y ∈ IN}.

(b) For each term t of PA we define a g-term int_{P A,ML}(t), short t̂and a term of L_{KP} int_{P A,KP} (t), short t̃, such that, if F V (t) = {v_{i_{1}}^{P A}, …, v_{i_{n}}^{P A}} (i_{1} < ⋯ i_{n}) (v_{i}^{P A} as in definition 7.1 of V ar_{P A}) then F V (t̂) ⊂ {z_{i_{1}}^{ML}, …, z_{i_{n}}^{ML}}, F V (t̃) ⊂ {u_{i_{1}}^{KP} , …, u_{i_{n}}^{KP} } and ML_{1}^{e}W_{T} ⊢ z_{i_{1}}^{ML} : N, …, z_{i_{n}}^{ML} : N ⇒ t̂: N, and KP i^{+} ⊢ ∀u_{i_{1}}^{KP} , …, u_{i_{n}}^{KP} ∈ IN.(t̃ ∈ IN). Case t = v_{i}^{P A}: t̂:= z_{i}^{ML}, t̃:= u_{i}^{KP}. Case t = 0: t̂:= 0, t̃:= 0. Case t = gt_{1} ⋯ t_{n}: t̂:= ĝt̂_{1} ⋯ t̂_{n}, t̃:= g̃(t̃_{1}, …, t̃_{n}).

(c) For each formula A of PA we define a g-type int_{P A,ML}(A), short Â, and a formula of L_{KP} int_{P A,KP} (A), short Ã, such that in KP i^{+} Ã is equivalent to a Δ_{0}-formula, and if F V (A) = {v_{i_{1}}^{P A}, …, v_{i_{n}}^{P A}} (i_{1} < ⋯ i_{n}) then F V (Â) ⊂ {z_{i_{1}}^{ML}, …, z_{i_{n}}^{ML}}, F V (Ã) ⊂ {u_{i_{1}}^{KP} , …, u_{i_{n}}^{KP} } and ML_{1}^{e}W_{T} ⊢ z_{i_{1}}^{ML} : N, …, z_{i_{n}}^{ML} : N ⇒ Â type. Case A = (s = t): Â := I(N, ŝ, t̂), Ã := (s̃ = t̃). Case A = (B ∧ C): Â := (B̂ × Ĉ), Ã := B̃ ∧ C̃. Case A = (B ∨ C): Â := (B̂ + Ĉ), Ã := (B̃ ∨ C̃). Case A = (B → C): Â := (B̂ → Ĉ), Ã := (B̃ → C̃). Case A = ∀v_{i}^{P A}.B: Â := Πz_{i}^{ML} ∈ N.B̂, Ã := ∀u_{i}^{KP} ∈ IN.B̃. Case A = ∃v_{i}^{P A}.B: Â := Σz_{i}^{ML} ∈ N.B̂, Ã := ∃u_{i}^{KP} ∈ IN.B̃. Case A =⊥: Â := N_{0}, Ã := (0 ≠ 0).

Definition 7.3 (a) We define emb : IN → IN, embnat(n) := S^{n}0(=: n̂) (or more precisely ⌈S^{n}0⌉), a function definable in KP i^{+}.

(b) makepair(a) := pair(a, a). Lemma 7.4 (a) If g : IN^{k} → IN is primitive recursive, then

KP i^{+} ⊢ ∀t_{1}, …, t_{k} ∈ T erm_{Cl}.∀n_{1}, …, n_{k}.
(r^{1}→_{red}n̂_{1} ∧ ⋯ ∧ r^{k}→_{red}n̂_{k}) → ĝr_{1}, …, r_{k}→_{red}emb(g̃(n_{1}, …, n_{k})).

(b) If t is a term of PA, F V (t) ⊂ {v_{1}^{P A}, …, v_{n}^{P A}}, then

KP i^{+} ⊢ ∀r_{1}, …, r_{n} ∈ T erm_{Cl}.∀n_{1}, …, n_{k}.(r^{1}→_{red}n̂_{1} ∧ ⋯ ∧ r^{k}→_{red}n̂_{k}) →
t̂[z_{1}^{ML}/r_{1}, …, z_{n}^{ML}/r_{n}]→_{red}emb(t̃[u_{1}^{KP} /n_{1}, …, u_{n}^{KP} /n_{n}]).

Proof: (a) Case g = S: ĝt_{1}→_{red}(λx.Sx)n̂_{1}→_{red}Sn̂_{1} = emb(Sn_{1}). Case g = P roj_{i}n: ĝt_{1}, …, t_{j}→_{red}λx_{i+1}, …, x_{n}.x_{i} for j < i, λx_{i+1}, …, x_{n}.n̂_{i} for i ≤ j. Case g = Cons_{c}^{n}: trivial. Case g(x⃗) = h(g_{1}(x⃗), …, g_{m}(x⃗)), l_{i} := g̃_{i}(n_{1}, …, n_{n}):

ĝ_{i}(n̂_{1}, …, n̂_{n})→_{red}emb(g̃_{i}(n_{1}, …, n_{n})) = emb(l_{i}),

therefore

gr_{1}, …, r_{n}
→_{red} hg_{1}(n̂_{1}, …, n̂_{n}), …, g_{m}(n̂_{1}, …, n̂_{n})
→_{red} emb(h̃(l_{1}, …, l_{m}))
= embnat(g(n_{1}, …, n_{n}))

Case g(x_{1}, …, x_{n}, 0) = h(x_{1}, …, x_{n}), g(x⃗, x_{n+1}) = k(x⃗, y, g(x⃗, x_{n+1}))). Induction on n_{n+1}. If n_{n+1} = 0,

g(r_{1}, …, r_{n}, r_{n+1})
→_{red} P (n_{n+1}, (ĥn̂_{1}, …, n̂_{n}), (λu, v.k̂n̂_{1}, …, n̂_{n}uv))
→_{red} P (n_{n+1}, emb(h̃(n_{1}, …, n_{n})), (λu, v.k̂n̂_{1}, …, n̂_{n}uv))
→_{red} emb(h̃(n_{1}, …, n_{n}))

If n_{n+1} = k + 1,

g(t_{1}, …, t_{n}, t_{n+1})
→_{red} P (Sk̂, (ĥn̂_{1}, …, n̂_{n}), (λu, v.k̂n̂_{1}, …, n̂_{n}uv))
→_{red} P (Sk̂, emb(h̃(n_{1}, …, n_{n})), (λu, v.k̂n̂_{1}, …, n̂_{n}uv))
→_{red} (λu, v.k̂n̂_{1}, …, n̂_{n}uv)k̂P (k̂, emb(h̃(n_{1}, …, n_{n})), (λu, v.k̂n̂_{1}, …, n̂_{n}uv))
→_{red} k̂n̂_{1}, …, n̂_{n}k̂embnat(g̃(n_{1}, …, n_{n}, k))
→_{red} emb(k̃(n_{1}, …, n_{n}, k, g̃(n_{1}, …, n_{n}, k))) = emb(g(n_{1}, …, n_{n}, n_{n+1}))

(b): If t = v_{i}^{P A}, 0, this is trivial, and if t = gt_{1}, …, t_{n} follows

t_{i}[r⃗]→_{red}emb(t̃_{i}[n⃗]),

by (a) therefore

t[r⃗]→_{red}g̃(t̃_{1}[r⃗], …, t̃_{n}[r⃗]) = t̃[r⃗]

Next task would now be to prove, that, when we first interpret a formula of L_{P A} in L_{ML} and then use the interpretation, as we have done in chapter 5, we get an equivalent formula to the one, we get by directly interpreting L_{P A} in L_{KP}. But in this formulation, this is not correct, here is the place, where we need to extend the set of term constructors by the constructors A_{i}: Assume, that we had only interpreted Πx ∈ A.B as a set of closed terms, which all represent recursive functions. Let A(x, y) be a prime formula of L_{P A}, such that, KP i^{+} ⊢ int_{P A,KP} (∀x.∃y.A(x, y)) and such that there is no recursive function g such that ∀x.A(x, g(x)). Then we had int_{P A,ML}(∀x.∃y.A(x, y))^{∗}[ ] = ∅, so for every t ∈ T erm, F V (t) ⊂ {x} pair(λx.t, λx.t) ∈ int_{P A,ML}((∀x.∃y.A(x, y)) →⊥)^{∗}[ ], although KP i^{+} ⊢ ¬int_{P A,KP} ((∀x.∃y.A(x, y)) →⊥). But when we add a constructor A_{i} together with its interpretation A_{i}^{∗} such that ∀x.(∃y.A(x, y)) → A(x, A_{i}^{∗}(x)), we have int_{P A,ML}(∀x.∃y.A(x, y))^{∗}[ ] ≠ ∅ and int_{P A,ML}(∀x.∃y.A(x, y) →⊥)^{∗}[ ] = ∅. Following this idea we can define for every formula a finite sequence of functions f_{1}, …, f_{n}, such that the two interpretations are equivalent, if we have A_{i} with A_{i}^{∗} = f_{i}, as will be stated in the next lemma: Lemma 7.5 For every formula A of PA with

F V (A) ⊂ {v_{i_{1}}^{P A}, …, v_{i_{l}}^{P A}}

with i_{1} < ⋯ i_{l} there exist a finite set of functions F, with ∀f ∈ F.∃n.f : IN^{n} → IN, definable in KP i^{+}, such that for every interpretation relative to an extending set of constructors (A_{i})_{i∈I} with interpretations (A_{i}^{∗})_{i∈I}, such that F ⊂ {A_{i}^{∗}|i ∈ I} there exists a h ∈ T erm_{(A_{i},A_{i}^{∗})_{i∈I}} F V (t) ⊂ {z_{i_{1}}^{ML}, …, z_{i_{l}}^{ML}}, (z⃗ := z_{i_{1}}^{ML}, …, z_{i_{l}}^{ML}, u⃗ := u_{i_{1}}^{KP} , …, u_{i_{l}}^{KP}) such that

KP i_{n}^{+} ⊢ ∀n_{1}, …, n_{l} ∈ IN.∀r_{1}, …, r_{l} ∈ T erm_{Cl}.
(r_{1}→_{red}n̂_{1} ∧ ⋯ ∧ r_{l}→_{red}n̂_{l}) →
((∃r ∈ T erm_{nf}.h[z⃗/r⃗]→_{red}r) ∧
(Ã[u⃗/n⃗] ↔ makepair(h[z⃗/r⃗]) ∈ Â^{∗}[z⃗/r⃗]) ∧
(Ã[u⃗/n⃗] ↔ Â^{∗}[z⃗/r⃗] ≠ ∅))

Proof: by induction on the definition of the formulas. Again we will not mention explicitly Variables, that occur in subterms, or do not occur at all. Case A =⊥: Choose F := ∅, h := 0 We have ¬Ã[n⃗], Â^{∗}[r⃗] = ∅.

Case A = (s = t): Choose as F := ∅, h := \underline{r} ∈ T erm_{nf}. We have, using that for r→_{red}s ∈ T erm_{nf} s is unique, and 7.4

Ã[n⃗] ↔ s̃[n⃗] = t̃[n⃗]
↔ (∃n ∈ IN.ŝ[r⃗]→_{red}S^{n}0 ∧ t̂[r⃗]→_{red}S^{n}0)
↔ pair(ŝ[r⃗], t̂[r⃗]) ∈ N^{∗}
↔ pair(\underline{r}, \underline{r}) ∈ Â^{∗}[r⃗]
↔ Â^{∗}[r⃗] ≠ ∅

Case A = (B ∧ C): Let F_{i} for A_{i} chosen, F := F_{1} ∪ F_{2}. If F ⊂ {A_{i}^{∗}|i ∈ I}, and we have h_{i} according to the assertion of the lemma chosen. Then we define h := p(h_{1}, h_{2}). Then for n⃗ ∈ IN, r⃗ ∈ T erm_{Cl} as in the lemma, exist s_{1}, s_{2} ∈ T erm_{nf}, such that r_{i}[r⃗]→_{red}s_{i}, h[r⃗]→_{red}p(s_{1}, s_{2}) ∈ T erm_{nf}.

Ã[n⃗] ↔ B̃_{1}[n⃗] ∧ B̃_{2}[n⃗]
↔ makepair(s_{1}) ∈ B̂_{1}^{∗}[n⃗] ∧ makepair(s_{2}) ∈ B̂_{2}^{∗}[n⃗]
↔ makepair(h[r⃗]) ∈ Â^{∗}[n⃗]
↔ B̂_{1}^{∗}[r⃗] ≠ ∅ ∧ B̂_{2}^{∗}[r⃗] ≠ ∅
↔ Â^{∗}[n⃗] ≠ ∅

Case A = (B_{1} ∨ B_{2}): Let F_{i} for B_{i} chosen. Let

f_{3} := {pair(tupel(n_{1}, …, n_{l}), i)|(i = 0 ∧ B̃_{0}[n⃗]) ∨ (i = 1 ∧ ¬B̃_{0}[n⃗])}

F := F_{1} ∪ F_{2} ∪ {f_{3}}, and if F ⊂ {A_{i}^{∗}|i ∈ I}, further A_{k}^{∗} = f_{3}, h_{i} chosen according to the IH for F and define h := P (A_{k}(z⃗), h_{1}, (u, v)h_{2}) (u, v new variables). Then if n⃗, r⃗ are chosen as in the assertion, there exist s_{i} such that h_{i}[r⃗]→_{red}s_{i} ∈ T erm_{nf}

A_{k}(r_{1}, …, r_{n})→_{red}A_{k}(n̂_{1}, …, n̂_{i})→_{red}S^{i}0

for i = f_{3}(n⃗) ∈ {0, 1}, if i = 0,

h[r⃗]→_{red}P (S^{i}0, s_{1}, (u, v)h_{2}[r⃗])→_{red}s_{1} ∈ T erm_{nf},

and if i = 1,

h[r⃗]→_{red}s_{2}.

We have

Ã[n⃗] ↔ B̃_{1}[n⃗] ∨ (¬B̃_{1}[n⃗] ∧ B̃_{2}[n⃗])
↔ (f_{3}(n⃗) = 0 ∧ makepair(s_{1}) ∈ B̂_{1}^{∗}[r⃗]) ∨ (f_{3}(n⃗) = 1 ∧ makepair(s_{2}) ∈ B̂_{2}^{∗}[r⃗])
↔ makepair(h[r⃗]) ∈ Â^{∗}[r⃗]
↔ B̂_{1}^{∗}[r⃗] ≠ ∅ ∨ B̂_{2}^{∗}[r⃗] ≠ ∅
↔ Â^{∗}[r⃗] ≠ ∅

Case A = (B_{1} → B_{2}). Let F_{i} for B_{i} chosen, F := F_{1} ∪ F_{2}, F ⊂ {A_{i}^{∗}|i ∈ I}, and assume that h_{i} are chosen for A_{i}. Define h := λx.h_{2} Then for r⃗, n⃗ as in the assertion h[r⃗] ∈ T erm_{nf}, By IH h_{2}[r⃗]→_{red}s_{2} for some s_{2} ∈ T erm_{nf}. Subcase Ã[n⃗]. If B̃_{1}[n⃗] is false, then by IH B̂_{1}^{∗}[r⃗] = ∅, therefore

∀pair(r, r′) ∈ B̂_{1}^{∗}[r⃗].pair(h_{2}[x/r, r⃗], h_{2}[x/r′, r⃗]) ∈ B̂_{2}^{∗}[r⃗]

therefore makepair(h[r⃗]) ∈ Â^{∗}[r⃗]. If B̃_{1}[n⃗] is true, then B̃_{2}[r⃗] is true, therefore makepair(s_{2}) ∈ B̂_{2}^{∗}[r⃗],

∀pair(r, r′) ∈ B̂_{1}^{∗}[r⃗].h_{2}[x/r, r⃗]→_{red}s_{2} ∧ h_{2}[x/r′, r⃗]→_{red}s_{2} ∧ makepair(s_{2}) ∈ B̂_{2}^{∗}[n⃗],

h[r⃗] ∈ Â^{∗}[n⃗]. Subcase ¬Ã[n⃗]. Then by IH exists s_{1} such that h_{1}[r⃗]→_{red}s_{1} ∈ T erm_{nf} and we have makepair(s_{1})inB̂_{1}^{∗}[r⃗] and, if we had pair(s, s′) ∈ Â^{∗}[r⃗], then makepair(s) ∈ Â^{∗}[r⃗], s→_{red}λx.t for some t, makepair(t[x/s_{1}]) ∈ B̂_{2}^{∗}[r⃗] = ∅, a contradiction, therefore Â^{∗}[r⃗] = ∅. Case A = ∀v_{i}^{P A}.B: Let F_{1} for B be chosen, F := F_{1}, F ⊂ {A_{i}^{∗}|i ∈ I}, h_{1} be chosen for B, and let h := λv_{i}^{P A}.h_{1}. Assume n⃗, r⃗ as in the assertion, h[r⃗] ∈ T erm_{nf}. Assume pair(r, r′) ∈ Â^{∗}[r⃗], then pair(r, r) ∈ Â^{∗}[r⃗], r→_{red}λx.t and

∀k ∈ IN.makepair(r[x/k̂, r⃗]). ∈ B̂^{∗}[z_{i}^{ML}/k̂, r⃗],

by IH follows ∀k ∈ IN.B̃[u_{i}^{KP} /k, n⃗], therefore Ã[n⃗]. Assume Ã[n⃗]. Then for all k ∈ IN B̃[v_{i}^{P A}/k, r⃗], therefore by I.H. makepair(h[v_{i}^{P A}/r, r⃗]) ∈ B̂^{∗}[z_{i}^{ML}/r, r⃗], whenever r→_{red}S^{k}0, therefore makepair(h[r⃗]) ∈ Â[r⃗]. Case A = ∃v_{i}^{P A}.B, F_{1} be chosen for B,

f_{2} := {pair(tupel(n⃗), k)| (B̃[u_{i}^{KP} /k, n⃗] ∧ ∀k′ < k.¬(B̃[u_{i}^{KP} /k, n⃗])) ∨
(k = 0 ∧ ∀k ∈ IN.¬(B̃[u_{i}^{KP} /k, n⃗]))}

Let F := F_{1} ∪ {f_{2}}. Let F ⊂ {A_{i}^{∗}|i ∈ I}, h_{1} be chosen for B and A_{k}^{∗} = f_{2}. h := p(A_{k}(z⃗), h_{1}[z_{i}^{ML}/A_{k}(z⃗)]) Assume n⃗, r⃗ as in the assertion, k := f_{2}(n⃗).

A_{i}(r⃗)→_{red}A_{i}(n̂_{1}, …, n̂_{n})→_{red}S^{k}0.

By IH we have h_{1}[z_{i}^{ML}/A_{k}(z⃗)][r⃗] = h_{1}[z_{i}^{ML}/A_{i}(r⃗), r⃗]→_{red}t_{1} for for some t_{1} ∈ T erm_{nf}, therefore h[r⃗]→_{red}p(S^{k}0, t_{1}). Assume pair(r, r′) ∈ Â^{∗}[r⃗]. Then pair(r, r) ∈ Â^{∗}[r⃗], r→_{red}p(S^{l}0, r′′) ∈ T erm_{nf}. Then makepair(r′′) ∈ B̂^{∗}[z_{i}^{ML}/S^{l}0, r⃗], by I.H. B̃[u_{i}^{KP} /l, n⃗], therefore Ã[n⃗]. Assume Ã[n⃗]. Then by definition B̃[u_{i}^{KP} /k, n⃗] and by IH

makepair(t_{1}) ∈ B̂^{∗}[z_{i}^{ML}/k̂, r⃗] = B̂^{∗}[z_{i}^{ML}/A_{k}(r⃗), r⃗] = B̂[z_{i}^{ML}/A_{k}(z⃗)]^{∗}[z⃗/r⃗],

(we use that ML_{1}^{e}W_{T} ⊢ z⃗ : N ⇒ A type, pair(A_{k}(r⃗), k̂) ∈ N^{∗} and the Main Lemma), therefore makepair(h[r⃗]) ∈ Â^{∗}[r⃗].

Lemma 7.6 If A is a formula in PA, ML_{1}^{e}W_{T,U} ⊢ s : Â, then KP i^{+} ⊢ Ã. Proof: lemma 7.5

Now we observe, that, for a particular proof, the interpretation of ML_{1}^{e}W_{T,U} can be carried out in KP i_{n}^{+} for some n: In definition 5.8, we can define A^{∗} in KP i_{n}^{+} for n > level(A), where level(A) counts the nesting of W -types (we replace a(u) by a(u)^{n}, α(u) by α(u)^{n}). (In fact, we only had to count nestings of W -types above U, but this is technically more complicated.) Û can be defined, by replacing again a(u), α(u) by a(u)^{n}, α(u)^{n}, in KP i_{n}^{+} for n ≥ 1. The small lemmata of chapter 6 can be proven in KP i_{n}^{+} for n ≥ 1 and in lemma 6.18, we can for each judgement prove the conclusion in KP i_{n}^{+} for n > n_{0}, where n_{0} is the maximum of the levels of types occurring in the proof in ML_{1}^{e}W_{T}. All arithmetical formulas have level 0. So we have the following stronger lemma: Lemma 7.7 If A is a formula of PA ML_{1}^{e}W_{T,U} ⊢ s : Â, then there exists n < IN, such that KP i_{n}^{+} ⊢ Ã. Theorem 7.8 |ML_{1}^{e}W_{T,U}|, |ML_{1}^{e}W_{T} |, |ML_{1}^{e}W_{R,U}|, |ML_{1}^{e}W_{R}|, |ML_{1}^{i}W_{T} |, |ML_{1}^{i}W_{R}| ≤ ψ_{Ω_{1}}(Ω_{I+ω}), where the ordinal denotation is as in [Buc92b]. Proof: We show, that |KP i_{n}^{+}| ≤ ψ_{Ω_{1}}(ϵ_{Ω_{I+n}+1}) ≤ ψ_{Ω_{1}}(Ω_{I+ω}). By lemma 7.6 follows the assertion. We follow the lines of [Buc92b]. First observe, that we can prove as in theorem 2.9 there, using several applications of ∃_{κ′′}, (⋀)^{∗}, and ⊢^{∗} Ad(L_{κ′′}) for κ′′ ∈ {κ, …, κ_{n1}}, that if:

(∗)  λ ∈ Lim ∧ ∃κ_{1}, …, κ_{n−1}, κ ∈ R.(∀α < κ.∃κ′ ∈ R.α < κ′ < κ) ∧

κ ∈ κ_{1} ∧ κ_{1} ∈ κ_{2} ∧ ⋯ ∧ κ_{n−2} ∈ κ_{n−1} ∧ κ_{n−1} ∈ λ. (∗)

and if we extend X^{∗} by κ, κ_{1}, …, κ_{n−1} follows

⊢_{λ}^{∗} (KP i_{n}^{+})^{λ}.

We can adjust theorem 3.12 of [Buc92b] to obtain, if we have (∗), λ ∈ R, and λ, κ, κ_{1}, …, κ_{n−1} ∈ H, and H closed under ξ ↦ ξ^{R}, then:

For each theorem ϕ of KP i_{n}^{+} exist k ∈ IN such that H ⊢_{λ+k}^{ω^{λ+k}} ϕ^{λ}.

Now observe, that H_{γ} in [Buc92b] has the desired properties (with λ := Ω_{I+n}, κ_{i} := Ω_{I+i}) and we conclude as in theorem 4.9, Corollary, with ν := ψ_{Ω_{1}}(ϵ_{Ω_{I+n}})

|KP i_{n}^{+}| ≤ ψ_{Ω_{1}}(ϵ_{Ω_{I+n}})

Part III

A lower bound for the proof theoretical strength of Martin-Löf’s

type theory A well-ordering proof in Martin-Löf’s type theory

In this chapter we will prove

ML_{1}^{i}W_{R} ⊢ r : ∀ϕ ∈ N → U.( ∀y ∈ N.(∀x ≺ y.ϕx) → ϕy)
→ ∀y ≺ D_{Ω_{1_{OT}}} Ω_{I+n}.ϕy

for some term r, that is ML_{1}^{i}W_{R} proves the well-ordering up to every ordinal less than our desired proof theoretical strength of ML_{1}^{i}W_{R}, ML_{1}^{i}W_{T} and ML_{1}^{e}W_{R}. To prove this, we need in some sense to write a computer program, the term r. To do this we will introduce some useful abbreviations. We will define in Martin-Löf’s type theory the analogue of many constructions known from analysis, such that we can later on more or less follow the lines of a well-ordering proof in Δ_{2}^{1} − CA + BI. After giving some general abbreviations (8.1), we define the type of truth values B (8.2), the natural numbers (8.3), lists (8.4) and state the obvious properties (lemma 8.5). On page 67 we introduce the four kinds of candidates for the power set of the natural numbers we have, and introduce them in definition 8.6. Then we introduce the subtree ordering (definition 8.8, properties are shown in 8.9), which, although looking very simple, was one of the key ideas in this proof, since it allowed to define W (X) (definition 8.11). On page 73, we explain the general method for the well-ordering proof. We assume some properties for the ordinals (general assumption 8.10) define now W (X) (definition 8.11). We prove, that we have induction over W (X) (8.12), and show some easy properties of W (X) (8.13 and 8.14. Next we define the “ausgezeichnete Mengen” (definition 8.16). We show, that ausgezeichnete Mengen are segments of each other in the sense of lemma 8.19, define W (definition 8.20), which is the union of all ausgezeichnete Mengen, and the segments of which are exactly the ausgezeichnete Mengen (lemma 8.21). We show that W is closed under + and the step to the next cardinal (8.24), contains the type of every element (8.25) and that W is a “ausgezeichnete Klasse” (definition 8.27). To prove that W is closed under certain ordinal functions, we need the fundamental sequences of it, stated in 8.10. We prove that, if I η X and Ag(X), then W (X) is closed under λz.Ω_{z} (z ≺ I), (8.30), that W (X) is closed under Veblen-function (8.29) and collapsing function (8.31). Now we define classes, which are syntactical increasing ausgezeichnete Klassen (definition 8.32) and show that we can prove using W_{n+1} transfinite induction up to D_{Ω_{1}}(Ω_{I+n} (theorem 8.34).

Preliminaries 8.1 In this part we use the formulation à là Russell for the Universe, where the elements of the universe are types, rather than terms representing indices for types, as in the formulation à là Tarski. In the first part, we preferred the formulation à là Tarski, since indices for the elements of the Universe are better for the interpration in KP i^{+}(we could interpret the Universe as a subset of the natural numbers), whereas the Russel-formulation is more suitable for actual using it (we do not have to apply the type constructor T to make a type out of an element of the Universe). The author thinks that it would have been able to carry out the well ordering proof in ML_{1}^{i}W_{T} rather than ML_{1}^{i}W_{R} — this way he would have proven the proof theoretical equivalence of four, and not only three theories. (we have not proven a lower bound for ML_{1}^{i}W_{T}) Unfortunately, there was at the end no time to check this any more.

We use the following abbreviations: r0 := p_{0}(r), r1 := p_{1}(r), < r, s >:= p(r, s), rs := Ap(r, s), (r =_{A} s) := I(A, r, s) ∀x ∈ A.B := Πx ∈ A.B, ∃x ∈ A.B := Σx ∈ A.B, we will use this, if the intended meaning of the type is a formula. B := N_{2}, ⊥:= N_{0}, ⊤ := N_{1}. A ∨ B := A + B, A ↔ B := (A → B) ∧ (B → A), ¬A := A →⊥, (r ≠_{A} s) := ¬(r =_{A} s). ∧, ∨, ∀, ∃ are used for talking of types, which have formulas as the inteded meaning, whereas the use of ×, +, Π, Σ, indicates, that we are talking of functions and sets. We will write λx, y.t for λx.λy.t, ∀x, y ∈ A.B for ∀x ∈ A.∀y ∈ A.B, similarly for ∃, Π, Σ, W and for more than two variables.

We will in the following argue informally, especially, if we say: “we have A”, or shorter “A”, for some g-type A, we mean, there exists a term r such that ML_{1}^{i}W_{R} ⊢⇒ r : A. We write r : A for ML_{1}^{i}W_{R} ⊢ r : A and r, s : A for ML_{1}^{i}W_{R} ⊢ r : A, further ML_{1}^{i}W_{R} ⊢ s : A, Γ ⇒ r : A for ML_{1}^{i}W_{R} ⊢ Γ ⇒ r : A, etc.

We will not be very restrictive in the choice of variables, so we will use x, y, z, a, b, c and sometimes capital letters such that A, B, C and X, Y, Z for them. We will prefer i, j, k, n, m to indicate natural numbers (considered as natural numbers and not as elements of the subsets of N T′′, T′, OT, A′′ etc. which are denotations for ordinals), α, β, γ for trees (elements of a W -type) but sometimes as well for elements of OT. Elements of OT are usually denoted by a, b, c or x, y, z. Definition 8.2 We define some functions corresponding to the type B, the type of truth values. For t : B we define \underline{if} t \underline{then} A \underline{else} B := C_{2}(t, A, B). tt := 0_{2}, ff := 1_{2}. ∧_{B} := λx, y. \underline{if} x \underline{then} (\underline{if} y \underline{then} tt \underline{else} ff) \underline{else} ff, ∨_{B} := λx, y. \underline{if} x \underline{then} tt \underline{else} (\underline{if} y \underline{then} tt \underline{else} ff), ∧_{B}, ∨_{B} : B → B → B, written infix, ∧_{B} is the boolean conjunction, ∨_{B} the disjunction, ¬_{B} := λx. \underline{if} x \underline{then} ff \underline{else} tt, ¬_{B} : B → B, the boolean negation, atom := λx. \underline{if} x \underline{then} ⊤ \underline{else}⊥, atom : B → U. (atom transfers a boolean value to a formula, atom(tt) is a true formula — a type having one element, and atom(ff) is a false formula — the type N_{0} =⊥, that has as elimination rule the rule which corresponds to “ex falsum quodlibet”)

Note, that for decidable predicates p we have ∀x, y ∈ N.atom(p(x, y)) ∨ ¬(atom(p(x, y))). The next task is to introduce certain functions and relations on N. Definition 8.3 (a) 1 := S0, 2 := SS0, etc.

(b) pred_{N} := λx.P (x, 0, (u, v)u), pred_{N} : N → N, the predecessor on N that is pred_{N}0 =_{N} 0, pred_{N}(Sx) =_{N} x.

(c) zero? := λx.P (x, tt, (u, v)ff), zero? : N → B, the test for 0, that is zero?0 =_{B} tt, zero?(Sx) =_{B} ff,

(d) <_{N,B}:= λx.P (x, λy.¬_{B}(zero?y),

(u, v)λy. \underline{if} (zero?y) \underline{then} ff \underline{else} (v(pred_{N}y))), <_{N,B}: N → B, the <-relation on N, which we write infix, and have: (0 <_{N,B} 0) =_{B} ff, (0 <_{N,B} St) =_{B} tt, (Sx <_{N,B} 0) =_{B} ff, (Sx <_{N,B} Sy) =_{B} (x <_{N,B} y). <_{N}:= λx, y.atom(x <_{N,B} y), written infix, too. <_{N}: N → U is the <-relation, seen as a formula.

(e) ≤_{N,B}:= λx.P (x, λy.tt,

(u, v)λy. \underline{if} (zero?y) \underline{then} ff \underline{else} (v(pred_{N}y))), <_{N,B}: N → B, the leq-relation on N and we write ≤_{N,B} infix. We have: (0 ≤_{N,B} x) =_{B} tt, (Sx ≤_{N,B} 0) =_{B} ff, (Sx ≤_{N,B} Sy) =_{B} (x ≤_{N,B} y). ≤_{N}:= λx, y.atom(x ≤_{N,B} y), written infix, too. ≤_{N}: N → U. We define ∀x <_{N} t.ϕ := ∀x ∈ N.x <_{N} t → ϕ, similarly for ∃ and ≤_{N}.

(f) =_{N,B}:= λx.P (x, λy.zero?y,

(u, v)λy. \underline{if} (zero?y) \underline{then} ff \underline{else} (v(pred_{N}y))),

=_{N,B}: N → B, the decidable equality on N. We write =_{N,B} infix, and have: (0 =_{N,B} 0) =_{B} tt, (0 =_{N,B} St) =_{B} ff, (Sx =_{N,B} 0) =_{B} ff, (Sx =_{N,B} Sy) =_{B} (x =_{N,B} y). ≠_{N,B}:= λx, y.¬_{B}x =_{N,B} y.

(g) +_{N} := λx, y.P (y, x, (u, v)Sv), +_{N} : N → N → N, the addition on N. We write +_{N} infix, and have: x +_{N} 0 =_{N} x, x +_{N} Sy =_{N} S(x +_{N} y).

(h) ˙− := λx, y.P (y, x, (u, v)(pred_{N}v)), ˙− : N → N → N, which is the minus-function on N, we write ˙− infix, and have: x ˙−0 =_{N} x, x ˙−Sy =_{N} pred_{N}y(x ˙−y).

(i) max_{N} := λx, y. \underline{if} x <_{N} y \underline{then} y \underline{else} x, min_{N} := λx, y. \underline{if} x <_{N} y \underline{then} x \underline{else} y, max_{N}, min_{N} : N → N → N, the maximum and minimum of two natural numbers and we write max_{N}{a, b}, min_{N}{a, b} for max_{N}ab, min_{N}ab.

(j) ∀_{B,<} := λn, ϕ.P (n, tt, (u, v)ϕ(u) ∧_{B} v), ∀_{B,<} : N → (N → B) → B, that is (∀_{B,<}0ϕ) =_{B} tt, (∀_{B,<}(Sn)ϕ) =_{B} (ϕn) ∧_{B} ∀_{B,<}nϕ).

∃_{B,<} := λn, ϕ.P (n, ff, (u, v)ϕ(u) ∨_{B} v), that is

(∃_{B,<}0ϕ) =_{B} ff,

∃_{B,<}(Sn)ϕ =_{B} ((ϕn) ∨_{B} ∃_{B,<}nϕ).

Further ∀_{B,≤} := λn, ϕ.∀_{B,<}(Sn)ϕ, ∃_{B,≤} := λn, ϕ.∃_{B,<}(Sn)ϕ.

We write ∀_{B}x <_{N} n.ϕ for ∀_{B,<}n(λx.ϕ), ∀_{B}x ≤_{N} n.ϕ for ∀_{B,≤}n(λx.ϕ), ∃_{B}x <_{N} n.ϕ for ∃_{B,<}n(λx.ϕ) and ∃_{B}x ≤_{N} n.ϕ for ∃_{B,≤}n(λx.ϕ). ∀_{B}, ∃_{B} are used for bounded universal quantification over a decidable predicate, which can be decided (see 8.5 (e)).

(k) We write = for =_{N}. We will now define lists as a pair < n, f >, where n is the length of the list, and (f(i))_{i<n} are the elements of the list. Definition 8.4 Let A be a g-type, and assume (we will need this for stating the typing judgements), x, y, z, u, v, w ∈ V ar_{ML}, ML_{1}^{i}W_{R} ⊢ Γ ⇒ A′ type, X, Y variables that do not occur free or bounded in Γ, A′. Alist := N × (N → A). lh := λx.x0, and have Γ ⇒ lh : A′list → A′. (·)_{·} := λx, y.(x1)y, and have Γ ⇒ (·)_{·} : A′list → N → A′, and write (a)_{i} for (·)_{·}i nil_{y} :=< 0, λx.y >, Γ, y : A′ ⇒ nil_{y} : A′list, (we omit the index y, if we have any usual dummy element, for instance 0 in case of A = N). cons := λx, y. < S(lh(y)), λz. \underline{if} z = 0 \underline{then} x \underline{else} (y)_{pred_{N} (z)} >,

Γ ⇒ cons : A′ → A′list → A′list, car := λx.(x)_{0},

Γ ⇒ car : A′list → A′ cdr := λx. < pred(lh(x)), λy.(a)_{Sy} >.

cdr : A′list → A′list append := λx, y. < lh(x) +_{N} lh(y), λz \underline{if} z <_{N,B} lh(x) \underline{then} (x)_{z} \underline{else} (y)_{lh(x)+_{N} z} >.

Γ ⇒ append : A′list → A′list → A′list ≅_{Nlist,B}:= λx, y.lh(x) =_{N,B} lh(y) ∧_{B} ∀_{B}z <_{N} lh(x).(x)_{z} =_{N,B} (y)_{z}), ≅_{Nlist,B}: Nlist → Nlist → B, the equivalence of two lists, which we write infix, and A ≅_{Alist} B := lh(B) = lh(C) ∧ ∀z <_{N} lh(B).(B)_{z} =_{A} (C)_{z}. (where z is a variable, that does not occur free in B or C) (for A : U, we can define a function λX, Y.X ≅_{Alist} Y : Alist → Alist → U and have then: Γ, X, Y : Alist ⇒ X ≅_{Alist} Y type) and for fixed n < a_{0}, …, a_{n} >_{List}:=< Sn, f >, where f := λx. \underline{if} x =_{N,B} 0 \underline{then} a_{0} \underline{else}

\underline{if} x =_{N,B} 1 \underline{then} a_{1} \underline{else} ⋯

⋯ \underline{if} x = n \underline{then} a_{n} \underline{else} a_{0}, and if x_{i} are new variables, then ML_{1}^{i}W_{R} ⊢ Γ, x_{1} : A, …, x_{n} : A ⇒< x_{1}, …, x_{n} >_{List}: A′list Sublist x y will be the list of the first y elements of the list x: Sublist := λx, y. \underline{if} y <_{N} lh(x) \underline{then}< y, x1 >\underline{else} x, Γ ⇒ Sublist : A′list → N → A′list, we write Sublist(a, b) for Sublist ab. We can easily prove the following properties: Lemma 8.5 (a) atom(tt), ¬(atom(ff)). (which means: there exist g-terms r, s, such that

ML_{1}^{i}W_{R} ⊢ r : atom(tt), ML_{1}^{i}W_{R} ⊢ s : ¬(atom(ff))).

∀x ∈ B.atom(x) ∨ ¬atom(x), (which means that for all decidable predicates (functions with codomain B) we have tertium non datur).

(b) ∀x, y ∈ B.atom(x ∧_{B} y) ↔ atom(x) ∧ atom(y), ∀x, y ∈ B.atom(x ∨_{B} y) ↔ atom(x) ∨ atom(y), ∀x ∈ B.atom(¬_{B}x) ↔ ¬atom(x).

(c) ∀x, y ∈ N.x <_{N} y ∨ x = y ∨ y <_{N} x, (which means: there exist a g-term r, such that ML_{1}^{i}W_{R} ⊢ r : ∀x, y ∈ N.x <_{N} y ∨ x = y ∨ y <_{N} x), ∀x, y ∈ N.atom(x =_{N,B} y) ↔ x = y. ∀x, y ∈ N.x <_{N} y → (x ≠ y ∧ ¬(y <_{N} x)), ∀x, y ∈ N.x = y → (¬(x <_{N} y) ∧ ¬(y <_{N} x)), ∀x, y ∈ N.x ≤_{N} y ↔ (x <_{N} y ∨ x = y),

(d) ∀x ∈ N.x ≠ 0 → x = S(pred_{N}(x)).

(e) ∀y ∈ N.∀ϕ ∈ N → B.atom(∀_{B}x <_{N} y.ϕ) ↔ ∀x <_{N} y.atom(ϕy), ∀y ∈ N.∀ϕ ∈ N → B.atom(∃_{B}x <_{N} y.ϕ) ↔ ∃x <_{N} y.atom(ϕy), similarly for ≤_{N}.

(f) Let Γ ⇒ A : type, Γ ⇒ d : A. ∀x, y ∈ Nlist.atom(x ≅_{Nlist,B} y) ↔ x ≅_{Nlist} y. ∀x ∈ Alist.x ≅_{Alist} nil_{d} ↔ lh(x) = 0. Γ ⇒ ∀x ∈ Alist.¬(x ≅_{Alist} nil_{d}) → x ≅_{Alist} cons(car(x), cdr(x)). Γ ⇒ ∀x ∈ Alist.append(nil_{d}, x) ≅_{Alist} x, Γ ⇒ ∀x, y ∈ Alist, z ∈ A.append(cons(z, x), y) ≅_{Alist} cons(z, append(x, y))

Proof:

(a) is obvious, (b) follows by boolean induction. In the second assertion of (c), “→” follows by induction on x, y : N, for “←” we use again the same induction, and argue, if 0 = Sy, then 0_{1} ∈ N_{1} = P (Sy, N_{0}, (u, v)N_{1}) = P (0, N_{0}, (u, v)N_{1}) = N_{0} = atom(0 =_{N,B} Sy), and if Sx = Sy, then x = pred_{N}(Sx) = pred_{N}(Sy) = y. The other assertions in (c), (d), (e) follow by induction on N. In (f), the first assertion follows (b) and (e), by induction on the length of the lists, the other one follow easily.

The most complicated definition is to introduce the analogue of the subsets of the natural numbers. We will use in this proof four different possibilities for introducing the power set of N, which we present in increasing order of their complexity: The finite sets P^{fin}(N), defined as lists of natural numbers, the decidable sets P^{dec}(N), defined as boolean functions on N, which decide, which elements belong to the set, the ordinary power set P(N), defined as a functions f : N → U, where the elements should be those n : N, such that fn is not an empty type, and the subclasses of N Cl_{y}(N), which are types (seen as properties) with free variable y. The subsets with lower complexity have, roughly speaking, the advantage of better degree of decidability, whereas we need the subsets of high complexity, to introduce concepts which correspond to the highly impredicative definitions, possible in the corresponding systems of analysis. For each power set we will introduce the relation η, which stands for the element relation (for the first two power sets additionally the decidable element relation η_{B}), the types ∀x η A.B, (with intended meaning for all x in A we have B), ∃x η A.B, the relations the relations ⊂ for subset and ≅ for “have the same elements” between power sets, union (∪), intersection (∩) and difference (\). Further we lift an element of one power set to the next complicated power set (operations ·^{+}, ·^{P}, ·^{+,Cl,y}, and we define the finite sets {a_{0}, …, a_{n}}, the empty set ∅, and, except in the finite case the set corresponding to N, namely N_{dec}, N_{P}, N_{Cl,y}. We will give, whenever possible, closed terms, which we can do, as long as the types are constructed of elements of U or elements of some fixed type related to U. For the case that the types are no longer elements of this type, we need another definition, which is for each type a new syntactical object. We will use capital letters like A, B, C and X, Y, Z to indicate elements of the four power sets of N. Definition 8.6 (a) The four kinds of subsets are:

The finite subsets are the elements of P^{fin}(N) := Nlist, the decidable subsets are those of P^{dec}(N) := N → B, the ordinary subsets are the elements of P(N) := N → U. For the subclasses of N we have no type containing all classes, but we have only the following meta statement: Γ ⇒ A ∈ Cl_{y}(N) is defined as ML_{1}^{i}W_{R} ⊢ Γ, y : N ⇒ A type.

The finite subsets can be seen as a list of the elements of the set, the decidable subsets are functions of boolean value, which decide for each element, if it belongs to the set. The ordinary subsets are functions, that assign an element of U to each natural number. If we see this element as a formula, the set is supposed to contain all elements, for which this formula is true (although this sentence is just some heuristic, we cannot say in type theory that an element of U is true, only that we have some t : U). To exhaust the full proof theoretical strength of Martin-Löf’s type theory we need classes, which correspond to classes in systems of the analysis. A class is a type, depending on one free variable y (and eventually on a bigger context), and the class A ∈ Cl_{y}(N) stands for {x ∈ IN|A[y/x]}. When using classes, we have to be careful with the use of free and bounded variables.

(b) In P^{fin}(N) and P^{dec}(N) we have decidable element relations η_{fin,B}, η_{dec,B}, which distinguish these from P(N), Cl_{y}(N):

η_{fin,B}:= λy, X.∃_{B}x <_{N} lh(X).y =_{N,B} (X)_{x}, η_{fin,B}: N → P^{fin}(N) → B;

η_{dec,B}:= λy, X.Xy, η_{dec,B}: N → P^{dec}(N) → B.

Both relations are written infix.

(c) The element relations η_{fin}, η_{dec}, η and η_{Cl,y} are defined as follows:

η_{fin}:= λy, X.atom(y η_{fin,B} X), η_{fin}: N → P^{fin}(N) → U;

η_{dec}:= λy.X.atom(y η_{dec,B} X), η_{dec}: N → P^{dec}(N) → U;

η:= λy, X.Xy, η: N → P(N) → U.

all these three relations are written infix, and y η_{·} X can be seen heuristically as a formula.

t η_{Cl,y} B := B′[y/t], where B′=_{α}B such that B′[y/t] is an allowed substitution (we can do this for all the finitely many types occurring in the proof we construct)

We further define η̸_{fin}:= λy, X.¬y η_{fin} X, η̸_{fin}: N → P^{fin}(N) → U, similarly η̸_{dec}, η̸, and t η̸_{Cl,y} B := ¬(t η_{Cl,y} B).

(d) P^{fin}(N) is distinguished from the other kinds of subsets, by having decidable quantification over boolean predicates:

∀_{B} := λX, f.∀_{B}x <_{N} lh(X).f((X)_{x}), ∃_{B} := λX, f.∃_{B}x <_{N} lh(X).ϕ((X)_{x}). ∀_{B}, ∃_{B} : P^{fin}(N) → (N → B) → B,

and we write ∀_{B}x η B.ϕ for ∀_{B}B(λx.ϕ) and ∃_{B}x η B.ϕ for ∃_{B}b(λx.ϕ).

We can prove: ∀X ∈ P^{fin}(N).∀f ∈ X → B.atom(∀_{B}x η X.f) ↔ ∀x ∈ N.x η_{fin} X → atom(fx), ∀X ∈ P^{fin}(N).∀f ∈ X → B.atom(∃_{B}x η X.f) ↔ ∃x ∈ N.x η_{fin} X ∧ atom(fx).

(e) We define the quantification over subsets of N first, for functions with values in U as closed terms:

Σ_{fin} := λX, Y.Σx ∈ N.x η_{fin} X ∧ (Y x), Π_{fin} := λX, Y.Πx ∈ N.x η_{fin} X → (Y x), Σ_{fin}, Π_{fin} : P^{fin}(N) → (N → U) → U;

Σ_{dec} := λX, Y.Σx ∈ N.x η_{dec} X ∧ (Y x), Π_{dec} := λX, Y.Πx ∈ N.x η_{dec} X → (Y x), Σ_{dec}, Π_{dec} : P^{dec}(N) → (N → U) → U;

Σ_{P} := λX, Y.Σx ∈ N.Σy ∈ (x η X).(Y xy), Π_{P} := λX, Y.Πx ∈ N.Πy ∈ (x η X).(Y xy), Σ_{P}, Π_{P} : ΠX ∈ P(N).(Πx ∈ N.x η X → U) → U.

We write ∀ for Π, ∃ for Σ, (when our heuristic is to speak of formulas rather than functions), and write Σ_{B}x η B.ϕ for Σ_{B}B(λx.ϕ), similarly for Π, ∀, ∃, and for indices dec, and in the case of ordinary subsets Σx η A.B or ∃x η A.B for Σ_{P}A.(λx, y.B), ∀x η A.B or Πx η A.B for Π_{P}A(λx, y.B), omitting the index y if the variable y does not occur free in B (we assume that in this case we choose a new variable y).

(f) If we have no function with value in U, but just a type with a free variable, quantification is defined as follows (where x, z are always new variables):

Σx η_{fin,typ} A.B := Σx ∈ N.x η_{fin} A ∧ B, Πx η_{fin,typ} A.B := Πx ∈ N.x η_{fin} A → B;

Σx η_{dec,typ} A.B := Σx ∈ N.x η_{dec} A ∧ B, Πx η_{dec,typ} A.B := Πx ∈ N.x η_{dec} A → B;

Σx η_{typ} A.B := Σx ∈ N.x η A ∧ B, Πx η_{typ} A.B := Πx ∈ N.x η A → B.

Since we have A : P(N), B : N → U ⇒ r : Πx η A.B ↔ Πx η_{typ} A.B, the same for Σ and in the cases fin and dec, we will omit the index typ, if there is no confusion.

Σx η_{Cl,y} A.B := Σx ∈ N.x η_{Cl,y} A ∧ B Πx η_{Cl,y} A.B := Πx ∈ N.x η_{Cl,y} A → B

(g) For the finite subsets we have a decidable subset relation:

⊂_{fin,B}:= λX, Y.∀_{B}x η X.x η_{fin,B} Y, ⊂_{fin,B}: P^{fin}(N) → P^{fin}(N) → B, written infix.

For all sets we define the subset relation by:

⊂_{fin}:= λX, Y.atom(X ⊂_{fin,B} Y), ⊂_{fin}: P^{fin}(N) → P^{fin}(N) → U;

⊂_{dec}:= λX, Y.∀x ∈ N.x η_{dec} X → x η_{dec} Y, ⊂_{dec}: P^{dec}(N) → P^{dec}(N) → U;

⊂:= λX, Y.∀x ∈ N.x η X → x η Y, ⊂: P(N) → P(N) → U; We write all these relations infix.

B ⊂_{Cl,y} C := ∀x ∈ N.x η_{Cl,y} B → x η_{Cl,y} C, where x is a new variable. If Γ ⇒ B, C ∈ Cl_{y}(N), then Γ ⇒ B ⊂_{Cl,y} C type.

(h) For the finite subsets we have a decidable relation ≅, for “have the same elements”: ≅_{fin,B}:= λX, Y.X ⊂_{fin,B} Y ∧_{B} Y ⊂_{fin,B} X, ≅_{fin,B}: P^{fin}(N) → P^{fin}(N) → B, written infix.

For all subsets we define ≅ as follows:

≅_{fin}:= λX, Y.atom(X ≅_{fin,B} Y), ≅_{fin}: P^{fin}(N) → P^{fin}(N) → U;

≅_{dec}:= λX, Y.X ⊂_{dec} Y ∧ Y ⊂_{dec} X, ≅_{dec}: P^{dec}(N) → P^{dec}(N) → U;

≅:= λX, Y.X ⊂ Y ∧ Y ⊂ X, ≅: P(N) → P(N) → U;

We write all these relations infix.

B ≅_{Cl,y} C := B ⊂_{Cl,y} C ∧ C ⊂_{Cl,y} B. If Γ ⇒ B, C ∈ Cl_{y}(N), then Γ ⇒ B ≅_{Cl,y} C type,

(i) We define union ∪, intersection ∩ and set difference \ as follows:

∪_{fin}:= λX, Y. < lh(X) +_{N} lh(Y), λx. \underline{if} (x <_{N,B} lh(X)) \underline{then} ((X)_{x}) \underline{else} (Y_{x ˙−lh(X)}) >, ∩_{fin}:= λX, Y.P (lh(X), ∅_{fin}, (u, v) \underline{if} ((X)_{u}) η_{fin,B} Y \underline{then}

(v ∪_{fin} \{(X)_{u}\}) \underline{else} v), \_{fin}:= λX, Y.P (lh(X), ∅_{fin}, (u, v) \underline{if} ((X)_{u}) η_{fin,B} Y \underline{then} v \underline{else} (v ∪_{fin} \{(X)_{u}\})), ∪_{fin}∩_{fin}, \_{fin}: P^{fin}(N) → P^{fin}(N) → P^{fin}(N);

∩_{dec} := λX, Y.λy.y η_{dec,B} X ∧_{B} y η_{dec,B} Y, ∪_{dec} := λX, Y.λy.y η_{dec,B} X ∨_{B} y η_{dec,B} Y, \_{dec} := λX, Y.λy.y η_{dec,B} X ∧_{B} ¬_{B}(y η_{dec,B} Y), ∩_{dec}, ∪_{dec}, \_{dec} : P^{dec}(N) → P^{dec}(N) → P^{dec}(N);

∩ := λX, Y.λy.y η X ∧ y η_{dec} Y, ∪ := λX, Y.λy.y η X ∨ y η Y, \ := λX, Y.λy.y η X ∧ ¬(y η Y), ∩, ∪, \ : P(N) → P(N) → P(N);

all these functions are written infix.

B ∩_{Cl,y} C := B ∧ C, B ∪_{Cl,y} C := B ∨ C, B \_{Cl,y} C := B ∧ ¬(C). If Γ ⇒ B, C ∈ Cl_{y}(N), then Γ ⇒ B ∪_{Cl,y} C, B ∩_{Cl,y} C, B \_{Cl,y} C ∈ Cl_{y}(N).

We can easily prove ∀X, Y ∈ P^{fin}(N).∀x ∈ N.x η_{fin} X ∪_{fin} Y ↔ (x η_{fin} X ∨ x η_{fin} Y), ∀X, Y ∈ P^{fin}(N).∀x ∈ N.x η_{fin} X ∩_{fin} Y ↔ (x η_{fin} X ∧ x η_{fin} Y), ∀X, Y ∈ P^{fin}(N).∀x ∈ N.x η_{fin} X \_{fin} Y ↔ (x η_{fin} X ∧ x η̸_{fin} Y), similarly for P^{dec}(N), P(N), and for the classes we prove, that, if Γ ⇒ A ∈ Cl_{y}(N), Γ ⇒ B ∈ Cl_{y}(N), then Γ ⇒ ∀x ∈ N.x η_{Cl,y} A ∪_{Cl,y} B ↔ (x η_{Cl,y} A ∨ x η_{Cl,y} B), similarly for the other functions.

(j) We define the lifting from P^{fin}(N) to P^{dec}(N), from P^{dec}(N) to P(N) and from P(N) to Cl_{y}(N):

·^{+} := λY.λy.y η_{fin,B} Y, ·^{+} : P^{fin}(N) → P^{dec}(N), and we write B^{+} for ·^{+}B;

·^{P} := λX.λy.y η_{dec} X, ·^{P} : P^{dec}(N) → P(N), written B^{P} for ·^{P}B;

·^{+,Cl,·} := λy.λB.y η B, written B^{+,Cl,y} for ·^{+,Cl,·}yB;

and we have B : P(N) ⇒ B^{+,Cl,y} ∈ Cl_{y}(N).

We see easily, that ∀X ∈ P^{fin}(N).∀x ∈ N.x η_{fin,B} X = x η_{dec,B} X^{+}, ∀X, Y ∈ P^{fin}(N).(X ∪_{fin} Y)^{+} ≅_{dec} (X^{+} ∪_{dec} Y^{+}) ∀X ∈ P^{dec}(N).∀Y ∈ N → U.(∀_{P}x ∈ X^{P}.(Y x)) ↔ (∀_{dec}x ∈ X.(Y x)) etc. similarly for ·^{P}, ·^{+,Cl,y}, therefore we will, if there is no confusion, omit these superscripts, and subscripts dec, fin, P, Cl, y.

(k) We define the empty set:

∅_{fin} := nil, ∅_{dec} := ∅_{fin}^{+}, ∅ := ∅_{dec}^{P}, ∅_{Cl,y} := ∅^{+,Cl,y}.

We define the finite sets:

{a_{0}, …, a_{n}}_{fin} :=< a_{0}, …, a_{n} >_{List}, λx_{0}, …, x_{n}.{x_{0}, …, x_{n}}_{fin} : N → N → ⋯ N → P^{fin}(N);

{a_{0}, …, a_{n}}_{dec} := {a_{0}, …, a_{n}}_{fin}^{+};

{a_{0}, …, a_{n}} := {a_{0}, …, a_{n}}_{dec}^{P};

{a_{0}, …, a_{n}}_{Cl,y} := {a_{0}, …, a_{n}}^{+,Cl,y}.

The set of natural numbers can be represented as follows:

N_{dec} := λx.tt, N_{dec} : P^{dec}(N); N_{P} := N_{dec}^{P}; N_{Cl,y} := N^{+,Cl,y}.

Remark 8.7 In the following, we will often have statements, which can be stated for all elements of P(N) and for all classes, and have similar proofs. So we have the following convention: The statement “for Y : P(N) or Y ∈ Cl_{y}(N) holds ϕ” stands for “ML_{1}^{i}W_{R} ⊢ ∀Y ∈ P(N).ϕ and, if ML_{1}^{i}W_{R} ⊢ Γ, y ∈ N ⇒ Y type, then ML_{1}^{i}W_{R} ⊢ Γ ⇒ ϕ′”, where in ϕ′ we replace η by η_{Cl,y}, and rename all bounded variables, such that they are different from all variables in Γ, similarly for the other operations on classes. The statement “for X, Y : P(N) or X, Y ∈ Cl_{y}(N) stands for “for X : P(N) or X ∈ Cl_{y}(N). for Y : P(N) or Y ∈ Cl_{z}(N) we have ϕ”, that is, unfolding it we have four statements for X, Y : P(N), X ∈ Cl_{y}(N), Y : P(N), Y ∈ Cl_{z}(N), X : P(N) and X ∈ Cl_{y}(N), Y ∈ Cl_{z}(N).

The next task is to define the subtree ordering ≺ on the W -type. An element sup(a, f) of a type W x ∈ A.B is a tree, having immediate subtrees (fz)_{z∈A}. By iterating the step to the immediate subtree, we get all subtrees. So we define: α ≺ β iff we can get from β to α by always going to an immediate subtree. We need two definitions, one where we quantify over all A : U and B : A → U, and another definition, suitable for A : type and x : A ⇒ Btype, so we have statements only for one special type A and B.

Definition 8.8 (a) Assume A, B g-types, α, β g-terms, x a variable.

The immediate subtree relation for arbitrary trees: α ≺_{A,x,B}^{1} β := ∃x ∈ A.∃f ∈ B → (W x ∈ A.B).∃z ∈ B.

β =_{W x∈A.B} sup(x, f) ∧ α =_{W x∈A.B} fz

(where f, z are new variables) The subtree relation on arbitrary trees: α ≺_{A,x,B} β :=

∃n ∈ N.0 <_{N} n ∧ ∃f ∈ (N → (W x ∈ A.B)). (f0) =_{W x∈A.B} β ∧ (fn) =_{W x∈A.B} α∧ ∀i <_{N} n.(fi) ≺_{A,x,B}^{1} (f(Si)). (where n, i, f are new variables).

α ⪯_{A,x,B} β := α ≺_{A,x,B} β ∨ α =_{W x∈A.B} β. We have, if ML_{1}^{i}W_{R} ⊢ Γ ⇒ A type, ML_{1}^{i}W_{R} ⊢ Γ, x : A ⇒ B type, α, β are new variables, then

ML_{1}^{i}W_{R} ⊢ Γ ⇒ ∀α, β ∈ W x ∈ A.B. ⇒ α ≺_{A,B} β type

the same for ⪯_{A,B}.

(b) The immediate subtree relation for trees “in U”:

≺_{Univ}^{1}:= λA, B.λα, β.∃x ∈ A.∃f ∈ (Bx) → (W x ∈ A.(Bx)).∃z ∈ (Bx).

β =_{W x∈A.(Bx)} sup(x, f) ∧ α =_{W x∈A.(Bx)} fz, (where A, B, α, β, x, f are different variables)

ML_{1}^{i}W_{R} ⊢≺_{Univ}^{1}: ΠA ∈ U.ΠB ∈ A → U.W x ∈ A.(Bx) → W x ∈ A(Bx) → U

We write α ≺_{Univ,A,B}^{1} β for ≺_{Univ} ABαβ. The subtree relation on these trees: ≺_{Univ}:= λA.λB.λα, β.

∃n ∈ N.0 <_{N} n ∧ ∃f ∈ N → (W x ∈ A.(Bx)). (f0) =_{W x∈A.(Bx)} α ∧ (fn) =_{W x∈A.(Bx)} β∧ ∀i <_{N} k.(fi) ≺_{Univ,A,B}^{1} (f(Si)).

Further: ⪯_{Univ}:= λA.λB.λα, β.α ≺ β ∨ α =_{W x∈A.(Bx)} β.

ML_{1}^{i}W_{R} ⊢≺_{Univ}: ΠA ∈ U.ΠB ∈ A → U.W x ∈ A.(Bx) → W x ∈ A.(Bx) → U

the same for ⪯_{Univ} and again write α ≺_{Univ,A,B} β. We will usually omit the Index Univ or even A, B, so we write α ≺ β for ≺_{Univ} ABαβ if this does not cause any confusion.

Lemma 8.9 Assume Γ ⇒ A type, Γ, x ∈ A ⇒ B type. Then:

(a) ∀α, β, γ ∈ (W x ∈ A.B).α ⪯ β ↔ (α ≺ β ∨ α =_{W x∈A.B} β)

(b) ∀α, β, γ ∈ (W x ∈ A.B).((α ≺ β ∧ β ⪯ γ) → α ≺ γ) ∧ ((α ⪯ β ∧ β ≺ γ) → α ≺ γ)

(c) ∀α ∈ (W x ∈ A.B).¬(α ≺ α).

(d) ∀α ∈ W x ∈ A.B.∀x ∈ A.∀s ∈ (B → W x ∈ A.B). α ≺ sup(r, s) ↔ (∃y ∈ B.α ⪯ sy).

(e) (a) - (d) are valid, if we quantify over all X : U, Y : X → U, e.g. in (a):

∀X ∈ U.∀Y ∈ (X → U).∀α, β, γ ∈ (W x ∈ A.(Bx)). (α ⪯ β ↔ α ≺ β ∨ α =_{W x∈A.(Bx)} β)

Proof: (a), (b), (d) are trivial.

(c) We prove by Induction on α : W x ∈ A.B.¬(α ≺ α): Assume f, n as in the Definition. We have 0 <_{N} n, f0 =_{W x∈A.B} sup(r, s) ∧ f1 =_{W x∈A.B} sy for some r, s, y. Define g := λx. \underline{if} x <_{N,B} n \underline{then} f(S(x)) \underline{else} sy, g : N → W x ∈ Ay.B. Then using g we conclude sx ≺ sx, a contradiction to I.H.

We will now explain the way of carrying out the well-ordering proof. A first attempt to do this, is to define operations like +, ω^{·}, Ω_{·}, and the collapsing function λx, y.D_{x}y on some huge W -type W x ∈ A.B, and then map some ordinal denotation system OT on this W -type, using the recursion on W x ∈ A.B to prove transfinite induction on OT. This can be done, as long as we restrict ourselves to ordinals of O_{n}, the n-th number class. For bigger number classes, we need for the definition of D_{x}y the relation τ(y) ≺ x: we need some relation between the elements of a tree and its branching type. When we think of them as elements of two trees, and of ≺ as the subtree ordering (modulo isomorphisms between trees), we have the problem, that this ordering is not decidable, we can not define a function D_{x}y. The next idea is, to use the decidable ordering ≺ on OT, together with fundamental sequences (a[x])_{x≺τ(a)}, where τ(a) is the type of the ordinal a, and besides some properties we have for limes ordinals a = sup{a[x]|x ≺ τ(a)}. The ordinals a[x] correspond to immediate subtrees, we had before. Let W_{1} := W x ∈ N.(Σy ∈ N.y ≺ τ(a)). If sup(r, s) ∈ W_{1}, then r should be the label, an ordinal, of this tree. (We will use arbitrary labels in N for technical reasons) Then we distinguish those trees of W_{1}, where the fundamental sequences of the labels of its subtrees correspond to the ordering in W_{1}, that is we define Correct(α), (in this informal part we omit some bounds for quantifiers, so that the formula can be easier read)

Cor(α) := ∀r, s.sup(r, s) ⪯ α → ∀x ∈ (Σy ∈ N.y η τ̂(r).∃s′.sx = sup(r[x0], s′)

(where τ̂(r) is some set related to τ(r).) Now to define the functions we need, we need some well-ordering on {x ∈ OT |x ≺ τ(a)}, which we originally want to prove by this method. The idea which helps, which is the first step towards the “ausgezeichnete Mengen”, is to replace τ(x) by τ(x) ∩ X, where we assume, that we know the well-ordering of X. Actually we will use τ(x)^{X} := (τ(x)^{≺} ∩ X) ∪ τ′(x) where τ′(x) contains some elements of τ(x)^{≺}, which we want to include in any case. Now we define W_{1}, Cor(α) as before and let W (X) be the natural numbers, that are labels of correct trees of W_{1}. The last problem is, that now τ(x)^{X} occurs negatively in the definition of W (X), that is, the bigger X, the smaller W (X). To get sets, which contain Ω_{α} for finite α or α in some ordering, which is already proven to be a well-ordering, we could construct O_{0} := W (∅) ∩ Ω_{1}, O_{1} := W (O_{0}) ∩ Ω_{2} and so on, so that at every step we know in some sense, that O_{n} is complete. To get sets which contain Ω_{Ω_{1}} and stronger cardinals and give very strong well ordering proofs, Buchholz has introduced the concept of the “ausgezeichnete Mengen”, the distinguished sets, which is the property, which all the O_{α} we constructed before have in common, and which we needed. The property is some sort of stability in the sense that the segment formed by the set does not grow by forming W (X). We define X is a “ausgezeichnete Menge” (Ag(X)), if X ⊑ W (X), that is: ∀x η X.X|x ≅ W (X)|x Now these sets allow induction (since we have a corresponding tree α ∈ W_{1}(X)) and we can define all the functions we need.

To get the full power of Martin-Löf’s type theory, in a next step we built “ausgezeichnete Klassen”. First we built the union of all “ausgezeichnete Mengen”, which is a class, closed under Ω_{·}. Then we built W_{0} := W ∩ I, W_{i+1} := W (W_{i}) ∩ Ω_{I+i+1}, which are “ausgezeichnete Klassen”, such that I η W_{1}, Ω_{I+i} η W_{i+1}. Historically, these steps are not the way, the author found this attempt. They are only a way to motivate it. Actually, after trying the first attempt, the author tried to adopt the paper of Buchholz [Buc90], the until now clearest version of the method of the “ausgezeichnete Mengen”, to Martin-Löf’s type theory. The method of “ausgezeichnete Mengen” goes back to Buchholz ([Buc75a]), who first needed Π_{2}^{1} − CA for introducing them and where further developed by Schütte, who discovered, how to define these concepts in weaker subsystems of analysis.

But before introducing all these concepts, we need some ordinal denotation system OT. We will introduce this in the next chapter 9, here we only assume, that we have a system OT, with certain subsets an ordering ≺, functions like +, that has certain properties, is given: General Assumption 8.10 We assume an ordinal denotation system

OT : P^{dec}(N),

which we will introduce in chapter 9, together with elements

0_{OT} , 1_{OT} , ω, I η OT

(I should be a representative for the first weakly inaccessible cardinal or its recursive analogue) sets

Lim, A, R : P^{dec}(N)

(where Lim will be the denotations for limit ordinals, A the additive principal numbers and R the regular cardinals) such that

R ⊂ A ⊂ Lim ⊂ OT , I η R,

and the following functions:

≺_{OT} , ⪯_{OT}: N → N → U

( written in Infix, we will, if there is no confusion, omit the index OT), We further assume

τ : N → N

such that ∀x η OT .τ(x) η R ∪ {0, 1_{OT} , ω},

·[·] : N → N → N

(written a[b] for [·] · ab)

+ : N → N → N

(written infix)

NF_{+,B} : N → N → B

(for + normal form), we write NF_{+,B}(x, y) for NF_{+,B}xy and define

NF_{+} := λx, y.atom(NF_{+,B}(x, y)),

written in pthe same form),

Alength : N → N,

(the length of the Cantor normal-form of an ordinal)

· : N → N → N,

written infix, (for multiplication of an ordinal by a natural number)

Ω_{·} : N → N

(the enumeration of the infinite cardinals)

·^{−} : N → N,

(written a^{−} for ·^{−}a, which we fuse only for cardinals, the most important definition is Ω_{a+1}^{−} = Ω_{a}).

We define a ⪯ b ≺ c := a ⪯ b∧b ≺ c the same for similar situations, and ·^{≺} := λx.λy.(y ≺ x ∧_{B} y η OT), ·^{≺} : N → P^{dec}(N), we write a^{≺} for ·^{≺}a. We have a^{≺} ⊂ OT and will usually omit the superscript ≺. Similar to the subsets of the natural numbers, we define abbreviations for the quantification over elements of OT ∀_{≺} := λx, f.(∀y η x^{≺}.fx), ∃_{≺} := λx, f.(∃y η x^{≺}.fx), ∀_{≺}, ∃_{≺} : N → (N → U) → U, and we write ∀x ≺ a.ϕ for ∀_{≺}a(λx.ϕ), ∀_{typ}x ≺ a.A := ∀x ∈ N.x η a^{≺} ∧ A, similarly for ∃, and have x : N → A type, then ∀_{typ}x ≺ a.A, ∃x ≺_{typ} a.A, type. ∀x ⪯ a.ϕ := ∀x η (a + 1_{OT})^{≺}.ϕ, similarly for ∃.

We assume that

(a) ∀x, y η OT .∀z ∈ N.Ω_{x}, τ(x), x + y, x · z η OT ∧ (y η τ(x)^{−} → x[y] η OT)

(b) ∀x, y, z η OT .¬(x ≺ x) ∧ (x ≺ y → y ≺ z → x ≺ z)∧

(x ≺ y ∨ x = y ∨ y ≺ x), ∀x, y η OT .x ⪯ y ↔ (x = y ∨ x ≺ y)

(c) ∀x, y, z η OT .x + (y + z) = (x + y) + z.

(d) ∀x η OT .x η̸ A ↔ (x = 0_{OT} ∨ ∃y, z η OT .z ≠ 0_{OT} ∧ NF_{+}(y, z) ∧ x = y + z), ∀x, y η OT .NF_{+}(x, y) → y ≠ 0_{OT} → (Alength(x) <_{N} Alength(x + y) ∧ Alength(y) <_{N} Alength(x + y)),

∀x η OT , y η OT . NF_{+}(x, y) ∨ x + y = y ∨

∃z, z′ η OT .(x = z + z′ ∧ z′ ≠ 0_{OT} ∧ NF_{+}(z, z′) ∧ x + y = z + y).

(e) ∀x η OT .x η Lim ∨ (∃y η OT .x = y + 1_{OT}) ∨ x = 0_{OT} ∀x η OT .x + 1_{OT} η̸ Lim ∧ x + 1_{OT} ≠ 0_{OT} ∧ 0_{OT} η̸ Lim.

(f) ∀x η OT .¬(x ≺ 0_{OT}), ∀x η OT .0_{OT} + x = x + 0_{OT} = x ∀x η OT .x ≺ y + 1_{OT} ↔ x ⪯ y, ∀x, y, z η OT .y ≺ z → x + y ≺ x + z. ∀x η OT .NF_{+}(x, 1_{OT})

(g) ∀x η OT .x ≺ ω ↔ ∃n ∈ N.x = 1_{OT} · n ∀x η OT .x · 0 = 0_{OT}, x · (Sn) = (x · n) + x.

(h) Ω_{0_{OT}} = ω, Ω_{I} = I, ∀x, y η OT .x ≺ y → Ω_{x} ≺ Ω_{y}

(i) ∀x η OT .Ω_{x+1_{OT}}^{−} = Ω_{x}, I^{−} = ω^{−} = 1_{OT}^{−} = 0_{OT}^{−} = 0_{OT}.

(j) ∀x η OT .x η R ↔ x = I ∨ ∃y η OT .x = Ω_{y+1_{OT}}, ∀x, y η R.x ≺ y ≠ I → x ⪯ y^{−}

(k) ∀x η OT .τ(x) ⪯ x ∧ τ(x) η R ∪ {0_{OT} , 1_{OT} , ω}

(l) ∀x η R ∪ {0_{OT} , 1_{OT} , ω}.∀y η OT .(y ≺ x → x^{−} + y ≺ x) ∧ NF_{+}(x, x).

(m) ∀x, y, z η OT .z ≺ y → NF_{+}(x, y) → NF_{+}(x, z).

Further we assume the following laws (which are minor modifications to [Buc90]):

(F1) ∀x η Lim.∀y, z η τ(x)^{≺}.(y ⪯ x[y] ≺ x) ∧ (y ≺ z → x[y] ≺ x[z])

(F2) ∀x η Lim.∀y η OT .x[0_{OT}] ⪯ y ≺ x → ∃z η τ(x)^{≺}.x[z] ⪯ y ≺ x[z + 1_{OT}]

(F3) ∀x η R ∪ {0_{OT} , 1_{OT} }.τ(x) = x ∧ ∀y η τ(x)^{≺}.x[y] = x^{−} + y

(F4) ∀x η Lim.∀y η τ(x)^{≺}.∀z η Lim.x[y] ≺ z ⪯ x[y + 1_{OT}] → x[y] ⪯ z[0_{OT}]

(F5) ∀x η Lim.∀y η τ(x)^{≺} ∩ Lim.τ(x[y]) = τ(y)∧

((∀z η τ(y)^{≺}.x[y[z]] = x[y][z])

∨(τ(y) = ω ∧ ∀z ≺ ω.x[y[z]] = x[y][z + 1_{OT}]))

(F6) ∀x η OT .τ(x) ⪯ x

(F7) ∀x η OT .(NF_{+}(x, y) ∧ y ≠ 0_{OT}) → τ(x + y) = τ(y) ∧ ∀z η τ(y)^{≺}.(x + y)[z] = x + (y[z])

(F8) τ(1_{OT}) = 1_{OT} ∧ 1_{OT} [0_{OT}] = 0_{OT}.

So (F1), (F2) express that the fundamental sequences are ascending and approximate the ordinal, (F3) tells, that for regular cardinals, the fundamental sequence is more or less the identity (we start with τ(a)^{−} to avoid fixed points of λx.Ω_{x}). (F4) is the Bachmann property, together with (F5) it guarantees, that the correct trees are highly uniform. These properties allow to prove lemma 8.15, especially 8.15 (a), which expresses, that for these uniform trees, there is some correspondence between the trees in W (X) and the number of the branch, they belong to. (F6)-(F8) are usual properties of fundamental sequences. We will in the following, if there is no confusion, write 0, 1 instead of 0_{OT} , 1_{OT}. Now we are ready to define W (X): Definition 8.11 (a) We define the sets τ′(a), , which we will include in τ(a)^{X} in any case.

τ′(a) will contain τ(a)^{−} except in the case τ(a) = 0, to guarantee some information on τ(a), and in the case τ(a) = ω, τ′(a) will be the whole set τ(a)^{≺}, which makes sense, since ω^{≺} is trivially well-ordered. τ′ := λx. \underline{if} (x =_{N,B} 0 ∨ x η̸_{B} OT) \underline{then} ∅_{P} \underline{else} \underline{if} τ(x) =_{N,B} ω \underline{then} ω^{≺} \underline{else} \{τ(x)^{−}\}, that is, in a form, that can be read more easily,

τ′(a) = \{ \begin{matrix}∅_{P} & if a =_{N,B} 0 ∨ a η̸_{B} OT \\ ω^{≺} & if τ(x) = ω \\ \{τ(a)^{−}\} & otherwise. \end{matrix}

τ′ : N → P(N) (note that we could define τ′ : N → P^{dec}(N), but do not need this).

(b) We first give the definition of W (X) for X a subclass. The possibility to define W (X) for classes (in Δ_{2}^{1} − CA + BI this can not be generally be done) is the reason, why ML_{1}^{i}W_{R} has strength ψ_{Ω_{1}}(Ω_{I+ω}), rather than ψ_{Ω_{1}}(Ω_{I})): After having constructed ausgezeichnete Mengen, which exhaust the ordinals up to I, we can form “ausgezeichnete Klassen”, distinguished classes, which exhaust the ordinals up to Ω_{I+n} for every n : N, by building the union of all ausgezeichnete Mengen and iterating the step to W (X) finitely many times.

x, y, z are variables

τ_{Cl,y}^{A,z}(x) := (y η_{Cl,z} A ∧ y η OT ∧ y ≺ τ(x)) ∨ y η τ′(x)

that is, if z = y

τ_{Cl,y}^{A,y}(x) = (A ∩_{Cl,y} τ(x)^{≺}) ∪_{Cl,y} τ′(x)

We will usually not mention the variable z, sometimes not the variable y. We have, if Γ ⇒ A ∈ Cl_{z}(N) for some variable z, and x ≠ y is a new variable, then

Γ, x : N ⇒ τ_{Cl,y}^{A,z} ∈ Cl_{y}(N)

and

Γ, x : N ⇒ p : τ_{Cl,y}^{A,z}(x) ⊂_{Cl,y} OT for some p
Γ ⇒ τ_{Cl,y}^{A,z}(0) ≅ ∅,

therefore

Γ ⇒ ∀y ∈ N.y η̸ τ_{Cl,y}^{A,z}(0),

B_{1,Cl,y,x,z}(A) := Σy ∈ N.y η τ_{Cl,y}^{A,z}(x)

(if x, y are variables)

W_{1,Cl,y,z}(A) := W x ∈ N.B_{1,Cl,y,x,z}(A),

where x is a new variable, y is a new variable. and we define

Cor_{Cl,y,z}(A)(α) := ∀x ∈ N.∀u ∈ B_{1,Cl,y,x,z}(A) → W_{1,Cl,y,z}(A).
sup(x, u) ⪯_{N,B_{1,Cl,y,x,z}(A)} α → ∀v ∈ B_{1,Cl,y,x,z}(A).label(uv) = x[v0]

where u, v, x are new variables, and

label := λα.R(α, (u, v, w)u),

label(sup(r, s)) = r, label : W_{1,Cl,y}(A) → N. and have, if Γ ⇒ A ∈ Cl_{z}(N) and α is a new variable, then

Γ, α : W_{1,Cl,y,z}(A) ⇒ Cor_{Cl,y,z}(N)(α) type.

W_{Cl,y,z}(A) := ∃α ∈ W_{1,Cl,y,z}(A).Cor_{Cl,y,z}(α) ∧ label(α) = y ∧ label(α) η OT

(where α is a new variable). Therefore, if we have sup(r, s) : W_{1,Cl,y,z}(A) such that Cor_{Cl,y,z}(sup(r, s)), then the predecessors of sup(r, s) form a tree, verifying, then r is in the least set X with ∀γ η OT .τ_{Cl,y,z}^{A}(γ) ⊂ X → γ η X. For X, Y ∈ Cl_{z}(N), z′, z′′ are new variable, we define A_{Cl,y,z}^{X}(Y) := y η OT ∧ ∀z′ η τ_{Cl,z′′}^{X,z}(y).y[z′] η Y, and have A_{Cl,y,z}^{X}(Y) ∈ Cl_{y}(N). We will usually omit the Indices Cl and indices and superscripts for variables, assuming that for the z we choose the variable for which we habe proven Γ ⇒ A ∈ Cl_{z}(N), and that y is the variable usually taking for classes, and choosing the bounded variables, such that they do not cause any problems.

(c) Now we define W (X) for X : P(N).

τ := λX, x.((X ∩ τ(x)^{≺}) ∪ τ(x)′). τ : P(N) → N → P(N). and ∀X ∈ P(N).∀x ∈ N.τXx ⊂ OT We write τ(a)^{X} for τXa.

We define B_{1} := λX.λx.Σz ∈ N.z η τ(x)^{X} (written B_{1}(A) for B_{1}A) B_{1} : P(N) → N → P(N) W_{1} := λX.W x ∈ N.B_{1}(X)x W_{1} : P(N) → U

indexCor Cor_{:} = λX, α.∀r ∈ N.∀s ∈ (B_{1}(A)r → W_{1}(A)).

sup(r, s) ⪯ α → ∀z ∈ B_{1,Cl}(A)r.label(sz) = r[z0],

where label := λX.λα.R(α, (u, v, w)u), label(sup(r, s)) = r, label : ΠX ∈ P(N).W_{1}(A) → N. Cor : ΠX ∈ P(N).W_{1}X → U,

W := λX.λy.∃α ∈ W_{1}X.CorXα ∧ (labelα) = y ∧ y η OT W : P(N) → N → P(N), and we have ∀X ∈ P(N), y ∈ OT .W Xy ⊂ OT. We write W_{1}(X) for W_{1}X, W (X) for W X.

We define A := λX, Y, y.y η OT ∧ ∀z η τ(y)^{X}.y[z] η Y, A : P(N) → P(N) → P(N). We write A^{X}(Y) for AXY.

Lemma 8.12 Now we have induction over W (X), which we prove by using the recursion over W_{1}(X):

(a) If Γ ⇒ X, Y ∈ Cl_{y}(N), then Γ ⇒ W (X) ∈ Cl_{y}(N), Γ ⇒ A^{X}(Y) ∈ Cl_{y}(N), Γ ⇒ W (X), A^{X}(Y) ⊂ OT.

∀X, Y ∈ P(N).W (X) ⊂ OT ∧ A^{X}(Y) ⊂ OT.

(b) For X ∈ Cl_{y}(N) or X : P(N) follows A^{X}(W (X)) ⊂ W (X)

(c) If X, Y ∈ Cl_{y}(N) or X, Y : P(N), then if Γ ⇒ A^{X}(Z) ∩ W (X) ⊂ Z, then Γ ⇒ W (X) ⊂ Z. More precisely we have If Γ ⇒ X ∈ Cl_{y}(N), Γ ⇒ Z ∈ Cl_{z}(N)and Γ ⇒ A^{X}(Z) ∩ W (X) ⊂ Z, then Γ ⇒ W (X) ⊂ Z.

If ∀X, Z ∈ P(N).A^{X}(Z) ∩ W (X) ⊂ Z → W (X) ⊂ Z,

If Γ ⇒ Z ∈ Cl_{y}(N), then Γ ⇒ ∀X ∈ P(N).A^{X}(Z) ⊂ Z → W (X) ⊂ Z. If Γ ⇒ X ∈ Cl_{y}(N),

then Γ ⇒ ∀Z ∈ P(N).A^{X}(Z) ⊂ Z → W (X) ⊂ Z.

Proof: (a) is obvious. (Note, that we really have, if Γ ⇒ X ∈ Cl_{y}(N), then Γ ⇒ W (X) ∈ Cl_{y}(N) with the same y.

(b) statement for classes: Let a be a fresh variables. Assume a η A^{X}(W (X)), that is

a η OT ∧ ∀x η τ(a)^{X}.a[x] η W (X)

Therefore

∀x ∈ (B_{1,Cl}(X)[x/a]).∃β ∈ W_{1}(X).Cor(β) ∧ label(β) = a[x0]

Let α := sup(a, s) where for x : (B_{1,Cl}(X)[x/a]) we have sx =_{W_{1}(X)} β such that Cor(β) ∧ label(β) =_{W_{1}(X)} a[x0]. Since we have Cor(sx) for x : (B_{1}(X)[x/a]), ∀x ∈ (B_{1}(X)[x/a]).label(sx) = a[x0] and γ ⪯ α ↔ γ =_{W_{1}(X)} α ∨ ∃x ∈ (B_{1}(X)[x/a]).γ ⪯ sx follows Cor(α). The statement for subsets follows similarly.

(c) statement for classes: Assume X, Z ∈ Cl_{y}(N), A^{X}(Z) ∩ W (X) ⊂ Z. We show ∀α ∈ W_{1}(X).Cor(α) → label(α) η Z by Induction on W_{1}(X). Let α =_{W_{1}(X)} sup(r, s), Cor(α).Then r η OT, ∀z ∈ B_{1}(X)[x/r].Cor(sz). By IH we have, for z : N, p : z η τ(a)^{X}, since < z, p >η B_{1}(X)[x/r], label(α)[z0] = s < z, p >η Z, therefore r = label(α) η A^{X}(Z) ∩ W (X) ⊂ Z. The statements for subsets follows similarly. Lemma 8.13 Assume X ∈ Cl_{y}(N) or X : P(N). Then ∀x, y η W (X).x + y η W (X). Proof: Statement for X : P(N). We first prove (NF_{+}(a, b) ∧ a, b η W (X)) → a + b η W (X) Let a η W (X), and Y := λy.NF_{+}(a, y) → a + y η W (X), and have Y : P(N). We show A^{X}(Y) ⊂ Y (then follows W (X) ⊂ Y). Assume b η A^{X}(Y), NF_{+}(a, b). If b = 0, then a + b η W (X), b η Y. Otherwise τ(a + b) = τ(b) and ∀z η τ(b).(a + b)[z] = a + (b[z]), and ∀z η τ(b)^{X}.b[z] η Y, e.g. since NF_{+}(a, b[z]), ∀z η τ(b)^{X} ≅ τ(a + b)^{X}.(a + b)[z] = a + (b[z]) η W (X), a + b η W (X). Now we prove a + b η W (X) → NF_{+}(a, b) → a η W (X). Let Y := λy.∀x, z η OT .NF_{+}(x, z) ∧ x + z = y → x η W (X). Let c η A^{X}(Y) ∩ W (X), c = a + b, NF_{+}(a, b). If b = 0, then a = c η W (X), and if b ≠ 0, then τ(c) = τ(b) and ∀ξ η τ(b)^{≺}.c[ξ] = a + b[ξ]. By (F2), (F7) and (F8) follows τ(b) ≠ 0, therefore a + b[0] η Y, NF_{+}(a, b[0]), a η W (X). Now follows the assertion, by Induction on Alength(a): If NF_{+}(a, b) follows a+b η W (X), if a + b = b the assertion is trivial and if a = c + d such that NF_{+}(c, d), a + b = c + b and Alength(c) <_{N} Alength(a) and c η W (X), and the assertion follows by IH. Lemma 8.14 We state some easily proven results on W (X). Assume X, X′ ∈ Cl_{y}(N) or X, X′ : P(N), a, b : N.

(a) 0 η W (X) ∧ ∀x η W (X).x + 1 η W (X) ∧ ∀x ⪯ ω.x η W (X).

(b) (X ∩ a ≅ X′ ∩ a ∧ (∀x ≺ b.τ(x) ⪯ a)) → W (X) ∩ b ≅ W (X′) ∩ b.

(c) If X ⊂ W (X), then ∀x η R.x^{−} η W (X) → x η W (X).

(d) X ⊂ W (X) → {ω, Ω_{1}, I} ⊂ W (X).

Proof: (a) 0 η A^{X}(W (X)) ⊂ W (X) and, if a η W (X), then a + 1 η A^{X}(W (X)) ⊂ W (X), therefore by induction on N

∀n ∈ N.1 · n η W (X)

and

∀x ≺ ω.x η W (X)

It follows ω η A^{X}(W (X)) ⊂ W (X), therefore ∀x ⪯ ω.x η W (X).

(b) Let Y := λy.y ≺ b → y η W (X′), in the case of X, X′ : P(N), Y := y ≺ b → y η W (X′) in the case of classes. We proof A^{X}(Y) ⊂ Y. This implies W (X) ∩ b ⊂ W (X′). Assume ∀x η τ(y)^{X}.y[x] η Y, y ≺ b. Then τ(y)^{X} ≅ τ(y)^{X′}, ∀x η τ(y)^{X}.y[x] η W (X′), therefore y η A^{X′}(W (X′)) and thus y η W (X′).

(c) If x^{−} η W (X), then x[τ(x)^{−}] = x^{−} + x^{−} η W (X) (using lemma 8.13) and since τ(x)^{X} ⊂ X ∪ {x^{−}} ⊂ W (X) follows by 8.13 ∀ξ η τ(x) ∩ X.x[ξ] = x^{−} + ξ η W (X).

(d) ω η W (X), therefore follows the assertion by Ω_{1}^{−} = ω, I^{−} = 0. Lemma 8.15 Let X : P(N) or X ∈ Cl_{y}(N), a, b η OT.

(a) a η Lim ∧ x ≺ τ(a) ∧ a[x] ⪯ b ⪯ a[x + 1] ∧ b η W (X)) → x η W (X), esp. ∀a η Lim, x η τ(a)^{≺}.a[x] η W (X) → x η W (X).

(b) a η Lim ∧ a[0] ⪯ b ≺ a ∧ b η W (X)) → a[0] η W (X).

(c) a η Lim → b η W (X) ∩ τ(a)^{≺} → ∃x η (W (X) ∩ τ(a)^{≺}).b ⪯ a[x].

(d) X ⊂ W (X) → ∀x η W (X).τ(x) η W (X).

Proof: (a) Let Y := ∀x η τ(a)^{≺}.a[x] ⪯ y ≺ a[x + 1] → y η W (X) → x η W (X), Y ∈ Cl_{y}(N). We show A^{X}(Y) ⊂ Y, therefore by lemma 8.12 (c) W (X) ⊂ Y Let b η A^{X}(Y). Case 1: a[x] ≺ b ⪯ a[x + 1]: Then a[x] ⪯ b[0] ⪯ b[τ(b)^{−}] ≺ a[x + 1] and by b[τ(b)^{−}] η Y, x η X. Case 2: ¬(a[x] ≺ b). Then a[x] = b. Subcase x = 0: x η W (X) Subcase x = x′ + 1: Then τ(a)^{−} ⪯ x′, a[x′] ≺ b ⪯ a[x′ + 1] and by Case 1 x′ η W (X) and by 8.14 (a) follows x η W (X). Subcase x η Lim: Then τ(b) = τ(x) ∧ ((∀y ≺ τ(x).a[x[y]] = b[y]) ∨ (τ(b) = ω ∧ ∀y ≺ ω.a[x[y]] = b[y + 1])). Therefore τ(b)^{X} = τ(x)^{X} and ∀y η τ(b)^{X}.b[y] η W (X). By b η A^{X}(Y), since τ(b) = τ(x) = ω → ∀y ≺ ω.y + 1 η τ(b)^{X} follows ∀y η τ(x)^{X}.x[y] η W (X), therefore x η W (X).

(b) Let Y := a[0] ⪯ y ≺ a → a[0] η W (X), Y ∈ Cl_{y}(N). We show A^{X}(Y) ∩ W (X) ⊂ Y, therefore by lemma 8.12 (c) W (X) ⊂ Y. Let b η A^{X}(Y) ∩ W (X). If b = a[0] follows a[0] η W (X) and if a[0] ≺ b follows a[x] ≺ b ⪯ a[x + 1] for some x η τ(a)^{≺}, a[0] ⪯ a[x] ⪯ b[0] ⪯ b[τ(b)^{−}] η Y, a[0] η W (X)

(c) If b ⪯ a[0] we chose x := 0 η W (X) If a[0] ≺ b we have by (F2) a[x] ⪯ b ≺ a[x + 1] for some x η τ(a)^{≺}. By (a) we obtain x η (W (X) ∩ τ(a)^{≺}).

(d) If a η W (X), then a[τ(a)^{−}] η W (X), τ(a)^{−} η W (X) by (a), τ(a) η W (X) by 8.14 (a),

(c) and (d). Definition 8.16 Now we define the meaning of “X is ausgezeichnete Menge”: Let X, Y : P(N) or X, Y ∈ Cl_{y}(N). X|a := X ∩ (a + 1). X ⊑ Y := ∀x η X.X|a ≅ Y |a (this is equivalent to X ⊂ Y ∧ ∀x η X.Y ∩ x ⊂ X, X is a segment of Y). Ag(X) := X ⊂ OT ∧ X ⊑ W (X), X is an “ausgezeichnete Menge”, a “distinguished set”. P rog(X, Y) := ∀x η X.X ∩ x ⊂ Y → x η Y. P rog(Y) := ∀x ≺ Ω_{1}.OT ∩ x ⊂ Y → x η Y. Lemma 8.17 From the induction over W (X) we conclude induction over ausgezeichnete Mengen: Assume X, Y : P(N) or X, Y ∈ Cl_{y}(N), such that Ag(X).

(a) X ⊑ A^{X}(X).

(b) X ∩ A^{X}(Y) ⊂ Y → X ⊂ Y.

(c) P rog(X, Y) → X ⊂ Y.

(d) X ∩ Ω_{1} ⊑ OT.

(e) P rog(Y) → X ∩ Ω_{1} ⊂ Y.

Proof: We treat here the case X, Y η Cl_{y}(N).

(a) a η X → X|a ≅ W (X)|a → A^{X}(X)|a ≅ A^{X}(W (X))|a ≅ W (X)|a ≅ X|a.

(b) Let Y′ := y η X → y η Y ∈ Cl_{y}(N) and assume X ∩ A^{X}(Y) ⊂ Y. Then, since X ⊂ A^{X}(X), ∀y η (X ∩A^{X}(Y′)).∀z η τ(y)^{X}.y[z] η Y therefore X ∩A^{X}(Y′) ⊂ X ∩A^{X}(Y) ⊂ Y, therefore A^{X}(Y′) ⊂ Y′, X ⊂ W (X) ⊂ Y′, X ⊂ Y.

(c) Let Y′ := X ∩ y ⊂ Y, we have Γ ⇒ Y′ ∈ Cl_{y}(N) ( more precisely Y′ := ∀u ∈ N.u η X ∧ u ≺ y ∧ u η OT → u η Y, where u is a new variable). We show X ∩_{Cl} A_{Cl}^{X}(Y′) ⊂_{Cl} Y′ (∗)

Proof: Let a η X ∩ A^{X}(Y′) and assume b η X ∩ a. From a η X ⊑ W (X) and b η X ∩ a follows by lemma 8.15 (a) b ⪯ a[x] for some x η W (X) ∩ τ(a)^{≺} ≅ X ∩ τ(a)^{≺}, (since τ(a) ⪯ a η X ⊑ W (X). By a η A^{X}(Y′) follows a[x] η Y′, X ∩ b ⊂ X ∩ a[x] ⊂ Y, and by P rog(X, Y) and b η X follows b η Y for arbitrary b η X ∩ a, therefore X ∩ a ⊂ Y, a η Y′, and we have (∗). By (b) and (∗) follows X ⊂ Y′, that is ∀x η X.X ∩ x ⊂ Y and using again P rog(X, Y), ∀x η X.x η Y.

(d) Let Y := y ≺ Ω_{1} → ∀z ≺ y.z η X (z a new variable). We prove X ∩ A^{X}(Y) ⊂ Y. By

(b) follows the assertion. Assume a η X ∩ Ω_{1}, a η A^{X}(Y). From ∀x η τ(a)^{X}.a[x] η Y we conclude a[x] ⊂ X, a[x] η W (X), and by a[x] ≺ a η X ⊑ W (X), a[x] η X, therefore ∀x η τ(a)^{X}.OT |a[x] ⊂ X. Since τ(a) η {0, 1, ω}, follows τ(a)^{X} ≅ τ(a), a ⊂ X.

(e) Assuming P rog(Y) follows by (d) ∀x η X ∩ Ω_{1}.X ∩ x ⊂ Y → x η Y, therefore with Y′ := y ≺ Ω_{1} → y η Y P rog(X, Y′) and by (c) X ∩ Ω_{1} ⊂ Y.

Lemma 8.18 Let Γ ⇒ X ∈ Cl_{y}(N) or X : P(N), Ag(X). Then ∀x η X.τ(x) η X. Proof: If a η X, then a η W (X), τ(a) η W (X) by 8.15 (d), and, since τ(a) ⪯ a η X ⊑ W (X), τ(a) η X. Lemma 8.19 We state, that ausgezeichnete Mengen are unique, in the following sense: Let X_{i} ∈ Cl_{y}(N) or X_{i} : P(N). If X_{i} ≅ W (X_{i}) ∩ a (i = 1, 2), then X_{0} ≅ X_{1}. Proof: We treat the case X_{i} ∈ Cl_{y}(N). We have

(1) If U ∈ Cl_{y}(N) and P rog(X_{0} ∪ X_{1}, U) then X_{0} ∪ X_{1} ⊂ U

Proof: Assume P rog(X_{0} ∪ X_{1}, U). Assume b such that X_{0} ∩ b ⊂ U. Then ∀c ≺ b.(c η X_{1}∧X_{1}∩c ⊂ U) → c η U. Therefore we have P rog(X_{1}∩b, U) and by lemma 8.17 (c), since we have Ag(X_{1}) X_{1} ∩ b ⊂ U. By P rog(X_{0} ∪ X_{1}, U) we have b η U for b with X_{0} ∩ b ⊂ U. Therefore we have P rog(X_{0}, U) and using Ag(X_{0}) and lemma 8.17 (c) X_{0} ⊂ U. Similarly we conclude X_{1} ⊂ U. and we have (1). We show P rog(X_{0} ∪ X_{1}, X_{0} ∩ X_{1}) (2)

Proof: Assume b η X_{0} ∪ X_{1} and (X_{0} ∪ X_{1}) ∩ b ⊂ X_{0} ∩ X_{1}. Then we have b ≺ a and X_{0} ∩ b ≅ X_{1} ∩ b and therefore X_{0}|b ≅ W (X_{0})|b ≅ W (X_{1})|b ≅ X_{1}|b, and b η X_{0} ∩ X_{1}. By (1) and (2) follows the assertion.

Next step is to define W as the union of all ausgezeichnete Mengen. It is itself a ausgezeichnete Klasse, closed under λx.Ω_{x} for x ≺ I, and will exhaust the ordinals up to I: Definition 8.20 W := ∃X ∈ P(N).Ag(X) ∧ y η X, W ∈ Cl_{y}(N). Lemma 8.21 ∀X ∈ P(N).Ag(X) ↔ X ⊑ W, that is: the ausgezeichnete Mengen are just segments of W. Proof: “⇒”: X ⊂ W is clear. Assume a η X, b η W ∩ a. Then there exists Y ∈ P(N) with b η Y and Ag(Y). X′ := X|b, Y′ := Y |b. Then

W (X′)|b ≅ W (X)|b  by lemma 8.14 (b)
≅ X|b  X ⊑ W (X), a η X, b ≺ a
≅ X′

W (Y′)|b ≅ W (Y)|b  by lemma 8.14 (b)
≅ Y |b  Y ⊑ W (Y), b η Y
≅ Y′

Therefore by lemma 8.19 X′ ≅ Y′, b η Y′ ≅ X′ ⊂ X. “⇐”If a η X, then there exists Y : P(N) such that a η Y and Ag(Y). By the proof of “⇒” follows Y ⊑ W, therefore X|a ≅ W|a ≅ Y |a ≅ W (Y)|a, therefore W (X)|a ≅ W (Y)|a ≅ X|a, and we have Ag(X). Lemma 8.22

Ag(W)

Proof: Let a η W. Then a η X ⊑ W for some X : P(N). It follows W|a ≅ X|a, therefore W (W)|a ≅ W (X)|a ≅ X|a ≅ W|a. Lemma 8.23 ∀X ∈ P(N).∀x η OT .(X ∩ x ≅ W (X) ∩ x ∧ x η W (X)) → x η W. Proof: Assume X : P(N), a η OT, X ∩a ≅ W (X)∩a∧a η W (X), and let Y := W (X)|a. Then X ∩ a ≅ Y ∩ a and therefore Y ≅ W (X)|a ≅ W (Y)|a, therefore Ag(Y). Lemma 8.24 (a) (∀x ⪯ ω.x η W) ∧ ∀x η W.x + 1 η W.

(b) ∀x η OT .Ω_{x} η W → Ω_{x+1} η W. Proof:

(a) X := λy.y ⪯ ω : P(N), and we have Ag(X). further, if X : P(N), Ag(X), a η X, then X ∩(a+1) ≅ W (X)∩(a+1) and a+1 η W (X), by 8.23 a + 1 η W. (b): Let Ω_{s} η X, Ag(X) for some X : P(N), Y := W (X), Y : P(N), a := Ω_{s+1}. Then Ω_{s+1} η Y. X ∩ Ω_{s} ≅ Y ∩ Ω_{s}, therefore Y ∩ a ≅ W (X) ∩ a ≅ W (Y) ∩ a, by 8.23 follows Ω_{s+1} η W. Lemma 8.25 ∀x η A^{W}(W) ∩ I.τ(x) η W. Proof: If 0 ≠ a η A^{W}(W), then a[τ(a)^{−}] η W, therefore a[τ(a)^{−}] η X for some X : P(N) with Ag(X). Therefore by lemma 8.15 τ(a)^{−} η W (X) and, since τ(a)^{−} ⪯ a[τ(a)^{−}] η X ⊑ W (X), τ(a)^{−} η X. We have τ(a)^{−} η (R \ {I}) ∪ {0, 1, ω}, so by lemma 8.24 (b) (or trivially in the cases τ(a) η {0, 1, ω}) follows τ(a) η W. Lemma 8.26 ∀x η A^{W}(W) ∩ I.(∃X ∈ P(N).X ≅ W ∩ x) → x η W. Proof: Assume a η A^{W}(W) ∩ I. By lemma 8.25 τ(a)ηW. Let τ(a) ≺ a and X ≅ W ∩ a. Then τ(a)ηX and Ag(X). We prove:

W (X) ∩ a ⊂ X (∗)

Let bηW (X) ∩ a. Then there exists by lemma 8.15 xηW (X) ∩ τ(a)^{≺} such that b ⪯ a[x]. Since τ(a)ηX ⊑ W (X), we conclude xηW ∩ τ(a)^{≺} and a[x]ηW ∩ a ≅ X. Since bηW (X)|a[x] and a[x]ηX follows bηX, and we have (∗). Now aηA^{W}(W) and X ≅ W ∩ a, therefore aηA^{X}(X) ⊂ A^{X}(W (X)) ≅ W (X). Since X ≅ W (X) ∩ a and aηW (X) follows aηW by lemma 8.23. Lemma 8.27 W (W) ∩ I ≅ A^{W}(W) ∩ I ≅ W ∩ I. Proof: We prove A^{W}(W) ∩ I ≅ W ∩ I: “⊃”: Ag(W), therefore W ⊑ A^{W}(W). “⊂”: Let a η A^{W}(W) ∩ I. Then τ(a) η W ∧ ∀x η W ∩ τ(a)^{≺}.a[x] η W. Therefore exists Q : P(N) such that Ag(Q) ∧ τ(a) η Q and with B := Σx ∈ N.x η Q ∩ a^{≺} exists s : B → P(N) such that ∀x ∈ B.Ag(sx) ∧ a[x0] η sx. Let P := λy.y η Q ∨ ∃x ∈ B.y η (sx), P : P(N). Then, since P is union of distinguished sets, follows Ag(P), P ⊑ W. Let c ≺ a, c η W, then there exists x η W (W) ∩ τ(a)^{≺} (≅ W ∩ τ(a)^{≺} ≅ Q ∩ τ(a)^{≺}) such that c ≺ a[x], therefore, since a[x] η P ⊑ W,c η P. We have therefore P ∩ a ≅ W ∩ a and by lemma 8.26 a η W. Now, with Y := y ≺ I → y η W, follows A^{W}(Y) ∩ I ≅ A^{W}(W) ∩ I ≅ W ∩ I, W (W) ⊂ Y, W (W) ∩ I ⊂ W ∩ I. W ⊂ W (W) follows by Ag(W).

General Assumption 8.28 s To show, that the ausgezeichnete Mengen and Klassen are closed under the Veblen-function ϕ, the collapsing function D, and that W is closed under λx.Ω_{x}, (x ≺ I), we need some assumptions about the fundamental sequences, and need to introduce some more functions:

(a) We assume functions

ϕ̂, D : N → N → N

(ϕ̂ is a version free of fixed points of the Veblen-function, D the collapsing function)

Dicom, NF_{D,B} : N → N → B

(Dicom will be the largest D_{I}a, contained in an ordinal. Dicom stands for “D_{I}-component”. NF_{D,B}(a, b) stands for D_{a}b is in normal-form and we define NF_{D} := λx, y.atom(NF_{D,B}xy))

Cr : N → P^{dec}(N)

(Cr(a) will be the set of a-critical ordinals)

length : N → N

(the length of an ordinal term), subsets

G, F i : P^{dec}(N)

(G will be the Gamma ordinals, the a such that ϕ_{a}0 = a, ϕ being the usual version of the Veblen-function (with fixed points). F i will be the fixed points of λx.Ω_{x}, that is I and the ordinals D_{I}a)

We assume the following list of properties:

F i ⊂ G ⊂ A ⊂ OT, R ⊂ G

∀x, y η OT , n ∈ N.ϕ̂_{x}y η OT ∧ (NF_{D}(x, y) → D_{x}y η OT)

We want, that every element of OT is constructed by elements of smaller length:

∀x η OT .∃y, z η OT .length(y), length(z) <_{N} length(x)∧ (NF_{+}(y, z) ∧ z η A ∧ x = y + z)∨

x = ϕ̂_{y}z∨
(NF_{D}(y, z) ∧ x = D_{y}z)∨
x = Ω_{y}∨
x = 0 ∨ x = I,

We demand some properties of the D-normal-form: ∀x η R.NF_{D}(x, 0) ∧ ∀y η OT .NF_{D}(x, y) → NF_{D}(x, y + 1) ∀x, y η OT .NF_{D}(x, y) → D_{x}y ≺ x ∀n ∈ N.NF_{D}(Ω_{1}, Ω_{I+1·n}) ∀x, y η OT .NF_{D}(x, y) → x η R.

We claim that Dicom(x), is the largest x ∈ F i below x:

∀x η OT .D_{I}0 ⪯ x ≺ I → ∃y η OT .NF_{D}(I, y) ∧ Dicom(x) = D_{I}y ∧ D_{I}y ⪯ x ≺ D_{I}(y + 1)

And we characterize the critical numbers:

∀x, y, z η OT .(NF_{+}(x, y) → x + y η̸ Cr(z))∧ (ϕ̂_{x}y η Cr(z) ↔ z ≺ x) ∧ (x η G → (x η Cr(z) ↔ z ≺ x)), characterize the additive principal numbers: ∀x, y η OT .(NF_{+}(x, y) → x + y η̸ A) ∧ ϕ̂_{x}y η A ∧ 0 η̸ A∧

(NF_{D}(x, y) → D_{x}y η A) ∧ Ω_{x} η A ∧ I η A. the Gamma numbers: ∀x, y η OT .(NF_{+}(x, y) → x + y η̸ G) ∧ ϕ̂_{x}y η̸ G ∧ 0 η̸ G

∧(NF_{D}(x, y) → D_{x}y η G) ∧ (x ≠ 0 → Ω_{x} η G) ∧ I η G. the fixed points of λx.Ω_{x}: ∀x, y η OT .(NF_{+}(x, y) → x + y η̸ F i) ∧ ϕ̂_{x}y η̸ F i∧

0 η̸ F i ∧ (NF_{D}(x, y) → (D_{x}y η F i ↔ x = I)) ∧ (Ω_{x} η F i ↔ x η F i) ∧ I η F i. and the regular cardinals: ∀x, y η OT .(NF_{+}(x, y) → x + y η̸ R) ∧ ϕ̂_{x}y η̸ R ∧ 0 η̸ G∧

(NF_{D}(x, y) → D_{x}y η̸ R)∧ (Ω_{x} η R ↔ (∃z η OT .x = z + 1)) ∧ I η R.

Further we assume ω = ϕ̂_{0}1.

(b) We assume, that the fundamental sequences are built as follows:

(A[ ].0) τ(0) = ∅. (A[ ].1) If NF_{+}(b, c), a = b + c, b η OT, c ≠ 0, then

τ(a) = τ(c), (b + c)[z] = b + (c[z])

(A[ ].2) Case a = ϕ̂_{b}c: (A[ ].2.1) Case b = 0: (A[ ].2.1.1) c = 0 ⇒ τ(a) = 1, a[z] = 0 (A[ ].2.1.2) c = c′ + 1 ⇒ τ(a) = ω,

a[1 · n] = ϕ̂_{0}(c′) · (SSn).

(A[ ].2.1.3) (c η Cr(b)) ⇒ τ(a) = ω,

a[1 · n] = c · (Sn).

(A[ ].2.1.4) (c η Lim ∧ (c η̸ Cr(b))) ⇒ τ(a) = τ(c),

a[z] = ϕ̂_{0}(c[z])

(A[ ].2.2) Case b = b′ + 1: (A[ ].2.2.1) c = 0 ⇒ τ(a) = ω,

a[1 · n] = ρ_{SSSn} where ρ_{0} = 0, ρ_{Sn} = ϕ̂_{b′}ρ_{n}

(A[ ].2.2.2) c = c′ + 1 ⇒ τ(a) = ω,

a[1 · n] = ρ_{n} where ρ_{0} = ϕ̂_{b}c′, ρ_{Sn} = ϕ̂_{b′}ρ_{n}

(A[ ].2.2.3) (c η Cr(b)) ⇒ τ(a) = ω,

a[1 · n] = ρ_{Sn}, where ρ_{0} = c, ρ_{Sn} = ϕ̂_{b′}ρ_{n}

(A[ ].2.2.4) (c η Lim ∧ c η̸ Cr(b)) ⇒ τ(a) = τ(c),

a[z] = ϕ̂_{b}(c[z])

(A[ ].2.3) Case b η Lim: (A[ ].2.3.1) (c = 0 ∧ b η OT \ G) ⇒ τ(a) = τ(b),

a[z] = ϕ̂_{b[z]}0

(A[ ].2.3.2) (c = 0 ∧ b η G) ⇒ τ(a) = τ(b),

a[z] = ϕ̂_{b[z]}b

(A[ ].2.3.3) c = c′ + 1 ⇒ τ(a) = τ(b),

a[z] = ϕ̂_{b[z]}(ϕ̂_{b}c′)

(A[ ].2.3.4) (c η Cr(b)) ⇒ τ(a) = τ(b),

a[z] = ϕ̂_{b[z]}c

(A[ ].2.3.5) (c η Lim ∧ c η̸ Cr(b)) ⇒ τ(a) = τ(c),

a[z] = ϕ̂_{b}(c[z])

(A[ ].3) Case a = D_{b}c: (A[ ].3.1) Case c = 0: (A[ ].3.1.1) b ≠ I ⇒ τ(a) = ω,

a[1 · n] = ρ_{Sn}, where ρ_{0} = b^{−}, ρ_{Sn} = ϕ̂_{ρ_{n}}0.

(A[ ].3.1.2) b = I ⇒ τ(a) = ω,

a[1 · n] = ρ_{SSn}, where ρ_{0} = 0, ρ_{Sn} = Ω_{ρ_{n}}.

(A[ ].3.2) Case c = c′ + 1: (A[ ].3.2.1) b ≠ I ⇒ τ(a) = ω,

a[1 · n] = ρ_{n}, where ρ_{0} = D_{b}c′, ρ_{Sn} = ϕ̂_{ρ_{n}}0.

(A[ ].3.2.2) b = I ⇒ τ(a) = ω,

a[1 · n] = ρ_{n}, where ρ_{0} = D_{I}c′, ρ_{1} = Ω_{ρ_{0}|1}, ρ_{SSn} = Ω_{ρ_{Sn}}.

(A[ ].3.3) Case c η Lim, τ(c) ≺ b:

τ(a) = τ(c), a[z] = D_{b}(c[z]).

(A[ ].3.4) Case c η Lim, b ⪯ τ(c):

Then τ(a) = ω, a[1 · n] = D_{b}c[ζ_{n}], where ζ_{n} is defined by:

(A[ ].3.4.1) τ(c) ≠ I ∨ b ≺ D_{I}c ∨ b = I ⇒

ζ_{0} = 0, ζ_{Sn} = D_{π}(c[ζ_{n}]), π := τ(c).

(A[ ].3.4.2) (τ(c) = I ∧ D_{I}c ⪯ b ≺ I) ⇒

ζ_{0} = Dicom(b), ζ_{1} = Ω_{ζ_{0}+1}, ζ_{SSn} = Ω_{ζ_{Sn}}.

(A[ ].4) Case a = Ω_{b}, b ≠ Ω_{b}: (A[ ].4.1) Case a η R:

τ(a) = a, a[z] = a^{−} + z

(A[ ].4.2) Case a η̸ R:

τ(a) = τ(b), a[z] = Ω_{b[z]} or a[z] = Ω_{b[z]+1}

(A[ ].5) Case a = I:

τ(a) = I, a[z] = a^{−} + z.

Further we assume, that in (A[ ].1) we have NF_{+}(b, c[z]) and in the cases (A[ ].3.4) we have ∀n ∈ N.ζ_{n} ≺ τ(c).

The next lemma is one of the most important, it shows that W ∩ I is closed under Ω_{·}. Lemma 8.29 (a) ∀X ∈ P(N).∀a, b η W (X).ϕ̂_{a}b η W (X).

(b) If X ∈ Cl_{y}(N), then ∀a, b η W (X).ϕ̂_{a}b η W (X). Proof:

(a) Let Y := λx.∀y η W (X).ϕ̂_{x}y η W (X), we show A^{X}(Y) ∩ W (X) ⊂ Y, and then follows assertion (a). Assume x η A^{X}(Y) ∩ W (X), let Y′ := λy.ϕ̂_{x}y η W (X). We show A^{X}(Y′) ∩ W (X) ⊂ Y′, and then follows the assertion. Assume y η A^{X}(Y′) ∩ W (X), and let a := ϕ̂_{x}y. Case y η Lim \ Cr(x): Then ∀z η τ(y)^{X} = τ(a)^{X}.y[z] η Y′, therefore ϕ̂_{x}y[z] η W (X), a η W (X). Assume therefore y η̸ Lim \ Cr(x): Case x = 0: Subcase y = 0: a[z] = 0 η W (X).

Subcase y = y′ + 1: y′ η Y′ since 0 η τ(y)^{X} and y η A^{X}(Y′). Therefore ϕ̂_{x}y′ · n η W (X) and ϕ̂_{x}y η W (X). Subcase y η Cr(x): y η W (X), therefore ϕ̂_{x}y[1·n] = y·Sn η W (X), therefore ϕ̂_{x}y η W (X). Case x = x′ + 1: Then x′ η Y, ∀y η W (X).ϕ̂_{x′}y η W (X). Subcase y = 0: ρ_{0} = 0 η W (X), and if ρ_{n} η W (X), then ρ_{Sn} = ϕ̂_{x′}ρ_{n} η W (X), therefore a η W (X). Subcase y = y′ + 1: y′ η Y′, therefore ρ_{0} = ϕ̂_{x}y′ η W (X), and if ρ_{n} η W (X), then ρ_{Sn} = ϕ̂_{x′}ρ_{n} η W (X), therefore a η W (X). Subcase y η Cr(x): ρ_{0} = y η W (X), and if ρ_{n} η W (X), then ρ_{Sn} = ϕ̂_{x′}ρ_{n} η W (X), therefore a η W (X). Case x η Lim: Subcase y = 0, x η̸ G: ∀z η τ(x)^{X} = τ(a)^{X}.ϕ̂_{x[z]}0 η W (X), a η W (X). Subcase y = 0, x η G: ∀z η τ(x)^{X} = τ(a)^{X}.ϕ̂_{x[z]}x η W (X), a η W (X). Subcase y = y′ + 1: y′ η Y′, therefore ϕ̂_{x}y′ η W (X), therefore ∀z η τ(x)^{X} = τ(a)^{X}.ϕ̂_{x[z]}(ϕ̂_{x}y′) η W (X), a η W (X). Subcase y η Cr(x): y η W (X), therefore ∀z η τ(x)^{X} = τ(a)^{X}.ϕ̂_{x[z]}y η W (X), a η W (X).

(b) follows as (a) . Lemma 8.30 Let X : P(N) or X ∈ Cl_{y}(N).

(a) X ⊂ W (X) → ∀x η X.Ω_{x} ∈ W (X).

(b) (I η X ∧ Ag(X)) → ∀a η X ∩ I.Ω_{a} η X.

Proof:

(a) Let Y := Ω_{y} η W, Y ∈ Cl_{y}(N). We show A^{X}(Y) ∩ W (X) ⊂ Y, then follows W (X) ⊂ Y. Assume a η A^{X}(Y) ∩ X. Case a = 0: Ω_{0} = ω η W (X) by 8.14 (a). Case a = a′ + 1: Ω_{a′} η W (X), by lemma 8.14 (c) Ω_{a} η W (X). Case Ω_{a} = a: a η W (X). Case a η Lim, a ≠ Ω_{a}: Then ∀x η τ(a)^{X}.Ω_{a}[x] = Ω_{a[x]} ∨ Ω_{a}[x] = Ω_{a[x]+1}, by assumption Ω_{a[x]} η W (X) and by 8.14 (c) Ω_{a[x]+1} η W (X) for x η τ(a)^{X}, therefore Ω_{a} η A^{X}(W (X)) ≅ W (X).

(b) If a ∈ X ∩ I follows Ω_{a} ∈ W (X) ∩ I, Ω_{a} ∈ X. Lemma 8.31 W (X)!closed under D Let X : P(N) or X ∈ Cl_{y}(N), Ag(X). Then ∀a, b η X.NF_{D}(a, b) → D_{a}b η X. Proof: We consider the case X : P(N): Let Y := λy.y η X → ∀x η X.D_{x}y η W (X), (in case X ∈ Cl_{y}(N) we would choose Y := y η X → ∀x η X.D_{x}y η W (X) for x a fresh variable). We show A^{X}(Y) ∩ W (X) ⊂ Y, and then follows W (X) ⊂ Y and, since X ⊑ W (X) and D_{a}b ≺ a the assertion. Assume x η A^{X}(Y) ∩ X, y η X, and let a := D_{x}y. Case y = 0: If x ≠ I, then x η W (X), therefore ρ_{0} = x^{−} = x[0] η W (X) by 8.15 (b) and if ρ_{n} η W (X) follows ρ_{Sn} η W (X), therefore a η W (X). If x = I, then ρ_{0} η W (X) and if ρ_{n} η W (X), then, since ρ_{n} ≺ D_{I}0 ≺ I η X, X ⊑ W (X), ρ_{n} η X, ρ_{Sn} η W (X) by lemma 8.30 (b), therefore a η W (X).

Case y = y′ + 1: If x ≠ I, then, since y′ η Y follows ρ_{0} = D_{x}y′ η W (X), and by induction ρ_{n} η W (X), a η W (X). If x = I, then ρ_{0} η W (X) since y′ η Y, and by induction on n : N, since ρ_{n} ≺ a ≺ I and, if ρ_{0} η W (X) then ρ_{0} + 1 η W (X), follows by lemma 8.30 ρ_{n} η W (X) and therefore a η W (X). Case y η Lim, τ(y) ≺ x: Since y[z] η Y for z η τ(y)^{X} = τ(a)^{X} follows a[z] η W (X) for these z and therefore a η W (X). Case y η Lim, x ⪯ τ(y): In this case we always show ∀n ∈ N.ζ_{n} η W (X). Since ζ_{n} ≺ y η X, ζ_{n} η τ(y)^{≺}, follows ζ_{n} η τ(y)^{X}, y[ζ_{n}] η Y, a[1 · n] = D_{x}(y[ζ_{n}]) η W (X) If τ(y) ≠ I or x ≺ D_{I}y or x = I, follows, since by y η W (X) we conclude by 8.15 (d) τ(y) η W (X), t := τ(y) η X, and, since y η X, if ζ_{n} η W (X), then ζ_{n} η X, x[ζ_{n}] η Y, ζ_{Sn} = D_{t}(a[ζ_{n}]) η W (X), by induction therefore ∀n ∈ N.ζ_{n} η W (X). If τ(y) = I ∧ D_{I}y ⪯ x ≺ I we have Dicom(x) = D_{I}a for some y ⪯ a η OT, such that NF_{D}(I, a). Since D_{I}(a + 1)[0] = D_{I}a ⪯ x ≺ D_{I}(a + 1), follows by lemma 8.15 (c) ζ_{0} = Dicom(x) η W (X), and because ζ_{n} ≺ I, by an immediate induction ζ_{n} η W (X).

Now we want to exhaust the full power of ML_{1}^{i}W_{R}, by forming the classes W_{i}, that are syntactically increasing objects (so we can not define them internal in Martin-Löf’s type theory, but only by external meta induction), and where W_{Si} allows to prove transfinite induction up to D_{Ω_{1}}(Ω_{I+1·i}). Definition 8.32 W_{0} := W ∩ I, W_{Si} := W (W_{i}) ∩ Ω_{I+1·Si}. Lemma 8.33 For all i ≺ ω Ag(W_{i}) ∧ Ω_{I+1·i} η W_{Si} ∧ W_{i} ≅ W_{Si} ∩ Ω_{I+1·i}. Proof: Meta Induction on i : N: i = 0: By lemma 8.27 W_{0} ≅ W (W) ∩ I, and since W ∩ I ≅ W_{0} ∩ I follows W_{0} ≅ W (W_{0}) ∩ I ≅ W_{1} ∩ I. Therefore Ag(W_{0}) ∧ W_{0} ≅ W_{1} ∩ Ω_{I}. Further I^{−} = 0 η W_{0} ∧ ∀x η τ(I)^{W_{0}}.I[x] = x η W_{0} ⊂ W (W_{0}), therefore Ω_{I} = I η W (W_{0}) ∩ Ω_{I+1} ≅ W_{1}. i = Sj: W_{j} ≅ W_{Sj} ∩ Ω_{I+1·j}. Therefore W_{Sj} ≅ W (W_{j}) ∩ Ω_{I+1·(Sj)} ≅ W (W_{Sj}) ∩ Ω_{I+1·(Sj)} ≅ W_{SSj} ∩ Ω_{I+1·(Sj)}, therefore Ag(W_{i}). Further Ω_{I+1·(Sj)}^{−} = Ω_{I+1·j} η W_{Sj} ∧ ∀x η τ(Ω_{I+1·(Sj)})^{W_{Sj}}.Ω_{I+1·(Sj)}[x] = Ω_{I+1·j} + x η W_{Sj} ⊂ W (W_{Sj}), and we have Ω_{I+1·(Sj)} η W (W_{Sj}) ∩ Ω_{I+1·(SSj)} ≅ W_{SSj}. Theorem 8.34 For all n ∈ IN we have: ML_{1}^{i}W_{R} ⊢ ∀X ∈ P(N).(∀y ∈ N.(∀x ∈ N.x ≺ y → x η X) → y η X) → ∀y ∈ N.y ≺ D_{Ω_{1}}Ω_{I+1·n} → y η X. Proof: Assume the premise of the assertion. Then X : P(N) and P rog(X), therefore by lemma 8.17 (e) and 8.33 W_{Sn} ∩ Ω_{1} ⊂ X. By lemma 8.33 Ω_{I+1·n} η W_{Sn} and Ω_{1} η W ∩ R ∩ I ⊂ W_{Sn} ∩ R, therefore by lemma 8.31 D_{Ω_{1}}Ω_{I+1·n} η W_{Sn}. Since W_{Sn} ∩ Ω_{1} ⊑ OT follows ∀y ∈ N.y ≺ D_{Ω_{1}}Ω_{I+1·n} → y η X.

Properties of the ordinals

In this chapter will now do all the technical work for the introduction of the ordinal denotation system OT. All the work done in this chapter could be carried out as well in HA, that is, we define primitive recursive functions and relations and only need induction over natural numbers in the proofs. To get unique terms, we will use in this chapter fix point-free versions for the Veblenfunction and the enumeration of the infinite cardinals. The analysis follows well known proof theoretical techniques, which can be found in [BS88], [Buc86], and on which the author did a lot of work in his Diplomarbeit [Set90]. Old versions of this system where studied for instance in [Buc75b] ,[BS76] We start to introduce sets T′′, T′ and OT (definitions 9.2, 9.7 and 9.10). We use for introducing sums of additive terms a construction ˜+, which is the sum of an ordinal with an additive principal number, to avoid the introduction of lists. We define the ordering (definition 9.4) and introduce its properties (9.5). Further we define the set of coefficients G_{π}a needed for the definition of OT. Now we introduce functions like +, Ω_{·}, and the set of critical numbers (definition 9.11), show some properties for the lists P l(a) (in some sense here we switch back to the ordinary introduction of T′ — 9.12). We show some easy properties on T′′ (9.13, 9.16), closure properties of T′ and OT (9.14), some ordering properties of the introduced functions (9.15, 9.17, 9.20), properties of the critical numbers (9.18) and Gamma numbers (9.19). Now we define the fundamental sequences, first a version which we will afterwards modify a little bit for the well-ordering proof (definition 9.21; essentially the difference is that in 9.21 for regular cardinals a[z] := z, later we will replace this by a[z] = a^{−} + z to avoid fixed points of Ω_{·}). We show some easy properties (9.22 and 9.23). In definition 9.24 we introduce a^{∗}, some predecessor relation, having the property that for a modified length lh′ we have lh′(a^{∗}) < lh′(a) (9.25), that s^{∗} ⪯ s[[τ(s)^{−}]] (9.26) s^{∗} ≺ t ⪯ s → s^{∗} ⪯ t^{∗} (9.28). We get no new critical numbers, Gamma numbers, regular cardinals or fixed points of Ω̂ between s^{∗} and s (9.27). If we introduce t ≪ s ⇔

∃n ∈ N.0 <_{N} n ∧ s\underbrace{∗ ⋯ ∗}_{ntimes} =_{N} t (definition 9.29), we can show that it exchanges with many functions (9.30), and that, if ξ ≪ ρ, then s[[ξ]] ≪ s[[ρ]] (9.31), and conclude the Bachmann property (lemma 9.32). To show that OT is closed under forming the fundamental sequences and that a = sup{a[[ξ]]|ξ ≺ τ(a) ∧ ξ η OT } for a η OT, we motivate on page 109 and introduce in definition 9.33 the relation <_{ξ}, which exchanges with many functions (9.35), allows to deduce properties of G_{π}a (9.36). In lemma 9.37, we show that s^{∗} <_{0} s, and in lemma 9.38, that a[[z]] <_{z} a. With these properties we can show that we have real fundamental sequences (theorem 9.39) and that OT is closed under fundamental sequences (lemma 9.40). At the end we modify the fundamental sequences, as stated before. (definition 9.43), prove that essentially we have a[b[z]] = a[b][z] (lemma 9.44) and prove all other properties (lemma 9.45).

Remark 9.1 We define some Gödel-numbering for the terms: ˜+ := λx, y. < ⌈ ˜+⌉, < x, y >>, ˜+N → N → N written as a ˜+b for ˜+ab, which corresponds to the sum of an ordinal with an additive principal number in normal-form, ϕ̂ := λx, y. < ⌈ϕ̂⌉, < x, y >>, ϕ̂ : N → N → N, written as ϕ̂_{a}b for ϕ̂ab, which corresponds to the fixed point free version of the Veblen-function Ω̂ := λx. < ⌈Ω̂⌉, x >, Ω̂ : N → N, written as Ω̂_{a} for Ω̂a, which corresponds to the version of the enumeration of the infinite cardinals, which is free of fixed points D := λx, y. < ⌈D⌉, < x, y >>, D : N → N → N, written D_{a}b for Dab, which corresponds to the collapsing function I :=< ⌈I⌉, 0 >, (which corresponds to the first weakly inaccessible cardinal) 0_{OT} :=< ⌈0⌉, 0 >, (for 0), where ⌈D⌉, ⌈Ω̂⌉, ⌈ϕ̂⌉, ⌈ ˜+⌉, ⌈I⌉, ⌈0⌉ are different natural numbers, and < ·, · > is some primitive recursive pairing operation on N. We will introduce three different sets of terms, T′′, T′, and OT. T′′ will be essentially all objects, we can construct by the operations 9.1, demanding only very small normality properties of the terms, without reference to the ordering. T′ will be a subset of T′′, by restricting the terms to those, which have better properties with respect to ≺, which was introduced on T′′. OT demands some more conditions for the D_{x}y. It is quite difficult, to prove, that the for a, b η OT a[b] η OT (b ≺ τ(a)), one of the technically difficult tasks of this chapter, at first hand we only get this closure property for T′. Besides T′′ we define sets Suc′′, A′′, G′′, R′′, F i′′, the restriction of them to T′ are Suc′, A′, G′, R′, F i′, and, restricted to OT, we get Suc, A, G, R, F i where (where Lim will be the limit ordinals, A the additive principal numbers, G Gamma ordinals, F i will be the fixed points of λx.Ω_{x}, that is I and the ordinals D_{I}a, R the regular cardinals). Definition 9.2 We give an inductive definition of sets T′′, Suc′′, A′′, G′′, R′′ of terms together with length(a) for a η T′′ ∪ Suc′′ ∪ A′′ ∪ G′′ ∪ R′′ ∪ F i′′, such that we can in an immediate way define T′′, Suc′′, A′′, G′′, R′′, F i′′ as elements of P^{dec}(N) and length as an element of N → N. (T′′ is the set of terms, Suc′′ the set of successor numbers, A′′ the additive principal numbers (except 0_{OT}), G′′ the Gamma numbers, R′′ the regular cardinals, F i′′ the set of fixed points of the function Ω_{·}.)

(T”1) 0_{OT} η T′′, length(0_{OT}) := 0.

(T”2) If a η T′′ \ {0_{OT} }, b η A′′, then a ˜+b η T′′, a ˜+1_{OT} η Suc′′,
length(a ˜+b) := max_{N}{length(a), length(b)} +_{N} 1.
(T”3) If a, b η T′′, then ϕ̂_{a}b η A′′, ϕ̂_{0_{OT}} 0_{OT} η Suc′′,
length(ϕ̂_{a}b) := max_{N}{length(a), length(b)} +_{N} 1.
and we define 1_{OT} := ϕ̂_{0_{OT}} 0_{OT}.
(T”4) If b η T′′, then D_{I}b η F i′′,
and if π η R′′ and b η T′′, then D_{π}b η G′′,
length(D_{π}b) := max_{N}{length(π), length(b)} +_{N} 1.
(T”5) If a η F i′′ ∪ Suc′′ ∪ {0_{OT} }, then Ω̂_{a} η R′′,
if a η T′′, then Ω̂_{a} η G′′,
length(Ω̂_{a}) := length(a) +_{N} 1,
(T”6) I η F i′′ ∪ R′′, length(I) := 0.

(T”7) R′′ ⊂ G′′ ⊂ A′′ ⊂ T′′, F i′′ ⊂ G′′, Suc′′ ⊂ T′′. Lim′′ := T′′ \ ({0_{OT} } ∪ Suc′′). We will write 0, 1 for 0_{OT} , 1_{OT}. Definition 9.3 We define first, last, Alength : N → N, where first will be the first, last the last additive principal number of the sequence of principal numbers, the ordinal term is built of, and Alength be the length of the sequence. If a η̸ T′′ last(a) := first(a) := Alength(a) := 0, last(0_{OT}) := first(0_{OT}) := 0_{OT}, Alength(0_{OT}) := 0, if a η T′′ ∧ b η A′′, last(a ˜+b) := b, first(a ˜+b) := first(a), Alength(a ˜+b) := S(Alength(a)) and for a η A′′ last(a) := first(a) := a, Alength(a) := 1. Definition 9.4 Definition of a ≺_{B} c for a, c η T′′. The definition will be in such a way, that we can define it as an infix written function ≺_{B}: N → N → bool. We define a ≺_{B} b by recursion on length(a) +_{N} length(b), using in the definition a ⪯_{B} b as an abbreviation for a ≺_{B} b ∨_{B} a =_{N,B} b. a ≺_{B} b := ff, if a η̸ T′′ ∨ b η̸ T′′.

(≺ 1) (0_{OT} ≺_{B} c) := ¬_{B}(0_{OT} =_{N,B} c).

c ≺_{B} 0_{OT} := ff. (≺ 2) a, c η T′′, b, d η A′′, then (a ˜+b ≺_{B} c ˜+d) :=

((Alength(a) <_{N,B} Alength(c)) ∧_{B} (a ˜+b ⪯_{B} c))∨_{B} ((Alength(a) =_{N,B} Alength(c)) ∧_{B} (a ≺_{B} c) ∨_{B} (a =_{N,B} c ∧_{B} b ≺_{B} d))∨_{B} ((Alength(c) ≺_{B} Alength(a)) ∧_{B} (a ≺_{B} c ˜+d))

(≺ 3) a, b η T′′, c η A′′ \ {0_{OT} }, then (a ˜+b ≺_{B} c) := a ≺_{B} c and (c ≺_{B} a ˜+b) := c ⪯_{B} a

(≺ 4) If a, b, c, d η T′′, then (ϕ̂_{a}b ≺_{B} ϕ̂_{c}d) := ((a ≺_{B} c ∧_{B} b ≺_{B} ϕ̂_{c}d) ∨_{B} (a =_{N,B} c ∧_{B} b ≺_{B} d)∨_{B} (c ≺_{B} a ∧_{B} ϕ̂_{a}b ⪯_{B} d)).

(≺ 5) If a, b η T′′, c η G′′, then (ϕ̂_{a}b ≺_{B} c) := (a ≺_{B} c ∧_{B} b ≺_{B} c) and (c ≺_{B} ϕ̂_{a}b) := (c ⪯_{B} a ∨_{B} c ⪯_{B} b)

(≺ 6) π, ρ η R′′,b, d η T′′, then

(D_{π}b ≺_{B} D_{ρ}d) := (π =_{N,B} ρ ∧_{B} b ≺_{B} d) ∨_{B} (ρ ≠_{N,B} I ∧ π ≠_{N,B} I ∧ π ≺ ρ)∨_{B} (I =_{N} π ∧ π ≠_{N} I ∧ D_{π}b ≺ ρ)∨_{B} (π ≠_{N,B} ρ ∧_{B} ρ =_{N,B} I ∧_{B} π ≺_{B} D_{ρ}d)

(≺ 7) If π η R′′, ρ =_{N} Ω̂_{c} or ρ =_{N} I, π ≠_{N} I, b η T′′, then (D_{π}b ≺_{B} ρ) := π ⪯_{B} ρ and (ρ ≺_{B} D_{π}b) := ρ ≺_{B} π.

(≺ 8) b, c η T′′, then (D_{I}b ≺_{B} Ω̂_{c}) := D_{I}b ⪯_{B} c and Ω̂_{c} ≺_{B} D_{I}b := c ≺_{B} D_{I}b.

(≺ 9) b η T′′, then D_{I}b ≺_{B} I := tt I ≺_{B} D_{I}b := ff

(≺ 8) If a, c η T′′, then

(Ω̂_{a} ≺_{B} Ω̂_{c}) := (a ≺_{B} c)

(≺ 9) If a η T′′, then

(Ω̂_{a} ≺_{B} I) := (a ≺_{B} I) and
(I ≺_{B} Ω̂_{a}) := (I ⪯_{B} a).

Let ⪯_{B}:= λx, y.x ≺_{B} y ∨_{B} x =_{N,B} y, ≺:= λx, y.atom(x ≺_{B} y), ⪯:= λx, y.atom(x ⪯_{B} y), all written infix. Lemma 9.5 (≺) is a linear ordering on T′′, that is: Assume a, b, c η T′′. Then:

(a) ¬(a ≺ a).

(b) a ≺ b ∨ b ≺ a ∨ a =_{N} b.

(c) a ≺ b → b ≺ c → a ≺ c.

Proof: (a) and (b) are easily checked by induction on length(a) and length(a) +_{N} length(b). (c): Long and tedious induction on length(a) +_{N} length(b) +_{N} length(c): Case a =_{N} 0_{OT}. Then c ≠_{N} 0_{OT}, a ≺ c. Case b =_{N} 0_{OT} or c =_{N} 0_{OT} are not possible. Case a = d ˜+e, b = f ˜+g, c = h ˜+k: (the case distinction is in the ordering Alength(f), Alength(h), Alength(d): If Alength(d), Alength(f) <_{N} Alength(h) follows a ≺ b ⪯ h, therefore a ≺ h, the assertion. If Alength(f) <_{N} Alength(h) =_{N} Alength(d) follows d ≺ b ≺ h, therefore a ≺ c. If Alength(f) <_{N} Alength(h) <_{N} Alength(d) follows d ≺ b ≺ c, therefore a ≺ c. If Alength(d) <_{N} Alength(f) =_{N} Alength(h) follows a ⪯ f ⪯ h, therefore a ⪯ h, the assertion. If Alength(d) =_{N} Alength(f) =_{N} Alength(h) follows d ⪯ f ⪯ h, therefore d ≺ h or d =_{N} f =_{N} h, e ≺ g ≺ k and by IH e ≺ k. If Alength(h) =_{N} Alength(f) ≺ Alength(d) follows d ≺ b ⪯ c. If Alength(d), Alength(h) <_{N} Alength(f) follows a ⪯ f ≺ c. If Alength(h) <_{N} Alength(f) =_{N} Alength(d) follows d ⪯ f ≺ c. If Alength(h) <_{N} Alength(f) <_{N} Alength(d) follows d ⪯ b ≺ c. Case ¬(a η A′′ ∧ b η A′′ ∧ c η A′′) ∧ (a η A′′ ∨ b η A′′ ∨ c η A′′): If a η A′′, b =_{N} f ˜+g, c =_{N} h ˜+k follows a ≺ f ⪯ h or a ≺ f ≺ c or a ≺ b ≺ h, and the assertion. If a =_{N} d ˜+e, b η A′′, c =_{N} g ˜+h, follows d ≺ b ≺ c or d ≺ b ⪯ g or a ≺ b ≺ g and the assertion. If a =_{N} d ˜+e, b =_{N} f ˜+g, c η A′′, follows a ⪯ f ≺ c or d ⪯ f ≺ c or d ≺ b ≺ c and the assertion. If a, b η A′′, c = h ˜+k follows a ≺ b ≺ h and the assertion. If a, c η A′′, b = f ˜+g follows a ⪯ f ≺ c and the assertion. If b, c η A′′, a = d ˜+e follows d ≺ b ≺ c and the assertion. Case a =_{N} ϕ̂_{d}e, b =_{N} ϕ̂_{f}g, c =_{N} ϕ̂_{h}k: As [BS88], theorem 14.2. Case a, b, c η A′′ \ {0_{OT} }, ¬(a, b, c η G′′), a η G′ ∨ b η G′′ ∨ c η G′′: If a =_{N} ϕ̂_{d}e, b =_{N} ϕ̂_{f}g, c η G′′, follows, if we had c ⪯ d, f ≺ c ⪯ d, ϕ̂_{d}e ≺ g ≺ c by IH, ϕ̂_{d}e ≺ c, d ≺ c. If d ≺ c ⪯ e follows in case of d ≺ f e ≺ ϕ̂_{f}g ≺ c, e ≺ c a contradiction, in case of d =_{N} f, e ≺ g ≺ c, e ≺ c a contradiction, and in case of f ≺ d, ϕ̂_{d}e ⪯ g ≺ c, ϕ̂_{d}e ≺ c, e ≺ c, again a contradiction. Therefore d ≺ c, e ≺ c and the assertion.

If a =_{N} ϕ̂_{d}e, b η G′′, c =_{N} ϕ̂_{f}g follows, if b ⪯ f, d ≺ f, e ≺ b ≺ c, therefore e ≺ c and the assertion. If f ≺ b ⪯ g, follows, if d ≺ f, e ≺ b ≺ c, therefore e ≺ c and the assertion, if d =_{N} f, e ≺ b ⪯ g, e ≺ g and a ≺ c, and if f ≺ d, a ≺ b ⪯ g, a ≺ g, a ≺ c. If a η G′′, b =_{N} ϕ̂_{d}e, c =_{N} ϕ̂_{f}g, follows, if d ≺ f, e ≺ c, therefore, if a ≺ d, a ≺ f and if a ≺ e, a ≺ c. If d =_{N} f follows e ≺ g, max_{T} {d, e} ⪯ max_{T} {f, g}. If f ≺ d, follows a ≺ b ⪯ g, a ≺ g. If a =_{N} ϕ̂_{d}e, b, c η G′′, follows max_{T} {d, e} ≺ b ≺ c, max_{T} {d, e} ≺ c. If a, c η G′′, b =_{N} ϕ̂_{d}e, follows a ⪯ max_{T} {d, e} ≺ c. If a, b η G′′, c =_{N} ϕ̂_{d}e, follows a ≺ b ⪯ max_{T} {d, e}. Case a =_{N} D_{d}e, b =_{N} D_{f}g, c =_{N} D_{h}k: If d, f, h ≠_{N} I or d =_{N} f =_{N} h =_{N} I follows d ⪯ f ⪯ h, d ≺ h or e ≺ g ≺ k and the assertion. If d =_{N} I, f, h ≠_{N} I follows a ≺ f ⪯ h. If f =_{N} I, d, h ≠_{N} I follows d ≺ b ≺ h. If h =_{N} I, d, f ≠_{N} I follows d ⪯ f ≺ c, d ≺ c. If d =_{N} h =_{N} I, f ≠_{N} I, follows a ≺ f ≺ c. If f =_{N} h =_{N} I, d ≠_{N} I, follows d ≺ b ⪯ c. If d =_{N} f =_{N} I, h ≠_{N} I, follows a ≺ b ≺ h. Case a =_{N} D_{d}e, b =_{N} Ω̂_{f} or b =_{N} I, c =_{N} D_{h}k: If d, h ≠_{N} I follows d ⪯ b ≺ h, if d =_{N} I, h ≠_{N} I follows a ≺ b ≺ h, If d ≠_{N} I, h =_{N} I follows d ⪯ b ≺ c, and if d =_{N} h =_{N} I follows b ≠_{N} I, a ⪯ f ≺ b. Case a =_{N} D_{d}e, b =_{N} D_{e}f c =_{N} Ω̂_{k} or c =_{N} I: If d, e ≠_{N} I follows d ⪯ e ⪯ c, if d ≠_{N} I, e =_{N} I follows d ⪯ b ≺ c, d ≺ c, if d =_{N} I, e ≠_{N} I follows a ≺ e ⪯ c, and if d =_{N} e =_{N} I, follows k ≠_{N} I, a ≺ b ≺ k. Case a =_{N} Ω̂_{d}, b =_{N} D_{e}f, c =_{N} D_{g}h: If e, g ≠_{N} I follows a ≺ e ⪯ g, if e =_{N} I ∧ g ≠_{N} I follows a ≺ b ≺ g, if e ≠_{N} I ∧ g =_{N} I follows a ≺ e ≺ c, and if a =_{N} g =_{N} I follows a =_{N} Ω̂_{d}, d ≺ b ≺ c, d ≺ c. Case a =_{N} Ω̂_{d} or a =_{N} I, b =_{N} Ω̂_{e} or b =_{N} I, c =_{N} D_{g}h: If g ≠_{N} I follows a ≺ b ≺ g, and if g =_{N} I follows b =_{N} Ω̂_{e}, e ≺ I therefore a ≠_{N} I, otherwise I ≺ I, d ≺ e ≺ c, d ≺ c. Case a =_{N} Ω̂_{d} or a =_{N} I, b =_{N} D_{e}f, c =_{N} Ω̂_{g} or c =_{N} I: If e ≠_{N} I follows a ≺ e ⪯ c, if e =_{N} I follows a =_{N} Ω̂_{d}, a ≺ c =_{N} I or a ≺ b ⪯ g. Case a =_{N} D_{d}e, b =_{N} Ω̂_{e} or b =_{N} I, c =_{N} Ω̂_{g} or c =_{N} I: If e ≠_{N} I follows d ⪯ b ≺ c, if d =_{N} I follows, if c =_{N} I, a ≺ c, if c =_{N} Ω̂_{g} with I ⪯ g a ≺ c, and if c =_{N} Ω̂_{g} with g ≺ I, b =_{N} Ω̂_{e}, a ≺ e ≺ g. Case a =_{N} Ω̂_{d}, b =_{N} Ω̂_{f}, c =_{N} Ω̂_{h}: d ≺ f ≺ h. Case a =_{N} I, b =_{N} Ω̂_{f}, c =_{N} Ω̂_{h}: a ⪯ f ≺ h. Case a =_{N} Ω̂_{d}, b =_{N} I, c =_{N} Ω̂_{h}: d ≺ b ⪯ h. Case a =_{N} Ω̂_{d}, b =_{N} Ω̂_{f}, c =_{N} I: d ≺ f ≺ c. The Cases a =_{N} I =_{N} b or b =_{N} I =_{N} c are not possible. Case a =_{N} I, b =_{N} Ω̂_{f}, c =_{N} I: a ⪯ f ≺ c. Definition 9.6 Let a η T′′, M, M′ : P^{fin}(N):

M ⪯_{B} M′  :=  ∀_{B}x η M∃_{B}y η M′(x ⪯_{B} y),
M ≺_{B} M′  :=  ∀_{B}x η M∃_{B}y η M′(x ≺_{B} y),
M ≺_{B} a := M ≺_{B} {a}_{fin},
a ⪯_{B} M := {a}_{fin} ⪯_{B} M.

M ⪯ M′ := atom(M ⪯_{B} M′), similarly for ≺. Definition 9.7 We define a set T′ ⊂ T′′ of restricted terms, and define:

Suc′ := Suc′′ ∩ T′, A′ := A′′ ∩ T′, G′ := G′′ ∩ T′, R′ := R′′ ∩ T′. Lim′ := Lim′′ ∩ T′, F i′ := F i′′ ∩ T′.

(T’1) 0_{OT} η T′.

(T’2) If a η T′ \ {0_{OT} }, b η A′′ ∩ T′, b ⪯ last(a), then a ˜+b η T′.
(T’3) If a, b η T′, then ϕ̂_{a}b η T′.
(T’4) If π η R′′ ∩ T′ and b η T′, then D_{π}b η T′.
(T’5) If a η T′, then Ω̂_{a} η T′.
(T’6) I η T′.

We want to construct a system, which has precisely one term for each α ≺ ψ_{Ω_{1}}(Ω_{I+ω}). In the set theoretical ordinals it happens, that for regular cardinals κ and certain β′ < β, ψ_{κ}β′ = ψ_{κ}β. Only if β′ ∈ C_{κ}(β′), β ∈ C_{κ}(β) we have ψ_{κ}β′ = ψ_{κ}β → β = β′. Since we want to assign (and will do this in chapter 10) for every ordinal term t a unique ordinal o(t) by o(D_{a}b) = ψ_{o(a)}o(b), we need to select those ordinal terms D_{a}b, such that o(b) ∈ C_{o(a)}(o(b)). We will introduce therefore finite sets of ordinals G_{b}a for b η R′′, a η T′′ such that for a, b, c η T′ we have o(c) ∈ C_{o(b)}(o(a)) ↔ G_{b}c ≺ a, and define OT as the subset of T′, where only subterms D_{b}c with G_{b}c ≺ c occur. G_{b}c will be the set of the e such that D_{d}e occurs as a problematic subterm in c. Problematic is D_{d}e, if to conclude o(c) ∈ C_{o(b)}(α) for an arbitrary α, we need to know o(D_{d}e) ∈ C_{o(b)}(α), and additionally C_{o(b)}(α) is only closed under ρ ↦ ψ_{o(d)}ρ for ρ ≺ α.

Definition 9.8 Inductive Definition of G_{π}a : P^{fin}(N), G_{π}a ⊂ T′′, such that for a η T′, π η R′, G_{π}a ⊂ T′, and we can define it as a function G : N → N → P^{fin}(N). The definition is by recursion on length(a). G_{π}a := ∅_{fin}, if π η̸ R′ ∨ a η̸ T′′.

(G1) G_{π}0_{OT} := ∅.

(G2) If a, b η T′′, then G_{π}(a ˜+b) := G_{π}(ϕ̂_{a}b) := G_{π}a ∪ G_{π}b.
(G3) If ξ η R′′, b η T′′, then

G_{π}D_{ξ}b := \{ \begin{matrix}\{b\} ∪ G_{π}ξ ∪ G_{π}b, & if π ⪯ ξ ≠_{N} I∨ \\ & ξ =_{N} I ∧ (π ⪯ D_{I}b ∨ π =_{N} I), \\ G_{π}ξ & if ξ ≺ π =_{N} I \\ ∅, & if ξ ≺ π ≠_{N} I or \\ & ξ =_{N} I ∧ D_{I}b ≺ π ≺ I. \end{matrix}

(G4) If a η T′′, then G_{π}(Ω̂_{a}) := G_{π}a.
(G1) G_{π}I := ∅.

G_{π}^{0}a := G_{π}a ∪ {0_{OT} }. Lemma 9.9 (a) a η G_{π}b → G_{π}a ⊂ G_{π}b.

(b) a η G_{π}b ⇒ length(a) <_{N} length(b). Proof: Easy Induction on length(a). Definition 9.10 (a) Inductive Definition of a set of terms OT (OT ⊂ T′), such that OT can be

defined (in an obvious way) as an element P^{dec}(N):

(OT1) 0_{OT} η OT.
(OT2) If a η OT \ {0_{OT} }, b η OT ∩ A′, b ⪯ last(a),
then a ˜+b η OT

(OT3) If π η R′ ∩ OT, b η OT, G_{π}b ≺ b, then D_{π}b η OT.

(OT4) If a, b η OT, then ϕ̂_{a}b η OT
(OT5) If a η OT, then Ω̂_{a} η OT
(OT6) I η OT.

The elements of OT are called ordinal terms.

(b) R := R′ ∩ OT (≅ R′′ ∩ OT), F i := F i′ ∩ OT (≅ F i′′ ∩ OT), G := G′ ∩ OT (≅ G′′ ∩ OT), A := A′ ∩ OT (≅ A′′ ∩ OT), Suc := Suc′ ∩ OT (≅ Suc′′ ∩ OT).

We will now introduce further functions on T′′: Definition 9.11 (a) ω := ϕ̂_{0_{OT}} 1_{OT}, ω η OT.

(b) Definition of ˆ+ : N → N → N, written infix. We define a ˆ+b by recursion on length(b): If a η̸ T′′ or b η̸ T′′, a ˆ+b := 0_{OT}. a ˆ+0_{OT} := a, a ˆ+b := a ˜+b for b η A′′, and a ˆ+(b ˜+c) := (a ˆ+b) ˜+c. We have ∀x, y η T′′.x ˆ+y η T′′.

(c) Definition of + : N → N → N, written infix. We define a + b by recursion on length(a), side recursion on length(b): If a η̸ T′′ or b η̸ T′′, a + b := 0_{OT}. If b =_{N} 0_{OT}, then a + b := a. If a =_{N} 0_{OT}, then a + b := b. Let a, b ≠_{N} 0_{OT}. If b η A′′, b ⪯ last(a), then a + b := a ˜+b. If b η A′′, last(a) ≺ b, then, if a η A′′, then a + b := b, and if a =_{N} e ˜+f, then a + b := e + b. If b =_{N} e ˜+f, e η T′′, f η A′′, then a + b := (a + e) ˜+f. We have ∀x, y η T′′.x + y η T′′.

(d) Definition of · : N → N → N, the multiplication of an ordinal with a natural number, written infix. We define a · n by recursion on n : N: a · 0 := 0_{OT}, a · Sb := (a · b) + a. We have ∀x η T′′.z ∈ N.x · y η T′′.

(e) Definition of · − ω, Rest(·, −ω), pred : N → N,

where we write a − ω for (· − ω)a (the largest limes-number below a) Rest(a, −ω) for Rest(·, −ω)a, (the difference between a and a − ω) pred(a) for pred a (the predecessor of a successor ordinal).

We define a − ω Rest(a, −ω) pred(a) by recursion on length(a):

If a η̸ T′′, then a − ω := pred(a) := 0_{OT}, Rest(a, −ω) := 0, 1_{OT} − ω := 0_{OT}, Rest(1_{OT} , −ω) := 1, pred(1_{OT}) := 0_{OT}, for a η T′′, (a ˜+1_{OT}) − ω := a − ω, Rest(a ˜+1_{OT} , −ω) := S(Rest(a, −ω)), pred(a ˜+1_{OT}) := a, and if a η A′′ ∪ {0_{OT} } or a =_{N} b ˜+c with c ≠_{N} 1_{OT}, then a − ω := a, Rest(a, −ω) := 0_{OT}, pred(a) := a.

We prove easily

∀x η T′′. x − ω η T′′ ∧ pred(x) η T′′ ∧ x =_{N} a − ω ˆ+(1_{OT} · Rest(x, −ω)) ∧
x − ω η Lim′′ ∪ {0_{OT} } ∧ (x η Suc′′ → x =_{N} pred(x) ˜+1_{OT})).

(f) Definition of ·^{−} : N → N, written as a^{−}, by recursion on length(a) (the cardinal-predecessor of a cardinal): a^{−} := 0_{OT} for a η̸ R′′ ∪ {0_{OT} , 1_{OT} , ω}, 0_{OT}^{−} := 0_{OT}, 1_{OT}^{−} := 0_{OT}, ω^{−} := 0_{OT}, Ω̂_{0_{OT}}^{−} := ω, if a η Suc′′, then Ω̂_{a}^{−} := Ω̂_{P red(a)}, if a η F i′′, then Ω̂_{a}^{−} := a, and I^{−} := 0_{OT}. We have ∀x η T′′.x^{−} η T′′.

(g) Definition of Ω_{·} : N → N, written as Ω_{a} for Ω_{·}a.

Ω_{a} := \{ \begin{matrix}a & if a η F i′′ \\ Ω̂_{pred(a)} & if a − ω η F i′′ and a η̸ F i′′ \\ 0_{OT} & if a η̸ T′′ \\ Ω̂_{a} & otherwise. \end{matrix}

We have ∀x η T′′.Ω_{x} η T′′.

(h) max_{T} := λx, y. \underline{if} x ≺ y \underline{then} y \underline{else} x, min_{T} := λx, y. \underline{if} x ≺ y \underline{then} x \underline{else} x, max_{T} , min_{T} : N → N → N, and we write max_{T} {a, b}, min_{T} {a, b} for max_{T} ab, min_{T} ab. We have ∀x, y η T′′.max_{T} {x, y}, min_{T} {x, y} η T′′.

(i) Definition of NF_{+,B} : N → N → B:

NF_{+,B} := λx, y.x =_{N,B} 0_{OT} ∨_{B} first(y) ⪯_{B} last(x), NF_{+} := λx, y.atom(NF_{+,B}(x, y)), NF_{+} : N → N → U We write NF_{+,B}(a, b), for NF_{+,B}ab, similarly of NF_{+}.

(j) Definition of P l : N → Nlist, which is the list of the additive principal numbers, an ordinal is sum of:

We define P l(a) by recursion on length(a):

P l(a) := nil, if a η̸ T′′ or a =_{N} 0_{OT}, P l(a ˜+b) := append(P l(a), P l(b)), and for a η A′′ P l(a) := cons(a, nil).

(k) Definition of Cr : N → P^{dec}(N), where Cr(a) should be the set of a-critical ordinals. We define Cr ab by recursion on length(a) +_{N} length(b):

Cr ab:= false if a η̸ T′′ ∨ b η̸ T′′, and for a, b, c η T′′, Cr a0_{OT} := ff, Cr a(b ˜+c) := ff, Cr aϕ̂_{b}c := a ≺_{B} b, and for b η G′′, Cr ab := a ≺_{B} b.

We write Cr(a) for Cr a.

(l) Definition of Dicom : N → N such that Dicom(a) η T′ for a η T′, Dicom(a) := 0_{OT} otherwise. Dicom(a) should be the largest component D_{I}c contained in a Dicom(a) := 0_{OT} if a η̸ T′′, Dicom(0_{OT}) := 0_{OT}, Dicom(a ˜+b) := Dicom(ϕ̂_{a}b) :=

max_{T} {Dicom(a), Dicom(b)}.

Dicom(D_{a}b) := Dicom(a) for a ≠_{N} I, Dicom(D_{I}a) := D_{I}a, Dicom(Ω̂_{a}) := Dicom(a), Dicom(I) := I.

(m) Definition of NF_{D,B} : N → N → B written as NF_{D,B}(a, b): NF_{D,B}(a, b) := a η R′′ ∧ b η T′′ ∧ G_{a}b ≺_{B} b.

We will now characterize the ordering of ordinals, which are not principal additive numbers by the lexicographic ordering of P list. Lemma 9.12 (a) ∀x η T′′.Alength(x) =_{N} lh(P l(x)) ∧ (x ≠_{N} 0_{OT} → (first(x) =_{N} (P l(a))_{0} ∧

last(x) =_{N} (P l(x))_{Alength(x) ˙−1})) ∧ ∀i <_{N} lh(P l(x)).(P l(x))_{i} ∈ A′′.

(b) ∀x η T′′.x η T′ ↔ ((∀i ∈ N.Si <_{N} Alength(x) → (x)_{Si} ⪯ (x)_{i}) ∧ ∀i <_{N} Alength(x).(x)_{i} η T′.

(c) ∀x, y η T′′.P l(x ˆ+y) ≅_{Nlist} append(P l(x), P l(y)).

(d) ∀x η T′.x η OT ↔ ∀i ≺ Alength(x).(P l(x))_{i} η OT.

(e) If a, b η T′′, la := P l(a), lb := P l(b), then

a ≺ b ↔ (∃i ≤_{N} min_{N}{lh(la), lh(lb)}.(la)_{i} ≺ (lb)_{i}∧

∀j <_{N} i.(la)_{j} =_{N} (lb)_{j}) ∨ (lh(la) <_{N} lh(lb) ∧ ∀i <_{N} lh(la).(la)_{i} =_{N} (lb)_{i}).

(f) If a, b η T′′, la ≅_{Nlist} P l(a), lb ≅_{Nlist} P l(b), then a =_{N} b ↔ (lh(la) =_{N} lh(lb) ∧ ∀i <_{N} lh(la).(la)_{i} =_{N} (lb)_{i}).

(g) Let

Q(a, b, i) := i ≤_{N} Alength(a) ∧ (∀j <_{N} Alength(a).i ≤_{N} j → (P l(a))_{0} ≺ (P l(b))_{i}) ∧
(i ≠ 0_{OT} → (P l(b))_{i ˙−1} ⪯ (P l(a))_{0})

Then ∀a, b η T′′.b ≠ 0_{OT} → (∃i ∈ N.Q(a, b, i))∧

(∀i, j ∈ N.Q(a, b, i) ∧ Q(a, b, j) → i =_{N} j)∧
∀i.Q(a, b, i) → P l(a + b) ≅ append(Sublist(P l(a), i), P l(b))

(h) ∀x ∈ T′′.G_{π}x = ⋃_{i=1}^{Alength(x)} G_{π}P l((x)_{i}).

(or the formalization of this theorem). Proof:

(a) follows by an easy induction on Alength(x)

(b) Induction on Alength(x): If x =_{N} 0_{OT}, then x η T′ and P l(x) ≅_{Nlist} nil. If x η A′′, then P l(x) ≅_{Nlist} cons(x, nil), x η T′ ↔ (P l(x))_{0} η T′ and we have the assertion. If x =_{N} b ˜+c, then x η T′ ↔ (b η T′ ∧ c η A′ ∧ c ⪯ last(a) =_{N} (a)_{Alength(a) ˙−1}) ↔ (((∀i ∈ N.Si <_{N} Alength(b) → (b)_{Si} ⪯ (b)_{i}) ∧ ∀i <_{N} Alength(b).(b)_{i} η T′) ∧ c η A′ ∧ c ⪯ (b)_{Alength(b) ˙−1}) ↔ ((∀i ∈ N.Si <_{N} Alength(a) → (a)_{Si} ⪯ (a)_{i}) ∧ ∀i <_{N} Alength(a).(a)_{i} η T′).

(d) follows in a similar way.

(c) follows by an easy induction on Alength(y) (e), (f): Let Less(a, b) := (∃i <_{N} min_{N}{lh(P l(a)), lh(P l(b))}.(P l(a))_{i} ≺ (P l(b))_{i}∧

∀j <_{N} i.(P l(a))_{j} =_{N} (P l(b))_{j})∨

(lh(P l(a)) <_{N} lh(P l(b)) ∧ ∀i <_{N} lh(P l(a)).(P l(a))_{i} =_{N} (P l(b))_{i}), Eq(a, b) := (lh(P l(a)) =_{N} lh(P l(b)) ∧ ∀i <_{N} lh(P l(a)).(P l(a))_{i} =_{N} (P l(b))_{i}), and m := max_{N}{lh(la), lh(lb)}. We show ∀a, b η T′′.(a ≺ b ↔ Less(a, b)) ∧ (a = b ↔ Eq(a, b)) by induction on Alength(a) + Alength(b). Assume a, b η T′′, and let la := P l(a), lb := P l(b). If b =_{N} 0_{OT}, follows ¬(a ≺ b), ¬Less(a, b), a =_{N} b ↔ b =_{N} 0_{OT} ↔ lh(lb) = 0 ↔ Eq(a, b). If a =_{N} 0_{OT}, b ≠_{N} 0_{OT} follows a ≺ b, and lh(la) <_{N} lh(lb) ∧ ∀i <_{N} lh(la).(la)_{i} =_{N} (lb)_{i}, Less(a, b), further a ≠_{N} b, lh(la) ≠_{N} lh(lb), ¬(Eq(a, b)). Case a =_{N} c ˜+d, b =_{N} e ˜+f. If Alength(c) <_{N} Alength(e) follows m =_{N} Alength(a) =_{N} min_{N}{Alength(a), Alength(e)}, a ≺ b ↔ a ⪯ e ↔ (Less(a, e) ∨ Eq(a, e)) ↔ Less(a, b), and a ≠_{N} b, ¬(Eq(a, b). If Alength(c) =_{N} Alength(e) follows m =_{N} Alength(a) =_{N} S min{Alength(c), Alength(e)}, a ≺ b ↔ (c ≺ e ∨ (c =_{N} e ∧ d ≺ f)) ↔ (Less(c, e) ∨ (Eq(c, e)∧d ≺ f))) ↔ Less(a, b). Further a =_{N} b ↔ (c =_{N} e∧d =_{N} f) ↔ (Eq(c, e)∧d =_{N}

f) ↔ Eq(a, b). If Alength(e) <_{N} Alength(c) follows the assertion symmetrically to the case Alength(c) <_{N} Alength(e). Case a =_{N} c ˜+d, b η A′. Then lh(lb) = 1 <_{N} lh(la), m = 1, a ≺ b ↔ c ≺ b ↔ Less(c, b) ↔ Less(a, b), a ≠_{N} b, ¬(Eq(a, b)). Case a η A′, b =_{N} c ˜+d. Then lh(la) = 1 <_{N} lh(lb), m = 1, a ≺ b ↔ a ⪯ c ↔ (Less(a, c) ∨ Eq(a, c)) ↔ Less(a, b), a ≠_{N} b, ¬(Eq(a, b)). Case a, b η A′. Then a ≺ b ↔ Less(a, b), a =_{N} b ↔ Eq(a, b). (g): The existence and uniqueness of i is easy. We show Q(a, b, i) ∧ b ≠_{N} 0_{OT} → P l(a +

b) ≅_{Nlist} append(Sublist(a, i), b) by main induction on Alength(a), side induction on Alength(b): Case a =_{N} 0_{OT}. Then i =_{N} 0 and we have the assertion. Case a ≠_{N} 0_{OT}: Case b η A′′: If b ⪯ last(a), then i =_{N} Alength(a), a + b =_{N} a ˜+b, P l(a + b) = append(P l(a), P l(b)). If a η A′′, a ≺ b, then i =_{N} 0, Sublist(a, i) ≅_{Nlist} nil, a + b =_{N} b. If a =_{N} c ˜+d, d ≺ b, then Q(c, b, i), a + b =_{N} c + b and we have the assertion. Case b =_{N} e ˜+f. Then Q(a, e, i), P l(a + b) ≅_{Nlist} P l((a + e) ˜+f) ≅_{Nlist}

append(append(Sublist(P l(a), i), P l(e)), P l(f)) ≅_{Nlist}
append(Sublist(P l(a), i), P l(b))

(h): Easy induction on Alength(x).

Lemma 9.13 Let a, b, c, d η T′.

(a) a =_{N} 0_{OT} ∨ (∃a′ η T′.a =_{N} a′ ˜+1_{OT}) ∨ a η Lim′

(b) a =_{N} 0_{OT} → ¬(a η Suc′ ∨ a η Lim′) and a η Suc′ → a η̸ Lim′.

(c) a =_{N} a′ ˜+1_{OT} ↔ (a η Suc′ ∧ a′ =_{N} pred(a))

(d) a, b η T′′, then b ≠_{N} 0_{OT} → last(a ˆ+b) =_{N} last(b), a ≠_{N} 0_{OT} → first(a ˆ+b) =_{N} first(a), Alength(a ˆ+b) =_{N} Alength(a) +_{N} Alength(b).

(e) ∀x, y, z η T′′.(x ˆ+y) ˆ+z =_{N} x ˆ+(y ˆ+z)

(f) ∀x, y, z η T′.(x + y) + z =_{N} x + (y + z).

(g) ∀x, y η T′.x ˆ+y η T′ ↔ (x =_{N} 0_{OT} ∧ first(y) ⪯ last(x)) (↔ NF_{+}(x, y)). ∀x, y η T′.NF_{+}(x, y) → x + y =_{N} x ˆ+y.

(h) (a, b, c η T′ ∧ c ≺ b) → (NF_{+}(a, b) → NF_{+}(a, c)) ∧ a + b ≺ a + c.

Proof: (a) - (e) are easy.

(f) as usual using 9.12 (b) and (g). (g): Use 9.12 (b) and (c). (h): first(c) ⪯ first(b), the second assertion follows as usually.

Lemma 9.14 (a) ∀x, y η T′.∀z ∈ N.x + y, x · z, x − ω, pred(x), x^{−}, Ω_{x}, max_{T} {x, }, min_{T} {x, y}, Dicom(x) η T′ ∧ (y η R′ → G_{y}x ⊂ T′) ∧ (NF_{+}(x, y) → x ˆ+y η T′).

(b) ∀x, y η OT .∀z ∈ N.x + y, x · z, x − ω, pred(x), x^{−}, Ω_{x}, max_{T} {x, }, min_{T} {x, y}, Dicom(x) η OT ∧ (y η R → G_{y}x ⊂ R) ∧ (NF_{+}(x, y) → x ˆ+y η OT).

Proof: (a): for + and ˆ+ the assertion follows by 9.12 (b), (c) and (g). The other assertions are easy.

(b) Use 9.12 (d), 9.12 (c), 9.12 (g), (a) for +, ˆ+. For the other functions the assertion is easy. Lemma 9.15 (a) If a, a′, b, b′ η T′′, Alength(a) =_{N} Alength(a′), then a ˆ+b ≺ a′ ˆ+b′ ↔ (a ≺

a′ ∨ (a =_{N} a′ ∧ b ≺ b′), and a ˆ+b =_{N} a′ ˆ+b′ ↔ (a =_{N} a′ ∧ b =_{N} b′).

(b) If a, b, b′, c η T′′, a ˆ+b ⪯ c ⪯ a ˆ+b′, then c =_{N} a ˆ+c′ for some c′ η T′′ such that b ⪯ c′ ⪯ b′.

(c) a ˆ+b η T′ → a ⪯ a ˆ+b ∧ b ⪯ a ˆ+b.

(d) ∀x, y ∈ T′′, π ∈ R′′.G_{π}y ⊂ G_{π}(x + y) ⊂ G_{π}x ∪ G_{π}y ∧ G_{π}(x ˆ+y) = G_{π}x ∪ G_{π}y

(e) ∀π η R′.π ≠_{N} I → length(π^{−}) <_{N} length(π).

(f) ∀a, b η T′.a ≺ ϕ̂_{a}b ∧ b ≺ ϕ̂_{a}b.

(g) ∀a η T′.a ≺ Ω̂_{a}.

(h) ∀π η R′, a η T′.a ≺ π → a ≺ D_{π}a. Proof: (a), (b), (c): Use 9.12, and the common known properties of the lexicographic ordering.

(d) follows by 9.12 (c), (g) and (h).

(e) Trivial. (f) (i) We prove b ≺ ϕ̂_{a}b by induction on length(b): If b =_{N} 0_{OT} this is trivial, if b =_{N} c ˜+d follows c ≺ ϕ̂_{a}c ≺ ϕ̂_{a}b by IH and (c). Case b =_{N} ϕ̂_{c}d: If c ≺ a follows d ≺ ϕ̂_{a}d ≺ ϕ̂_{a}(ϕ̂_{c}d), therefore the assertion. If c =_{N} a follows d ≺ ϕ̂_{c}d =_{N} b and the assertion. If a ≺ c follows ϕ̂_{c}d ⪯ b, ϕ̂_{c}d ≺ ϕ̂_{a}b. If b η G′ follows the assertion by definition. (ii) Now we prove a ≺ ϕ̂_{a}b by induction on length(a): If a =_{N} 0_{OT} this follows by definition, if a =_{N} c ˜+d follows c ≺ ϕ̂_{c}b ≺ ϕ̂_{a}b by IH, therefore a ≺ ϕ̂_{a}b. If a =_{N} ϕ̂_{c}d follows by IH c ≺ a, and since d ≺ a, b ≺ ϕ̂_{a}b by (i), d ≺ ϕ̂_{d}b ≺ ϕ̂_{a}b. If a η G′ follows the assertion by definition.

(g) By Induction on length(a): The cases a =_{N} 0_{OT}, a =_{N} I follow by definition. If a =_{N} b ˜+c, follows by IH b ≺ Ω̂_{b}, therefore a ≺ Ω̂_{b} ≺ Ω̂_{a}. If a =_{N} ϕ̂_{b}c, follows b ≺ Ω̂_{b} ≺ Ω̂_{_{ϕ_{b}}c}, c ≺ Ω̂_{c} ≺ Ω̂_{_{ϕ_{b}}c}, therefore ϕ̂_{b}c ≺ Ω̂_{a}.

If a =_{N} D_{Ω_{b_{+}1_{OT}}} c, follows Ω̂_{b} ≺ Ω̂_{Ω_{b}}, Ω̂_{b_{+}1_{OT}} ≺ Ω̂_{Ω_{b}}, a ≺ Ω̂_{b_{+}1_{OT}} ≺ Ω̂_{Ω_{b}} ≺ Ω̂_{a}.

If a =_{N} D_{Ω_{b}}c, b =_{N} D_{I}d or b =_{N} I, follows D_{Ω_{b}}c ≺ Ω̂_{b} ≺ Ω̂_{D_{Ω_{b}}c}.

If a =_{N} D_{Ω_{0_{OT}}} c, follows D_{Ω_{0_{OT}}} c ≺ Ω̂_{0_{OT}} ≺ Ω̂_{D_{Ω_{0_{OT}}} c}.

If a =_{N} D_{I}b, follows a ≺ Ω̂_{a} by definition. If a =_{N} Ω̂_{b} follows by IH b ≺ Ω̂_{b}, therefore a ≺ Ω̂_{a}.

(h) If a =_{N} 0_{OT} this is trivial, if a =_{N} I follows a ⪯ π^{−} ⪯ D_{π}a.

If a =_{N} b ˜+c follows b ≺ D_{π}b ⪯ D_{π}a, therefore a ⪯ D_{π}a, if a =_{N} ϕ̂_{b}c follows b, c ≺ D_{π}max_{T} {b, c} ⪯ D_{π}a. If a =_{N} D_{ρ}b follows if ρ ≺ π and π ≠_{N} I, a ≺ π^{−} ⪯ D_{π}a, if ρ ≺ π =_{N} I follows by IH, since length(ρ^{−}) <_{N} length(ρ), ρ^{−} ≺ D_{π}ρ^{−} ⪯ D_{π}a, therefore a ≺ ρ ≺ D_{π}a, further if π ≺ I =_{N} ρ follows from a ≺ π a ⪯ π^{−} ≺ D_{π}a, and if ρ =_{N} π follows by IH b ≺ D_{π}b, a =_{N} D_{π}b ≺ D_{π}a. If a =_{N} Ω̂_{b} follows, in case of π =_{N} I b ≺ D_{π}b ⪯ D_{π}a, a ≺ D_{π}a, and if π ≠_{N} I, a ⪯ π^{−} ≺ D_{π}a. Lemma 9.16 (a) ∀x η T′′.¬(x ≺ 0_{OT}).

(b) ∀x η T′′.x ≺ 1_{OT} ↔ x =_{N} 0_{OT}.

(c) ∀x η T′.x ≺ ω → ∃n ∈ N.x =_{N} 1_{OT} · n.

(d) a ≠_{N} 0_{OT}, a η T′ → 1_{OT} ⪯ a. Proof: All by 9.12 (b) and (e) and ∀x η A′.x =_{N} 0_{OT} ∨ x =_{N} 1_{OT} ∨ x =_{N} ω ∨ ω ≺ x. Lemma 9.17 (a) ∀x η R′ \ {I}.∀y η (R′ ∪ F i′).y ≺ x → y ⪯ x^{−}.

(b) ∀x η R′ \ {I}.∀y η (R′ ∪ F i′).x^{−} ⪯ y ⪯ x → y η {x^{−}, x}.

(c) ∀x η R′.∀y η A′.y ≺ x → x^{−} + y ≺ x. Proof:

(a) x =_{N} Ω̂_{0_{OT}} is not possible. If x =_{N} Ω̂_{a_{+}1_{OT}}, y =_{N} Ω̂_{e} follows e ≺ a ˜+1_{OT} therefore e ⪯ a, y ⪯ Ω̂_{a} =_{N} x^{−}. If in the same case y =_{N} I, follows I ⪯ a ˜+1_{OT}, I ⪯ a, I ≺ Ω̂_{a}, and if y =_{N} D_{I}d follows y ≺ a ˜+1_{OT}, y ⪯ a ≺ Ω̂_{a}. If x =_{N} Ω̂_{a} x^{−} =_{N} a, x =_{N} D_{I}f or x =_{N} I, y =_{N} Ω̂_{e}, follows by e ≺ a, Ω̂_{e} ≺ a, and if y =_{N} I or y =_{N} D_{I}d, follows y ⪯ a.

(b) by (a).

(c) If y η A′, y ⪯ x^{−} this follows by x^{−} ≺ x, if x^{−} ≺ y η A′ this is trivial, and if y = y_{1} ˜+y_{2} follows x^{−} + y_{1} ≺ x and the assertion.

Lemma 9.18 Assume a, b, c η T′.

(a) a η Cr(b) → b ≺ a.

(b) a η Cr(b) → ((a ≺ ϕ̂_{b}c ↔ a ⪯ c) ∧ (ϕ̂_{b}c ⪯ a ↔ c ≺ a)).

(c) b η Cr(a) → ϕ̂_{a_{+}1_{OT}} 0_{OT} ⪯ b.

(d) ϕ̂_{a}b ≺ c ⪯ ϕ̂_{a}(b ˜+1_{OT}) → c η̸ Cr(a) ∪ G′.

(e) (b η Cr(a) ∧ b ≺ c ⪯ ϕ̂_{a}b) → c η̸ Cr(a) ∪ G′.

(f) a ≺ ϕ̂_{b}0_{OT} → a η̸ Cr(b).

Proof:

(a) If a =_{N} ϕ̂_{c}d with b ≺ c follows b ≺ c ≺ a, and if a η G′ we have b ≺ a.

(b) If a =_{N} ϕ̂_{f}g with b ≺ f follows the assertion by definition, and if b ≺ a =_{N} G′ follows in the first part of the assertion a ≺ ϕ̂_{b}c ↔ a ⪯ max_{T} {b, c} ↔ a ⪯ c and in the second part ϕ̂_{b}c ⪯ a ↔ max_{T} {b, c} ≺ a ↔ c ≺ a.

(c) If a =_{N} ϕ̂_{c}d with a ≺ c, follows a ˜+1_{OT} ⪯ c, ϕ̂_{a_{+}1_{OT}} 0_{OT} ⪯ a, and if a ≺ b η G′ follows

max_{T} {a ˜+1_{OT} , 0_{OT} } ≺ b, ϕ̂_{a_{+}1_{OT}} 0_{OT} ≺ b.

(d) Assume c η Cr(a). By (b) follows b ≺ c ⪯ b ˜+1_{OT}, c =_{N} b ˜+1_{OT} a contradiction. If c η G′ follows c η Cr(a).

(e) Assume c η Cr(a). By (b) follows b ≺ c ⪯ b ˜+1_{OT}, c =_{N} b ˜+1_{OT} a contradiction. If c η G′ follows c η Cr(a).

(f) Assume a η Cr(b). By (b) follows a ⪯ 0_{OT}. Lemma 9.19 (a) ∀x η R′ \ {I}.∀z η T′.x^{−} ≺ z ≺ D_{x}0_{OT} → z η̸ G′.

(b) ∀x η R′ \ {I}.∀y, z η T′.D_{x}y ≺ z ≺ D_{x}(y ˜+1_{OT}) → z η̸ G′. Proof: (a), (b): Assume z η G′. If z =_{N} D_{I}d follows x^{−} ≺ z ≺ z, which yields a contradiction. If z =_{N} D_{d}e with d ≠_{N} I follows in (a) x^{−} ≺ d ⪯ x, in (b) x ⪯ d ⪯ x, in both cases therefore d =_{N} x, further in (a) e ≺ 0_{OT} and in (b) y ≺ e ≺ y ˜+1_{OT}, both a contradiction. If z =_{N} Ω̂_{d} follows in (a) x^{−} ≺ z ≺ x, in (b) x ⪯ z ≺ x, both yields a contradiction.

Here follow some properties of Dicom(a): Lemma 9.20 (a) Let a η T′, ζ_{0} := Dicom(a), ζ_{Sn} := Ω̂_{ζ_{n}}.

Then Dicom(a) =_{N} 0_{OT} and a ≺ ζ_{n} ≺ D_{I}0_{OT} for some n : N, or Dicom(a) =_{N} D_{I}b for some b and D_{I}b ⪯ a ≺ ζ_{n} ≺ D_{I}(b ˜+1_{OT}) for some n : N, or Dicom(a) =_{N} I and I ⪯ a, especially we habe ∀a η T′.Dicom(a) ⪯ a.

(b) If π η R′ and a η T′, then G_{π}Dicom(a) ⊂ G_{π}a. Proof:

(a) By induction on length(a): If a =_{N} 0_{OT} , b ˜+c, ϕ̂_{b}c, D_{I}c, Ω̂_{b}, I this follows trivially or by IH, if a =_{N} D_{b}c with I ≺ b follows Dicom(a) =_{N} I ≺ a, and if a =_{N} D_{b}c with b ≺ I follows Dicom(a) =_{N} D_{I}d with D_{I}d ⪯ b ≺ ζ_{n} ≺ D_{I}(d ˜+1_{OT}), D_{I}d ≺ b since b η R′, and D_{I}d ≺ D_{b}c ≺ ζ_{n} ≺ D_{I}(d ˜+1_{OT}).

(b) By induction on length(a): If a =_{N} 0_{OT} , b ˜+c, ϕ̂_{b}c, D_{I}c, Ω̂_{b}, I follows the assertion trivially or by IH, if a =_{N} D_{b}c with I ≺ b follows G_{π}Dicom(a) =_{P^{fin}(N)} G_{π}I =_{P^{fin}(N)} ∅ and if a =_{N} D_{b}c with b ≺ I follows, if π ⪯ Dicom(a) ⪯ a or π =_{N} I, G_{π}Dicom(a) ⊂ G_{π}b ⊂ G_{π}D_{b}c by IH, and if Dicom(a) ≺ π ≠_{N} I follows G_{π}(Dicom(a)) =_{P^{fin}(N)} ∅.

Now we are ready to define the fundamental sequences in OT. We will start with a version, where a[[z]] =_{N} z for a η R′′, where the ordinals only behave well as long as τ(a)^{−} ≺ z (in fact they behave well as long as Dicom(a) ≺ z, the problem occurs only when we have fixed points of Ω_{·}, but the approach starting with τ(a)^{−} seems to be a little bit more uniform). In the well-ordering proof we use a[z] := a[[τ(a)^{−} + z]]. For the analysis of the sequences, it seems to be easier to introduce and analyze first ·[[·]], where we do not have problems with the sum and in a second step to define later ·[·] and transfer the properties proven. Definition 9.21 Definition of τ(a) und a[[ξ]] for a η T′′, ξ η T′′, ξ ≺ τ(a), such that we can define it as primitive recursive functions, have τ : N → N ·[[·]] : N → N → N, and we have ∀x η T′′.τ(x) η {0_{OT} , 1_{OT} , ω} ∪ R′′ and ∀x η T′′.∀ξ η T′′.ξ ≺ τ(x) → x[[ξ]] η T′′. If τ(a) =_{N} ω we define only a[[1_{OT} · n]], a[[x]] := a[[1_{OT} · length(x)]] for arbitrary x ≺ ω which is for x =_{N} 1_{OT} · n consistent with the first definition. For x η̸ T′′ τ(x) := 0_{OT}, and for x η̸ T′′ ∨ ξ η̸ T′′ ∨ ¬(ξ ≺ τ(x)) x[[ξ]] := 0_{OT}.

([[ ]].0) τ(0_{OT}) := ∅.

([[ ]].1) If a =_{N} b ˜+c, b η T′′, c η A′′, then
τ(a) := τ(c), (b ˜+c)[[ξ]] := b ˆ+(c[[ξ]])
([[ ]].2) Case a =_{N} ϕ̂_{b}c:

([[ ]].2.1) Case b =_{N} 0_{OT}: ([[ ]].2.1.1) c =_{N} 0_{OT} ⇒ τ(a) := 1_{OT}, a[[ξ]] := 0_{OT} ([[ ]].2.1.2) c =_{N} c′ ˜+1_{OT} ⇒ τ(a) := ω,

a[[n]] := ϕ̂_{0_{OT}} (c′) · SSn.

([[ ]].2.1.3) (c η Cr(b)) ⇒ τ(a) := ω,

a[[n]] := c · Sn.

([[ ]].2.1.4) (c η Lim′′ ∧ c η̸ Cr(b)) ⇒ τ(a) := τ(c),

a[[ξ]] := ϕ̂_{0_{OT}} (c[[ξ]])

([[ ]].2.2) Case b =_{N} b′ ˜+1_{OT}: ([[ ]].2.2.1) c =_{N} 0_{OT} ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{SSSn} where ρ_{0} := 0_{OT}, ρ_{Sn} := ϕ̂_{b′}ρ_{n}

([[ ]].2.2.2) c =_{N} c′ ˜+1_{OT} ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{n} where ρ_{0} := ϕ̂_{b}c′, ρ_{Sn} := ϕ̂_{b′}ρ_{n}

([[ ]].2.2.3) (c η Cr(b)) ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{Sn}, where ρ_{0} := c, ρ_{Sn} := ϕ̂_{b′}ρ_{n}

([[ ]].2.2.4) (c η Lim′′ ∧ (c η̸ Cr(b)) ⇒ τ(a) := τ(c),

a[[ξ]] := ϕ̂_{b}(c[[ξ]])

([[ ]].2.3) Case b η Lim′′: ([[ ]].2.3.1) (c =_{N} 0_{OT} ∧ b η T′′ \ G′′) ⇒ τ(a) := τ(b),

a[[ξ]] := ϕ̂_{b[[ξ]]}0_{OT}

([[ ]].2.3.2) (c =_{N} 0_{OT} ∧ b η G′′) ⇒ τ(a) := τ(b),

a[[ξ]] := ϕ̂_{b[[ξ]]}b

([[ ]].2.3.3) c =_{N} c′ ˜+1_{OT} ⇒ τ(a) := τ(b),

a[[ξ]] := ϕ̂_{b[[ξ]]}(ϕ̂_{b}c′)

([[ ]].2.3.4) (c η Cr(b)) ⇒ τ(a) := τ(b),

a[[ξ]] := ϕ̂_{b[[ξ]]}c

([[ ]].2.3.5) (c η Lim′′ ∧ c η̸ Cr(b)) ⇒ τ(a) := τ(c),

a[[ξ]] := ϕ̂_{b}(c[[ξ]])

([[ ]].3) Case a =_{N} D_{b}c: ([[ ]].3.1) Case c =_{N} 0_{OT}: ([[ ]].3.1.1) (b ≠_{N} I ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{Sn}, where ρ_{0} := b^{−}, ρ_{Sn} := ϕ̂_{ρ_{n}}0_{OT}.

([[ ]].3.1.2) b =_{N} I ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{SSn}, where ρ_{0} := 0_{OT}, ρ_{Sn} := Ω̂_{ρ_{n}}.

([[ ]].3.2) Case c =_{N} c′ ˜+1_{OT}: ([[ ]].3.2.1) (b ≠_{N} I ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{n}, where ρ_{0} := D_{b}c′, ρ_{Sn} := ϕ̂_{ρ_{n}}0_{OT}.

([[ ]].3.2.2) b =_{N} I ⇒ τ(a) := ω,

a[[1_{OT} · n]] := ρ_{n}, where ρ_{0} := D_{I}c′, ρ_{Sn} := Ω̂_{ρ_{n}}.

([[ ]].3.3) Case c η Lim′′, τ(c) ≺ b:

τ(a) := τ(c), a[[ξ]] := D_{b}c[[ξ]].

([[ ]].3.4) Case c η Lim′′, b ⪯ τ(c):

Then τ(a) := ω, a[[n]] := D_{b}c[[ζ_{n}]], where ζ_{n} is defined by:

([[ ]].3.4.1) (τ(c) ≠_{N} I ∨ b ≺ D_{I}c ∨ b =_{N} I ⇒

ζ_{0} := π^{−}, ζ_{Sn} := D_{π}(c[[ζ_{n}]]), π := τ(c).

([[ ]].3.4.2) (τ(c) =_{N} I ∧ D_{I}c ⪯ b ≺ I) ⇒ ζ_{0} := Dicom(b), ζ_{Sn} := Ω̂_{ζ_{n}}.

([[ ]].4) Case a =_{N} Ω̂_{b}:
([[ ]].4.1) Case a η R′′:
τ(a) := a, a[[ξ]] := ξ
([[ ]].4.2) Case a η̸ R′′:
τ(a) := τ(b), a[[ξ]] := Ω̂_{b[[ξ]]}
([[ ]].5) Case a =_{N} I:
τ(a) := I, a[[ξ]] := ξ

We define τ̂(:) = λy.λξ.ξ η T′′ ∧ ξ ≺ τ(y) ∧ τ(y)^{−} ⪯ ξ, and have τ̂(:)N → P^{dec}(N), ∀x η T′′.τ̂(x) ⊂ T′′. Lemma 9.22 (a) ∀x, ξ η T′.ξ ≺ τ(x) → x[[ξ]] η T′ ∧ x[[ξ]] ≺ x ∧ τ(x) ⪯ x.

(b) In the situation of [[ ]].3.4 we have ∀n ∈ N.ζ_{n} ≺ τ(c), and if b, c ∈ T′ then ∀n ∈ N.ζ_{n} ∈ T′.

(c) ∀x, y η T′.y ≠_{N} 0_{OT} → NF_{+}(x, y) → (τ(x ˆ+y) =_{N} τ(y) ∧ ∀ξ η τ̂(y).(x ˆ+y)[[ξ]] =_{N} x ˆ+(y[[ξ]])).

Proof: (a), (b): (a) follows by an easy induction on length(x), using in ([[]].1) c[[ξ]] ≺ c → a ˜+c η T′ → a ˆ+c[[ξ]] η T′ by 9.13 (h) and proving in the situation of [[ ]].3.4 (b) by side induction on N.

(c) Easy induction on Alength(y). Lemma 9.23 Assume a, ξ, ρ η T′.

(a) ξ ≺ ρ ≺ τ(a) → a[[ξ]] ≺ a[[ρ]].

(b) (ω ≺ τ(a) ∧ ξ ⪯ τ(a)) → (ξ ⪯ a[[ξ]] ∧ ∀π η R′.(π ≠ I ∨ π ⪯ τ(a)) → G_{π}ξ ⪯ G_{π}a[[ξ]]) (where a[[τ(a)]] := a).

(c) (0 ≠ a ∧ a η̸ R′) ↔ τ(a) ≺ a.

(d) (a η OT → τ(a) η OT)∧.

(e) a η̸ G′ → a η Lim′ → 1 ≺ x[[τ(a)^{−}]].

Proof:

(a) By an induction on length(a) in an easy way, where in the cases τ(a) = ω we show first by induction n := length(ξ) length(ρ) = Sn → a[[ξ]] ≺ a[[ρ]] and the assertion follows by side induction on length(ρ) − length(ξ).

(b) Case a = b ˜+c: ξ ⪯ c[[ξ]] ⪯ b ˆ+c[[ξ]] Case a = ϕ̂_{b}c: If c η Lim′ \ Cr(b) follows ξ ⪯ c[[ξ]] ⪯ ϕ̂_{b}c[[ξ]] by IH and G_{π}(ξ) ⪯ G_{π}c[[ξ]] ⪯ G_{π}a[[ξ]]. If b η Lim′ ∧ ((c η̸ Lim′ ∨ c η Cr(b)) follows ξ ⪯ b[[ξ]] ⪯ ϕ̂_{b[[ξ]]}e = a[[ξ]] for some e and G_{π}(ξ) ⪯ G_{π}b[[ξ]] ⪯ G_{π}a[[ξ]]. If a = D_{b}c c η Lim′, τ(c) ≺ b follows ξ ≺ D_{b}(ξ) ⪯ D_{b}c[[ξ]] by lemma 9.15 (h) and ξ ⪯ c[[ξ]]. Further, if π ⪯ b ≠ I or b = Ianddπ ≺ D_{I}c[[ξ]] follows G_{π}ξ ⪯ G_{π}c[[ξ]] ⪯ G_{π}(D_{b}c[[ξ]]). If b ≺ π ≠ I ∨ (I = b ∧ D_{I}c[[ξ]] ≺ π ≺ I) follows, since ξ ⪯ τ(c) ≺ b ≺ π ≠ I or ξ ≺ D_{I}ξ ⪯ D_{I}c[[ξ]] ≺ π ≠ I G_{π}(ξ) ≅ ∅. b ⪯ π = I is not possible by assumption. If a = Ω̂_{b}, and if a[[ξ]] = a^{−} + ξ, follows the assertion immediately and if a[[ξ]] = Ω̂_{b[[ξ]]}

follows ξ ⪯ b[[ξ]] ≺ Ω̂_{b[[ξ]]}, G_{π}(ξ) ⪯ G_{π}b[[ξ]] ≅ G_{π}a[[ξ]].

(c) Trivial. (d): Induction on length(c): The cases τ(a) = 0, 1, ω or τ(a) = a are trivial, otherwise the assertion follows by IH.

(e) If x = b ˜+c, then ω ⪯ c ⪯ b ⪯ x[[τ(x)^{−}]]. If x = ϕ̂_{b}c, follows, if b = 0, c = c′ ˜+1, 1 ≺ ϕ̂_{0}c′ · SSn, if c η Lim′ \ Cr(b) ∧ b = 0 follows ¬(c η G), 1 ≺ c[[τ(c)^{−}]], 1 ≺ x[[τ(x)^{−}]], and if c η Lim′ \ Cr(b) ∧ 0 ≺ b, 1 ≺ ϕ̂_{b}c[[τ(b)^{−}]] = x[[τ(x)^{−}]]. If b = 0, c η G′ follows 1 ≺ c · Sn. If b = b′ ˜+1, c η̸ Lim′ \ C(b), follows x[[n]] = ϕ̂_{b′}d for some d ≠ 0 or x[[n]] = ϕ̂_{b}c′ and we have the assertion. If b η Lim′ \ G′, c = 0 follows 1 ≺ b[[τ(b)^{−}]] ≺ x[[τ(x)^{−}]]. If b η G′, c = 0 follows 1 ≺ ϕ̂_{b[[τ(b)^{−}]]}b.

If β η Lim′, c = c′ ˜+1, follows 1 ≺ ϕ̂_{b[[τ(b)^{−}]]}ϕ̂_{b}c′, since 0 ≺ ϕ̂_{b}c′.

If c η Cr(b), b η Lim′, follows 1 ≺ ϕ̂_{b[[τ(b)^{−}]]}c = x[[τ(x)^{−}]].

We will now introduce a^{∗} for a η T′, which will be some sort of predecessor of a. a^{∗}

has the property, that lh′(a^{∗}) <_{N} lh′(a), so, we have ∀a η T′.∃n ∈ N.s\underbrace{∗ ⋯ ∗}_{ntimes} = 0, and we can use induction over the length of the descending sequences. We will use ·^{∗} to prove the property a[[x]] ≺ b ⪯ a[[x ˜+1]] → a[[x]] ⪯ b[[τ(b)^{−}]].

Definition 9.24 (a) Definition of ·^{∗} : N → N such that s^{∗} η T′ for s η T′, s^{∗} η T′′ for s η T′′ (s^{∗} := 0 for s η̸ T′′). 0^{∗} := I^{∗} := 0. (a ˜+b)^{∗} := a ˆ+(b^{∗}).

(ϕ̂_{a}b)^{∗} := \{ \begin{matrix}ϕ̂_{a}(b^{∗}) & if b η̸ Cr(a) ∪ \{0\}, \\ b & if a = 0 ∧ b η Cr(a) ∪ \{0\} \\ a & if b = 0 ∧ a η G′ \\ ϕ̂_{0}0 & if b = 0 ∧ a = 1 \\ ϕ̂_{a^{∗}}b & if (b η Cr(a) ∧ a ≠ 0) ∨ (b = 0∧ \\ & a η̸ G′ ∪ \{0, 1\}) \end{matrix}.

Ω̂_{0}^{∗} := ω, Ω̂_{a}^{∗} := a for a η F i′, Ω̂_{a}^{∗} := Ω̂_{a^{∗}} for a η̸ F i′ ∪ {0}

(D_{π}b)^{∗} := \{ \begin{matrix}D_{π}(b^{∗}) & if b ≠ 0 \\ π^{−} & if b = 0 ∧ π ≠ I. \\ Ω̂_{0} & if b = 0 ∧ π = I \end{matrix}.

(b) Definition of lh′ : N → N. lh′(n) := 0 for n η̸ T′. lh′(0) := 0, lh′(I) := 1, lh′(a ˜+b) := lh′(a) +_{N} lh′(b) +_{N} 1, lh′(ϕ̂_{a}b) := lh′(a) +_{N} lh′(a) +_{N} lh′(b) +_{N} 1, lh′(D_{a}b) := lh′(a) +_{N} lh′(b) +_{N} 3, lh′(Ω̂_{a}) := lh′(a) +_{N} 3.

Lemma 9.25 Let a, b, c η T′.

(a) (c ˜+1)^{∗} = c, ω^{∗} = 1, ϕ̂_{a}(b ˜+1)^{∗} = ϕ̂_{a}b.

(b) c ≠ 0 → lh′(c^{∗}) <_{N} lh′(c).

(c) If n : N, follows (1 · (Sn))^{∗} = 1 · n.

(d) If b ≠ 0, then (a ˆ+b)^{∗} = a ˆ+(b^{∗}).

(e) a ≠ 0 → a^{∗} ≺ a.

(f) b^{∗} ⪯ (ϕ̂_{a}b)^{∗}.

Proof:

(a) By definition.

(b) Easy induction on length(c), using that for a η R′ a^{−} = a^{∗}.

(c) By induction on n′ : N.

(d) By induction on Alength(b).

(e) Immediate by induction on length(a).

(f) Immediate, using b^{∗} ⪯ b. Lemma 9.26 s η T′ → s^{∗} ⪯ s[[τ(s)^{−}]]. Proof: Induction on length(s). If s = 0, the assertion is trivial. If s η R′ follows s^{∗} = s[[τ(s)^{−}]]. If s = a ˜+b follows b^{∗} ⪯ b[[τ(b)^{−}]], a ˆ+(b^{∗}) ⪯ a ˆ+b[[τ(b)^{−}]]. Case s = ϕ̂_{a}b: If (a = 0∨a = a′ ˜+1) and (b = 0∨b = b′ ˆ+1∨b η Cr(a)) follows immediately s^{∗} ⪯ s[[τ(s)^{−}]]. If b η Lim′ \ Cr(a) follows b^{∗} ⪯ b[[τ(b)^{−}]] by IH therefore ϕ̂_{a}(b^{∗}) ⪯ ϕ̂_{a}(b[[τ(b)^{−}]]. If a η Lim′ \ G′, b = 0 follows s^{∗} = ϕ̂_{a^{∗}}0 ⪯ ϕ̂_{a[[τ(a)^{−}]]}0 = s[[τ(s)^{−}]] since a^{∗} ⪯ a[[τ(a)^{−}]] by

IH and 0 ⪯ ϕ̂_{a[[τ(a)^{−}]]}0.

If a η G′, b = 0 follows s^{∗} = a ⪯ ϕ̂_{a[[τ(a)^{−}]]}a = s[[τ(s)^{−}]].

If a η Lim′, b = b′ ˜+1 follows s^{∗} = ϕ̂_{a}b′ ⪯ ϕ̂_{a[[τ(a)^{−}]]}(ϕ̂_{a}b′) = s[[τ(s)^{−}]].

If a η Lim′, b η Cr(a), follows s^{∗} = ϕ̂_{a^{∗}}b ⪯ ϕ̂_{a[[τ(a)^{−}]]}b = s[[τ(s)^{−}]]. Case s = D_{a}b: If b η̸ Lim′ follows s[[τ(s)^{−}]] ⪯ s^{∗} immediately. If b η Lim′, τ(b) ≺ a or a ⪯ τ(b) ≠ I or a ⪯ τ(b) = I ∧ (a ≺ D_{I}b ∨ a = I) follows s^{∗} = D_{a}(b^{∗}) ⪯ D_{a}(b[[τ(b)^{−}]]) = s[[τ(s)^{−}]]. If b η Lim′, a ⪯ τ(b) = I, D_{I}b ⪯ a ≺ I follows s^{∗} = D_{a}b^{∗} ⪯ D_{a}(b[[τ(b)^{−}]]) ⪯ D_{a}(b[[Dicom(b)]]) = s[[τ(s)^{−}]]. Case s = Ω̂_{a}, s η̸ R′: a^{∗} ⪯ a[[τ(a)^{−}]], Ω̂_{a}^{∗} ⪯ Ω̂_{a[[τ(a)^{−}]]}.

Lemma 9.27 Assume a, b, c η T′′.

(a) (a η̸ Cr(c) ∧ a^{∗} ≺ b ⪯ a) → b η̸ Cr(c).

(b) (a η̸ G′ ∧ a^{∗} ≺ b ⪯ a) → b η̸ G′.

(c) (a η̸ F i′ ∪ R′ ∧ a^{∗} ≺ b ⪯ a) → b η̸ F i′ ∪ R′.

(d) (a η̸ F i′ ∧ a^{∗} ≺ b ⪯ a) → b η̸ F i′.

(e) b η Cr(a) ∪ {0} → (b ≺ c ⪯ ϕ̂_{a}b → b ⪯ c^{∗})

(f) b η {1, ω, Ω̂_{0}, } → (b ≺ a ≺ I → b ⪯ a^{∗}).

∀a η T′.I ≺ a → I ⪯ a^{∗}.

Proof:

(a) Induction on length(a), side induction on length(b). Assume a, b, c η T′, such that a η̸ Cr(c) and a^{∗} ≺ b ⪯ a, b η Cr(c) Case a = d ˜+e: Then b = d ˆ+f such that e^{∗} ≺ f ⪯ e, especially f ≠ 0. Case a = ϕ̂_{d}e: Then d ⪯ c.

Case e η̸ Cr(d) ∪ {0}: Then ϕ̂_{d}e^{∗} ≺ b ⪯ ϕ̂_{d}e, and since b η Cr(c) ⊂ Cr(d), e^{∗} ≺ b ⪯ e, and, since e η̸ Cr(d), e η̸ Cr(c), b η̸ Cr(c).

Case d = 0 and e η Cr(d) ∪ {0}. Then e ≺ b ⪯ ϕ̂_{d}e, and by lemma 9.18 (c) and (e) follows b η̸ Cr(d), b η̸ Cr(a).

Case e = 0 ∧ d η G′: Then by lemma 9.18 (c) follows the assertion.

Case (e η Cr(d) ∧ d ≠ 0) ∨ (e = 0 ∧ d η T′ \ (G′ ∪ {0, 1}). Then ϕ̂_{d^{∗}}e ≺ b ⪯ ϕ̂_{d}e, and since b η Cr(c) ⊂ Cr(d), e ≺ b ⪯ e, a contradiction. Case d = 1 ∧ e = 0: 9.18 (c). Case a η G′: b ⪯ a ⪯ c, b η̸ Cr(c).

(b) Induction on length(a), side induction on length(b). Assume a, b η T′, such that a η̸ G′ and a^{∗} ≺ b ⪯ a, b η G′ Case a = d ˜+e: Then b = d ˆ+f such that e^{∗} ≺ f ⪯ e, especially f ≠ 0. Case a = ϕ̂_{d}e: If e η̸ Cr(d) ∪ {0} or d = 0 or e = 0 ∧ d η G′ follows d ⪯ (ϕ̂_{d}e)^{∗} ≺ b, and if we had b η G′ we had b η Cr(d) contradicting (a). If e η Cr(d) ∧ d ≠ 0 follows d ⪯ e ⪯ ϕ̂_{d^{∗}}e ≺ b and by the same argument the assertion, and if e = 0 ∧ d η̸ G′ ∪ {0, 1} follows from b η G′, d^{∗} = max_{T} {d^{∗}, e} ≺ b ⪯ max_{T} {d, e} = d, contradicting d η̸ G′ and the IH. If d = 1∧e = 0 follows b ≺ ϕ̂_{1}0 ≺ d_{Ω_{0}}0, b η̸ G′.

(c) Induction on length(a), side induction on length(b). Assume a, b η T′, such that a η̸ F i′ ∪ R′ and a^{∗} ≺ b ⪯ a, b η G′ Case a = 0, a = d ˜+e, ϕ̂_{d}e: Then b η̸ G′ by (b). Case a = D_{d}e, d ≠ I: Then d^{−} ⪯ a^{∗} ≺ b ≺ d which is not possible.

(d) Induction on length(a), side induction on length(b). Assume a, b η T′, such that a η̸ F i′ and a^{∗} ≺ b ⪯ a, b η F i′ Case a = 0, a = d ˜+e, ϕ̂_{d}e : Then b η̸ G′ by (b). Case a = D_{d}e, d ≠ I: Then b η̸ F i′ by (c). Case a = Ω̂_{d}. If d η F i′ follows d = a^{∗} ≺ b ⪯ a, d ≺ b ⪯ d, not possible, and if d η̸ F i′∪{0} follows a^{∗} = Ω̂_{d^{∗}}, d^{∗} ≺ b ⪯ d and the assertion by IH. If d = 0 follows a ≺ D_{I}0, a η̸ F i′.

(e) The case b = 0 is trivial. Let b η Cr(a). Induction on length(c): By lemma 9.18 (e) c η̸ Cr(a) ∪ G′, therefore c = d ˜+e or c = ϕ̂_{d}e with d ⪯ a. If c = d ˜+e follows b ⪯ d ⪯ c^{∗}. Case c = ϕ̂_{d}e. Subcase d ≺ a: Then b ⪯ e ⪯ ϕ̂_{a}b. If b = e η Cr(d) follows c^{∗} = e or c^{∗} = ϕ̂_{d^{∗}}e, b ⪯ c^{∗}. If b ≺ e follows by IH b ⪯ e^{∗}, c^{∗} = ϕ̂_{d}e^{∗} or c^{∗} = e or c^{∗} = ϕ̂_{d^{∗}}e, b ⪯ c^{∗}. Subcase d = a: Then b ⪯ e ⪯ b, c = ϕ̂_{a}b, c^{∗} = b or c^{∗} = ϕ̂_{a^{∗}}b or c^{∗} = ϕ̂_{0}1∧d = 1∧e = 0 = b.

(f) Immediate. Lemma 9.28 ∀s, t η T′.s^{∗} ≺ t ⪯ s → s^{∗} ⪯ t^{∗}. Proof: Induction on length(s) +_{N} length(t). Case s = a ˜+b: a ˆ+b^{∗} ≺ t ≺ a ˜+b, therefore t = a ˆ+c for some b^{∗} ≺ c ⪯ b, b^{∗} ⪯ c^{∗}, s^{∗} = a ˆ+b^{∗} ⪯ t^{∗}. Case s = ϕ̂_{a}b: Then by lemma 9.27 (a), (b) t η̸ Cr(a) ∪ G′, therefore t = a ˜+b (in which case, since s^{∗} η A, s^{∗} ⪯ a ⪯ t^{∗}), or t = ϕ̂_{c}d with c ⪯ a. Let t = ϕ̂_{c}d. Subcase b ≠ 0, b η T′ \ Cr(a): ϕ̂_{a}b^{∗} ≺ t ⪯ ϕ̂_{a}b. If c ≺ a ϕ̂_{a}b^{∗} ⪯ d ⪯ ϕ̂_{a}b. If d = s^{∗}, follows d η Cr(c) \ G′, t^{∗} = s^{∗} or t^{∗} = ϕ̂_{c^{∗}}d, therefore s^{∗} ⪯ t^{∗}. If s^{∗} ≺ d follows by IH s^{∗} ⪯ d^{∗}, d ≠ 0, therefore t^{∗} η {d, ϕ̂_{c}d^{∗}, ϕ̂_{c^{∗}}d}, s^{∗} ⪯ t^{∗}. If c = a, follows b^{∗} ≺ d ⪯ b, by IH b^{∗} ⪯ d^{∗}, and, since b η̸ Cr(a), d η̸ Cr(a) and d ≠ 0. Therefore s^{∗} ⪯ ϕ̂_{a}b^{∗} = t^{∗}. Subcase a = 0 ∧ b η Cr(a) ∪ {0}: by lemma 9.27 (e).

Subcase b = 0 ∧ a η G′. If c ≺ a follows a η Cr(c), a ⪯ d ⪯ ϕ̂_{a}b. If d = a follows t^{∗} = d or t^{∗} = ϕ̂_{c^{∗}}d, a ⪯ t^{∗}, and if a ≺ d, a ⪯ d^{∗} ⪯ t^{∗}. If c = a follows d ⪯ b = 0, s = t. Subcase (b η Cr(a) ∧ a ≠ 0) ∨ (b = 0 ∧ a η G′ \ {0, 1}): Then ϕ̂_{a^{∗}}b ≺ t ⪯ ϕ̂_{a}b. Subsubcase c ≺ a^{∗}: Then s^{∗} ⪯ d ⪯ s. If in this case d = s^{∗}, follows t^{∗} = d or t^{∗} = ϕ̂_{c^{∗}}d, s^{∗} ⪯ t^{∗}, and if s^{∗} ≺ d follows s^{∗} ⪯ d^{∗} ⪯ t^{∗}. Subsubcase c = a^{∗}: b ≺ d ⪯ ϕ̂_{a}b, by lemma 9.27 (e) b ⪯ d^{∗}. If then t^{∗} = ϕ̂_{c}d^{∗}, follows s^{∗} ⪯ t^{∗}. If t^{∗} = d, d η Cr(c) ∪ {0} follows d η Cr(a^{∗}), by b ≺ d, ϕ̂_{a^{∗}}b ⪯ d. The case t^{∗} = c does not occur, because d ≠ 0. The case t^{∗} = ϕ̂_{c^{∗}}d occurs only, in case of d η Cr(c), c ≠ 0, and by b ≺ d, follows ϕ̂_{a^{∗}}b ≺ d ⪯ ϕ̂_{c^{∗}d}. The case t = ϕ̂_{1}0 is not possible. Subsubcase a^{∗} ≺ c ⪯ a: If b = 0 ∧ a η G′ \ {0, 1} follows a^{∗} ⪯ c^{∗}, s^{∗} = ϕ̂_{a^{∗}}0 ⪯ ϕ̂_{c}d^{∗} = t^{∗} or s^{∗} = ϕ̂_{a^{∗}}0 ⪯ c = t^{∗} where c η G′, or s^{∗} ⪯ ϕ̂_{c^{∗}}d ⪯ t^{∗}. If b η Cr(a) ∧ a ≠ 0 follows from s^{∗} ≺ t ⪯ s, b ≺ ϕ̂_{c}d ⪯ ϕ̂_{a}b, b ⪯ d ⪯ ϕ̂_{a}b, b ⪯ d^{∗} ⪯ t^{∗} or b = d η Cr(c), s^{∗} = ϕ̂_{a^{∗}}0 ⪯ ϕ̂_{c^{∗}}d = t^{∗}. Subcase a = 1 ∧ b = 0: ω ≺ s ≺ I, by 9.27 (f) follows ω ⪯ s^{∗}. Case s = D_{a}b: s^{∗} η G′, so in case of t η̸ G′ follows by lemma 9.27 (b) s^{∗} ⪯ t^{∗}. Let therefore t η G′. Subcase b ≠ 0: Then D_{a}(b^{∗}) ≺ t ⪯ D_{a}b. Subsubcase a ≠ I: Then t = D_{a}c with b^{∗} ≺ c ⪯ b, b^{∗} ⪯ c^{∗}, s^{∗} ⪯ D_{a}c^{∗} = t^{∗}. Subsubcase a = I: If t = D_{c}d with c ≠ I follows D_{a}b^{∗} ⪯ c^{−} ⪯ t^{∗}. If c = I follows the assertion as in the case b ≠ I. If t = Ω̂_{c} follows s^{∗} ⪯ c ⪯ s, if c = s^{∗} = D_{I}b^{∗} follows t^{∗} = s^{∗}, if s^{∗} ≺ c, s^{∗} ⪯ c^{∗} ⪯ t^{∗}. Subcase b = 0 ∧ a ≠ I: a^{−} ≺ t ≺ a, t = D_{a}d, d ⪯ 0, therefore t = s. Subcase b = 0 ∧ a = I: Then Ω̂_{0} ≺ t ⪯ D_{I}0, and by lemma 9.27 (f) Ω̂_{0} ⪯ t^{∗}. Case s = Ω̂_{a}: If a = 0 follows the assertion by lemma 9.27 (f). Otherwise follows s η G′ and the assertion in case t η̸ G′. Let t η G′. Subcase s^{∗} = s^{−}. Then t = D_{s}c, s^{−} ⪯ t^{∗}. Subcase s η̸ R′, s^{∗} = Ω̂_{a^{∗}}. If t = D_{c}d, with c ≠ I follows Ω̂_{a^{∗}} ≺ c, Ω̂_{a^{∗}} ⪯ c^{∗} ⪯ t^{∗}. If t = D_{I}c follows a^{∗} ≺ t ⪯ a, since t η F i′, a η F i′, contradicting t^{∗} = Ω̂_{a^{∗}}. If t = Ω̂_{c} follows by IH a^{∗} ≺ c ⪯ a, a^{∗} ⪯ c^{∗} and the assertion. Case s = I: Then the assertion is trivial. Definition 9.29 t ≪ s :⇔ ∃l ∈ Nlist.0 <_{N} lh(l) ∧ (∀i <_{N} lh(l).(l)_{i} η T′) ∧ (l)_{0} = t ∧ (l)_{pred_{N} (lh(l))} = s ∧ ∀i ∈ N.Si <_{N} lh(l) → (l)_{Si} = (l)_{i}^{∗}, that is, informally written, t ≪ s ⇔ ∃n ∈ N, ∃s_{0}, …, s_{Sn} η T′.s_{0} = s ∧ s_{Sn} = t ∧ ∀i ≤ n.s_{Si} = s_{i}^{∗}, or even more informal:

t ≪ s ⇔ ∃n ∈ N.0 <_{N} n ∧ s\underbrace{∗ ⋯ ∗}_{ntimes} = t.

In the following lemmata we will refer to the informal definition, and all assertions can easily be transformed into formal proofs, which can not be read any more. In the situation of the second definition we define lh_{≪}(s, t) := n. Lemma 9.30 Let r, s, t η T′.

(a) For r η {0, I} we have r ≺ s → r ≪ s, and for r η {1, ω, Ω̂_{0}} we have r ≺ s ≺ I → r ≪ s.

(b) r ≪ r ˆ+s, and r ≪ s → t ˆ+r ≪ t ˆ+s.

(c) r ≪ s ∧ (∀x η T′.r ≺ x ⪯ s → (x η̸ Cr(t)) → ϕ̂_{t}r ≪ ϕ̂_{t}s.

(d) (r ≪ s ∧ (∀x η T′.r ≺ x ⪯ s → (x η̸ G′)) → ϕ̂_{r}0 ≪ ϕ̂_{s}0, (s η Cr(r) → s ≪ ϕ̂_{r}s), (t η Cr(s) ∧ r ≪ s) → ϕ̂_{r}t ≪ ϕ̂_{s}t.

(e) If r η G′, s η T′, s ≺ ϕ̂_{r_{+}1}0, then r ≪ ϕ̂_{r}s.

(f) (r ≪ s ∧ π η R′) → D_{π}r ≪ D_{π}s.

(g) r^{−} ≺ s ⪯ r, r η R′, then r^{−} ≪ s.

(h) (r ≪ s ∧ ∀x η T′.r ≺ x ⪯ s → c η̸ F i′) → Ω̂_{r} ≪ Ω̂_{s}).

Proof: We use lh′(s^{∗}) <_{N} lh′(s). (a) By induction on lh′(s) since r ≺ s → r ⪯ s^{∗} by lemma 9.27 (a).

(b) by induction on lh′(s) follows the first, by induction on lh_{≪}(r, c) the second assertion.

(c) Induction on lh_{≪}(r, s). (ϕ̂_{t}s_{Si})^{∗} = ϕ̂_{t}(s_{Si})^{∗} = ϕ̂_{t}s_{i} since s_{Si} η̸ Cr(t) ∧ s_{Si} ≠ 0.

(d) Induction on lh_{≪}(r, s), lh′(r), lh_{≪}(r, s).

(e) Induction on lh′(s). (f), (h) Induction on lh_{≪}(r, s).

(g) Induction on lh′(s). If s = r follows s^{∗} = r^{−}. Otherwise follows if r^{−} = ω, by lemma 9.27 (f), if r = I trivially, and if r^{−} η (R′ ∪ F i′) and ∀x η T′.r^{−} ≺ x ≺ r → x η̸ R′ ∪ F i′ by lemma 9.27 (c) r^{−} ⪯ s^{∗}.

(h) Induction on lh_{≪}(r, s). Lemma 9.31 If s, ξ, ρ η T′, τ(s)^{−} ⪯ ξ ≪ ρ ≺ τ(s), then s[[ξ]] ≪ s[[ρ]]. Proof: by Induction on length(s). In case of τ(s) = ω, it is sufficient to show ∀n ∈ N.s[[1 · n]] ≪ s[[1 · Sn]]. If s[[1 · n]] := ρ_{S^{i}n} for some i in Definition 9.21, we will prove ρ_{n} ≪ ρ_{Sn} by side induction on n : N. If s = 0, the premise cannot be fulfilled. If s = a ˜+b, the assertion follows by b[[ξ]] ≪ b[[ρ]] and by lemma 9.30. Case s = ϕ̂_{a}b: Subcase a = b = 0 is not possible. Subcase a = 0, b = b′ ˜+1: s[[1 · n]] = ϕ̂_{a}b′ · SSn ≪ ϕ̂_{a}b′ · SSSn = s[[1 · (Sn)]]. Subcase b η Lim′ \ Cr(a): b[[ξ]] ≪ b[[ρ]], b^{∗} ⪯ b[[τ(b)^{−}]] ⪯ b[[ξ]] ≺ b[[ρ]] ⪯ b, therefore by lemma 9.27 (a) and 9.30 (c) ϕ̂_{a}(b[[ξ]]) ≪ ϕ̂_{a}(b[[ρ]]). Subcase a = 0, b η Cr(a): s[[1 · n]] = b · (Sn) ≪ b · (SSn) = s[[1 · Sn]]. Subcase a = a′ ˜+1, b = 0: if n = 0 we have ρ_{0} = 0 ≪ ϕ̂_{a′}0 = ρ_{1}. In the step from n to

Sn we have ρ_{n} ≪ ρ_{Sn} ≺ ϕ̂_{a′_{+}1}0, therefore ∀c η T′.ρ_{n} ≺ c ≺ ρ_{Sn} → c η̸ Cr(a′), therefore

ρ_{Sn} = ϕ̂_{a′}ρ_{n} ≪ ϕ̂_{a′}ρ_{Sn} = ρ_{SSn}. Subcase a = a′ ˜+1, b = b′ ˜+1: ρ_{0} = ϕ̂_{a}b′, ∀c η T′.s[[τ(s)^{−}]] ≺ c ⪯ s → c η̸ Cr(a). By lemma 9.30 (d) ρ_{0} = ϕ̂_{a}b′ ≪ ϕ̂_{a′} ϕ̂_{a}b′ = ρ_{1}, and in the side induction step by lemma 9.30

(c) ρ_{Sn} = ϕ̂_{a′}ρ_{n} ≪ ϕ̂_{a′}ρ_{Sn} = ρ_{SSn}. Subcase a = a′ ˜+1, b η Cr(a). By lemma 9.30 (d) ρ_{0} = b ≪ ρ_{1}, by lemma 9.18 (e) and in the side induction step follows, since ∀x η T′.ρ_{0} ⪯ ρ_{n} ≺ x ⪯ s → x η̸ Cr(a), by lemma 9.30 (c) ρ_{Sn} = ϕ̂_{a′}ρ_{n} ≪ ϕ̂_{a′}ρ_{Sn} = ρ_{SSn}. Subcase a η Lim′ \G′, b = 0: Then ∀c η T′.a^{∗} ≺ c ⪯ a → c η̸ G′ by lemma 9.27 (b), by IH a^{∗} ⪯ a[[τ(a)^{−}]] ⪯ a[[ξ]] ≪ a[[ρ]] ≺ a, therefore by lemma 9.30 (d) s[[ξ]] = ϕ̂_{a[[ξ]]}0 ≪ s[[ρ]].

Subcase a η G′, b = 0: a[[ξ]] ≪ a[[ρ]], a η Cr(a[[ρ]]), therefore ϕ̂_{a[[ξ]]}a ≪ ϕ̂_{a[[ρ]]}a.

Subcase a η Lim′, b = b′ ˜+1. a[[ξ]] ≪ a[[ρ]], ϕ̂_{a}b′ η Cr(a[[ρ]]), therefore ϕ̂_{a[[ξ]]}(ϕ̂_{a}b′) ≪

ϕ̂_{a[[ρ]]}(ϕ̂_{a}b′).

Subcase a η Lim′, b η Cr(a): a[[ξ]] ≪ a[[ρ]], ϕ̂_{a[[ξ]]}b ≪ ϕ̂_{a[[ρ]]}b. Case s = D_{a}b. Subcase a ≠ I, b = 0: By lemma 9.30 (e) ρ_{0} = a^{−} ≪ ϕ̂_{a^{−}}0 = ρ_{1}, and in the Step from n to Sn follows from ρ_{n} ≪ ρ_{Sn}, for a^{−} ⪯ ρ_{n} ≺ c ⪯ ρ_{Sn} ≺ D_{a}0 by lemma 9.19 (a) c η̸ G′, ϕ̂_{ρ_{n}}0 ≪ ϕ̂_{ρ_{Sn}}0. Subcase a = I, b = 0: ∀c η T′.c ≺ s → c η̸ F i′. ρ_{0} = 0 ≪ Ω̂_{0} = ρ_{1} by lemma 9.30 (a) and in the induction step follows from ρ_{n} ≪ ρ_{Sn} ≺ s, since ∀c η T′.0 ≺ c ⪯ D_{I}0 → c η̸ F i′, ρ_{Sn} = Ω̂_{ρ_{n}} ≪ Ω̂_{ρ_{Sn}} = ρ_{SSn}. Subcase a ≠ I, b = b′ ˜+1: By lemma 9.19 (a) ∀c η T′.ρ_{0} = D_{a}b′ ≺ c ≺ s → c η̸ G′. By lemma 9.30 (d) ρ_{0} ≪ ϕ̂_{ρ_{0}}0 = ρ_{1}, and in the Step from n to Sn follows ρ_{n} ≪ ρ_{Sn}, ρ_{Sn} = ϕ̂_{ρ_{n}}0 ≪ ϕ̂_{ρ_{Sn}}0 = ρ_{SSn}. Subcase a = I, b = b′ ˜+1: ∀c η T′.ρ_{0} = D_{I}b′ ≺ c ≺ s → c η̸ F i′. ρ_{0} = D_{I}b′ ≪ Ω̂_{D_{I}d′} = ρ_{1}, and in the induction step follows from ρ_{n} ≪ ρ_{Sn} ≺ s ρ_{Sn} = Ω̂_{ρ_{n}} ≪ Ω̂_{ρ_{Sn}} = ρ_{SSn}. Subcase b η Lim′, τ(b) ≺ a: b[[ξ]] ≪ b[[ρ]], therefore D_{a}b[[ξ]] ≪ D_{a}b[[ρ]]. Subcase b η Lim′, a ⪯ τ(b) =: π: We show ζ_{n} ≪ ζ_{Sn} by induction on n : N. Then b[[ζ_{n}]] ≪ b[[ζ_{Sn}]] by IH and D_{a}(b[[ζ_{n}]]) ≪ D_{a}(b[[ζ_{Sn}]]). Subsubcase π ≠ I ∨ a ≺ D_{I}b ∨ a = I. By lemma 9.30 (h) ζ_{0} = π^{−} ≪ D_{π}(b[[ζ_{0}]]) and in the induction step follows from ζ_{n} ≪ ζ_{Sn} b[[ζ_{n}]] ≪ b[[ζ_{Sn}]], ζ_{Sn} = D_{π}b[[ζ_{n}]] ≪ D_{π}b[[ζ_{Sn}]] = ζ_{SSn}. Subsubcase π = I, D_{I}b ⪯ a ≺ I. Then by lemma 9.20 and since D_{I}b ⪯ a ≺ I follows Dicom(a) = D_{I}c for some c and D_{I}c ⪯ a ≺ D_{I}(c ˜+1), therefore (by an immediate induction) D_{I}c ⪯ ζ_{n} ≺ ζ_{Sn} ≺ D_{I}(c ˜+1), ∀c η T′.ζ_{n} ≺ c ⪯ ζ_{Sn} → c η̸ F i′. Thereforeζ_{0} = Dicom(a) ≪ Ω̂_{Dicom(a)} = ζ_{1} and in the induction step follows from ζ_{n} ≪ ζ_{Sn} ζ_{Sn} = Ω̂_{ζ_{n}} ≪ Ω̂_{ζ_{Sn}} = ζ_{SSn}. Case s = Ω̂_{a} ∨ s = I: If s[[z]] = z the assertion is trivial, and if s[[z]] = Ω̂_{a[[z]]} follows for a^{∗} ≺ c ⪯ a c η̸ F i′ since a η̸ F i′ and, since a^{∗} ⪯ a[[ξ]] ≪ a[[ρ]] ≺ a, follows by lemma 9.30

(h) Ω̂_{a[[ξ]]} ≪ Ω̂_{a[[ρ]]}.

Lemma 9.32 If s, t, ξ η T′,ξ ˜+1 ≺ τ(s), s[[ξ]] ≺ t ⪯ s[[ξ ˜+1]], then s[[ξ]] ⪯ t[[τ(t)^{−}]]. Proof: s[[ξ]] ≪ s[[ξ ˜+1]], therefore exist s_{0}, …, s_{Sn} such that s_{0} = s[[ξ]], s_{Sn} = s[[ξ ˜+1]] and s_{i} = s_{Si}^{∗}. Then s_{i} ≺ t ⪯ s_{Si} for some i <_{N} n, s[[ξ]] ⪯ s_{i} = s_{Si}^{∗} ⪯ t^{∗} ⪯ t[[τ(t)^{−}]].

The most complicated task in this chapter is, to prove, that for a, z η OT, z η τ(a) \ τ(a)^{−} we have a[[z]] η OT, and that sup{a[[z]]!z η (τ(a) \ τ(a)^{−}) ∩ OT } = a for Limes ordinals. To prove the latter in the case of (D_{a}b)[[1 · n]] = D_{a}(b[[ζ_{n}]]), we will argue, that by knowing D_{a}b[[ζ_{0}]] ≺ D_{a}f ≺ D_{a}b, and D_{a}f η OT we know f ≺ G_{a}f, and have therefore some information on the components of f, which we want to relate to the components of b to have D_{a}f ≺ D_{a}b[[ζ_{n}]] for some n. The components of D_{a}b[[ζ_{n}]] are built by iterated applications of the collaps and building b[[z]]. Therefore we need some relation, that interchanges in some way with the collapsing functions, allows to relate z to a[[z]] and gives some informations on the G_{π} sets. To prove a[[z]] η OT we need again such a relation. The relation that solves this problem is a<_{ξ} b, which relates G_{π}a to the G_{π}c for a ⪯ c ⪯ b, and allows to prove lemma 9.38, that relates a[[z]] to z. Lemma 9.36 allows to infer from D_{π}b η OT to D_{π}a η OT in by <-controlled situations. The relation interchanges with some ordinal functions (see lemma 9.35). Theorem 9.39 and lemma 9.40 are the desired lemmata we need. Definition 9.33 Let s, t η T′. s <_{ξ} t :⇔ s ≺ t ∧ ∀π, r η T′.s ⪯ r ⪯ t → G_{π}s ⪯ G_{π}r ∪ G_{π}^{0}ξ, and this is equivalent to ∀πr η T′.s ≺ r ⪯ t → G_{π}s ⪯ G_{π}r ∪ G_{π}^{0}ξ. Lemma 9.34 Let a, b, c η T′, π η R′.

(a) ∀x η G_{π}b.G_{π}x ⊂ G_{π}b ∧ length(x) <_{N} length(b).

(b) a ≺ D_{π}0 → G_{π}a ≅ ∅.

(c) G_{π}π = 0

(d) (τ(b) = π η R′ ∧ G_{π}c ≺ a ∧ c ≺ b) → c ≺ b[[D_{π}a]].

(e) If a ≺ π, a η OT, then a ≺ D_{π}b ↔ G_{π}a ≺ b.

(f) π ⪯ ρ ≠ I ⇒ G_{ρ}a ⊂ G_{π}a.

(g) π ≺ D_{I}a, G_{π}b ≺ a, b η OT ⇒ G_{I}b ≺ a.

(h) If ρ η R′, G_{π}ρ^{−} ≅ G_{π}ρ.

Proof:

(a) Immediate by induction on length(a).

(b) Induction on length(a). The cases a = b ˜+c and a = ϕ̂_{b}c follow by IH. If a = D_{b}c with (b ≠ I ∨ b ⪯ π ≠ I), follows b ≺ π, G_{π}a ≅ ∅. If a = D_{I}c, π ≺ I follows a ≺ π, G_{π}a ≅ ∅. If a = D_{b}c, b ≺ I = π follows G_{π}a = ∅.

(d) Induction on length(b), Side induction on length(c): Case b η R′: Then b[[D_{π}a]] = D_{π}a. If c = d ˜+e, ϕ̂_{d}e follows the assertion by side IH for d, e. Subcase c = D_{ξ}e. Then ξ ⪯ π∨ξ = I. If π = ξ follows e η G_{π}c ≺ a, c ≺ D_{π}a, if ξ ≺ π ≠ I follows c ≺ D_{π}a, if ξ ≺ π = I follows ξ ≺ b, G_{π}ξ ≅ G_{π}c ≺ a, therefore ξ ≺ D_{π}a, c ≺ D_{π}a, and if π ≺ ξ = I follows D_{ξ}e ≺ π → D_{ξ}e ≺ D_{π}b. Subcase c = Ω̂_{e}: Then if π ≠ I follows from c ≺ π, c ≺ D_{π}a, and if π = I follows by IH e ≺ D_{π}a, c ≺ D_{π}a. Subcase c = I, 0: trivial. Case b = f ˜+g: if c ≺ f the assertion is trivial, otherwise c = f ˆ+d with d ≺ g and by IH follows the assertion. Case b = ϕ̂_{f}g, b[[z]] = ϕ̂_{f}(g[[z]]). If c = d ˜+e the assertion follows by IH for d. If c = ϕ̂_{d}e follows if d ≺ f e ≺ b, by IH e ≺ b[[D_{π}a]], c ≺ b[[D_{π}a]], if f = d e ≺ g, by IH e ≺ g[[D_{π}a]] and the assertion, if f ≺ d c ⪯ g, c ≠ g since g η̸ Cr(f), c ≺ g[[D_{π}a]] and the assertion. If c η G′ follows c ≺ f or c ≺ g and therefore c ≺ g[[D_{π}a]], c ≺ b[[D_{π}a]]. Case b = ϕ̂_{f}g, b[[z]] = ϕ̂_{f[[z]]}g′, g′ = g = 0 or g′ = f η G′ and g = 0 or g′ = ϕ̂_{f}g′′ and

g = g′′ ˜+1 or g′ = g η Cr(f). If c = d ˜+e follows the assertion by IH for d. If c = ϕ̂_{d}e follows, if d ≺ f, d ≺ f[[D_{π}a]], further e ≺ b, e ≺ b[[D_{π}a]], therefore c ≺ b[[D_{π}a]], if d = f follows e ≺ g, either e ⪯ g′′ ∧ g′′ ˜+1 = g therefore c ≺ g′, c ≺ ϕ̂_{f[[D_{π}a]]}g′, or e ≺ g η Cr(f)

therefore c ≺ g′, c ≺ ϕ̂_{f[[D_{π}a]]}g′, and if f ≺ d follows c ⪯ g ⪯ g′, c ⪯ b[[D_{π}a]]. If c η G′ follows by c ≺ b c ⪯ max_{T} {f, g}, g ≺ c = f is not possible, therefore c ⪯ g or c ≺ f, by IH c ⪯ max_{T} {f[[D_{π}a]], g} c ≺ a[[D_{π}a]].

Case b = D_{ρ}f, π = τ(f) ≺ ρ. If c = d ˜+e, ϕ̂_{d}e follows the assertion by side IH for d, e. If c = D_{ξ}d with ξ ≺ ρ ≠ I, the assertion is trivial. If c = D_{ξ}d with ξ ≺ ρ = I follows ξ ≺ D_{ρ}f. If π ⪯ ξ or ξ ≺ π = I follows G_{π}ξ ⊂ G_{π}D_{ξ}d ≺ a, if ξ ≺ π ≠ I follows ξ ⪯ π^{−}, G_{π}ξ ≅ ∅ ≺ a, therefore by side IH ξ ≺ D_{ρ}f[[D_{π}a]], and the assertion. If c = D_{I}d, ρ ≺ I, follows c ≺ ρ, c ≺ D_{ρ}f[[D_{π}a]]. If c = D_{ξ}d with ξ = ρ follows G_{π}d ⊂ G_{π}c ≺ a, d ≺ f, d ≺ f[[D_{π}a]] by IH, c ≺ b[[D_{π}a]]. If c = Ω̂_{d} follows, if ρ ≠ I, c ≺ ρ, c ≺ D_{ρ}f[[D_{π}a]], and if ρ = I by IH d ≺ D_{ρ}f[[D_{π}a]], c ≺ D_{ρ}f[[D_{π}a]]. If c = 0, I, the assertion is trivial. Case b = Ω̂_{e}, b η̸ R′. If c = d ˜+e, ϕ̂_{d}e follows the assertion by side IH. If c = D_{ξ}d, ξ ≠ I follows ξ ⪯ b, since b η̸ R′, ξ ≺ b. If π ⪯ ξ or ξ ≺ π = I follows G_{π}ξ ⊂ G_{π}b ≺ a, and if ξ ≺ π ≠ I follows G_{π}ξ = ∅ ≺ a, by side IH ξ ≺ b[[D_{π}a]]. If c = D_{I}d follows c ⪯ e, c ≠ e, therefore c ≺ e[[D_{π}b]]. If c = Ω̂_{d}, d ≺ f follows G_{π}d ≺ a, by IH d ≺ f[[D_{π}a]], c ≺ b[[D_{π}a]]. If c = 0, I the assertion ist trivial.

(e) “←” follows by (d), with b := π. Proof of “→” by Induction on length(a). If a = c ˜+d, ϕ̂_{b}c, Ω̂_{c}, the assertion follows by IH, and if a = 0, I the assertion is trivial. Case a = D_{ρ}c. If ρ ≺ π ≠ I follows G_{π}a ≅ ∅. If ρ ≺ π = I follows ρ ≺ D_{π}b, G_{π}a ≅ G_{π}ρ ≺ b by IH. If π ≺ ρ = I follows a ≺ D_{π}0, G_{π}a ≅ ∅. If ρ = π follows c ≺ b, G_{π}a ≅ G_{π}π ∪ G_{π}c ∪ {c} ≺ b, since G_{π}c ≅ G_{ρ}c ≺ c, G_{π}π ≅ ∅.

(f) Induction on length(a). If a = b ˜+c, ϕ̂_{b}d, Ω̂_{b}, the assertion follows by IH, and if a = 0, I, the assertion is trivial. Case a = D_{ξ}b. If ρ ⪯ ξ ≠ I follows G_{ρ}a ≅ {b} ∪ G_{ρ}a ∪ G_{ρ}ξ ⊂ {b} ∪ G_{π}a ∪ G_{π}ξ ≅ G_{π}a. If ρ ⪯ D_{I}b ∧ ξ = I follows G_{ρ}a ≅ {b} ∪ G_{ρ}a ∪ G_{ρ}ξ ⊂ {b} ∪ G_{π}a ∪ G_{π}ξ ≅ G_{π}a. Otherwise follows G_{ρ}a ≅ ∅.

(g) Induction on length(b). If b = c ˜+d, ϕ̂_{c}d, Ω̂_{c} the assertion follows by IH, and if b = 0, I the assertion is trivial. Case b = D_{ρ}c. If ρ ≺ π follows b ≺ ρ ⪯ π ≺ D_{I}a, G_{I}b ≺ a. If π ⪯ ρ ≺ I follows G_{π}ρ ⊂ G_{π}b ≺ a, by IH G_{I}d ≅ G_{I}ρ ≺ c. Subcase ρ = I: If b ≺ D_{I}a follows c ≺ a and, if D_{I}a ≺ b, by G_{π}b ≅ G_{π}c ∪ G_{π}I ∪ {c} ≺ a again c ≺ a, therefore in both cases, since by b η OT, G_{I}c ≺ c, G_{I}b = G_{I}c ∪ G_{I}I ∪ {c} ⪯ c ≺ a. If I ≺ ρ follows G_{I}d ≅ ∅.

(h) If ρ = I this is trivial, if ρ = Ω̂_{c_{+}1} follows G_{π}ρ ≅ G_{π}(c ˜+1) ≅ G_{π}c ≅ G_{π}ρ, and if ρ = Ω̂_{c} with ρ^{−} = c follows G_{π}ρ ≅ G_{π}c. Lemma 9.35 Let a, b, c, ξ, ρ η T′, b <_{ξ} c.

(a) a η {0, 1, ω, Ω̂_{0}, I}, a ≺ b → a <_{0} b.

(b) If a <_{ξ} c, a ⪯ b ≺ d ≺ c, ∀π η R′.G_{π}b ⪯ G_{π}a ∪ G_{π}ξ^{0}, then b <_{ξ} d.

(c) If a ≺ c, then a <_{a} c.

(d) If a <_{ξ} b, ∀π η R′.G_{π}ξ ⪯ G_{π}ρ (e.g. ξ = 0), then a <_{ρ} b.

(e) If c ≠ 0, then a <_{ξ} a ˆ+c.

(f) NF_{+}(a, c) → a ˆ+b <_{ξ} a ˆ+c.

(g) ∀d η T′.b ≺ d ⪯ c → d η̸ Cr(a)) → ϕ̂_{a}b <_{ξ} ϕ̂_{a}c.

(h) b η Cr(a) ∪ {0}, then b <_{0} ϕ̂_{a}b.

(i) If b = 0 ∨ b η Cr(a) and a′ <_{ξ} a, then ϕ̂_{a′}b <_{ξ} ϕ̂_{a}b.

(j) If a η G′, a <_{0} ϕ̂_{a}0.

(k) If a η G′, a′ <_{ξ} a, then ϕ̂_{a′}a <_{ξ} ϕ̂_{a}0.

(l) a′ <_{ξ} a → ϕ̂_{a′} ϕ̂_{a}b <_{ξ} ϕ̂_{a}(b ˜+1).

(m) ϕ̂_{0}1 <_{0} ϕ̂_{1}0.

(n) D_{a}b <_{ξ} D_{a}c.

(o) Ω̂_{b} <_{ξ} Ω̂_{c}.

(p) If s η R′, then s^{−} <_{0} s.

(q) a^{−} <_{0} D_{a}b.

Proof:

(a) Trivial, since G_{π}a ≅ ∅.

(b) b ⪯ r ⪯ d, then G_{π}b ⪯ G_{π}a ∪ G_{π}^{0}ξ ⪯ G_{π}r ∪ G_{π}^{0}ξ. (c), (d): trivial.

(e) If a ⪯ d ⪯ a ˆ+c follows d = a ˆ+e for some e and we have the assertion.

(f) a ˆ+b ⪯ d ⪯ a ˆ+c, then d = a ˆ+e with b ⪯ e ⪯ c, G_{π}b ⪯ G_{π}e ∪ G_{π}^{0}ξ, G_{π}(a ˆ+b) ⪯ G_{π}a ∪ G_{π}e ∪ G_{π}^{0}ξ ≅ G_{π}(a ˆ+e) ∪ G_{π}^{0}ξ.

(g) s := ϕ̂_{a}b ⪯ d ⪯ ϕ̂_{a}c := t. Then d η̸ Cr(a) since otherwise b ≺ d ⪯ c. We show G_{π}s ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f follows s ⪯ max_{T} {e, f} ⪯ ϕ̂_{a}c, G_{π}s ⪯ G_{π}max_{T} {e, f} ∪ G_{π}^{0}ξ. If d = ϕ̂_{e}f with e ≺ a follows s ⪯ f ⪯ t and again the assertion by IH. If d = ϕ̂_{a}f follows b ⪯ f ⪯ c and by b <_{ξ} c the assertion. d η G′ is not possible, otherwise a ≺ ϕ̂_{a}b ≺ d, d η Cr(a).

(h) The case b = 0 is trivial. Let therefore b η Cr(a), b ≺ d ⪯ ϕ̂_{a}b. Then d η̸ Cr(a) since otherwise b ≺ d ⪯ b. We show G_{π}b ⪯ G_{π}^{0}d by induction on length(d). If d = e ˜+f follows b ⪯ e ⪯ ϕ̂_{a}b, G_{π}b ⪯ G_{π}^{0}e. If d = ϕ̂_{e}f with e ≺ a follows b ⪯ f ⪯ ϕ̂_{a}b and again the assertion by IH. If d = ϕ̂_{a}f follows b ⪯ f ⪯ b, b = f and the assertion trivially. d η G′ is not possible, otherwise a ≺ b ≺ d, d η Cr(a).

(i) Let ϕ̂_{a′}b ≺ d ⪯ ϕ̂_{a}b. Then d η̸ Cr(a) since otherwise b ≺ d ⪯ b. We show G_{π}ϕ̂_{a′}b ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f this follows by IH. Case d = ϕ̂_{e}f. If e ≺ a′ follows ϕ̂_{a}b′ ⪯ f ⪯ ϕ̂_{a}b and by IH the assertion. If a′ ⪯ e ⪯ a follows b ≺ ϕ̂_{a′}b ≺ ϕ̂_{e}f ⪯ ϕ̂_{a}b and by(h) G_{π}b ⪯ G_{π}ϕ̂_{e}f ∪G_{π}^{0}ξ, and with a′ <_{ξ} a, G_{π}a′ ⪯ G_{π}^{0}e∪G_{π}ξ, therefore G_{π}ϕ̂_{a′}b ⪯ G_{π}ϕ̂_{e}f ∪ G_{π}^{0}ξ. d η G′ ∪ {0} is not possible.

(j) Let a ≺ d ⪯ ϕ̂_{a}0. Then d η̸ Cr(a). We show G_{π}a ⪯ G_{π}d by induction on length(d). If d = e ˜+f this follows by IH. Case d = ϕ̂_{e}f. If e ≺ a follows a ⪯ f ⪯ ϕ̂_{a}0 and by IH the assertion. If e = a follows G_{π}a ⪯ G_{π}^{0}e. d η G′ is not possible.

(k) Let s := ϕ̂_{a′}a ≺ d ⪯ ϕ̂_{a}0 =: t. Then d η̸ Cr(a). We show G_{π}ϕ̂_{a′}a ⪯ G_{π}^{0}d by induction on length(d). If d = e ˜+f this follows by IH. Case d = ϕ̂_{e}f. If e ≺ a′ follows s ⪯ f ⪯ t and by IH the assertion. If a′ ⪯ e ⪯ a follows a ≺ ϕ̂_{a′}a ≺ ϕ̂_{e}f ⪯ t and by (j) G_{π}a ⪯ G_{π}^{0}ϕ̂_{e}f, and with a′ <_{ξ} a, G_{π}a′ ⪯ G_{π}e ∪ G_{π}^{0}ξ, therefore G_{π}ϕ̂_{a′}b ⪯ G_{π}ϕ̂_{e}f ∪ G_{π}^{0}ξ. d η G′ ∪ {0} is not possible.

(l) Let s := ϕ̂_{a′} ϕ̂_{a}b ≺ d ⪯ ϕ̂_{a}(b ˜+1) =: t. Then ϕ̂_{a}b ≺ d ⪯ t, therefore d η̸ Cr(a). We show G_{π}s ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f this follows by IH. Case d = ϕ̂_{e}f. If e ≺ a′ follows s ⪯ f ⪯ t and by IH the assertion. If a′ ⪯ e ⪯ a follows ϕ̂_{a}b ≺ ϕ̂_{a′} ϕ̂_{a}b ≺ ϕ̂_{e}f ⪯ t and by (g), since b <_{0} b ˜+1 G_{π}ϕ̂_{a}b ⪯ G_{π}^{0}ϕ̂_{e}f, and with a′ <_{ξ} a, G_{π}a′ ⪯ G_{π}e ∪ G_{π}^{0}ξ, therefore G_{π}ϕ̂_{a′} ϕ̂_{a}b ⪯ G_{π}ϕ̂_{e}f ∪ G_{π}^{0}ξ. d η G′ ∪ {0} is not possible.

(n) Let s := D_{a}b ≺ d ⪯ D_{a}c =: t. We show G_{π}s ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f or d = ϕ̂_{e}f follows s ⪯ max_{T} {e, f} ⪯ t, G_{π}b ⪯ G_{π}max_{T} {e, f} ∪ G_{π}^{0}ξ. Case d = D_{e}f: If a ≠ I ∨ e = I, follows e = a, b ⪯ f ⪯ c, and in all cases the assertion by b <_{ξ} c, b ⪯ f. If a = I ∧ e ≠ I follows e ≺ I and s ⪯ e ⪯ t, G_{π}s = ∅ or π ⪯ D_{I}b ⪯ e or π = I and G_{π}s ⪯ G_{π}e ∪ G_{π}^{0}ξ ⊂ G_{π}(D_{e}f) ∪ G_{π}^{0}ξ. Case d = Ω̂_{e}. This is only possible, if a = I, s ⪯ e ⪯ t. Then by IH G_{π}s ⪯ G_{π}e ∪ G_{π}^{0}ξ ≅ G_{π}d ∪ G_{π}^{0}ξ. Cases d = 0, I: not possible.

(m) G_{π}ϕ̂_{0}1 ≅ ∅.

(o) Let s := Ω̂_{b} ≺ d ⪯ Ω̂_{c} =: t. We show G_{π}s ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f, ϕ̂_{e}f follows s ⪯ max_{T} {e, f} ⪯ t, G_{π}b ⪯ G_{π}max_{T} {e, f} ∪ G_{π}^{0}ξ by IH or trivially (if s = max_{T} {e, f}). Case d = D_{e}f: If e = I, follows b ≺ d ⪯ c, G_{π}s ≅ G_{π}b ⪯ G_{π}d∪G_{π}^{0}ξ. Otherwise s ≺ e ⪯ t. If in this case π ⪯ s ≺ e or e ⪯ π = I follows G_{π}s ⪯ G_{π}e ∪ G_{π}^{0}ξ ⊂ G_{π}D_{e}f ∪ G_{π}^{0}ξ. If s ≺ π ≠ I follows G_{π}s ≅ ∅. Case d = Ω̂_{e}. Then the assertion follows immediately by b ≺ e ⪯ c and b <_{ξ} c.

(p) If s = Ω̂_{a_{+}1} this follows by a <_{0} a ˜+1, s^{−} = Ω̂_{a} and (o). If s η {Ω̂_{0}, I, Ω̂_{I}} follows G_{π}(s) ≅ ∅. Case s = Ω̂_{D_{I}b}, let s′ := D_{I}b ≺ d ⪯ Ω̂_{s′}. We show G_{π}s′ ⪯ G_{π}d ∪ G_{π}^{0}ξ by induction on length(d). If d = e ˜+f, ϕ̂_{e}f follows s′ ⪯ max_{T} {e, f} ⪯ s, G_{π}s′ ⪯ G_{π}^{0}max_{T} {e, f} by IH or trivially (if s = max_{T} {e, f}). Subcase d = D_{e}f: Then e ≠ I, since otherwise D_{I}b ≺ d ⪯ D_{I}b. Therefore s′ ≺ e ⪯ s, e = t. If now π ⪯ s = e or s ≺ π = I, follows G_{π}s ⪯ G_{π}e ∪ G_{π}^{0}ξ ⊂ G_{π}D_{e}f ∪ G_{π}^{0}ξ. If s ≺ π ≠ I, follows s ≺ D_{π}0, G_{π}s ≅ ∅. Case d = Ω̂_{e}. Then D_{I}b ⪯ e ⪯ D_{I}b.

(q) Follows by (p), (b). Lemma 9.36 If b <_{z} a, G_{π}a ≺ a, G_{π}z ≺ b, then G_{π}b ≺ b. Proof: Since G_{π}b is finite and ∀x, y η T′.x ≺ y ∨ x = y ∨ y ≺ x, follows G_{π}b ≺ b ∨ ∃c η G_{π}b.b ⪯ c. Assume c η G_{π}b with minimal length(c) such that b ⪯ c. Since G_{π}c ⊂ G_{π}b and for all d η G_{π}c, length(d) <_{N} length(c) follows G_{π}c ≺ b. Further G_{π}b ⪯ G_{π}a∪G_{π}^{0}z ≺ a therefore b ⪯ c ≺ a, by b <_{z} a follows c η G_{π}b ⪯ G_{π}c ∪ G_{π}^{0}z ≺ b, a contradiction. Lemma 9.37 (a) If s ≠ 0, then s^{∗} <_{0} s.

(b) r ≪ s ⇒ r <_{0} s. Proof: (a) By Induction on length(s). If s η R′ s^{−} = s^{∗} <_{0} s. If s = a ˜+b follows b^{∗} <_{0} b, therefore the assertion by lemma 9.35 (f). If s = ϕ̂_{a}b follows the assertion by IH, lemma 9.35 (g) and lemma 9.27 (a), or lemma 9.35 (h), or lemma 9.35 (j), or lemma 9.35 (i) and IH or lemma 9.35 (m). If s = D_{a}b follows the assertion by 9.35

(n) or 9.35 (q) and 9.35 (a). If s = Ω̂_{a}, s η̸ R′ follows by lemma 9.35 (o) the assertion.

(b) By (a). Lemma 9.38 Let a η T′.

(a) If z η τ̂(a) ∩ T′, then a[[z]] <_{z} a.

(b) a ≠ 0 → a[[τ(a)^{−}]] <_{0} a.

Proof of (a), (b) simultaneously by Induction on length(a). If τ(a) = ω we show (except in some cases where the proof is trivial) by side induction on n : N a[[1 · n]] <_{0} a, and (b) follows by (a). If a = 0 this is trivial. If a η R′, follows (a) by 9.35 (c), further a[[τ(a)^{−}]] = a^{−} <_{0} a by 9.35 (p). Case a = b ˜+c: by IH and 9.35 (f). Case a = ϕ̂_{b}c: Subcase b = c = 0: trivial. Subcase b = 0, c = c′ ˜+1: By lemma 9.35 (g) ϕ̂_{0}c′ <_{0} ϕ̂_{0}c, further G_{π}ϕ̂_{0}c′ · Sn = G_{π}ϕ̂_{0}c′, by 9.35 (b) follows ϕ̂_{0}c′ · SSn <_{0} ϕ̂_{0}c. Subcase c η Lim′, c η̸ Cr(b). c[[z]] <_{z} c, c[[τ(c)^{−}]] <_{0} c by IH, c^{∗} ⪯ c[[τ(c)^{−}]] ⪯ c[[z]] ≺ c, therefore by lemma 9.27 (a) and 9.35 (g) the assertion. Subcase b = 0, c η Cr(b): Lemma 9.35 (h) and (b). Subcase b = b′ ˜+1, c = 0: By lemma 9.35 (i) ρ_{1} <_{0} a, and G_{π}a[[1 · Sn]] ≅ G_{π}ρ_{1}, therefore a[[1 · n]] <_{0} a. Subcase b = b′ ˜+1, c = c′ ˜+1: By lemma 9.35 (g) ρ_{0} <_{0} a, and G_{π}a[[1 · Sn]] ≅ G_{π}ρ_{0}, since G_{π}a ≅ G_{π}a′ ∪ G_{π}0, therefore a[[1 · n]] <_{0} a. Subcase b = b′ ˜+1, c η Cr(b): By lemma 9.35 (i) ρ_{1} <_{0} a, and G_{π}a[[1 · Sn]] ≅ G_{π}ρ_{1}, therefore a[[1 · n]] <_{0} a. Subcase b η Lim′, c = 0, c′ ˜+1 ∨ c η Cr(b): b[[z]] <_{z} b, b[[τ(b)^{−}]] <_{0} b, lemma 9.35 (i), (k), (l). Case a = D_{b}c: Subcase c = 0: By lemma 9.35 (q), (a) follows ρ_{0} <_{0} a, and G_{π}a[[1 · n]] ≅ G_{π}ρ_{0}. Subcase c = c′ ˜+1: By lemma 9.35 (n) follows a[[0]] <_{0} a, and G_{π}a[[1 · n]] ≅ G_{π}a[[0]]. Subcase c η Lim′, τ(c) ≺ b: Lemma 9.35 (n) and IH. Subcase c η Lim′, b ⪯ τ(c): Then a[[1 · n]] = D_{b}(c[[ζ_{n}]]). By IH we have c[[ζ_{n}]] <_{ζ_{n}} c. Assume (D_{b}c)[[1 · n]] ⪯ d ⪯ D_{b}c, π η R′. We show G_{π}((D_{b}c)[[1 · n]]) ⪯ G_{π}^{0}d by side induction on length(d). If d = e ˜+f follows G_{π}((D_{b}c)[[1 · n]]) ⪯ G_{π}^{0}e ⊂ G_{π}^{0}d, if d = ϕ̂_{e}f follows the assertion similarly with e replaced by max_{T} {e, f}, and the case d = Ω̂_{e} is only possible if b = I, and the assertion follows by side IH for e. Subsubcase d = D_{e}f, e ≠ b: Then b = I, a[[1 · n]] ≺ e ⪯ a. If a[[1 · n]] ≺ π ≠ I follows G_{π}a[[1·n]] ≅ ∅, and if π ⪯ a[[1·n]]∨π = I follows by IH G_{π}a[[1·n]] ⪯ G_{π}e∪G_{π}^{0}ξ ⊂ G_{π}d∪G_{π}^{0}ξ. Subsubcase d = D_{b}f: If b ≺ π ≠ I or D_{b}(c[[ζ_{n}]]) ≺ π ≺ I ∧b = I follows G_{π}a[[1·n]] ≅ ∅. If b ≺ π = I follows G_{π}a[[1 · n]] ≅ G_{π}b ≅ G_{π}d. Therefore assume π ⪯ b ≠ I ∨ (b = I ∧ (π ⪯ D_{I}c[ζ_{n}] ∨ π = I)). Then G_{π}a[[1 · n]] ≅ G_{π}b ∪ G_{π}c[[ζ_{n}]] ∪ {c[[ζ_{n}]]}, and, since c[[ζ_{n}]] ⪯ f, G_{π}d ≅ G_{π}b ∪ G_{π}f ∪ {f}. We show by side induction on m : N

∀m <_{N} n.G_{π}c[[ζ_{m}]] ⪯ G_{π}^{0}d. (+)

We have by main IH c[[ζ_{m}]] <_{ζ_{m}} c ∧ c[[ζ_{m}]] ⪯ c[[ζ_{n}]] ⪯ f ⪯ c, therefore

G_{π}c[[ζ_{m}]] ⪯ G_{π}f ∪ G_{π}^{0}ζ_{m} ⊂ G_{π}d ∪ G_{π}^{0}ζ_{m} ⊂ G_{π}d ∪ G_{π}^{0}ζ_{m} (∗)

Case m = 0: If τ(c) = I ∧ D_{I}c ⪯ b ≺ I follows G_{π}ζ_{0} = G_{π}Dicom(b) ⊂ G_{π}b ⊂ G_{π}d by lemma 9.20 (b) and by (∗) follows the assertion, otherwise, we have ζ_{0} = τ(c)^{−}, c[[ζ_{0}]] <_{0} c ∧ c[[ζ_{0}]] ⪯ f ⪯ c, therefore G_{π}c[[ζ_{m}]] ⪯ G_{π}^{0}f ∪ G_{π}b. Case m = Sm′: If τ(c) ≠ I or b ≺ D_{I}c or b = I follows with t := τ(c):

G_{π}c[[ζ_{m}]] ⪯ G_{π}f ∪ G_{π}^{0}ζ_{m}
≅ G_{π}f ∪ G_{π}^{0}D_{t}c[[ζ_{m′}]]
⪯ G_{π}f ∪ G_{π}^{0}t ∪ G_{π}^{0}c[[ζ_{m′}]] ∪ {c[[ζ_{m′}]]}
⪯ G_{π}f ∪ G_{π}^{0}d ∪ G_{π}^{0}c[[ζ_{m′}]] ∪ {c[[ζ_{m′}]]}
⪯ G_{π}^{0}f ∪ G_{π}^{0}d ∪ {f}
≅ G_{π}^{0}d

since π ⪯ b ⪯ τ(c) and G_{π}t ≅ G_{π}ζ_{0} ⪯ G_{π}(c[[ζ_{0}]]) ⪯ G_{π}^{0}d by lemma 9.23 (b) or τ(c) = I and G_{π}t ≅ ∅. If D_{I}c ⪯ b ≺ τ(c) = I follows G_{π}^{0}ζ_{m} ≅ G_{π}^{0}ζ_{0}, G_{π}c[[ζ_{m}]] ⪯ G_{π}f ∪ G_{π}^{0}ζ_{0} ⪯ G_{π}^{0}d. Now we can conclude (+) G_{π}a[[1·n]] ≅ G_{π}b∪G_{π}c[[ζ_{n}]]∪{c[[ζ_{n}]]} ⪯ G_{π}b∪G_{π}^{0}d∪{f} ≅ G_{π}^{0}d. Case a = Ω̂_{b}, a η̸ R′. Then by IH b[[z]] <_{z} b, b[[τ(b)^{−}]] <_{0} b and by lemma 9.35 (o) follows the assertion. Theorem 9.39 ∀s, t η OT .s[[τ(s)^{−}]] ⪯ t ≺ s → ∃ξ η OT ∩ τ̂(s).s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Proof by Induction on length(s), side induction on length(t). If τ(s) = ω it is sufficient to prove ∃n ∈ N.t ≺ s[[1 · n]], or (if s[[1 · n]] = ρ_{S^{i}n} to prove ∃n ∈ N.t ≺ ρ_{n}) and for the minimal such n : N follows the assertion. Case s = 0: trivial. Case s η R′: Let ξ = t. Case s = b ˜+c. Then b ˆ+(c[[τ(c)^{−}]]) ⪯ t ≺ b ˆ+c, t = b ˆ+d with c[[τ(c)^{−}]] ⪯ d ≺ c, by IH c[[ξ]] ⪯ d ≺ c[[ξ ˜+1]] for some ξ η OT ∩ τ̂(c) and with this ξ we have the assertion. Case s = ϕ̂_{b}c: Then by lemma 9.27 (a) and (b) t η̸ Cr(b) ∪ G′ Subcase b = c = 0: Then t = 0, let ξ = 0. Subcase b = 0, c = c′ ˜+1: Then ϕ̂_{0}c′ ≺ t ≺ ϕ̂_{0}c. If t = ϕ̂_{d}e follows by t η̸ Cr(0), d = 0, c′ ≺ t ≺ c, which is not possible, therefore t = d ˜+e. Then by IH or trivially we have d ≺ s[[1 · n]] for some n : N, e ⪯ t ≺ ϕ̂_{0}c, e ⪯ ϕ̂_{0}c′, t ≺ s[[1 · Sn]]. Subcase b = 0, c η Cr(b): Similar to the case c = c′ ˜+1. Subcase c η Lim′ \ Cr(b): If t = d ˜+e follows the assertion by side IH for d. If t = ϕ̂_{d}e and d ≺ b follows the assertion by side IH for e, and if d = b follows by main-IH d[[ξ]] ⪯ e ≺ d[[ξ ˜+1]] and the assertion. Subcase b = b′ ˜+1, c = 0: If t = d ˜+e follows the assertion by side IH for d. Subsubcase t = ϕ̂_{d}e: If d ≺ b′ follows the assertion by side IH for e. If d = b′ follows by side IH (or trivially if e ≺ ρ_{0}) e ≺ ρ_{n}, ϕ̂_{d}e ≺ ϕ̂_{b′}ρ_{n} = ρ_{Sn}. If d = b follows e ≺ 0, a contradiction. Subcase b = b′ ˜+1, c = c′ ˜+1: If t = d ˜+e follows the assertion by side IH for e. Subsubcase t = ϕ̂_{d}e: If d ≺ b′ follows the assertion by side IH for e. If d = b′ follows by side IH (or trivially if e ≺ ρ_{0}) e ≺ ρ_{n}, ϕ̂_{d}e ≺ ϕ̂_{b′}ρ_{n} = ρ_{Sn}. If d = b follows e = c′, let ξ = 0. Subcase b = b′ ˜+1, c η Cr(b): As the last subcase. Subcase b η Lim′ \ G′, c = 0: The assertion follows by IH if t = d ˜+e or t = ϕ̂_{d}e with d ≺ b[[τ(b)^{−}]]. Otherwise follows t = ϕ̂_{d}e with b[[τ(b)^{−}]] ⪯ d ≺ b, b[[ξ]] ⪯ d ≺ b[[ξ ˜+1]] for some ξ η OT ∩ τ(b). By t ≺ b follows e ≺ ϕ̂_{b}0. If e ≺ ϕ̂_{b[[ξ_{+}1]]}0 follows s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Otherwise, s[[τ(s)^{−}]] ⪯ s[[ξ ˜+1]] ⪯ e ≺ s, by side IH s[[ρ]] ⪯ e ≺ s[[ρ ˜+1]] for some ξ ≺ ρ ≺ τ(s)^{−}. Since d ≺ b[[ρ ˜+1]] follows s[[ρ]] ⪯ t ≺ s[[ρ ˜+1]].

Subcase b η G′, c = 0: The assertion follows by IH if t = d ˜+e or t = ϕ̂_{d}e with d ≺ b[[τ(b)^{−}]]. Otherwise follows t = ϕ̂_{d}e with b[[τ(b)^{−}]] ⪯ d ≺ b, b[[ξ]] ⪯ d ≺ b[[ξ ˜+1]] for some ξ η OT ∩ τ(b). b ⪯ s[[τ(s)^{−}]] ⪯ ϕ̂_{d}e ≺ ϕ̂_{b}0, d ≺ b η G′, therefore b ⪯ e ⪯ ϕ̂_{b}0. If e ≺ ϕ̂_{b[[ξ_{+}1]]}b follows s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Otherwise, s[[τ(s)^{−}]] ⪯ s[[ξ ˜+1]] ⪯ e ≺ s, by side IH s[[ρ]] ⪯ e ≺ s[[ρ ˜+1]] for some ξ ≺ ρ. Since d ≺ b[[ρ ˜+1]] follows s[[ρ]] ⪯ t ≺ s[[ρ ˜+1]]. Subcase b η Lim′, c = c′ ˜+1: Again the only interesting case is t = ϕ̂_{d}e, b[[τ(b)^{−}]] ⪯ d ≺ b, b[[ξ]] ⪯ d ≺ b[[ξ ˜+1]] by IH. ϕ̂_{b[[τ(b)^{−}]]}ϕ̂_{b}c′ ⪯ ϕ̂_{d}e ≺ ϕ̂_{b}c, if b[[τ(b)^{−}]] = d, ϕ̂_{b}c′ ⪯ e ≺ c ⪯ ϕ̂_{b}c,

and if b[[τ(b)^{−}]] ≺ d, ϕ̂_{b}c′ ⪯ ϕ̂_{d}e ≺ ϕ̂_{b}c and, because d ≺ b, ϕ̂_{b}c′ ⪯ e ≺ ϕ̂_{b}c, in any case therefore ϕ̂_{b}c′ ⪯ e ≺ ϕ̂_{b}c. If e ≺ s[[ξ ˜+1]] follows s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Otherwise, s[[τ(s)^{−}]] ⪯ s[[ξ ˜+1]] ⪯ e ≺ s, by side IH s[[ρ]] ⪯ e ≺ s[[ρ ˜+1]] for some ξ ≺ ρ. Since d ≺ b[[ρ]] follows s[[ρ]] ⪯ t ≺ s[[ρ ˜+1]]. Subcase b η Lim′, c η Cr(b): Again the only interesting case is t = ϕ̂_{d}e, b[[τ(b)^{−}]] ⪯ d ≺ b, b[[ξ]] ⪯ d ≺ b[[ξ ˜+1]] by IH. ϕ̂_{b[[τ(b)^{−}]]}c ⪯ ϕ̂_{d}e ≺ ϕ̂_{b}c, therefore c ⪯ e ≺ ϕ̂_{b}c. If e ≺ s[[ξ ˜+1]] follows s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Otherwise, s[[τ(s)^{−}]] ⪯ s[[ξ ˜+1]] ⪯ e ≺ s, by side IH s[[ρ]] ⪯ e ≺ s[[ρ ˜+1]] for some ξ ≺ ρ. Since d ≺ b[[ρ]] follows s[[ρ]] ⪯ t ≺ s[[ρ ˜+1]]. Case s = D_{b}c: Then t η R′ only possible if b = I. Subcase c = 0, b ≠ I. If t = d ˜+e follows from the side IH for d or trivially d ≺ ρ_{n} for some n, t ≺ ρ_{max_{T} {n,1}}. If t = ϕ̂_{c}d follows c, d ≺ s, ∃n ∈ N.c, d ≺ ρ_{n}, ϕ̂_{c}d ≺ ϕ̂_{ρ_{n}}0. t = D_{d}e, Ω̂_{e}, I is not possible. Subcase c = 0, b = I: If t = d ˜+e, ϕ̂_{d}e follows by side IH or trivially d, e ≺ ρ_{n} for some n : N, t ≺ ρ_{n}. If t = D_{d}e follows a ≠ I, d ≺ ρ_{n} for some n, t ≺ ρ_{n}. If t = Ω̂_{e} follows e ≺ s, e ≺ ρ_{n} for some n : N, t ≺ ρ_{Sn}. Subcase c = c′ ˜+1, b ≠ I: If t = d ˜+e follows d ≺ ρ_{n} for some n, t ≺ ρ_{n}. If t = ϕ̂_{c}d follows c, d ≺ s, c, d ≺ ρ_{n} for some n : N, ϕ̂_{c}d ≺ ϕ̂_{ρ_{n}}0. If t = D_{d}e follows t = ρ_{0}, and t = Ω̂_{e}, I is not possible. Subcase c = c′ ˜+1, b = I: If t = d ˜+e, ϕ̂_{d}e follows by side IH or trivially d, e ≺ ρ_{n} for some n : N, t ≺ s[[1 · n]]. If t = D_{d}e follows d = I and t = ρ_{0} or d ≺ I, d ≺ ρ_{n}, d ≺ ρ_{n} for some n, t ≺ ρ_{n}. If t = Ω̂_{e} follows e ≺ s, e ≺ ρ_{n} for some n : N, t ≺ ρ_{Sn}. Subcase c η Lim′, τ(c) ≺ b: If t = d ˜+e follows by side IH s[[ξ]] ⪯ d ≺ s[[ξ ˜+1]], s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]], and if t = ϕ̂_{d}e follows similarly the assertion by side IH for max_{T} {d, e}. If t = D_{d}e follows, if b ≠ I ∨d = b, first d = b, by IH c[[ξ]] ⪯ e ≺ c[[ξ ˜+1]], s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]], and if b = I, d ≠ I, s[[ξ]] ⪯ d ≺ s[[ξ ˜+1]], s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. If t = Ω̂_{d} follows b = I, s[[ξ]] ⪯ d ≺ s[[ξ ˜+1]], and further s[[ξ]] ⪯ t ≺ s[[ξ ˜+1]]. Subcase c η Lim′, b ⪯ τ(c) =: π. If t = 0, I the assertion follows in all cases trivially, if t = d ˜+e ∨ t = ϕ̂_{d}e ∨ (t = D_{d}e ∧ d ≠ b) ∨ (t = Ω̂_{d} ∧ b = I) by side IH. Let t = D_{b}e. Subsubcase π ≠ I ∨ b ≺ D_{I}c ∨ b = I: We show by induction on length(f):

G_{π}f ≺ c → ∃n ∈ N.G_{π}f ≺ c[[ζ_{n}]] (∗)

If f = g ˜+h, ϕ̂_{g}h, Ω̂_{g} follows (∗) by IH. Let f = D_{ξ}h. If π ⪯ ξ ≠ I ∨ (ξ = I ∧ (π = I ∨ π ⪯ D_{I}b) follows G_{π}ξ ∪ G_{π}h ∪ {h} ≺ c, by IH G_{π}ξ ∪ G_{π}h ≺ c[[ζ_{n}]] for some n : N. Further G_{π}h ≺ c[[ζ_{n}]] ∧ τ(c) = π ∧ h ≺ c, therefore by lemma 9.34 (d) h ≺ c[[D_{π}c[[ζ_{n}]]]] = c[[ζ_{Sn}]]. Otherwise G_{π}f = ∅, or G_{π}f = G_{π}ξ ≺ c[[ζ_{n}]] for some n. Now by t η OT follows G_{b}e ≺ e ≺ c and since b ⪯ π ≠ I ∨ π = b ∨ (π = I ∧ b ≺ D_{I}c) follows by lemma 9.34 (f) and (g) G_{π}d ≺ c, by (∗) ∃n ∈ N.G_{π}d ≺ c[[ζ_{n}]]. τ(c) = π ∧ d ≺ c, therefore by lemma 9.34 (d) d ≺ c[[D_{π}c[[ζ_{n}]]]] = c[[ζ_{Sn}]], t ≺ D_{b}c[[ζ_{Sn}]] = a[[1 · Sn]].

Subsubcase π = I ∧ (D_{I}c ⪯ b ≺ I): We show by Induction on length(f)

G_{b}f ≺ c ∧ f ≺ I → ∃n ∈ N.f ≺ ζ_{n} (∗)

If f = g ˜+h, ϕ̂_{g}h, Ω̂_{g} the assertion follows by IH, if f = 0, I trivially. Case f = D_{ρ}g, ρ ≠ I: If ρ ⪯ b follows, since by lemma 9.20 (a) b ≺ ζ_{n} for some n, ρ ≺ ζ_{n}. If b ≺ ρ ≺ I follows G_{b}ρ ⊂ G_{b}f ≺ c ∧ ρ ≺ I, therefore f ≺ ρ ≺ ζ_{n} for some n. Case f = D_{I}g. If b ⪯ f, follows G_{b}f ≅ G_{b}I ∪ G_{b}g ∪ {g} ≺ c, f ≺ D_{I}c ⪯ b a contradiction. Therefore f ≺ b ≺ ζ_{n} for some n by lemma 9.20 (a). Now we have c[[0]] = c[[Ω̂_{0}]] ⪯ c[[ζ_{0}]] ⪯ e ≺ c, by IH we have c[[ξ]] ⪯ e ≺ c[[ξ ˜+1]]. By lemma 9.31 and 9.37 (b) follows c[[ξ]] <_{0} c[[ξ ˜+1]], therefore by lemma 9.23 (b) G_{b}ξ ⪯ G_{b}c[[ξ]] ⪯ G_{b}^{0}e ⪯ e ≺ c, ξ ≺ I, therefore by (∗) ξ ≺ ζ_{n} for some n, t = D_{I}d ≺ D_{I}d[[ξ ˜+1]] ≺ D_{I}d[[ζ_{Sn}]] = s[[1 · Sn]]. Lemma 9.40 If a η OT, z η OT ∩ τ̂(a), then a[[z]] η OT. Proof: Induction on length(a). Case a = 0: Obvious. Case a = b ˜+c. By IH c[[z]] η OT, c[[z]] ≺ c, b ˆ+c η OT, therefore by lemma 9.13 (h) b ˆ+c[[z]] η OT. Case a = ϕ̂_{b}c: The assertion follows in all cases by IH, in some cases by induction on length(z) (if τ(a) = ω). Case a = D_{π}c: Subcase c = 0: Obvious, since π^{−} η OT. Subcase c = c′ ˜+1: Since D_{π}(c′ ˜+1) η T′ follows G_{π}(c′ ˜+1) ≺ (c′ ˜+1), c′ <_{0} c′ ˜+1, by lemma 9.36 therefore G_{π}c′ ≺ c′, therefore a[[0]] η OT (since c′, π η OT). We conclude a[[1·n]] η OT by induction on n : N. Subcase ω ⪯ τ(c) ≺ π: c[[z]] η OT, c[[z]] <_{z} c, G_{π}c ≺ c. z ≺ τ(c) ≺ π, by lemma 9.15 (h) z ≺ D_{π}z ⪯ D_{π}c[[z]], by lemma 9.34 (e), since z η OT G_{π}z ≺ c[[z]], therefore by lemma 9.36 G_{π}c[[z]] ≺ c[[z]], a[[z]] η OT. Subcase π ⪯ τ(c) =: t, t ≠ I ∨ π ≺ D_{I}c ∨ π = I: We show by induction on m

∀m ≤_{N} n.G_{π}c[[ζ_{m}]] ≺ c[[ζ_{m}]] ∧ G_{t}c[[ζ_{m}]] ≺ c[[ζ_{m}]] ∧ c[[ζ_{m}]] η OT (∗)

If m = 0 follows c[[ζ_{0}]] = c[[τ(c)^{−}]] <_{0} c, G_{π}c ≺ c, by lemma 9.34 (f) and (g) G_{t}c ≺ c and therefore therefore by lemma 9.36 G_{π}c[[ζ_{0}]] ≺ c[[ζ_{0}]] and G_{t}c[[ζ_{0}]] ≺ c[[ζ_{0}]]. If m = Sm′, follows by IH c[[ζ′_{m}]] η OT, G_{π}c[[ζ_{m′}]] ≺ c[[ζ_{m′}]] and G_{t}c[[ζ_{m′}]] ≺ c[[ζ_{m′}]] and therefore ζ_{Sm′} η OT, c[[ζ_{m}]] η OT. Further c[[ζ_{Sm′}]] <_{ζ_{Sm′}} c, G_{π}ζ_{Sm′} ≅ G_{π}D_{t}c[[ζ_{m′}]] ⊂ G_{π}t∪G_{π}c[[ζ_{m′}]]∪{c[[ζ_{m′}]]}. G_{π}t ≅ G_{π}τ(c) ≅ G_{π}τ(c)^{−} ⪯ G_{π}c[[τ(c)^{−}]] ≅ G_{π}c[[ζ_{0}]] ≺ c[[ζ_{0}]] ⪯ c[[ζ_{m′}]], therefore G_{π}ζ_{Sm′} ⪯ c[[ζ_{m′}]] ≺ c[[ζ_{Sm′}]], G_{π}c ≺ c, therefore G_{π}c[[ζ_{Sm′}]] ≺ c[[ζ_{Sm′}]]. G_{t}ζ_{Sm′} ≅ G_{t}D_{t}c[[ζ_{m′}]] ⊂ G_{t}t ∪ G_{t}c[[ζ_{m′}]] ∪ {c[[ζ_{m′}]]} ⪯ c[[ζ_{m′}]] ≺ c[[ζ_{m}]], and we have (∗). From (∗) follows D_{π}c[[ζ_{n}]] η OT. Subcase D_{I}c ⪯ π ≺ I = τ(c): G_{π}ζ_{n} ≅ G_{π}ζ_{0} ≅ G_{π}Dicom(c), Dicom(c) = D_{c}d for some d, D_{I}d ⪯ c ≺ D_{I}(d ˜+1), D_{I}d <_{0} D_{I}(d ˜+1), therefore Dicom(c) <_{0} c, by G_{π}c ≺ c and lemma 9.36, G_{π}Dicom(c) ≺ Dicom(c). Since c[[ζ_{n}]] <_{ζ_{n}} c, G_{π}c ≺ c, G_{π}ζ_{n} = G_{π}ζ_{0} ≺ ζ_{0} ⪯ c[[ζ_{n}]], follows G_{π}c[[ζ_{n}]] ⪯ c[[ζ_{n}]] by lemma 9.36. From c η OT follows Dicom(c) η OT, therefore ζ_{n} η OT and it follows D_{π}c[[ζ_{n}]] η OT.

Now we prove the property (F5), first the corresponding property for ·[[·]]: Lemma 9.41 If a, b η Lim, b η τ̂(a) follows

τ(a[[b]]) = τ(b) ∧ ∀ξ η τ̂(b).a[[b]][[ξ]] = a[[b[[ξ]]]]

(Note that Lim ≅ Lim′ ∩ OT.

Proof: Induction on length(a). Since b η Lim, ω ≺ τ(a). Note that, if a η Lim ↔ τ(a) η Lim, therefore in the situation of this lemma follows, if τ(z[[b]]) = τ(b), z[[b]] η Lim. If a η R′ the assertion is trivial. If a = c ˜+d follows by IH ∀ξ η τ̂(b).a[[b[[ξ]]]] = c ˆ+(d[[b[[ξ]]]]) = c ˆ+(d[[b]][[ξ]]) = (c ˆ+d[[b]])[[ξ]] = a[[b]][[ξ]]. If a = ϕ̂_{c}d, d η Lim \ Cr(c) follows, since d^{∗} ⪯ d[[τ(d)^{−}]] ≺ d[[b]] ≺ d by lemma 9.27 (a) d[[b]] η̸ Cr(c), by IH d[[b]] η Lim, therefore by IH τ(a[[b]]) = τ(ϕ̂_{c}(d[[b]])) = τ(d[[b]]) = τ(b), and ∀ξ η τ̂(b).a[[b[[ξ]]]] = ϕ̂_{c}(d[[b[[y]]]]) = ϕ̂_{c}(d[[b]][[ξ]]) = (ϕ̂_{c}d[[b]])[[ξ]] = a[[b]][[ξ]]. If a = ϕ̂_{c}0, c η Lim \ G′ follows, since c^{∗} ⪯ c[[τ(c)^{−}]] ≺ c[[b]] ≺ c by lemma 9.27 (b) c[[b]] η̸ G′, by IH c[[b]] η Lim, therefore by IH τ(a[[b]]) = τ(ϕ̂_{c[[b]]}0) = τ(c[[b]]) = τ(b), and

∀ξ η τ̂(b).a[[b[[ξ]]]] = ϕ̂_{c[[b[[y]]]]}0 = ϕ̂_{c[[b]][[ξ]]}0 = (ϕ̂_{c[[b]]}0)[[ξ]] = a[[b]][[ξ]].

If a = ϕ̂_{c}0, c η G′ follows c η Cr(c[[b]]), by IH c[[b]] η Lim, therefore by IH τ(a[[b]]) = τ(ϕ̂_{c[[b]]}c) = τ(c[[b]]) = τ(b), and ∀ξ η τ̂(b).a[[b[[ξ]]]] = ϕ̂_{c[[b[[y]]]]}c = ϕ̂_{c[[b]][[ξ]]}c =

ϕ̂_{c[[b]]}c)[[ξ]] = a[[b]][[ξ]].

If a = ϕ̂_{c}(d′ ˜+1), c η Lim follows ϕ̂_{c}d′ η Cr(c[[b]]), by IH c[[b]] η Lim, therefore by IH τ(a[[b]]) = τ(ϕ̂_{c[[b]]}ϕ̂_{c}d′) = τ(c[[b]]) = τ(b), and ∀ξ η τ̂(b).a[[b[[ξ]]]] = ϕ̂_{c[[b[[y]]]]}ϕ̂_{c}d′ =

ϕ̂_{c[[b]][[ξ]]}ϕ̂_{c}d′ = (ϕ̂_{c[[b]]}ϕ̂_{c}d′)[[ξ]] = a[[b]][[ξ]].

If a = ϕ̂_{c}d, c η Lim, d η Cr(c) follows d η Cr(c[[b]]), by IH c[[b]] η Lim, therefore by IH τ(a[[b]]) = τ(ϕ̂_{c[[b]]}d) = τ(c[[b]]) = τ(b), and ∀ξ η τ̂(b).a[[b[[ξ]]]] = ϕ̂_{c[[b[[y]]]]}d = ϕ̂_{c[[b]][[ξ]]}d =

(ϕ̂_{c[[b]]}d)[[ξ]] = a[[b]][[ξ]]. If a = D_{π}c, τ(c) ≺ π, follows by IH τ(c[[b]]) = τ(b) ⪯ b ≺ τ(c) ≺ π, τ(a[[b]]) = τ(D_{π}(c[[b]])) = τ(c[[b]]) = τ(b), and ∀ξ η τ̂(b).a[[b[[ξ]]]] = D_{π}(c[[b[[y]]]]) = D_{π}(c[[b]][[ξ]]) = (D_{π}(c[[b]]))[[ξ]] = a[[b]][[ξ]]. If a = Ω̂_{c}, a η̸ R′, follows by IH c[[b]] η Lim, since c^{∗} ⪯ c[[τ(c)^{−}]] ≺ c[[b]] ≺ c by 9.27 (d) c[[b]] η̸ F i′, therefore by IH τ(a[[b]]) = τ(Ω̂_{c[[b]]}) = τ(c[[b]]) = τ(b) and ∀ξ η τ̂(b).a[[b[[ξ]]]] =

Ω̂_{c[[b[[y]]]]} = Ω̂_{c[[b]][[ξ]]} = Ω̂_{c[[b]]}[[ξ]] = a[[b]][[ξ]].

No we change from ·[[·]] to ·[·]. For the last property, we need to know something about (π + a)[[ξ]]: Lemma 9.42 If z η Lim′ ∩ T′, z ≺ π η R′ ∪ {ω, 1} follows

τ(π^{−} + z) = τ(z)
∧( ( ∀ξ η τ̂(z).(π^{−} + z)[[ξ]] = π^{−} + (z[[ξ]])) ∨
( τ(z) = ω ∧ ∀ξ ≺ τ(z).(π^{−} + z)[[ξ ˜+1]] = π^{−} + (z[[ξ]])).

Proof: Induction on length(z). If z = a ˜+b follows π^{−} + z = (π^{−} + a) ˜+b, τ(π^{−} + z) = τ(b) = τ(z), (π^{−} + z)[[ξ]] = (π^{−} + a) ˆ+b[[ξ]] = π^{−} + (a ˆ+b[[ξ]]) = π^{−} + z[[ξ]]. If z η A, z ⪯ π^{−} follows (π^{−} + z)[[ξ]] = (π^{−} ˆ+z)[[ξ]] = π^{−} ˆ+(z[[ξ]]) = π^{−} + (z[[ξ]]) by lemma 9.22 (c) and 9.13 (h). If π^{−} = 0, the assertion is trivial, let therefore π^{−} ≠ 0.

Case π^{−} ≺ z η A: Then π^{−} + z = z. If in this case z[[ξ]] η A and π^{−} ≺ z[[τ(c)^{−}]] follows (π^{−} + z)[[ξ]] = z[[ξ]] = π^{−} + (z[[ξ]]) and we have the assertion. Subcase z = ϕ̂_{a}b: If π ≠ Ω̂_{0} follows π^{−} η G′, π^{−} ⪯ max_{T} {a, b}, and if π = Ω̂_{0} follows a = 0 → 1 ≺ b. a = b = 0 is not possible. Subsubcase a = 0, b = b′ ˜+1: If π ≠ Ω̂_{0} follows π^{−} ⪯ b, π^{−} ⪯ b′, (π^{−} + z)[[1 · n]] = ϕ̂_{0}b′ · SSn = π^{−} + (ϕ̂_{0}b′ · SSn) = π^{−} + (z[[1 · n]]). If π = Ω̂_{0} follows, if 1 ≺ b′ π^{−} ≺ ϕ̂_{0}b′ and the assertion as before, and if b′ = 1, (π^{−} + z)[[n ˆ+1]] = ϕ̂_{0}b′ · SSSn = π^{−} + (ϕ̂_{0}b′ · SSn) = π^{−} + (z[[1 · n]]). Subsubcase a = 0, b η Cr(a): If π ≠ Ω̂_{0} follows π^{−} ⪯ b, and if π = Ω̂_{0}, we get π^{−} ⪯ b directly. If now π^{−} = b follows (π^{−}+z)[[1·n ˆ+1]] = b·SSn = π^{−}+(b·Sn) = π^{−}+(z[[1·n]]), and if π^{−} ≺ b follows (π^{−} + z)[[1 · n]] = b · Sn = π^{−} + (b · Sn) = π^{−} + (z[[1 · n]]). In the cases a ≠ 0 ∨ b η (Lim \ Cr(a)) we have z[[ξ]] η A and we show π^{−} ≺ z[[τ(z)^{−}]]. Subsubcase b η Lim \ Cr(a): If π ≠ Ω̂_{0} follows, if π^{−} ⪯ a, π^{−} ≺ ϕ̂_{a}(b[[τ(b)^{−}]]), and if a ≺ π^{−}, π^{−} ⪯ b, since b η̸ Cr(a) π^{−} ≺ b, therefore by lemma 9.27 (a), since π^{−} η Cr(a), π^{−} ⪯ b^{∗} ⪯ b[[τ(b)^{−}]] ≺ z[[τ(z)^{−}]]. If π = Ω̂_{0} follows if 0 ≺ a, π^{−} ≺ z[[τ(z)^{−}]], and if a = 0, b η Lim, therefore by lemma 9.23 (e) 1 ≺ b[[τ(b)^{−}]], π^{−} ≺ z[[τ(z)^{−}]]. Subsubcase a = a′ ˜+1, b = 0: If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b} = a, π^{−} ⪯ a′ ≺ z[[0]]. If π = Ω̂_{0} follows π^{−} = ω ≺ z[[0]]. Subsubcase a = a′ ˜+1, b = b′ ˜+1: If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b}, π^{−} ⪯ max_{T} {a, b′}, π^{−} ≺ z[[0]]. If π = Ω̂_{0} follows π^{−} = ω ≺ z[[0]]. Subsubcase a = a′ ˜+1, b η Cr(a): If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b}, π^{−} ⪯ max_{T} {a′, b}, π^{−} ≺ z[[0]]. If π = Ω̂_{0} follows, since 1 ≺ b, π^{−} = ω ≺ z[[0]]. Subsubcase a η Lim \ G′, b = 0: If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b} = a, π^{−} ≠ a, since a η̸ G′, π^{−} ≺ a, π^{−} ⪯ a[[τ(a)^{−}]] ≺ z[[τ(z)^{−}]]. If π = Ω̂_{0} follows, by lemma 9.23 (e) 1 ≺ a[[τ(a)^{−}]], π^{−} = ω ≺ z[[τ(z)^{−}]]. Subsubcase a η G′, b = 0: If π = Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b} = a ≺ z[[τ(z)^{−}]]. If π = Ω̂_{0} follows, by lemma 9.23 (e) 1 ≺ a[[τ(z)^{−}]], π^{−} = ω ≺ z[[τ(z)^{−}]]. Subsubcase a η Lim, b = b′ ˜+1: If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b}, π^{−} ⪯ max_{T} {a, b′}, π^{−} ≺ ϕ̂_{a}b′ ≺ z[[τ(z)^{−}]]. If π = Ω̂_{0} follows the assertion immediately. Subsubcase a =η Lim, b η Cr(a): If π ≠ Ω̂_{0} follows π^{−} ⪯ max_{T} {a, b} = b ≺ z[[τ(z)^{−}]], and if π = Ω̂_{0} follows the assertion by 1 ≺ b. Subcase z = D_{ρ}b. Then, since I^{−} = 0, ρ = π ≠ I, π^{−} ≺ z[[τ(z)^{−}]] and z[[ξ]] η A. Subcase z = Ω̂_{a}: Since π^{−} ≺ z ≺ π follows π = I, π^{−} = 0, (π^{−} + z)[[ξ]] = π^{−} + (z[[ξ]]) Subcases z = 0, I: not possible. Definition 9.43 Now we define the fundamental sequences, as we needed them in chapter 8: ·[·] := λx, y.x[[τ(x)^{−} + y]]. We write r[s] for (·[·])rs. Lemma 9.44 ∀x, y η Lim.y ≺ τ(x) → τ(x[y]) = τ(y)∧ (∀ξ ≺ τ(y).x[y][ξ] = x[y[ξ]])∨ τ(x) = ω ∧ ∀n η ω.x[y][n ˜+1] = x[y[n]]. Proof: τ(x[y]) = τ(x[[τ(x)^{−} + y]]) = τ(τ(x)^{−} + y) = τ(y) and for all z ≺ τ(y) x[y][z] = x[[τ(x)^{−} + y]][[τ(y)^{−} + z]] = x[[(τ(x)^{−} + y)[[τ(y)^{−} + z]]]] = x[[τ(x)^{−} + (y[[τ(y)^{−} + z]])]] = x[y[z]], or τ(y) = ω and x[y][z ˜+1] = x[[(τ(x)^{−} + y)[[τ(y)^{−} + z ˜+1]]]] = x[[τ(x)^{−} + (y[[τ(y)^{−} + z]])]] = x[y[z]]. Lemma 9.45 OT fulfills all the properties of General Assumption 8.10 and 8.28 Proof: This is all proven in this chapter or easy, only most properties of ·[·] are not immediately proven, but similar results are proven for ·[[·]] and can be easily transferred by using

∀x, z, z′ η OT .τ(x)^{−} η OT ∧ τ(x)^{−} + z η OT ∧ (z ≺ z′ → τ(a)^{−} + z ≺ τ(a)^{−} + z′)

Comparison of OT with the ordinals in [Buc92b]

In this chapter we prove that the ordinal denotation system OT is in accordance with the functions we used in 7.8, which correspond to the system of [Buc92b]. We follow the lines of [Buc86]and [BS88]. In this chapter we will use T, OT ≺ etc. for the sets, functions and relations, which are defined in HA and correspond to the definitions used in chapter 9. We introduce the ordinal functions as used in [Buc92b](definition 10.1, 10.2), and cite some lemmata (10.3, 10.4). Next we introduce ϕ̂ and Ω̂ as ordinal functions (definition 10.5), the interpretation of the ordinal denotations (definition 10.6). To prove the equivalence, we show that C(α, β) can be defined in a more restricted way (definition 10.8 and lemma 10.9), that we can invert functions like + and Ω_{·} (lemma 10.10), and that the ordinals are ordered as the denotations (lemma 10.12). Now we prove, that we can define C(α, β) in a way such that we allow ψ_{π}α only for α ∈ C_{π}(α). Now we can define G_{π} on ordinals (definition 10.19), and conclude, that the interpretation of ordinal denotation is correct (lemmata10.21, 10.22 and 10.7).

We first repeat the definitions of [Buc92b]: Definition 10.1 (variant of definition 4.1 of [Buc92b]) Let # be the direct sum on ordinals. Ω_{0} := 0, Ω_{σ} := ℵ_{σ} for σ > 0. We assume the existence of a weakly inaccessible cardinal, e. g. a regular fixed point of σ ↦ Ω_{σ} and define

I := min{σ|σregular Cardinal ∧ Ω_{σ} = σ}

I^{+} := sup{ζ_{n}|n < Ω}, where ζ_{0} := Ω_{I+1}, ζ_{n+1} := Ω_{ζ_{n}},

On := {α|α ordinal , α < I^{+}}

R := {σ ∈ On|ω < σ ∧ σregular} = {I} ∪ {Ω_{σ+1}|σ < I^{+}}

Let κ, π, τ denote elements of R, α, β, γ, δ elements of On. Let ϕ be the usual Veblen-function. Definition 10.2 (variant on definition 4.1 of [Buc92b]) By transfinite recursion on α, we define ordinals ψ_{κ}α and sets C(α, β) ⊂ On (κ ∈ R) as follows:

ψ_{κ}α := min{β|κ ∈ C(α, β) ∧ C(α, β) ∩ κ ⊂ β}

C(α, β) := \{ \begin{matrix}the closure of β ∪ \{0, I\} under the functions \\ +, ϕ, σ ↦ Ω_{σ}, (ξ, π) ↦ ψ_{π}ξ (ξ < σ, π ∈ R) \end{matrix}

(Note that by I.H. ψ_{π}ξ is already defined for all ξ < α, π ∈ R.) We define ψ_{κ} : On → On, ψ_{κ}(α) := ψ_{κ}α, C_{κ}(α) := C(α, ψ_{κ}α). Lemma 10.3 (Lemma 4.4 of [Buc92b])

(a) β < π ⇒ cardinality(C(α, β)) < π

(b) C(α, β) = ⋃_{η<β} C(α, η), for each limit ordinal β.

(c) κ ∈ C(α, κ).

(d) C_{κ}(α) ∩ κ = ψ_{κ}α. Proof: All statements are immediate consequences of definition 10.2. Lemma 10.4 (Lemma 4.5 of [Buc92b])

(a) ψ_{κ}α < κ ∧ ψ_{κ}α ∉ C_{κ}(α)

(b) (α_{0} < α ∧ α_{0} ∈ C_{κ}(α)) → ψ_{κ}α_{0} < ψ_{κ}α

(c) (α_{0} < α ∧ α_{0} ∈ C_{κ}(α) ⇒ ψ_{κ}α_{0} < ψ_{κ}α

(d) Ω_{σ} ∈ C(α, β) ⇒ σ ∈ C(α, β)

(e) Ω^{ξ_{0}}# ⋯ #Ω^{ξ_{n}} ∈ C(α, β) ⇒ {ξ_{0}, …, ξ_{n}} ⊂ C(α, β)

(f) κ = Ω_{σ+1} ⇒ Ω_{σ} < ψ_{κ}α < Ω_{σ+1}

(g) Ω_{ψ_{I}α} = ψ_{I}α

(h) (Ω_{σ} ≤ γ ≤ Ω_{σ+1} ∧ γ ∈ C(α, β)) ⇒ σ ∈ C(α, β).

(i) α_{0} ≤ α ⇒ (ψ_{κ}α_{0} ≤ ψ_{κ}α ∧ C_{κ}(α_{0}) ⊂ C_{κ}(α)) Proof: See [Buc92b] Definition 10.5 A := {α ∈ On|α principal additive number} = {α ∈ On|∀β, γ < α.β + γ < α}, Lim := {α ∈ On|αLimes ordinal}, Suc := {α + 1|α ∈ On}, G := {α ∈ On|αGamma ordinal} = {α ∈ On|α = ϕ_{α}0}, F i := {α ∈ On|αfixed point of σ ↦ Ω_{σ}} = {α ∈ On|α = Ω_{α}}. Cr(α) := {γ ∈ On|γ = ϕ_{α}γ} For α = ω^{α_{1}}+, …, +ω^{α_{n}}, α_{1} ≥ ⋯ α_{n}, we define: last(α) := ω^{α_{n}}, first(α) := ω^{α_{1}}, length(α) := n. last(0) := first(0) := 0, length(0) := 0.

ϕ̂_{α}β := \{ \begin{matrix}ϕ_{α}(β + 1) & if ∃n < ω, γ.β = γ + n∧ \\ & (γ = 0 ∧ α ∈ G) ∨ γ ∈ Cr(α) \\ ϕ_{α}β & otherwise \end{matrix}

Ω̂_{α} := \{ \begin{matrix}Ω_{α+1} & if ∃n < ω, β.α = β + n ∧ β ∈ F i ∪ \{0\}, \\ Ω_{α} & otherwise. \end{matrix}

Ω_{σ+1}^{−} := Ω_{σ}, I^{−} := 0.

NF_{+}(α, β) :⇔ α ≠ 0 ∧ β ∈ A ∧ β ≤ last(α). (Note, that this definition differs from definition 9.11 (i). Definition 10.6 For a ∈ OT we define an ordinal o(a) ∈ On: o(0) := 0, o(I) := I, o(a ˜+b) := o(a) + o(b), o(ϕ̂_{a}b) := ϕ̂_{o(a)}o(b), o(Ω̂_{a}) := Ω_{o(a)}, o(D_{a}b) := ψ_{o(a)}o(b). We will prove the following lemma: Lemma 10.7 (a) C_{Ω_{1}}(I^{+}) = {o(x) | x ∈ OT }.

(b) ∀a ∈ OT such that a ≺ Ω̂_{0}:

o(a) = ordertype({x ∈ OT | x ≺ a}, ≺).

(c) ψ_{Ω_{1}}I^{+} = ordertype{x ∈ OT | x ≺ Ω̂_{0}}, ≺). Proof: At the end of this chapter.

Definition 10.8 Let α, β ∈ On.

C^{0}(α, β) := β ∪ {0, I}
C^{n+1}(α, β) := C^{n}(α, β) ∪ {ϕ̂_{γ}δ, Ω̂_{γ}|γ, δ ∈ C^{n}(α, β)}
∪ {γ + δ|γ, δ ∈ C^{n}(α, β), NF_{+}(γ, δ)}
∪ {ψ_{π}ξ|π, ξ ∈ C^{n}(α, β), π ∈ R, δ < α}

C_{τ}^{n}(α) := C^{n}(α, ψ_{τ}α). Lemma 10.9 ⋃_{n<ω} C^{n}(α, β) = C(α, β). Proof: “⊃”: If δ ∈ C(α, β), then δ + 1 ∈ C(α, β), therefore C^{0}(α, β) ⊂ C(α, β), if γ, δ ∈ C(α, β), then ϕ̂_{γ}δ ∈ {ϕ_{γ}δ, ϕ_{γ}(δ + 1)} ⊂ C(α, β), and Ω̂_{γ} ∈ {Ω_{γ}, Ω_{γ+1}} ⊂ C(α, β), therefore C^{n}(α, β) ⊂ C(α, β). “⊃”: By induction on n follows easily, if ω^{γ_{1}} + ⋯ + ω^{γ_{n}} ∈ C^{n}(α, β), γ_{1} ≥ ⋯ ≥ γ_{n}, then ω^{γ_{i}} ∈ C^{n}(α, β), especially γ + 1 ∈ C^{n}α, β) → γ ∈ C^{n}(α, β). Therefore we have, if γ, δ ∈ C^{n}(α, β), then ϕ_{γ}δ ∈ {ϕ̂_{γ}δ, γ, δ} ⊂ C^{n+1}(α, β) or δ = δ′ + 1 and ϕ_{γ}δ = ϕ̂_{γ}δ′ ∈ C^{n+1}(α, β), in a similar way follows γ ∈ C^{n}(α, β) → Ω_{γ} ∈ C^{n+1}(α, β). If γ, δ ∈ C^{n}(α, β),

γ = ω^{γ_{1}} + ⋯ + ω^{γ_{n}}, γ_{1} ≥ ⋯ ≥ γ_{n},
δ = ω^{δ_{1}} + ⋯ + ω^{δ_{m}}, δ_{1} ≥ ⋯ ≥ δ_{m},

next

γ + δ = ω^{γ_{1}} + ⋯ + ω^{γ_{i}} + ω^{δ_{1}} + ⋯ + ω^{δ_{m}} ∈ C^{n+i}(α, β)

for some i. Therefore we have “⊂”. Lemma 10.10 (a) If γ + δ ∈ C(α, β), NF_{+}(γ, δ), then γ, δ ∈ C(α, β).

(b) ϕ̂_{γ}δ ∈ C(α, β) → γ, δ ∈ C(α, β).

(c) Ω_{γ} ∈ C(α, β) → γ ∈ C(α, β).

(d) ψ_{τ}γ ∈ C(α, β) → τ ∈ C(α, β). Proof: (a) - (c): We have: If NF_{+}(γ, δ), NF_{+}(γ′, δ′), γ + δ = γ′ + δ′, then γ = γ′, δ = δ′. Further follows in this case γ + δ ∉ A, 0, I, ϕ̂_{α}β, Ω̂_{α}, ψ_{π}ρ ∈ A. If ϕ̂_{γ}δ = ϕ̂_{γ′}δ′, then γ = γ′, δ = δ′, further ϕ̂_{γ}δ ∉ G ∪ {0}, I, Ω̂_{α}, ψ_{π}ρ ∈ G. If Ω̂_{γ} = Ω̂_{γ′}, then γ = γ′, further Ω̂_{γ} ∈ R \ {I}, ψ_{π}ρ ∉ R. Now we prove by induction on n, that if γ + δ ∈ C^{n}(α, β) γ, δ as before, then γ, δ ∈ C^{n}(α, β): If γ + δ ∈ C^{0}(α, β), then γ, δ < γ + δ < β, γ, δ < β, and in the induction step follows the assertion by the uniqueness above. Similarly follow assertions (b) and (c).

(d): If τ = I this is trivial. Otherwise let τ = Ω̂_{δ}. If ψ_{τ}γ < β, follows δ < β, δ ∈ C(α, β), τ ∈ C(α, β). Otherwise, if ψ_{τ}γ ∈ C^{n+1}(α, β)\C^{n}(α, β), ψ_{τ}γ = ψ_{τ}γ′ with τ, γ′ ∈ C^{n}(α, β). Lemma 10.11 (a) (ρ, π ∈ R ∧ ρ < π ≠ I) → ρ ≤ π^{−},

(b) ρ < I^{+} → ρ^{−} < ψ_{ρ}α < ρ,

(c) ψ_{I}α < ρ ≠ I → ψ_{I}α ≤ ρ^{−}. Proof:

(a) trivial.

(b) 10.4 (f) and (a).

(c) If ρ = Ω_{σ+1}, ψ_{I}α = Ω_{ψ_{I}α} < Ω_{σ+1}, then ψ_{I}α ≤ σ. Lemma 10.12 (a) 0 ≠ α → 0 < α.

(b) If NF_{+}(α, γ), NF_{+}(β, δ), then

α + γ < β + δ ↔

((Alength(α) < Alength(β) ∧ α + γ ≤ β) ∨(Alength(α) = Alength(β) ∧ α < β∨

(α = β ∧ γ < δ))

∨(Alength(β) < Alength(α) ∧ α < β + δ)

(c) If NF_{+}(α, β), γ ∈ A \ {0}, then α + β < γ ↔ α < γ, γ < α + β ↔ γ ≤ α.

(d)

ϕ̂_{α}β < ϕ̂_{γ}δ ↔ (α < γ ∧ β < ϕ̂_{γ}δ) ∨ (α = γ ∧ β < δ)
∨(γ < α ∧ ϕ̂_{α}β ≤ δ)

(e) If δ ∈ G, then:

ϕ̂_{α}β < δ ↔ max{α, β} < δ. δ < ϕ̂_{α}β ↔ δ ≤ max{α, β}.

(f) If π, ρ ∈ R, π ≠ ρ, then

ψ_{π}α < ψ_{ρ}β ↔ (I ≠ π < ρ ≠ I) ∨ (I ≠ ρ ∧ π = I ∧ ψ_{π}α < ρ) ∨ (ρ = I ∧ π < ψ_{ρ}β)

(g) If π, ρ ∈ R, α ∈ C_{π}(α), β ∈ C_{ρ}(β), then

ψ_{π}α < ψ_{ρ}β ↔ (π = ρ ∧ α < β) ∨ (I ≠ π < ρ ≠ I) ∨ (I ≠ ρ ∧ π = I ∧ ψ_{π}α < ρ) ∨ (ρ = I ∧ π < ψ_{ρ}β)

Further if α ∈ C_{ρ}(α), α < β follows ψ_{ρ}(α) < ψ_{ρ}(β).

(h) If ρ, π ∈ R, π ≠ I, then

ρ < ψ_{π}β ↔ (ρ < π) ψ_{π}β < ρ ↔ (π ≤ ρ)

(i) ψ_{I}α < Ω̂_{β} ↔ ψ_{I}α ≤ β, Ω̂_{β} < ψ_{I}α ↔ β < ψ_{I}α.

(j) ψ_{I}α < I.

(k) Ω̂_{α} < Ω̂_{β} ↔ α < β.

(l) Ω̂_{α} < I ↔ α < I.

I < Ω̂_{α} ↔ I ≤ α. Proof: (a): trivial. (b):Let α = ω^{α_{0}} + ⋯ + ω^{α_{n}}, α_{0} ≥ ⋯ ≥ α_{n}, n = Alength(α)n, β = ω^{β_{0}} + ⋯ + ω^{β_{m}}, β_{0} ≥ ⋯ ≥ β_{m}, m = Alength(β). Then

α + γ < β + δ ↔  ((n < m ∧ (∀i ≤ n.α_{i} = β_{i}) ∧ γ = ω^{β_{n+1}})
∨ (∃i ≤ min{n, m}.(∀j < i.α_{j} = β_{j}) ∧ α_{i} < β_{i})
∨ (n < m ∧ (∀j ≤ n.α_{j} = β_{j}) ∧ γ < ω^{β_{n+1}})
∨ (n = m ∧ (∀j ≤ n.α_{j} = β_{j}) ∧ γ < δ)
∨ (m < n ∧ (∀j ≤ m.α_{j} = β_{j}) ∧ ω^{α_{m+1}} < δ))

If n < m we have

α + γ ≤ β ↔  (∀i ≤ n.α_{i} = β_{i}) ∧ γ ≤ ω^{β_{n+1}})
∨ (∃i ≤ n.(∀j < i.α_{j} = β_{j}) ∧ α_{i} < β_{i})
↔  α + γ < β + δ

and the assertion for Alength(α) < Alength(β). The caseAlength(α) = Alength(β) is obvious and the case Alength(β) < Alength(α) follows as the case Alength(α) < Alength(β).

(c) Let α = ω^{α_{0}} + ⋯ + ω^{α_{n}}, α_{0} ≥ ⋯ ≥ α_{n}. Then α + β = ω^{α_{0}} + ⋯ + ω^{α_{n}} + β < γ ↔ ω^{α_{0}} < γ ↔ α < γ, and γ < α + β ↔ γ ≤ ω^{α_{0}} ↔ γ ≤ α.

(d) Let

β̃ := \{ \begin{matrix}β + 1 & if ∃n < ω, β′.β = β′ + n∧ \\ & (β′ ∈ Cr(α) ∨ (β′ = 0 ∧ α ∈ G)) \\ β & otherwise \end{matrix}

Analogously we define δ̃. Then

ϕ̂_{α} β < ϕ̂_{γ}δ
↔ ϕ_{α}β̃ < ϕ_{γ}δ̃
↔ (α < γ ∧ β̃ < ϕ_{γ}δ̃) ∨ (α = γ ∧ β̃ < δ̃) ∨ (α > γ ∧ ϕ_{α}β̃ < δ̃)

Further, if α = γ, then β̃ < δ̃ ↔ β < δ, if α < γ, then β̃ < ϕ_{γ}δ̃ ↔ β < ϕ_{γ}δ̃, and if γ < α, then ϕ_{α}β̃ < δ̃ ↔ ϕ_{α}β̃ ≤ δ (since if δ = ϕ_{α}β̃ → δ̃ = δ + 1), and we have the assertion.

(e) We have first ϕ_{α}β < δ = ϕ_{δ}0 ↔ ((α < δ ∧ β < ϕ_{δ}0 = δ) ∨ (α = δ ∧ β < 0) ∨ (δ < α ∧ ϕ_{α}β < 0)) ↔ max{α, β} < δ, Therefore (β̃ being defined as before) ϕ̂_{α}β < δ ↔ max{α, β̃} < δ ↔ max{α, β} < δ, and δ < ϕ̂_{α}β ↔ ¬(ϕ̂_{α}β < δ) ↔ δ ≤ max{α, β}.

(f) Case π, ρ ≠ I: Then π^{−} < ψ_{π}α < π, ρ^{−} < ψ_{ρ}β < ρ. Therefore if π < ρ, then π ≤ ρ^{−}, ψ_{π}α < ψ_{ρ}β. If ρ < π, then ρ ≤ π^{−}, ¬(ψ_{π}α < ψ_{ρ}β) and therefore the assertion in this case. Case ρ < π = I: Then ρ^{−} < ψ_{ρ}β < ρ. If now ρ ≤ ψ_{π}α, follows ψ_{ρ}β < ψ_{π}α, and if ψ_{π}α < ρ follows ψ_{π}α ≤ ρ^{−} < ψ_{ρ}β. Case I = π < ρ: Then ψ_{π}α < ρ, ψ_{ρ}β. Case π < ρ = I: Then if π < ψ_{ρ}β follows ψ_{π}α < π < ψ_{ρ}β and if ψ_{ρ}β ≤ π follows ψ_{ρ}β ≤ π^{−} < ψ_{π}α. Case ρ < π = I: Then ψ_{ρ}β, ρ < ψ_{π}α.

(g) If π ≠ ρ the assertion follows by (f). Case π = ρ. If α < β, α ∈ C_{π}(α) ⊂ C_{π}(β) follows by 10.4 (b) ψ_{π}α < ψ_{ρ}β, and if β ≤ α, ψ_{ρ}β ≤ ψ_{π}α.

(h) π^{−} < ψ_{π}α, and ρ < π ∧ ρ ≤ π^{−} or π ≤ ρ.

(i) Let Ω̂_{β} = Ω_{β}. Then ψ_{I}α = Ω_{ψ_{I}α} < Ω̂_{β} = Ω_{β} ↔ ψ_{I}α < β̃ ↔ ψ_{I}α ≤ β.

Ω̂_{β} < ψ_{I}α ↔ ¬(ψ_{I}α < Ω̂_{β}) ↔ β < ψ_{I}α.

(j) trivial. (k), (l) easy. Lemma 10.13 (Lemma 2.7 of [BS88]) If α < β and for all α ≤ δ < β we have δ ∉ C_{σ}(α), then C_{σ}(β) = C_{σ}(α) and ψ_{σ}β = ψ_{σ}α. Proof: “⊃” is trivial, for “⊂” we prove by induction on n for γ ∈ C_{σ}^{n}(β), that γ ∈ C_{σ}^{n}(α). The only difficult case is γ = ψ_{τ}δ, δ < α, τ, δ ∈ C_{σ}^{n−1}(β). But in this case δ < β, and we are done. Lemma 10.14 (Lemma 2.8 of [BS88]) If β = min{ξ|α ≤ ξ ∈ C_{σ}(α)}, then C_{σ}(α) = C_{σ}(β), ψ_{σ}α = ψσβ, and β ∈ C_{σ}(β). Proof: 10.13 Lemma 10.15 (Corresponds to lemma [BS88]2.11.) Let τ, γ, γ_{0} ∈ C_{σ}^{n}(α), σ ≤ τ ∧ β ≤ α. Then δ := min{ξ|γ ≤ ξ ∈ C_{τ}(β)} ∈ C_{σ}^{n}(α), δ′ := min{ξ|γ ≤ ϕ̂_{γ_{0}}ξ ∈ C_{τ}(β)} ∈ C_{σ}^{n}(α), Proof: Induction on n. Case n = 0, γ < ψ_{σ}α: If γ < ψ_{τ}β follows δ = γ, δ′ ≤ γ < ψ_{τ}β. If ψ_{τ}β ≤ γ, follows ψ_{τ}β ≤ γ < ψ_{σ}α ≤ σ ≤ τ. Since C_{τ}(β) ∩ τ = ψ_{τ}β, τ ∈ C_{τ}(β) follows δ = τ, τ ∈ C_{σ}^{n}(α). δ′ < ψ_{τ}β or δ′ = τ, δ′ ∈ C_{σ}^{n}(α), similarly for δ′′.

Case n = 0, γ = 0, I: δ = γ, δ′ ∈ {0, I} ⊂ C_{τ}(β). Case n = n′ + 1, γ = γ_{1} + γ_{2}, NF_{+}(γ_{1}, γ_{2}), γ_{i} ∈ C_{τ}^{n′}(β): Let δ_{i} be chosen for γ_{i}. If γ ≤ δ_{1} follows δ = δ_{1}. Otherwise γ_{1} ≤ δ_{1} < γ_{1} + γ_{2}, δ_{1} = γ_{1} + ρ ∈ C_{τ}(β), 0 < ρ < γ_{2} and by NF_{+}(γ_{1}, γ_{2}) γ_{1} ∈ C_{τ}(β). Therefore γ_{1} + γ_{2} ≤ δ ≤ γ_{1} + δ_{2}, δ = δ_{1} + ρ with γ_{2} ≤ ρ ≤ δ_{2}, ρ ∈ C_{τ}(β), ρ = δ_{2}, we see that δ_{2} ∈ A, therefore δ = δ_{1} + δ_{2} ∈ C_{τ}^{n}(β). δ′ = δ′_{1}, where δ′_{1} ∈ C_{σ}(α) by the second IH for γ_{1}. Case n = n′ + 1, γ = ϕ̂_{γ_{1}}γ_{2}, γ_{i} ∈ C_{τ}^{n′}(β): Let δ_{i} be determined for γ_{i}. If γ ≤ δ_{i} follows δ = δ_{i}. Let δ_{i} < γ (i = 1, 2). Then δ_{i} ≤ δ ≤ ϕ̂_{δ_{1}}δ_{2}, therefore δ ∉ G, otherwise δ = max{δ_{1}, δ_{2}}. If δ = δ_{3} + δ_{4}, NF_{+}(δ_{3}, δ_{4}), we had γ ≤ δ_{3} < δ, δ_{3} ∈ C_{τ}(β), a contradiction. Therefore δ = ϕ̂_{δ_{3}}δ_{4}, γ ≤ δ ≤ ϕ̂_{δ_{1}}δ_{2}. If δ_{3} < γ_{1}, we had γ ≤ δ_{4} < δ, δ_{4} ∈ C_{τ}(β), a contradiction. Therefore γ_{1} ≤ δ_{3} ∈ C_{τ}(β), δ_{1} ≤ δ_{3}. If δ_{1} < δ_{3}, we had γ ≤ δ ≤ δ_{2}, a contradiction, therefore δ_{1} = δ_{3}, δ_{4} = δ′_{2} ∈ C_{σ}^{n}(α) by the second IH for γ_{2}. Second part in this Case: If γ_{0} < γ_{1} follows δ′ = δ, if γ_{0} = γ_{1}, δ′ = δ_{2}, and if γ_{0} > γ_{1} chose δ′_{2} for γ_{0}, γ. If γ_{2} < ϕ̂_{γ_{0}}δ′_{2}, δ = δ′_{2}, otherwise δ = δ′_{2} + 1. In all cases, where γ ∈ G, for the second assertion we have δ′ = 0, if γ ≤ γ_{0}, δ′ = δ otherwise. Case γ = ψ_{γ_{1}}γ_{2}, γ_{i} ∈ C_{σ}^{n}(α). γ_{2} < α. Let δ_{i} be chosen for γ_{i}. If δ = δ_{3} + δ_{4} with NF_{+}(δ_{3}, δ_{4}) or δ = ϕ̂_{δ_{3}}δ_{4}, follows γ ≤ δ_{3} < δ or γ ≤ δ_{4} < δ. Therefore we have δ ∈ G. If γ_{1} < δ_{1} follows γ_{1} ≠ I δ = δ_{1}. If γ_{1} = δ_{1} = δ or γ = δ, we are finished, too. Therefore let γ_{1} = δ_{1}, γ < δ < γ_{1}: Subcase γ_{1} ≠ I: Then δ = ψ_{γ_{1}}δ_{3} γ_{2} < δ_{3} < β ≤ α δ_{3}^{−}, δ_{3} ∈ C_{τ}(β), therefore δ_{2} ≤ δ_{3}, and by minimality and since ψ_{τ}δ_{2} ≤ ψ_{τ}δ_{3}, δ = ψ_{τ}δ_{2} ∈ C_{σ}^{n}(α). Subcase γ_{1} = I. If δ = Ω̂_{δ_{3}} follows γ ≤ δ_{3} ∈ C_{τ}(β), a contradiction, and if δ = ψ_{δ_{3}}δ_{4} with δ_{3} ≠ I follows γ ≤ δ_{3}^{−} < δ, δ_{3} ∈ C_{τ}(β), therefore δ = ψ_{I}δ_{4}, and as in the Subcase before follows the assertion. Case γ = I: δ = I. Case γ = 0: δ = 0, δ′ = 0. Case γ = Ω̂_{γ_{1}}: Let δ_{1} be chosen for γ_{1}. If γ ≤ δ_{1}, we have δ = δ_{1}. Otherwise follows δ ∈ G, δ ≠ ψ_{δ_{3}}δ_{4} with δ_{3} ≠ I (otherwise γ ≤ δ_{3}^{−}). Therefore δ = I or = Ω̂_{δ_{3}} (therefore δ_{3} = δ_{1}) or δ = ψ_{I}δ_{3} (but in this case γ ≤ Ω̂_{δ_{1}} < δ, a contradiction). Lemma 10.16 Let

C′^{0}(α, β) := β ∪ {0, I}
C′^{n+1}(α, β) := C′^{n}(α, β) ∪ {ϕ̂_{γ}δ, Ω̂_{γ}|γ, δ ∈ C′^{n}(α, β)}
∪{γ + δ|γ, δ ∈ C′^{n}(α, β), NF_{+}(γ, δ)}
∪{ψ_{π}ξ|π, ξ ∈ C′^{n}(α, β), π ∈ R, δ < α, ξ ∈ C_{π}(ξ)}

C′(α, β) := ⋃_{n<ω} C′^{n}(α, β), C′_{τ}^{n}(α) := C′(α, ψ_{τ}α)). Then C′(α, ψ_{σ}(α)) = C(α, ψ_{σ}(α)) = C_{σ}(α). Proof: C′^{n}(α, β) ⊂ C^{n}(α, β), the only difficulty is, to show: C_{σ}^{n}(α) ⊂ C′_{σ}^{n}(α), and here the only difficulty is the case γ = ψ_{τ}β ∈ C_{σ}^{n+1}(α), τ, β ∈ C_{σ}^{n}(α), β < α. If τ ≤ σ or τ = I and ψ_{I}β < σ follows γ ≤ ψ_{σ}(α), otherwise follows by 10.15 β_{0} := min{ξ|β ⪯ ξ ∈ C_{τ}(β)} ∈ C_{σ}^{n}(α) ⊂ C′^{n}(α), by 10.14 ψ_{τ}β = ψ_{τ}β_{0}, β_{0} ∈ C_{τ}(β_{0}). If β = β_{0} follows β_{0} < α. Otherwise β ∉ C_{τ}(β_{0}) = C_{τ}(β), if τ ≠ I follows by σ < τ β ∉ C_{σ}(β_{0}), since β ∈ C_{σ}(α), β_{0} < α, and if τ = I we have σ < ψ_{I}β, and from β ∉ C_{τ}(β_{0}) follows by ψ_{σ}β_{0} < ψ_{τ}β_{0}, β ∉ C_{σ}(β_{0}) and again β_{0} < α. Therefore γ ∈ C_{σ}^{n+1}(α).

Lemma 10.17 If (γ ∈ C_{π}(β) ∧ ((π^{−} ≤ γ ∈ G ∧ π ≠ I) ∨ π = I ≤ γ)), then

∃δ ∈ C_{π}(β).γ = Ω̂_{δ}) ∨
(∃ρ, δ ∈ C_{π}(β).γ = ψ_{ρ}δ ∧ δ < β ∧ (π ≤ ρ ∨ ρ = I) ∧ δ ∈ C_{ρ}(δ)).

Proof: Case γ < ψ_{π}β: Then, since π^{−} ≤ γ < ψ_{π}β follows ∃ρ ≤ β.ψ_{π}ρ ≤ γ < ψ_{π}(ρ + 1), therefore γ = ψ_{π}ρ, ρ < β, and by 10.13 ρ ∈ C_{π}(ρ). Case π ≤ γ: 10.16. Lemma 10.18 (a) I ≠ κ ∈ R ∪ I^{+} → C_{κ}(α) = C′(α, κ^{−} + 1).

(b) C_{Ω_{1}}(I^{+}) = C′(I^{+}, 0) Proof:

(a) “⊃” is obvious. For “⊂” we prove by induction on α: ρ ∈ ψ_{κ}α → ρ ∈ C′(α, κ^{−} + 1) (⇒ ψ_{κ}α ⊂ C′(α, κ^{−} + 1) and further C′(α, κ^{−} + 1) = C′(α, ψ_{κ}α) = C_{κ}(α)). If ρ ≤ κ^{−}, this is obvious, and if ρ = ρ_{1} + ρ_{2}, NF_{+}(ρ_{1}, ρ_{2}), or ρ = ϕ̂_{ρ_{1}}ρ_{2} or ρ = Ω̂_{ρ_{1}} this follows by IH. Otherwise follows ∃δ.δ ∈ C_{κ}(δ) ∧ δ < α ∧ ρ = ψ_{κ}δ. Then δ ∈ C_{κ}(δ) = C′(δ, κ^{−} + 1) ⊂ C′(α, κ^{−} + 1) by IH, ψ_{κ}δ ∈ C′(α, κ^{−} + 1). (b): C_{Ω_{1}}(I^{+}) = C′(I^{+}, ω + 1) = C′(I^{+}, 0). Definition 10.19 Definition of G_{π}(α) for α ∈ C_{Ω_{1}}(I^{+}) = C′(I^{+}, 0) by recursion on the minimal n such that α ∈ C^{′n}(I^{+}, 0).

(G1) G_{π}0 := ∅.

(G2) NF_{+}(α, β) → G_{π}(α + β) := G_{π}α ∪ G_{π}β.
(G3) If ρ η R, β ∈ G_{ρ}(β), then

G_{π}ψ_{ρ}β := \{ \begin{matrix}\{β\} ∪ G_{π}ρ ∪ G_{π}β, & if π ≤ ρ ≠ I∨ \\ & ρ = I ∧ (π ≤ ψ_{I}β ∨ π = I), \\ G_{π}ρ & if ρ < π = I \\ ∅, & if ρ < π ≠ I or \\ & ρ = I ∧ ψ_{I}β < π < I. \end{matrix}

(G4) G_{π}(Ω̂_{α}) := G_{π}α.
(G1) G_{π}I := ∅.

Lemma 10.20 If α ∈ C_{Ω_{1}}(I^{+}), then α ∈ C_{π}(β) ↔ G_{π}(α) < β. Proof: Induction on n, such that α ∈ C′(I^{+}, 0). If α = γ +δ and NF_{+}(γ, δ) or α = ϕ̂_{γ}δ, Ω̂_{γ}, I, 0 the assertion follows by IH or immediately. Let α = ψ_{ρ}ξ, ξ ∈ C_{ρ}(ξ), ξ, ρ ∈ C′(I^{+}, 0). If π = ρ follows α ∈ C_{π}(β) ↔ α < ψ_{π}β ↔ ξ ∈ C_{π}(β) ∧ ξ < β (using ξ ∈ C_{π}(ξ)) ↔ ξ, π ∈ C_{π}(β) ∧ ξ < β ↔ G_{π}(α) < β. If ρ < π ≠ I follows G_{π}(α) = ∅, α ∈ C_{π}(β). If π < ρ ≠ I follows α ∈ C_{π}(β) ↔ ρ, ξ ∈ C_{π}(β) ∧ ξ < β. If π < ρ = I follows, if ψ_{ρ}ξ < π ψ_{ρ}ξ ∈ C_{π}(β), G_{π}(α) = ∅, and if π ≤ ψ_{ρ}ξ, ψ_{ρ}ξ ∈ C_{π}(α) ↔ ρ, ξ ∈ C_{π}(β) ∧ ξ < β ↔ G_{π}(α) < β. If ρ < π = I follows α ∈ C_{π}(β) ↔ ψ_{ρ}ξ < ψ_{π}β ↔ ρ < ψ_{π}β ↔ ρ ∈ C_{π}(β) ↔ G_{π}ρ < β ↔ G_{π}(α) < β. Lemma 10.21 Let a, u, c ∈ OT. Then follows:

(a) o(a) ∈ C_{Ω_{1}}(I^{+}).

(b) a ∈ G → o(a) ∈ G, similarly for Lim, Suc, A, R, F i.

(c) G_{o}(u)(o(a)) = {o(x) | x ∈ G_{u}a}.

(d) a ≺ d ⇒ o(a) < o(d).

Proof by induction on length(a) + length(u), simultaneously for (a) - (d):

1. a = D_{b}c: Then G_{b}c ≺ c and b, c ∈ OT.

(a) By IH o(b), o(c) ∈ C_{Ω_{1}}(I^{+}) und G_{o(b)}o(c) = {o(x) | x ∈ G_{b}c} < o(c). By lemmata 10.20 follows o(b) ∈ I^{+} ∩ C_{o(b)}(o(c)) and therefore o(a) = ψ_{o(b)}o(a) ∈ C_{Ω_{1}}(I^{+}).

(b) trivial.

(c) Immediate by IH and definition of G_{u}a.

(d) follows by side induction on length(d) using 10.12. The only difficulty is the case d = D_{e}f. If e ≠ b we use 10.12 (f), and if b = e we have a ≺ d ↔ c ≺ f → o(c) < o(f) → o(a) ≺ o(d) by 10.12 (g).

2. All other cases follow immediately, using in (c) again side induction on length(d) and 10.12. Lemma 10.22 For all α ∈ C′^{n}(I^{+}, 0) exists a ∈ OT α = o(a). Proof: If α = 0, I this is immediate, and if α = γ + δ with NF_{+}(γ, δ) or α = ϕ̂_{γ}δ, Ω̂_{γ} this follows by IH for γ, δ and if α = ψ_{ρ}δ, δ ∈ C_{ρ}(δ), that is G_{ρ}(δ) < δ follows ρ = o(r) for some r ∈ R, δ = o(d) for some d ∈ OT, G_{r}(d) < d by 10.21, α = o(D_{r}d) with D_{r}d ∈ OT.

Proof of lemma 10.7: (a) is proven. Further {o(x)|x ≺ Ω̂_{0} ∧ x ∈ OT } = C_{Ω_{1}}(I^{+}) ∩ Ω_{1} = ψ_{Ω_{1}}I^{+}, and o(·) is an order preserving map {x|x ≺ Ω̂_{0} ∧ x ∈ OT } → ψ_{Ω_{1}}I^{+}, and for a ≺ Ω̂_{1}, {o(x)|x ≺ a ∧ x ∈ OT } = C_{Ω_{1}}(I^{+}) ∩ o(a) = o(a), again o(·) is an order preserving isomorphism.

Bibliography

[Acz77] P. Aczel. The strength of Martin-Löf’s intuitionistic type theory with one universe. In S. Miettinen and J. Väänänen, editors, Proceedings of the symposium on mathematical logic (Oulu, 1974), University of Helsinki, 1977. Report no. 2, Dept. of philosophy.

[Ara91] T. Arai. A slow growing analogue to Buchholz’ proof. Ann. Pure a. Appl. Logic, 54:101 – 120, 1991.

[Bar75] J. Barwise. Admissible Sets and Structures. An Approach to Definability Theory. Omega-series. Springer, Berlin, Heidelberg, New York, 1975.

[BB85] E. Bishop and D. Bridges. Constructive Analysis. Springer, Berlin, Heidelberg, New York, 1985.

[Bee85] M. J. Beeson. Foundations of Constructive Mathematics. Metamathematical Studies. Springer, Berlin, Heidelberg, New York, 1985.

[BFPS81] W. Buchholz, S. Feferman, W. Pohlers, and W. Sieg. Iterated Inductive Definitions. Recent Prooftheoretical Studies, volume 897 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, New York, 1981.

[BP78] W. Buchholz and W. Pohlers. Provable wellorderings of formal theories for transfinitely iterated inductive definitions. J. Symb. Logic, 43:118 – 125, 1978.

[BS76] W. Buchholz and K. Schütte. Die Beziehungen zwischen den Ordinalzahlsystemen Σ und Θ̄(ω). Archiv math. Logik, 17:179 – 190, 1976.

[BS80] W. Buchholz and K. Schütte. Syntaktische Abgrenzung von formalen Systemen der Π_{1}^{1}-Analysis und (Δ_{2}^{1}-)Analysis. Sitzungsberichte der Bayerischen Akademie der Wissenschaften, Math.-Nat. Klasse, pages 1 – 35, 1980.

[BS88] W. Buchholz and K. Schütte. Proof Theory of Impredicative Subsystems of Analysis. Bibliopolis, Naples, 1988.

[Buc75a] W. Buchholz. Normalfunktionen und konstruktive Systeme von Ordinalzahlen. In J. Diller and G.H. Müller, editors, Proof Theory Symposion, Kiel 1974, volume 500 of Springer Lecture Notes in Mathematics, pages 4 – 25, Berlin, Heidelberg, New York, 1975. Springer.

[Buc75b] W. Buchholz. Über Teilsysteme von Θ̄({g}). Archiv math. Logik, 18:85 – 98, 1975.

[Buc86] W. Buchholz. A new system of proof-theoretic ordinal functions. Ann. Pure a. Appl. Logic, 32:195 – 207, 1986.

[Buc90] W. Buchholz. Wellordering proofs for systems of fundamental sequences. Draft, München, 1990.

[Buc91a] W. Buchholz. Approximating infinitary derivations of ω-arithmetic by finite inital segments. Draft, Feb., 1991.

[Buc91b] W. Buchholz. Notation systems for infinitary derivations. Archiv math. Logik, 30:277 – 296, 1991.

[Buc92a] W. Buchholz. Fragments of arithmetic. Handwritten notes of a lecture given in Pittsburgh, 1992.

[Buc92b] W. Buchholz. A simplified version of local predicativity. In P. Aczel, H. Simmons, and S. S. Wainer, editors, Proof Theory. A selection of papers from the Leeds Proof Theory Programme 1990, pages 115 – 147, Cambridge, 1992. Cambridge University Press.

[Buc93] Wilfried Buchholz. A note on the ordinal analysis of KPM. In J. Väänänen, editor, Proceedings of the Logic Colloquium’90, Helsinki, 1993. To appear.

[BW87] Wilfried Buchholz and Stan Wainer. Provably computable functions and the fast growing hierarchy. In Stephen Simpson, editor, Logic and Combinatorics, Proceedings of the AMS–IMS–SIAM joint summer conference held August 4–10, 1985, pages 179–198, Providence, 1987. Amer. Math. Soc., Contemporary Mathematics Series.

[DT88] D. van Dalen and A. Troelstra. Constructivism in Mathematics. An Introduction, Vol. II. North-Holland, Amsterdam, 1988.

[Fef70] S. Feferman. Formal theories for transfinite iterations of generalized inductive definitions and some subsytems of analysis. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Prooftheory, Proceedings Buffalo, 1968, pages 303 – 326, Amsterdam, 1970. North-Holland.

[Fef75] S. Feferman. A language and axioms for explicit mathematics. In J. N. Crossley, editor, Algebra and Logic. Proc. 1974, Monash Univ Australia, volume 450 of Springer Lecture Notes in Mathematics, pages 87 – 139, 1975.

[Fef79] S. Feferman. Constructive theories of functions and classes. In M. Boffa, D. v. Dalen, and K. McAloon, editors, Logic Colloquium 78, pages 159 – 224, Amsterdam, 1979. North-Holland.

[Fef82a] S. Feferman. Inductively presented systems and the formalization of meta-mathematics. In D. van Dalen, D. Lascar, and J. Smiley, editors, Logic Colloquium ’80, pages 95 – 128, Amsterdam, 1982. North-Holland.

[Fef82b] S. Feferman. Iterated inductive fixed-point theories: Application to hancock’s conjecture. In G. Metakides, editor, Patras Logic Symposium, pages 171 – 195, Amsterdam, 1982. North-Holland.

[Gen36] G. Gentzen. Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann., 112:493 – 565, 1936.

[Gen38] Gentzen. Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften. Neue Folge, 4:19 – 44, 1938.

[Gen43] G. Gentzen. Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Math. Ann., 119:149 – 161, 1943.

[Gir87] J.-Y. Girard. Proof Theory and Logical Complexity. Bibliopolis, Napoli, 1987.

[Jäg79] G. Jäger. Die konstruktible Hierarchie als Hilfsmittel zur beweistheoretischen Untersuchung von Teilsystemen der Analysis. PhD thesis, Universität München, 1979. Dissertation.

[Jäg83] G. Jäger. A well-ordering proof for Feferman’s theory T_{0}. Archiv math. Logik, 23:65 – 77, 1983.

[Jäg86] G. Jäger. Theories for Admissible Sets: A Unifying Approach to Proof Theory. Bibliopolis, Naples, 1986.

[JP82] G. Jäger and W. Pohlers. Eine beweistheoretische Untersuchung von (Δ_{2}^{1} − CA) + BI und verwandter Systeme. Sitzungsberichte der Bayer. Akad. d. Wiss., Math.-Nat. Kl., pages 1 – 28, 1982.

[Kle52] S. Kleene. Introduction to Metamathematics. North-Holland, Amsterdam, 1952.

[Kre68] G. Kreisel. A survey of proof theory. J. Symb. Logic, 33:321 – 388, 1968.

[Kre71] G. Kreisel. A survey of proof theory. part ii. In J. E. Fenstad, editor, Proc. Second Scandinavian Logic Symposium, pages 109 – 170, Amsterdam, 1971. North-Holland.

[ML75] P. Martin-Löf. An intuitionistic theory of types: predicative part. In H.E. Rose and J. Sheperdson, editors, Logic Colloquium ’73, pages 73 – 118, Amsterdam, 1975. North-Holland.

[ML82] P. Martin-Löf. Constructive mathematics and computer programming. In L.J. Cohen, J. Los, H. Pfeiffer, and K.-P. Podewski, editors, Logic, Methodology and Philosophy of Science VI, pages 153–175, Amsterdam, 1982. North-Holland.

[ML84] P. Martin-Löf. Intuitionistic Type Theory. Bibliopolis, Naples, 1984.

[Pal91] E. Palmgren. On Fixed Point Operators, Inductive Definitions and Universes in Martin-Löf’s Type Theoies. PhD thesis, University of Uppsala (Sweden), 1991. U.U.D.M. Report 1991:8.

[Pal92] E. Palmgren. Type-theoretic interpretation of iterated, strictly positive inductive definitions. Archiv math. Logik, 32:75–99, 1992.

[Poh82] W. Pohlers. Admissibility in proof theory, a survey. In Logic, Methodology and Philosphy of Science VI, Hannover 1979, pages 123 – 139, Amsterdam, 1982. North-Holland.

[Poh89] W. Pohlers. Proof Theory. An introduction, volume 1407 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, New York, 1989.

[Poh91] W. Pohlers. Proof theory and ordinal anlysis. Archiv math. Logik, 30:311 – 376, 1991.

[Poh92] W. Pohlers. A short course in ordinal analysis. In P. Aczel, H. Simmons, and S. S. Wainer, editors, Proof Theory. A selection of papers from the Leeds Proof Theory Programme 1990, pages 27 – 78, Cambridge, 1992. Cambridge University Press.

[Rat89] M. Rathjen. Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen Δ_{2}^{1} − CA und Δ_{2}^{1} − CA + BI liegenden Beweisstärke. PhD thesis, Universität Münster, 1989.

[Rat90] M. Rathjen. Ordinal notations based on a weakly Mahlo cardinal. Archiv math. Logik, 29:249 – 263, 1990.

[Rat91a] M. Rathjen. Proof-theoretical analysis of KPM. Archiv math. Logik, 30:377 – 403, 1991.

[Rat91b] Michael Rathjen. Admissible proof theory and beyond. To appear in D. Prawitz, B. Skyrms and D. Westerstahl (eds.): Proceedings of the 9th International Congress of Logic, Methodology and Philosophy of Science, Uppsala 1991, 1991.

[Rat92] M. Rathjen. Eine Ordinalzahlanalyse der Π_{3}–Reflexion. Habilitationsschrift, Westfälische Wilhelms–Universität Münster, 1992.

[Sch77a] K. Schütte. Proof Theory. Springer, Berlin, Heidelberg, New York, 1977.

[Sch77b] Helmut Schwichtenberg. Proof theory: Some applications of cut–elimination. In Jon Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter Proof Theory and Constructive Mathematics, pages 867–895. North–Holland Publishing Company, Amsterdam, 1977.

[Sch88a] K. Schütte. Eine beweistheoretische Abgrenzung des Teilsystems der Analysis mit Π_{2}^{1}- Separation und Barinduktion. Sitzungsberichte der Bayer. Akad. d. Wiss., 1988.

[Sch88b] Kurt Schütte. Ein Wohlordnungsbeweis für das Ordinalzahlensystem T (J). Archive for Mathematical Logic, 27:5–20, 1988.

[Sch91a] Kurt Schütte. Beweistheoretische Analyse von KPM, 1991. Nach M. Rathjen in modifizierter Fassung.

[Sch91b] Kurt Schütte. Ein Ordinalzahlensystem T (M), 1991. Nach M. Rathjen in modifizierter Fassung.

[Sch92a] Kurt Schütte. Beziehungen des Ordinalzahlensystems OT(ϑ) zur Veblen–Hierarchie, 1992.

[Sch92b] Kurt Schütte. Zur Beweistheorie von KPM, 1992. Draft.

[Sch92c] H. Schwichtenberg. Proofs as programs. In P. Aczel, H. Simmons, and S. S. Wainer, editors, Proof Theory. A selection of papers from the Leeds Proof Theory Programme 1990, pages 79 – 113, Cambridge, 1992. Cambridge University Press.

[Set90] A. Setzer. Ordinalzahlanalyse prädikativer Theorien unter Verwendung von Fundamentalfolgen. Master’s thesis, Universität München, January 1990.

[Sho67] J. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, Massachusetts, 1967.

[Smi84] J. Smith. An interpretation of Martin-Löf‘s type theory in a type-free theory of propositions. Theoretical Computer Science, 49:730 – 753, 1984.

[Tak87] Gaisi Takeuti. Proof Theory. North–Holland Publishing Company, Amsterdam, second edition, 1987.

[Tro73] A. Troelstra, editor. Metamathematical investigations of intuitionistic arithmetic and analysis, volume 344 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, New York, 1973.

[Tro87] A. Troelstra. On the syntax of Martin-Löf’s type theories. Theoretical Computer Science, 51:1 – 26, 1987.

[Wai89] S. S. Wainer. Slow growing versus fast growing. J. Symb. Logic, 54(2):608 – 614, June 1989.

[WW92] S. S. Wainer and L. Wallen. Basic proof theory. In P. Aczel, H. Simmons, and S. S. Wainer, editors, Proof Theory. A selection of papers from the Leeds Proof Theory Programme 1990, pages 1 – 26, Cambridge, 1992. Cambridge University Press.

Index

(+_{U}^{E}), 22 (A = B)^{∗}[x⃗/r⃗; s⃗], 44 (A type)^{∗}[x⃗/r⃗; s⃗], 44 (F i), 76 (I_{U}^{E}), 22 (W_{U}^{E}), 22 (Δ_{0} − coll), 28 (Δ_{0} − sep), 28 (Π^{E}), 22 (Σ_{U}^{E}), 22 (π^{E}), 22 ( ˜+^{E}), 22 (σ^{E}), 22 (a)_{i}, 66 (i^{E}), 22 (t : A)^{∗}[x⃗/r⃗, s⃗], 44 (t = t′ : A)^{∗}[x⃗/r⃗; s⃗], 44 (w^{E}), 22 (x)t, 11 (x, y), 11 (x, y, z), 11 +, 10, 95 0, 64 0_{OT}, 74 1, 64 < r, s >, 64 < x_{1}, …, x_{n} >_{List}, 66 =_{A}, 64 =_{N,B}, 65 A, 74, 95, 122 A′, 94 A′′, 90 A[ ].i, 85 A^{+}, 70 A^{∗}, 38 A^{P}, 71 A^{+,Cl,y}, 71 A^{basis}, 38 A_{i}, 34 A_{i}^{∗}, 34 Ad(u), 27

Ag(X), 81
Alength, 75, 91, 122
Alist, 66
Ap, 10
B_{1}(A), 78
B_{1,Cl,y,x,z}(A), 77
C′(α, β), 127
C^{′i}(α, β), 127
C(α, β), 121
C^{i}(α, β), 123
C_{τ}^{n}(α), 123
C_{κ}(α), 122
C_{n}, 10
Cl_{y}(N), 68
Compl(u), 32, 33, 37
Compl_{U}, 39
Compl_{U}(a), 39
Cor(α), 78
Cor(u), 42
Cor_{Cl,y,z}(A)(α), 77
Cr(a), 84, 96, 122

properties, 100 D, 10

D_{a}b, 84, 90
Dicom, 84, 96

properties, 101 ES, 28

F (n⃗, ·), 37
F V′, 34
F V (t), 10
F^{α}(v), 37
F i, 84, 95, 122
F i′, 94
F lex(A), 46
G, 84, 95, 122

properties, 101 G′, 94

G′′, 90
G_{π}a, 94, 128

properties, 110 I, 10, 28, 36, 55, 74, 90, 121 I^{+}, 121 KP, 28 KP ω, 28 KP i, 28 KP i^{+}, 28 KP i_{n}^{+}, 28

strength, 61 KP l, 28 L_{ML}, 10 L_{P A}, 56 Lim, 74, 122 Lim′, 94 Lim′′, 91 M ≺_{B} M′, 93 M ≺_{B} a, 93 M ⪯ M′, 94 M ⪯_{B} M′, 93 ML_{1}^{e}W_{R}, 13 ML_{1}^{e}W_{T}, 13 ML_{1}^{e}W_{R,U}, 22 ML_{1}^{e}W_{T,U}, 22 ML_{1}^{i}W_{R}, 13 ML_{1}^{i}W_{T}, 13 N, 10 NF_{+}, 96, 123 NF_{D}, 96 NF_{+,B}, 74, 96 NF_{D,B}, 96 N_{k}, 10 N_{Cl,y}, 71 N_{P}, 71 N_{dec}, 71 OT, 94

assumptions on, 74, 84, 119 closed under [[ ]], 117

On, 29, 121 P, 10 P l, 96

properties, 96 P roj_{i}^{n}, 56 R, 10, 74, 95, 121 R′, 94 R′′, 90 Rest(a, −ω), 95 S, 10 Stable(a), 46 Sublist, 66 Suc, 95, 122 Suc′, 94

Suc′′, 90
T, 10
T′′, 90
T erm, 34
T erm_{cl}, 34
T erm_{nf}, 35
U, 10
V ar_{KP}, 27
V ar_{ML}, 10
V ar_{P A}, 56
W x ∈ A.B, 11
W, 10
W (X), 78

closed under +, 79 closed under Ω_{·}, 87 closed under ϕ̂, 86 induction, 78

W_{1}(A), 78 W_{1,Cl,y,z}(A), 77 W_{Cl,y,z}(A), 78 X | a, 81 [x⃗/t⃗] ∩ X, 11 [x⃗/t⃗], 11 [x_{1}/t_{1}, …, x_{n}/t_{n}] \ X, 11 [x_{1}/t_{1}, …, x_{n}/t_{n}], 11 #, 121 Γ ⇒ A ∈ Cl_{y}(N), 68 Γ^{=}, 47 ML_{1}^{e}W_{R}, 13 ML_{1}^{i}W_{R}, 13 ML_{1}^{e}W_{T}, 13 ML_{1}^{i}W_{T}, 13 Ω_{·}, 75, 121, 123 Ω̂_{·}, 90, 122 Πx η_{Cl,y} A.B, 69 Πx η_{dec,typ} A.B, 69 Πx η_{fin,typ} A.B, 69 Πx η_{typ} A.B, 69 Πx ∈ A.B, 11 Π, 10 Π_{P}, 69 Π_{dec}, 69 Π_{fin}, 69 Σ, 10 Σx η_{Cl,y} A.B, 69 Σx η_{dec,typ} A.B, 69 Σx η_{fin,typ} A.B, 69 Σx η_{typ} A.B, 69 Σx ∈ A.B, 11 Σ function, 32

iteration, 37 Σ_{P}, 69 Σ_{dec}, 69 Σ_{fin}, 69 Û, 33, 39

correctness of, 44 Ũ, 39 Ũ^{basis}, 39 ∀x ∈ A.B, 64 ∀x ≺ a.A, 75 ∀x ⪯ a.A, 75 ∀_{≺}, 75 ∀_{B,<}, 65 ∀_{B,≤}, 66 ∀_{B}x ≤_{N} n.ϕ, 66 ∀_{B}x <_{N} n.ϕ, 66 ∀_{B}, 69 ∀_{typ}, 75 α-equality, 11

properties, 35 α(u)^{n}, 37 α^{+}, 37 α^{+,n}, 37 α_{I}, 36 α_{I,n}, 36 ∧_{B}, 64 →, 12 B

properties, 66 A^{X}, 78 A_{Cl,y,z}^{X}, 78 B, 64 W, 82

Ag(W), 82 W_{i}, 88 ∩, 70 ∩_{Cl,y}, 70 ∩_{dec}, 70 ∩_{fin}, 70 ·, 75, 95 ≅, 70 ≅_{Alist}, 66 ≅_{Cl,y}, 70 ≅_{Nlist,B}, 66 ≅_{dec}, 70 ≅_{fin,B}, 70 ≅_{fin}, 70 ×, 12

∪, 70
∪_{Cl,y}, 70
∪_{dec}, 70
∪_{fin}, 70
emb, 57
∅, 71
∅_{Cl,y}, 71
∅_{dec}, 71
∅_{fin}, 71
=_{α}, 11
→_{red}, 35

properties, 36 η, 68

η_{dec,B}, 68
η_{dec}, 68
η_{fin,B}, 68
η_{fin}, 68
∃x ∈ A.B, 64
∃x ≺ a.A, 75
∃x ⪯ a.A, 75
∃_{≺}, 75
∃_{B,<}, 65
∃_{B}x ≤_{N} n.ϕ, 66
∃_{B}x <_{N} n.ϕ, 66
∃_{B}, 69
∃_{typ}, 75
ĝ, 56
⌈ ⌉, 34, 90
g̃, 56
\underline{if} t \underline{then} A \underline{else} B, 64
→_{red,imm}, 34
ĩ, 10
L_{KP}, 27
<_{ξ}, 110, 111

and fundamental sequences, 114 properties, 113

≪, 107

≪

and functions, 107

and fundamental sequences, 108 ˙−, 65

μ, 23
\underline{n}, 10
\underline{n}_{k}, 10
¬, 64
¬_{B}, 64
≠_{N,B}, 65
η̸, 68
η̸_{Cl,y}, 68

η̸_{dec}, 68 η̸_{fin}, 68 ω, 95 ∨_{B}, 64 ∨, 64 ϕ, 21, 121 ϕ(r, x, s, r′, x′, s′, b, f, u), 39 ϕ̂, 84, 90, 122 π, 10 ˆ+, 95 +_{N}, 65 ˜+, 10, 90 P(N), 68 P^{dec}(N), 68 P^{fin}(N), 68 ≺, 74, 92

ordering, 92 ≺_{OT}, 74 ≺_{A,x,B}, 72 ≺_{A,x,B}^{1}, 72 ≺_{Univ}, 72 ≺_{Univ}^{1}, 72 ≺_{B}, 91 ⪯, 74, 92 ⪯_{OT}, 74 ⪯_{A,x,B}, 72 ⪯_{Univ}, 72 ⪯_{B}, 91 ≤_{N,B}, 65 <_{N,B}, 65 ψ, 23 ψ^{u}, 28 ψ_{+}(r, s, r′, s′, b, c, u), 39 ψ_{κ}, 122 ψ_{i}(r, s, t, r′, s′, t′, b, u), 39 \underline{r}, 10 ρ, 23 \, 70 \_{Cl,y}, 70 \_{dec}, 70 \_{fin}, 70 σ, 10 ⊂, 69 ⊂_{Cl,y}, 69 ⊂_{dec}, 69 ⊂_{fin,B}, 69 ⊂_{fin}, 69 τ′(a), 76 τ(a), 74, 101

properties, 103 τ(a)^{X}, 78 τ_{Cl,y}^{A,z}(x), 77 0, 90 {x_{0}, …, x_{n}}, 71 {x_{0}, …, x_{n}}_{Cl,y}, 71 {x_{0}, …, x_{n}}_{dec}, 71 {x_{0}, …, x_{n}}_{fin}, 71 a − ω, 95 a ⪯_{B} M, 93 a(u), 37 a(u)^{n}, 37 a[x], 74 a[[ξ]], 101 a^{−}, 75, 96

properties, 100 a^{≺}, 75

append, 66
arity(A_{i}), 34
atom, 64
car, 66
cons, 66
equiv(u), 42
etarel_{Cl,y}, 68
ff, 64
first, 91, 122
i, 10

if t \underline{then} A \underline{else} B, 64

infinite(u), 28
int_{P A,KP}, 56
int_{P A,ML}, 56
j, 10
label, 77, 78
last, 91, 122
length, 84, 122
length(a), 90
lh, 66
lh′, 104

makepair(a), 57

max_{T}, 96
max_{nat}, 65
min_{T}, 96
min_{nat}, 65
n_{k}, 10
nil, 66
nil_{y}, 66
o(a), 94, 123
p, 10
p_{0}, 10

p_{1}, 10 pair(x, y), 29 pred, 95 pred_{N}, 65 rs, 12 s[x⃗/t⃗], 11 s^{∗}

and critical ordinal, 105

and fundamental sequences, 105 definition, 104 nesting, 106

sub, 34 sup, 10 tran(u), 28 tripel(x, y, z), 29 tt, 64 u^{KP}, 27 u_{+}(b, c), 40 u_{+}^{basis}(b, c), 40 u_{π}(b, f), 40 u_{π}^{basis}(b, f), 40 u_{σ}(b, f), 40 u_{σ}^{basis}(b, f), 40 u_{i}(b, r, s), 40 u_{i}^{basis}(b, r, s), 40 u_{w}(b, f, v), 40 u_{w}^{basis}(b, f, v), 40 u_{fin}(k), 40 u_{fin}^{basis}(k), 40 u_{nat}, 40 v^{P A}, 56 w, 10 z^{ML}, 10 zero?, 65 (Δ_{0}-coll)

needed, 41, 55

power set, 68

ausgezeichnete Klasse, 63, 74, 77 ausgezeichnete Menge, 63, 73, 77, 81

comparability, 82

b-object, 10 b-substitution, 11 b-terms, 11

extended, 34 Bachmann property, 76, 109

classes, 68 collapsing function, 84 collection axiom, 28 Compl, 33, 37 constructor, 10 constructor symbol, 10

Dicom properties, 101

for Y : P(N) or Y ∈ Cl_{y}(N) holds ϕ, 71 fundamental sequences, 74, 101

and s^{∗}, 105

supremum property, 115

G

properties, 101 g-context, 12 g-context-piece, 12 g-judgement, 12 g-statement, 12 g-substitution, 12 g-terms, 12 Gödel-numbers, 34, 90

interpretation

P A in ML_{1}^{e}W_{T} correctness, 58

P A in KP i^{+}

correctness, 58 L_{P A} in ML_{1}^{e}W_{T}, 56 correctness, 57 L_{P A} in KP i^{+}, 56 correctness, 57 dependent types, 32 motivation, 31 types of ML in KP i^{+}, 37

judgements interpretation, 44

Klasse

ausgezeichnete, 63, 74, 77 Kripke-Platek set theory, 27

axioms, 28

label, 78 language

Kripke-Platek, 27 Martin-Löf, 10 Peano Arithmetic, 56

list, 66

properties, 66

Menge

ausgezeichnete, 63, 73, 77, 81 monotonicity of functions, 99

natural numbers, 65 properties, 66

On, 121 ordinal

A, 74 Lim, 74 OT, 94 A, 95, 122 A′, 94 A′′, 90 F i, 84, 95, 122 F i′, 94 G, 84, 95, 122 G′, 94 G′′, 90 Lim, 122 Lim′, 94 Lim′′, 91 R, 74, 95, 121 R′, 94 R′′, 90 Suc, 95, 122 Suc′, 94 Suc′′, 90 T′′, 90 ≺, 91 critical, 84, 96, 122

and s^{∗}, 105

properties, 100 fundamental sequences, 101 Gamma

properties, 101 segment, 81

Peano Arithmetic, 56

r-context, 13 r-context-piece, 13 r-judgement, 13 r-statement, 13 r-term, 12 r-type, 12 rules for Martin-Löf

definition, 13 derived, 23 weak inferences, 24

Russel-version, 9, 13 compared with Tarski, 22, 25

segment, 81

separation axiom, 28 set, 68

motivation, 67 substitution, 11 allowed, 11 substitution lemma, 44

Tarski-version, 9, 13

compared with Russel, 22, 25 term

closed, 34

interpretation, 34 normal-form, 35 term constructor, 10 introductory, 34

term constructors extended, 34

tree

subtree, 71

properties, 72 truth values, 64

properties, 66 type

interpretation F lex, 46 Stable, 46 closed under α-equality, 46 closed under →_{red}, 46 symmetric and transitive, 44

type constructor, 10

types dependent, 32

Veblen-function, 84, 121

well-ordering proof motivation, 73 result, 88

\underline{Lebenslauf}

14.1.1963  Geburtstag
Geburtsort:  Sindelfingen (Kreis Böblingen)
Staatsangehörigkeit: deutsch
Familienstand:  ledig

Schulbildung

1969 - 73  Grundschule  (Hildrizhausen, Kreis Böblingen)
1973 - 82  Gymnasium  (Herrenberg, Kreis Böblingen)
18.5.1982  Abitur
1982- 83  Wehrdienst

Hochschulbildung

1983 - 84  TU München  Mathematik (Diplom)
1984 - 90  Universität München Mathematik (Diplom)
27.8.1990  Diplom
Diplomarbeitsthema: ”Ordinalzahlanalyse prädikativer

Theorien unter besonderer Berücksichtigung von Fundamentalfolgen” (Prof. Dr. W. Buchholz)

\underline{Berufsweg}

1.10.90- 31.3.91 wissenschaftlicher  Institut für Informatik
Mitarbeiter  Univ. München
1.5.91 - 31.7.91 studentische  Institut für Informatik
Hilfskraft  Univ. München
1.11.91 - 29.2.92 wissenschaftliche  Institut für Informatik
Hilfskraft  Univ. München
1.3.92 - 31.8.92 Research-Assistant  Dep. of Pure

Mathematics. Univ. of Leeds, England

1.9.93 - 31.8.93 wissenschaftlicher  Institut für Mathematik
Mitarbeiter  Univ. München
ab 1.9.93 Research-Assistant Dep. of Pure

Mathematics.

München, 7. 9. 93

Anton Setzer