Classical System of Martin-Löf’s Inductive Definitions Is Not Equivalent to Cyclic Proof System

Abstract

A cyclic proof system, called $\mathtt{CLKID}^\omega$, gives us another way of representing inductive definitions and efficient proof search. The 2011 paper by Brotherston and Simpson showed that the provability of $\mathtt{CLKID}^\omega$ includes the provability of Martin-Löf’s system of inductive definitions, called $\mathtt{LKID}$, and conjectured the equivalence. Since then, the equivalence has been left an open question. This paper shows that $\mathtt{CLKID}^\omega$ and $\mathtt{LKID}$ are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in $\mathtt{CLKID}^\omega$, but the statement is not provable in $\mathtt{LKID}$, by constructing some Henkin model where the statement is false.

1 Introduction

An inductive definition is a way to define a predicate by an expression which may contain the predicate itself. The predicate is interpreted by the least fixed point of the defining equation. Inductive definitions are important in computer science, since they can define useful recursive data structures such as lists and trees. Inductive definitions are important also in mathematical logic, since they increase the proof theoretic strength. Martin-Löf’s system of inductive definitions given in [10] is one of the most popular system of inductive definitions. This system has production rules for an inductive predicate, and the production rule determines the introduction rule and the elimination rule for the predicate.

Brotherston [3] and Simpson [6] proposed an alternative formalization of inductive definitions, called a cyclic proof system. A proof, called a cyclic proof, is defined by proof search, going upwardly in a proof figure. If we encounter the same sequent (called a bud) as some sequent we already passed (called a companion) we can stop. The induction rule is replaced by a case rule, for this purpose. The soundness is guaranteed by some additional condition, called the global trace condition, which guarantees the case rule decreases some measure of a bud from that of the companion. In general, for proof search, a cyclic proof system can find an induction formula in a more efficient way than Martin-Löf’s system, since a cyclic proof system does not have to choose fixed induction formula in advance. A cyclic proof system enables us efficient implementation of theorem provers with inductive definitions [2, 4, 5, 7]. In particular, it works well for theorem provers of separation logic.

Brotherston and Simpson [6] investigated Martin-Löf’s system $\mathtt{LKID}$ of inductive definitions in classical logic for the first-order language, and the cyclic proof system $\mathtt{CLKID}^\omega$ for the same language, showed the provability of $\mathtt{CLKID}^\omega$ includes that of $\mathtt{LKID}$, and conjectured the equivalence. Since then, the equivalence has been left an open question. Simpson [11] submitted a proof of a particular case of the conjecture, for the theory of Peano Arithmetic.

This paper shows $\mathtt{CLKID}^\omega$ and $\mathtt{LKID}$ are indeed not equivalent. To this aim, we will consider the first-order language formed by 0, the successor ${ s}$, the natural number predicate $N$, and a binary predicate symbol p. We introduce a statement we call 2-Hydra, which is a miniature version of the Hydra problem considered by Laurence Kirby and Jeff Paris [9]: the proviso “2” means that we only have two “heads”. We define some statement, called the 2-Hydra statement, and we show that the 2-Hydra statement is provable in $\mathtt{CLKID}^\omega$ with the language, but the statement is not provable in $\mathtt{LKID}$ with the language. 2-Hydra is similar to a candidate for a counter-example proposed by S. Stratulat [12]. The unprovability is shown by constructing some model of $\mathtt{CLKID}^\omega$ where 2-Hydra is false.

For constructing the counter model ${\mathcal M}$ for the second result, we take both the universe of ${\mathcal M}$ and the interpretation of the predicate $N$ to be $\mathtt{Nat} + \mathtt{Z}$, where $\mathtt{Nat}$ is the set of natural numbers and $\mathtt{Z}$ is the set of integers, and some predicate p which is a counter-example of 2-Hydra. We prove that ${\mathcal M}$ is a model of $\mathtt{LKID}$ by using a set of partial bijections on ${\mathcal M}$ and a quantifier elimination result.

The quantifier elimination theorem for a theory of partial equivalence relations is new, as far as we know, and it may have some independent interest.

This model also shows that $\mathtt{LKID}$ is not conservative when we add inductive predicates, namely, it is not the case that for any language L, the system of $\mathtt{LKID}$ with language L and any additional inductive predicate is conservative over the system of $\mathtt{LKID}$ with L.

Section 2 describes Brotherston-Simpson conjecture. Section 3 defines the 2-Hydra statement and proves the 2-Hydra statement in $\mathtt{CLKID}^\omega$. Section 4 defines the counter model ${\mathcal M}$ and the proof outline of it. Section 5 introduces a family of partial bijections. Section 6 proves a quantifier elimination theorem for a theory of partial bijections. Section 7 proves that the 2-Hydra statement is not provable in $\mathtt{LKID}$. Section 8 shows non-conservativity of $\mathtt{LKID}$ with additional inductive predicates. We conclude in Sect. 9. Detailed proofs are in [1].

2 Brotherston-Simpson Conjecture

In this section we introduce Brotherston-Simpson Conjecture.

2.1 Martin-Löf’s Inductive Definition System $\mathtt{LKID}$

We briefly remind you of Martin-Löf’s inductive definition system $\mathtt{LKID}$, defined in detail in [6].

The language of $\mathtt{LKID}$ is determined by a first-order language with inductive predicate symbols. The logical system $\mathtt{LKID}$ is determined by production rules for inductive predicate symbols. These production rules mean that the inductive predicate denotes the least fixed point defined by these production rules.

We often abbreviate p(t), q(t, u) with pt, qtu. For example, for an inductive predicate symbol N, the production rules may be written as

These production rules mean that N denotes the smallest set closed under 0 and ${ s}$, namely the set of natural numbers. We call this set of production rules $\varPhi _N$.

The inference rules of $\mathtt{LKID}$ are standard inference rules in classical first-order logic $\mathtt{LK}$ with the introduction rules and the elimination rules for inductive predicates, determined by the production rules. These rules describe that the predicate actually denotes the least fixed point. In particular, the elimination rule describes the induction principle.

For example, the above production rules give the introduction rules

and the elimination rule

This elimination rule describes mathematical induction principle restricted to $N$. $\mathtt{LKID}$ is sound with respect to a class of models called Henkin models (Definition 2.10 of [6]). We omit the definition of Henkin models and we only use the following property: if a first order structure ${\mathcal M}$ satisfies the induction schema for $N$, then ${\mathcal M}$ is an Henkin model of $\mathtt{LKID}$ with the predicate $N$.

2.2 Cyclic Proof System $\mathtt{CLKID}^\omega$

A cyclic proof system $\mathtt{CLKID}^\omega$ [6] is defined as a system obtained from $\mathtt{LKID}$ by (1) replacing elimination rules by case rules, (2) allowing a bud as an open assumption and requiring a companion for each bud, (3) requiring the global trace condition.

The case rule is defined by unfolding the production rule in the antecedent. For example, the case rule for N is

In a cyclic proof, we can have open assumptions, called buds, but it is required that each bud has some corresponding sequent of the same form, called a companion, inside the proof figure.

An example of a cyclic proof is

where the mark (a) denotes the bud-companion pair. Remark that the companion (a) uses Nx, but the bud (a) uses Nx where x is $x'$, so their actual meanings are different even though they are of the same form.

A pre-proof of $\mathtt{CLKID}^\omega$ is obtained by recursively replacing every bud by the proof of its companion. A trace is a sequence of occurrences of an atom in a path of the proof tree, possibly moving to a case-descendant when passing through a case rule. Moving to a case-descendant is called a progress point of the trace (Definition 5.4 [6]). The global trace condition says that for every infinite path there is a trace with infinitely many progress points following some tail of the path (Definition 5.5 [6]). The global trace condition guarantees the soundness of a cyclic proof system for fixed-point models. $\mathtt{CLKID}^\omega$ is not known to be sound for Henkin models, and this leaves the possibility of having an Henkin counter-model for a theorem of $\mathtt{CLKID}^\omega$.

2.3 Brotherston-Simpson Conjecture

$\mathtt{LKID}$ has been often used for formalizing inductive definitions, while $\mathtt{CLKID}^\omega$ is another way for formalizing inductive definitions, and moreover $\mathtt{CLKID}^\omega$ is more suitable for proof search. This raises the question of the relationship between $\mathtt{LKID}$ and cyclic proofs: Brotherston and Simpson conjectured the equality for each inductive definition. The left-to-right inclusion is proved in [3], Lemma 7.3.1 and in [6], Theorem 7.6. Brotherston-Simpson conjecture (the Conjecture 7.7 in [6]) is that the provability $\mathtt{LKID}$ includes that of $\mathtt{CLKID}^\omega$. Simpson [11] submitted a proof of the conjecture in the case of Peano Arithmetic. The goal of this paper is to prove that it is false in general.

3 2-Hydra Problem

3.1 Hydra Problem

The Hydra of Lerna was a mythological monster, popping two smaller heads whenever you cut one. It was a swamp creature (its name means “water”) and possibly was the swamp itself, whose heads are the swamp plants, with two smaller plants growing whenever you cut one. The original Hydra was defeated by fire, preventing heads to grow again. In the mathematical problem of Hydra, we ask whether we may destroy an Hydra just by cutting heads.

Laurence Kirby and Jeff Paris [9] formulated the Hydra problem as a statement for mathematical trees. We are interested about making Hydra a problem for natural numbers, representing the length of a head, and restricting to the case when the number of heads is always 2. We call our statement 2-Hydra.

3.2 2-Hydra Statement

In this subsection we give the 2-Hydra statement, which is a formula saying that any 2-hydra eventually looses its two heads. This statement actually will give a counterexample to Brotherston-Simpson conjecture.

Let $\varSigma _ N$ be the signature $\{0, { s}, N , p\}$ of a first order language, where 0, the successor ${ s}$, an inductive predicate $N$ for natural numbers, and an ordinary binary predicate symbol p. The logical system $\mathtt{LKID} (\varSigma _ N ,\varPhi _ N )$ is defined as the system $\mathtt{LKID}$ with the signature $\varSigma _N$ and the production rules $\varPhi _N$.

We consider a formal statement of 2-Hydra. The number of head is always 2. Either both heads have positive length, you reduce the length of the first head by 1 unit, and of the second head by 2 units (if possible), or there is a unique head with positive length, you duplicate it and you reduce it by 1 and by 2 units (if possible). We may express H by the convergence of the following set of transformations on $n, m \in \mathtt{Nat}$: if $n \ge 1$ and $m \ge 2$ then $(n,m) \mapsto (n-1,m-2)$; if $n \ge 2$ then $(n,0) \mapsto (n-1,n-2)$; if $m \ge 2$ then $(0,m) \mapsto (m-1,m-2)$. When no transformation applies we stop. We may define H by a formula in the language $\varSigma _ N$: the intended meaning of p is the complement of the union of all infinite sequences of transformations. From now on, we write $A_1, \ldots , A_n \rightarrow B$ for $A_1 \wedge \ldots \wedge A_n \rightarrow C$ and $\forall x_1, \ldots , x_n \in N . \ A$ for $\forall x_1. \ \ldots . \ \forall x_n. \ N (x_1) \wedge \ldots \wedge N (x_n) \rightarrow A$.

Definition 1

(2-Hydra Statement ${{\varvec{H}}}$ ). We define $H = (H_a,H_b,H_c,H_d \rightarrow \forall x,y \in N . \ p(x,y))$, where $H_a,H_b,H_c,H_d$ are:

• $(H_a)$ $\forall x \in N . \ p(0,0) \wedge p({{ s}}0,0) \wedge p(x,{{ s}}0)$,

• $(H_b)$ $\forall x,y \in N . \ p(x,y) \rightarrow p({ s}x, { s}{ s}y)$,

• $(H_c)$ $\forall y \in N . \ p({ s}y,y) \rightarrow p(0,{ s}{ s}y)$,

• $(H_d)$ $\forall x \in N . \ p({ s}x,x) \rightarrow p({ s}{ s}x,0)$.

For all $n,m \in \mathtt{Nat}$ there is a unique formula among $H_a,H_b,H_c,H_d$ having some instance inferring p(n, m). The assumption $p(n',m')$ of such a formula, if any assumption exists, satisfies $\max (n',m') < \max (n,m)$. Thus, we may prove H in $\mathtt{PA}$ by induction on $\max (n,m)$. We could define p as an inductive predicate: however, we preferred having p just a predicate symbol, because in this way the definition of a counter-model does not require to check the inductive rule for p.

We will prove that $\mathtt{LKID} (\varSigma _ N , \varPhi _ N ) + (0,{ s})$-axioms does not prove 2-Hydra. We define the $(0,{ s})$-axioms as the axioms “0 is not successor” or $\forall x \in N . \ { s}x \not = 0$, and “successor is injective”, or $\forall x, y \in N . \ { s}x= { s}y \rightarrow x = y$. These axioms cannot be proved in $\mathtt{LKID} (\varSigma _ N , \varPhi _ N )$, because they fail, respectively, in the model of $\mathtt{LKID} (\varSigma _ N , \varPhi _ N )$ uniquely determined by ${\mathcal M} = N _{\mathcal M} = \{0\}$, ${ s}0=0$, in the model uniquely determined by ${\mathcal M} = N _{\mathcal M} = \{0, { s}0\}$, $0 \not = { s}0$ and ${ s}{ s}0 = { s}0$. Compared with $\mathtt{PA}$, in $\mathtt{LKID} (\varSigma _ N , \varPhi _ N ) + (0,{ s})$-axioms we do not have a sum nor a product on $N$, nor we have inductive predicate symbols for addition or multiplication.

3.3 2-Hydra Statement in Cyclic-Proof System

In this section, we give a cyclic proof of the 2-Hydra statement.

We define the logical system $\mathtt{CLKID}^\omega (\varSigma _N,\varPhi _N)$ as the system $\mathtt{CLKID}^\omega$ with the signature $\varSigma _N$ and the production rules $\varPhi _N$.

Theorem 1

The 2-Hydra statement H is provable in $\mathtt{CLKID}^\omega (\varSigma _ N , \varPhi _ N )$.

Proof. Let the 2-Hydra axioms $\hat{H}$ be $H_a,H_b,H_c,H_d$ defined in Definition 1.

For simplicity, we will write the use of 2-Hydra axioms by omitting (Cut), , , (Axiom), as in the following example.

We will also write a combination of (Case) and (=L) as one rule in the following example.

For saving space, we omit writing $\hat{H}$ in every sequent int the next proof figure. For example, actually denotes .

The following is a cyclic proof of .

The global trace condition holds for the following reason (the detailed proof is in [1]). We have three possible choices for constructing an infinite path in the proof: taking a bud of the left, middle, or right. For a given bud and $z_1,z_2 \in \{x,y\}$, we write $z_1 \leadsto z_2$ for a progressing trace from ${ N }z_1$ in the companion to ${ N }z_2$ in the bud. We write $z_1 \leadsto z_2,z_3$ for $z_1 \leadsto z_2$ and $z_1 \leadsto z_3$. For the left bud, there are $x \leadsto x,y$. For the middle bud, there are $y \leadsto x,y$. For the left bud, there are $x \leadsto x$ and $y \leadsto y$. Hence, given an infinite path, there is some tail of the path with an infinitely progressing trace, by cleverly choosing x and y possibly alternatively. Hence the global trace condition holds.   $\Box$

4 A Structure ${\mathcal M}$ for the Language $\varSigma _ N$ Falsifying 2-Hydra

In this section we define a structure ${\mathcal M}$ for the language $\varSigma _ N$, we prove that ${\mathcal M}$ falsifies the 2-Hydra statement H, and we characterize the subsets of ${\mathcal M}$ which satisfy the induction schema.

4.1 Outline of Proof of Non-Provability

In Sect. 4, we define a counter model ${\mathcal M}$. The most difficult point is to prove that ${\mathcal M}$ satisfies Definition 2.10 of [6] for having an Henkin model of $\mathtt{LKID} + \varSigma _ N$. We will prove a sufficient condition for it, the induction schema for $N$.

On one hand, we prove that in our structure ${\mathcal M}$ all unary definable predicates of ${\mathcal M}$ are sets whose measure is some dyadic rational number. This involves proving a quantifier-elimination result (Sect. 6) for a theory of partial equivalence relations (Sect. 5). This result is new, as far as we know: for an introduction to quantifier-elimination we refer to [8], Sects. 3.1 and 3.2). On the other hand, Sect. 4.3 shows that a definable set of ${\mathcal M}$ with dyadic measure satisfies the induction schema. Combining them, finally we will show that ${\mathcal M}$ satisfies the induction schema for $N$ and according to Definition 2.10 of [6] is an Henkin model of $\mathtt{LKID} + \varSigma _ N$.

4.2 Definition of the Structure ${\mathcal M}$

Let $\mathtt{Z}$ be the set of relative integers. ${\mathcal M}$ is $\mathtt{Nat} + \mathtt{Z}$: we represent $\mathtt{Nat} + \mathtt{Z}$ by $\{(1,x) \ |\ x \in \mathtt{Nat} \} \cup \{(2,x) \ |\ x \in \mathtt{Z} \}$. We first define the interpretations $0_{\mathcal M} , { s}_{\mathcal M} , N _{\mathcal M}$. $0_{\mathcal M}$ is 0 in the component $\mathtt{Nat}$ and ${{ s}}_{\mathcal M}$ is the successor on $\mathtt{Nat}$ and on $\mathtt{Z}$. We choose $N _{\mathcal M} = {\mathcal M}$: by construction, ${\mathcal M}$ satisfies the $(0,{{ s}})$-axioms. We abbreviate $x + n = {{ s}}_{\mathcal M} ^n(x)$, $\infty$ for the 0 in the component $\mathtt{Z}$, and $\infty - n$ for the relative integer $-n$ in the component $\mathtt{Z}$, for all $n \in \mathtt{Nat}$. We define the following subsets of ${\mathcal M}$: $\underline{ \mathtt{Nat} } = \{0_{\mathcal M} + n | n \in \mathtt{Nat} \}$ and $\underline{ \mathtt{Z} }^{-} = \{\infty - (n+1) | n \in \mathtt{Nat} \}$ and $\underline{ \mathtt{Z} }^{+}_0 = \{\infty + n | n \in \mathtt{Nat} \}$. The sets $\underline{ \mathtt{Nat} }$, $\underline{ \mathtt{Z} }^{-}$, $\underline{ \mathtt{Z} }^{+}_0$ are a partition of ${\mathcal M}$.

In order to complete the definition of ${\mathcal M}$ we have to choose the interpretation $p_{\mathcal M}$ of the binary predicate p. We first define $\pi = \{(n,2n) | n \in \mathtt{Nat} \} \subseteq \mathtt{Nat} \times \mathtt{Nat}$. $\pi$ is the set of points of the straight line $y = 2x$ whose coordinates are some pair of natural numbers. We imagine $\pi$ starting from the infinite, moving at each step from some $({{ s}}a,{ s}{ s}b)$ to (a, b), and ending in (0, 0). Given any $(m_1, m_2) \in {\mathcal M} \times {\mathcal M}$ we define $(m_1, m_2) + \pi = \{(m_1 + a, m_2 + b) | (a,b) \in \pi \}$ and $(m_1, m_2) - \pi = \{(m_1 - a, m_2 - b) | (a,b) \in \pi \}$. We define three paths in ${\mathcal M} \times {\mathcal M}$ by $\pi _1 = (0_{\mathcal M} , \infty ) + \pi$ and $\pi _2 = (\infty ,0_{\mathcal M} ) + \pi$ and $\pi _3 =(\infty -1, \infty -2) - \pi$. Eventually, we set $p_{\mathcal M} = {\mathcal M} ^2 \setminus (\pi _1 \cup \pi _2 \cup \pi _3)$. As explained in the figure below, we may move forever along $\pi _1 \cup \pi _2 \cup \pi _3$ (in red) while “cutting heads” as follows: $\ldots \mapsto (0_{\mathcal M} +2, \infty +4) \mapsto (0_{\mathcal M} +1, \infty +2) \mapsto (0_{\mathcal M} , \infty ) \mapsto (\infty -1, \infty - 2) \mapsto (\infty - 2, \infty - 4) \mapsto \ldots$.

Lemma 1

(The 2-Hydra Lemma). H is false in ${\mathcal M}$

${\mathcal M}$ satisfies by construction the closure of $N$ under 0 and ${ s}$, and the $(0,{{ s}})$-axioms. In order to prove that ${\mathcal M}$ is a model for $\mathtt{LKID} (\varSigma _ N , \varPhi _ N ) +(0,{ s})$-axioms, we have to prove that ${\mathcal M}$ satisfies Definition 2.10 of [6], Definition 2.10). Let ${\mathcal H}$ be the set of definable predicates of ${\mathcal M}$. A predicate $P \subseteq {\mathcal M} ^n$ is definable in ${\mathcal M}$ if for some formula A in the language $\varSigma _N$ plus constants denoting the elements of ${\mathcal M}$, and with free variables in $x_1, \ldots , x_n$, we have $P = \{(m_1, \ldots , m_n) \in {\mathcal M} ^n | {\mathcal M} \models A[m_1/x_1, \ldots , m_n/x_n]\}$. We write ${\mathcal H} _n$ for the subset of definable predicates of arity n: we call ${\mathcal H} _1$ the set of definable sets of ${\mathcal M}$. According to Definition 2.10 of [6], we have to prove that ${\mathcal M}$ is the smallest pre-fixed point in ${\mathcal H} _1$ for the inductive definition of N: a sufficient condition is to prove the induction schema, that all $X \in {\mathcal H} _1$ which are closed under 0 and ${ s}$ are equal to ${\mathcal M}$.

4.3 The Measure of the Subsets of ${\mathcal M}$ Closed Under 0 and ${ s}$

In this subsection we define a sufficient condition for a predicate on ${\mathcal M}$ to satisfy the induction schema, by using a finitely additive measure $\mu (X)$, defined on some subsets $X \subseteq {\mathcal M}$. We will prove that all definable subsets of ${\mathcal M}$ satisfy this condition.

Definition 2

(Measure of a Subset of ${\mathcal {M}}$ ). For any $X \subseteq {\mathcal M}$ we set: $\mu (X) = \lim _{x \rightarrow \infty } \frac{ \mathtt{card} (\{0_{\mathcal M} + n, \infty -n, \infty +n \in {\mathcal M} | n \in [0,x] \} \cap X)}{3(x+1)}$ whenever this limit exists.

For instance, $\mu (\underline{ \mathtt{Nat} }) = 1/3$ and if $E = \{0_{\mathcal M} , 0_{\mathcal M} + 2, \ldots , \infty -2, \infty , \infty + 2, \ldots \}$, then $\mu (E) = 1/2$. We may now provide a sufficient condition for a predicate to satisfy the induction rule.

Lemma 2

(Measure Lemma). If $\mu (P)$ is a dyadic rational, then P satisfies the induction schema.

An example: if $P = \underline{ \mathtt{Nat} } \cup \underline{ \mathtt{Z} }^+_0$, then P is closed under 0, ${{ s}}$ and $\infty -1 \not \in P$. P does not satisfy the induction schema and $\mu (P) = 2/3$ is not dyadic.

5 A Set ${\mathcal R}$ of Partial Bijections on ${\mathcal M}$

In this section we introduce some set ${\mathcal R}$ of partial bijections on ${\mathcal M}$, whose domain have measure some dyadic rational. In Sects. 6 and 7 we will prove that all definable predicates in ${\mathcal M}$ are a boolean combination of atomic formulas of the language ${\mathcal R}$, and that all definable sets in ${\mathcal M}$ are domains of bijections in ${\mathcal R}$, therefore all have measure some dyadic rational, and by Lemma 2 satisfy the induction schema. We will conclude that ${\mathcal M}$ is an Henkin model of $\mathtt{LKID} + \varSigma _ N$.

We say that a relation R is finite if there are finitely many pairs $(x,y) \in R$. For any set X and any binary relations R, S we write: $\mathtt{id} _X = \{(x,x) | x \in X\}$, $\mathtt{dom} (R) = \{x|\exists y.(x,y) \in R\}$, $\mathtt{codom} (R) = \{y|\exists x.(x,y) \in R\}$, $R^{-1} = \{(y,x) | (x,y) \in R\}$, $R {\circ } S = \{(x,z) | \exists y. ((x,y) \in S) \wedge ((y,z) \in R)\}$ and $R \lceil X = \{(x,y) \in R | x \in X\}$. Remark that we defined relation composition in the same order as function composition: the reason is that we will only consider relations which are partial bijections.

5.1 The Set ${\mathcal D}$ of Subsets of ${\mathcal M}$

In this subsection we propose a candidate ${\mathcal D}$ for the definable subsets of ${\mathcal M}$.

For any sets I, J we define $I { {}^{\subset }_{\sim } } J$ as “$(I \setminus J)$ is finite”: this means “$I \subseteq J$ up to finitely many elements. We define $I \sim J$ as $I { {}^{\subset }_{\sim } } J \wedge J { {}^{\subset }_{\sim } } I$: this means “I, J are equal up to finitely many elements”. $I \sim J$ is equivalent to: $(I \setminus J) \cup (J \setminus I)$ is finite.

For any $r, s \in \mathtt{Q}$ we introduce the formal notations $0_{\mathcal M} + r$, $\infty + s$: they denote elements of ${\mathcal M}$ if and only if $r \in \mathtt{Nat}$, $s \in \mathtt{Z}$. For any $z \in \mathtt{Z}$, $r \in \mathtt{Q}$, we define the set of formal notations $M(2^z,r) = \{0_{\mathcal M} + (2^z*n + r), \infty + (2^z*w + r) | (n \in \mathtt{Nat} ) \wedge (w \in \mathtt{Z} ) \}$. We denote with ${\mathcal B}$ the set of all sets $M(2^z,r)$, for some $z \in \mathtt{Z}$, $r \in \mathtt{Q}$.

We define ${\mathcal D}$ as the family of subsets of ${\mathcal M}$ which are equivalent, up to finitely many elements, to some finite union of sets in ${\mathcal B}$.

Definition 3

(The Family ${\mathcal D}$ ). $D \in {\mathcal D}$ if and only if $D \sim (B_1 \cup \ldots \cup B_n)$ for some $B_1, \ldots , B_n \in {\mathcal B}$. We call ${\mathcal D}$ the dyadic family.

Since $2^z > 0$, all sets $M(2^z,r)$ are infinite. We have $M(2^z,r) { {}^{\subset }_{\sim } } {\mathcal M}$ if and only if $(2^z*n + r) \in \mathtt{Nat}$ for all but finitely many $n \in \mathtt{Nat}$ and $(2^z*w + r) \in \mathtt{Z}$ for all but finitely many $w \in \mathtt{Z}$. We may check that this is equivalent to: $z \in \mathtt{Nat}$ and $r \in \mathtt{Z}$.

We prove that every set in ${\mathcal D}$ has measure some dyadic rational.

Lemma 3

( ${\mathcal D}$ -Lemma). Let $a_0, a \in \mathtt{Z}$ and $D \in {\mathcal D}$.

• 1. All finite subsets of ${\mathcal M}$ are in ${\mathcal D}$.

• 2. For all $a \ge a_0$ there are $0 \le b_1< \ldots< b_i < 2^a$ such that $M(2^{a_0},b) = (M(2^{a},b_1) \cup \ldots \cup M(2^{a},b_{i}))$.

• 3. For some $a_0$ and for all $a \ge a_0$ there are $0 \le b_1, \ldots , b_i < 2^a$ such that $D \sim (M(2^a,b_1) \cup \ldots \cup M(2^a,b_i))$.

• 4. $\mu (D)$ is some dyadic rational.

• 5. ${\mathcal D}$ is closed under $\sim$ and all boolean operations.

5.2 The Family ${\mathcal R}$ of Partial Bijections on ${\mathcal M}$

In this subsection we define a family ${\mathcal R}$ of partial bijections on ${\mathcal M}$ with domain in ${\mathcal D}$. The elements of ${\mathcal R}$ up to finitely many elements are empty or are some power of the complement of $p_{\mathcal M}$, restricted to some $D \in {\mathcal D}$.

We define first some set ${\mathcal F}$ of straight lines. ${\mathcal F}$ is the set of maps $\phi : \mathtt{Q} \rightarrow \mathtt{Q}$, defined by $\phi (x) = 2^z x + r$ for some $z \in \mathtt{Z}$ and some $r \in \mathtt{Q}$. ${\mathcal F}$ is closed under inverse: if $\phi (x) = 2^{z} x + r$, then $\phi ^{-1}(x) = 2^{-z} x -r/2^{z}$. ${\mathcal F}$ is closed under composition: if $\phi _i(x) = 2^{z_i} x + r_i$ for $i=1,2$, then $\phi _2(\phi _1(x)) = 2^{z_1 + z_2} x + (2^{z_2} r_1 + r_2)$.

Let $\mathtt{Q} + \mathtt{Q} = \{ (i,r) | i=1,2 \wedge r \in \mathtt{Q} \}$. We extend $\phi : \mathtt{Q} \rightarrow \mathtt{Q}$ to a map $:{\mathcal M} \rightarrow \mathtt{Q} + \mathtt{Q}$ by $\phi ((i,r)) = (i,\phi (r))$ (recall that each element of ${\mathcal M}$ is coded by some pair (i, r)). For any $D \subseteq {\mathcal M}$ we define $\phi (D) = \{\phi (d) | d \in D\} \subseteq \mathtt{Q} + \mathtt{Q}$. We provide a sufficient condition for having $\phi (D) \subseteq {\mathcal M}$ and $\phi (D) \in {\mathcal D}$.

Lemma 4

( $\phi$ -Lemma). Let $\phi (x) = 2^{z_1}x + r_1$ for some $z_1 \in \mathtt{Z}$ and some $r_1 \in \mathtt{Q}$ and all $x \in \mathtt{Q}$. Assume $M(2^z,r) \in {\mathcal B}$.

• 1. $\phi (M(2^z,r)) \in {\mathcal B}$.

• 2. If $D \in {\mathcal D}$ and $\phi (D) { {}^{\subset }_{\sim } } {\mathcal M}$ then $\phi (D) \in {\mathcal D}$.

Proof

• 1. We have $\phi (M(2^z,r)) = M(2^{z+z_1},2^{z_1}r + r_1) \in {\mathcal B}$.

• 2. If $D \in {\mathcal D}$ then $D \sim M(2^{a_1},b_1) \cup \ldots \cup M(2^{a_i},b_i)$ for some $a_1, \ldots , a_i \in \mathtt{Z}$ and some $b_1, \ldots , b_i \in \mathtt{Q}$. Then $\phi (D) \sim \phi (M(2^{a_1},b_1)) \cup \ldots \cup \phi (M(2^{a_i},b_i))$, and by point 1 above $\phi (M(2^{a_1},b_1)) \cup \ldots \cup \phi (M(2^{a_i},b_i)) \in {\mathcal B}$. Thus, $\phi (D) \in {\mathcal D}$.

A partial bijection on ${\mathcal M}$ is a bijection between two subsets of ${\mathcal M}$. We now define a family ${\mathcal R}$ of partial bijections on ${\mathcal M}$. For instance, one bijection in ${\mathcal R}$ is defined by $\phi (x) = 4x$, with domain ${\mathcal M}$ and codomain M(4, 0), mapping $0_{\mathcal M} +n \mapsto 0_{\mathcal M} +4n$ and $\infty +z \mapsto \infty + 4z$.

Let $\phi \in {\mathcal F}$, $\phi (x) = 2^z x + r$ with $z \in \mathtt{Z}$ and $r \in \mathtt{Q}$. We say that $\phi$ is even if z even, and that $\phi$ is odd if z is odd. We divide infinite bijections in ${\mathcal R}$ between “even” and “odd”. They will be restrictions of an even or odd power of the relation $Q = {\mathcal M} ^2 \setminus p_{\mathcal M}$, up to finitely many point. We will prove that the first order definable predicates of ${\mathcal M}$ are the propositional formulas of ${\mathcal R}$.

Definition 4

(Even Bijections). Let $D, E \in {\mathcal D}$ and $\phi \in {\mathcal F}$ be even. R is an even $(D,E,\phi )$-bijection if D, E are infinite, R is a bijection between D, E, and R is equal up to finitely many elements to the graph of $\phi$ restricted to D, E: $R \sim \{(x,y) \in D \times E | y = \phi (x) \}$. We denote the set of even bijections with ${\mathcal R} ^{+}$.

Q is a partial bijection on ${\mathcal M}$, and by definition Q maps $0_{\mathcal M} + n \mapsto \infty + 2n$ and $\infty + n \mapsto 0_{\mathcal M} + 2n$, and Q is associated to the odd map $\phi (x) = 2x$.

Definition 5

(Odd Bijections). Let $\phi \in {\mathcal F}$ be odd. An odd $(D,E,\phi )$-bijection is any bijection R between some infinite $D, E \in {\mathcal D}$, such that, up to finitely many points, R maps: (1) $\infty - n - 1 \mapsto \infty + \phi (-n-1)$, (2) $0_{\mathcal M} + n \mapsto \infty + \phi (n)$ and $\infty + n \mapsto 0_{\mathcal M} + \phi (n)$. We denote the set of odd bijections with ${\mathcal R} ^{-}$.

Q is an example of odd bijection. Let $\phi \in {\mathcal F}$ be even and $D, E\in {\mathcal D}$. A $(D,E,\phi )$-even bijection may alternatively be defined as any bijection between D, E such that, up to finitely many points: (1) $\infty - n-1 \mapsto \infty + \phi (-n-1)$, (2) $0_{\mathcal M} + n \mapsto 0_{\mathcal M} + \phi (n)$ and $\infty + n \mapsto \infty + \phi (n)$.

We define ${\mathcal R} _0$ as the set of all bijections between finite sets $D, E \in {\mathcal D}$. Eventually, we define a family ${\mathcal R}$ of partial bijections by ${\mathcal R} = {\mathcal R} ^{+} \cup {\mathcal R} _0 \cup {\mathcal R} ^{-}$.

If $R \in {\mathcal R} ^{+} \cup {\mathcal R} ^{-}$, associated to the map $\phi \in {\mathcal F}$ with domain D and codomain E, then we may prove that $E \sim \phi (D)$. ${\mathcal R}$ and ${\mathcal D}$ satisfy the following closure properties.

Lemma 5

(Partial bijections). Assume that $R, S \in {\mathcal R}$ and $D \in {\mathcal D}$.

• 1. $\mathtt{id} _D \in {\mathcal R}$

• 2. If $D \in {\mathcal D}$ then $R(D) \in {\mathcal D}$

• 3. $R {\circ } S \in {\mathcal R}$.

• 4. $R^{-1} \in {\mathcal R}$

• 5. ${\mathcal D}$ is closed under complement.

${\mathcal R}$ is closed under intersection.

Lemma 6

(Closure Under Intersection). Assume that $R, S \in {\mathcal R}$ are associated to $\phi , \psi \in {\mathcal F}$.

• 1. If $\phi = \psi$ then $R \cap S \in {\mathcal R}$

• 2. If $\phi \not = \psi$ then $R \cap S \in {\mathcal R}$

• 3. ${\mathcal R}$ is closed under intersections.

• 4. For all $R \in {\mathcal R}$ there is some $D \in {\mathcal D}$ such that $R \cap \mathtt{id} _{\mathcal M} = \mathtt{id} _D$.

Our goal is to prove that every first-order definable subset of ${\mathcal M}$ is in ${\mathcal D}$. Since the sets definable in the language of ${\mathcal R}$ include those definable in ${\mathcal M}$, it is enough to prove that any first-order definable set in language of ${\mathcal R}$ is in ${\mathcal D}$. To this aim, we need a quantifier-elimination result for the language of ${\mathcal R}$.

6 A Quantifier Elimination Result for Partial Bijections

In this section we prove a quantifier elimination result for a theory of partial equivalence relation, which is the abstract counterpart of the families ${\mathcal R}$ and ${\mathcal D}$ introduced in the previous section. This is a simple, self-contained result introducing a model-theoretical tool of some interest.

Theorem 2

(Quantifier Elimination for Partial Bijections). Let U be a set and ${\mathcal R}$ a set of partial bijections on U. Assume that all finite partial bijections on U are in ${\mathcal R}$, that ${\mathcal D} = \{ \mathtt{dom} (R) | R \in {\mathcal R} \}$ is closed under complement, and that for all $R,S \in {\mathcal R}$, $D \in {\mathcal D}$ we have $\mathtt{id} _{U}, R^{-1}, R {\circ } S \in {\mathcal R}$ and $R \cap S, R \lceil D \in {\mathcal R}$. Let ${\mathcal U}$ be the structure with universe U, one constant denoting each element of U, and one predicate symbol denoting each $R \in {\mathcal R}$. Then:

• 1. The theory of ${\mathcal U}$ has quantifier-elimination.

• 2. Any set definable in ${\mathcal U}$ is in ${\mathcal D}$ and is R(x, x) for some $R \in {\mathcal R}$.

In order to give a flavor of our quantifier elimination procedure, we give an example: detailed proofs are in [1].

$$ \exists x_4(R_1x_1x_4 \wedge R_2x_2x_4 \wedge \lnot Rx_3x_4) $$

is equivalent to

$$ \exists x_4(R_{1,4}x_1x_4 \wedge R_{2,4}x_2x_4 \wedge \lnot Rx_3x_4) $$

where $D_4 = \mathtt{codom} (R_1) \cap \mathtt{codom} (R_2)$, $D_1 = R^{-1}_1(D_4)$, $D_2 = R^{-1}_2(D_4)$, and $R_{1,4} = R_1 \cap (D_1 \times D_4)$, $R_{2,4} = R_2 \cap (D_2 \times D_4)$. Note domain restriction here. It is equivalent to

$$ \exists x_4(R_{1,4}x_1x_4 \wedge R_{2,4}x_2x_4 \wedge R_{1,2}x_1x_2 \wedge \lnot Rx_3x_4) $$

where $R_{1,2} = R_{2,4}^{-1} {\circ } R_{1,4}$. Note composition of relations here. It is equivalent to

$$ \exists x_4(R_{1,4}x_1x_4 \wedge R_{2,4}x_2x_4 \wedge R_{1,2}x_1x_2 \wedge \lnot R'x_3x_2) $$

where $R' = R_{2,4}^{-1} {\circ } R$. Note partial bijections here. It is equivalent to

$$ \exists x_4(R_{1,4}x_1x_4 \wedge R_{2,4}x_2x_4) \wedge R_{1,2}x_1x_2 \wedge \lnot R'x_3x_2 $$

Using the properties of partial bijections, finally this is equivalent to

$$ R_{1,2}x_1x_2 \wedge \lnot R'x_3x_2. $$

In the proof we identify each constant symbol with the element $c \in U$ it denotes, and each predicate symbol with the relation $R \in {\mathcal R}$ it denotes. We prove quantifier elimination for ${\mathcal U}$ as defined in [8], Sects. 3.1 and 3.2, namely that each formula A with possibly free variables in the language $U \cup {\mathcal R}$ is equivalent in ${\mathcal U}$ to some formula B in the same language but without quantifiers. We will in fact prove a little more, namely that B may be chosen without constants.

We derive some closure properties for ${\mathcal R}$. For any $D \in {\mathcal D}$ we have $\mathtt{id} _D = ( \mathtt{id} _{\mathcal M} \lceil D) \in {\mathcal R}$. We will abbreviate $\mathtt{id} _D(x,x)$ with $(x \in D)$. ${\mathcal D}$ is closed under intersection, because if $D, E \in {\mathcal D}$ then $\mathtt{id} _D, \mathtt{id} _E \in {\mathcal R}$, hence $\mathtt{id} _{D \cap E} = ( \mathtt{id} _D \cap \mathtt{id} _E) \in {\mathcal R}$ and $D \cap E \in {\mathcal D}$. ${\mathcal D}$ includes ${\mathcal M} = \mathtt{dom} ( \mathtt{id} _{\mathcal M} )$ and it is closed under complement, therefore ${\mathcal D}$ is closed under all boolean operations. For all $x \in {\mathcal M}$ the partial bijection $\mathtt{id} _{\{x\}}$ is finite and by assumption it is in ${\mathcal R}$: thus, all singletons of ${\mathcal M}$ are in ${\mathcal D}$. Assume $R \in {\mathcal R}$ and $D \in {\mathcal D}$: then $\mathtt{codom} (R) = \mathtt{dom} (R^{-1}) \in {\mathcal D}$ and $R(D) = (R\lceil D)(D) = \mathtt{codom} (R \lceil D) \in {\mathcal D}$. If $R \in {\mathcal R}$ and $D, E \in {\mathcal D}$, then $R \cap (D \times E) \in {\mathcal R}$ follows from $R \cap (D \times E) = (((R \lceil D)^{-1}) \lceil E)^{-1}$.

In the next statement, recall that we defined the relation composition in the same order as function composition.

Lemma 7

(Composition and Product). Let R, S be any relation and D, E, F be any sets. Assume $R(D) \cap S^{-1}(F) \subseteq E$. Then composition and Cartesian product commute: $(S \cap (E \times F)) {\circ } (R \cap (D \times E)) = (S {\circ } R) \cap (D \times F).$

6.1 A Notion of Normal Form for the Language ${\mathcal R}$

Let $n>0$ be any positive integer. Assume $R \in {\mathcal R}$ and $i, j \in \{1, \ldots , n\}$. We call any formula $R(x_i,x_j)$ a positive $( {\mathcal R} ,n)$-atom and any formula $\lnot R(x_i,x_j)$ a negative $( {\mathcal R} ,n)$-atom. A $( {\mathcal R} ,n)$-atom is either a positive or a negative $( {\mathcal R} ,n)$-atom. A $( {\mathcal R} ,n)$-propositional formula is any formula obtained from positive $( {\mathcal R} ,n)$-atoms by repeatedly applying disjunction and negation. Any $( {\mathcal R} ,n)$-propositional formula has free variables in $x_0, \ldots , x_{n-1}$. We denote by ${\mathcal A} _n$ the set of $( {\mathcal R} ,n)$-propositional formula, and by ${\mathcal H} _n$ the set of n-ary predicates definable in ${\mathcal U}$.

Our goal is to prove that for all $n \in \mathtt{Nat}$, any first-order predicate P of ${\mathcal R}$ is definable by some $A \in {\mathcal A} _n$, and if $n=1$ then $P \in {\mathcal D}$. We have to prove that formulas of ${\mathcal A} _n$ are closed under existential.

This is the plan of the proof. We will define a notion of $( {\mathcal R} ,n)$-normal form for formulas of ${\mathcal A} _n$, and prove that every $A \in {\mathcal A} _n$ has some $( {\mathcal R} ,n)$-normal form. Then we will prove that if $A \in {\mathcal A} _n$ is in $( {\mathcal R} ,n)$-normal form, then $\exists x_n.A$ (with possibly free variables) may be expressed in ${\mathcal A} _{n-1}$ in one of the following ways: either as some finite disjunction $A[c_1/x_1] \vee \ldots \vee A[c_k/x_n]$ for some constants $c_1, \ldots , c_k \in U$, or by the formula $B \in {\mathcal A} _{n-1}$, obtained from A by erasing all $( {\mathcal R} ,n)$-atoms including $x_n$.

Assume $n >0$ is any positive integer. Let ${\mathcal G}$ be any binary relation on $\{1, \ldots , n\}$. A $( {\mathcal G} ,n)$-family is any family ${\mathcal F} = \{R_{i,j}(x_i, x_j) | (i,j) \in {\mathcal G} \}$ of positive $( {\mathcal R} ,n)$-atoms such that $\mathtt{dom} (R_{i,j}) = \mathtt{dom} (R_{i,k})$ for all $i,j,k = 1, \ldots , n$. ${\mathcal F}$ is a symmetric family if ${\mathcal G}$ is a symmetric relation, and for all $(i,j) \in {\mathcal G}$ we have $R_{i,j} = R^{-1}_{i,j}$. ${\mathcal F}$ is a equivalence family if ${\mathcal G}$ is an equivalence relation, and for all $i = 1, \ldots , n$ we have $R_{i,i} = \mathtt{id} _{D_i}$ for some $D_i$, for all $(i,j) \in {\mathcal G}$ we have $R_{i,j} = R^{-1}_{i,j}$, for all $(i,j), (j,k) \in {\mathcal G}$ we have $R_{j,k} {\circ } R_{i,j} = R_{i,k}$. In this case $D_i = \mathtt{dom} (R_{i,j})$ for all $i,j = 1, \ldots , n$ and we call $D_1, \ldots , D_n$ the domains of the family.

A $( {\mathcal G} ,n)$-symmetric conjunction is any conjunction of a $( {\mathcal G} ,n)$-symmetric family. A $( {\mathcal G} ,n)$-equivalence conjunction is any conjunction of a $( {\mathcal G} ,n)$-equivalence family.

We recall some basic graph theory. We call an indirect, simple graph on $\{1, \ldots , n\}$ just a graph, and we represent it by any irreflexive and symmetric relation ${\mathcal G}$ on $\{1, \ldots , n\}$. A simple cycle in ${\mathcal G}$ is any sequence $\sigma = \{(i_0, i_1), (i_1, i_2), \ldots (i_{m-1},i_m), (i_m, i_0)\} \subseteq {\mathcal G}$ of pairwise distinct $i_0, \ldots , i_m$ with $m \ge 2$. ${\mathcal G}$ is acyclic if ${\mathcal G}$ has no simple cycle. A path is any sequence $\pi = \{(i_0, i_1), (i_1, i_2), \ldots (i_{m-1},i_m)\}$ with pairwise distinct $i_1, \ldots , i_m$, with possibly $m=0$. The connection relation on ${\mathcal G}$ is: “there is some path from i to j” In any acyclic graph ${\mathcal G}$ the path from i to j if it exists then it is unique. Given any equivalence relation ${\mathcal P}$, there is some minimal graph ${\mathcal G} \subseteq {\mathcal P}$ among those such that ${\mathcal P}$ is the smallest equivalence relation including ${\mathcal G}$. All these minimal graphs are acyclic.

Definition 6

( $( {\mathcal R} ,n)$ -Normal Forms). $C = C_1 \wedge C_2$ is a $( {\mathcal R} ,n)$-normal conjunction if $C_1$ is some conjunction of positive $( {\mathcal R} ,n)$-atoms, $C_2$ is some conjunction of negative $( {\mathcal R} ,n)$-atoms, and for some equivalence relation ${\mathcal P}$

• 1. $C_1$ is some $( {\mathcal P} ,n)$-equivalence conjunction

• 2. for any $\lnot S(x_i,x_j)$ in $C_2$ we have $i<j$

• 3. if $[n]_ {\mathcal P} \not = \{n\}$ then $x_n$ does not occur in $C_2$

Any $A \in {\mathcal A} _n$ is an $( {\mathcal R} ,n)$-normal form if A is some disjunction of $( {\mathcal R} ,n)$-normal conjunctions.

We first prove that any $( {\mathcal G} ,n)$-symmetric conjunction, with ${\mathcal G}$ some acyclic graph, is equivalent to some $( {\mathcal P} ,n)$-equivalence conjunction, where ${\mathcal P}$ is the reflexive and transitive closure of ${\mathcal G}$.

Lemma 8

(Transitive Closure Lemma). Let $n>0$ be any positive integer. Assume ${\mathcal G}$ is any acyclic graph on $\{1, \ldots , n\}$ and $A = \wedge _{i,j \in {\mathcal G} } R_{i,j}(x_i,x_j)$ is any $( {\mathcal G} ,n)$-symmetric conjunction. Let ${\mathcal P}$ be the reflexive, symmetric and transitive closure of ${\mathcal G}$. Then A is equivalent to some unique $( {\mathcal P} ,n)$-equivalence conjunction B whose family of atoms extends the family of atoms of A.

Now we prove that the $( {\mathcal P} ,n)$-equivalence conjunctions are closed under conjunction with a positive $( {\mathcal R} ,n)$-atom $R(x_i,x_j)$. For all $i = 1, \ldots , n$, we denote with $[i]_ {\mathcal P}$ the equivalence class of i in ${\mathcal P}$.

Lemma 9

(Partition Lemma). Assume $A = \wedge _{i,j \in {\mathcal P} } R_{i,j}(x_i,x_j)$ is any $( {\mathcal P} ,n)$-equivalence conjunction, and $i,j \in \{1, \ldots , n\}$. Assume $D \in {\mathcal D}$ and $R \in {\mathcal R}$.

• 1. $A \wedge (x_i \in D)$ is equivalent to some $( {\mathcal P} ,n)$-equivalence conjunction

• 2. Assume $[i]_ {\mathcal P} = [j]_ {\mathcal P}$. Then $A \wedge R(x_i, x_j)$ is equivalent to some $( {\mathcal P} ,n)$-equivalence conjunction

• 3. Assume $[i]_ {\mathcal P} \not = [j]_ {\mathcal P}$ and $\mathtt{dom} (R) = \mathtt{dom} (R_{i,i})$ and and $\mathtt{codom} (R) = \mathtt{dom} (R_{j,j})$. Then $A \wedge R(x_i, x_j)$ is equivalent to some $( {\mathcal P} ,n)$-equivalence conjunction

• 4. Any $A \wedge R(x_i, x_j)$ is equivalent to some $( {\mathcal P} ,n)$-equivalence conjunction

6.2 A Quantifier Elimination Result for ${\mathcal R}$

Now we prove a quantifier-elimination result for the language with symbols the binary predicates in ${\mathcal R}$, using Lemmas 6 and 9.

Lemma 10

(Quantifier Elimination for ${\mathcal R}$ ). Let $n >0$ be any positive integer.

• 1. Any finite conjunction of positive $( {\mathcal R} ,n)$-atoms has some $( {\mathcal P} ,n)$-equivalence form.

• 2. Any finite conjunction of positive and negative $( {\mathcal R} ,n)$-atoms has some $( {\mathcal P} ,n)$-equivalence form.

• 3. If A is some finite conjunction of positive and negative $( {\mathcal R} ,n)$-atoms, then $\exists x_n.A$ is equivalent to some $B \in {\mathcal A} _{n-1}$.

• 4. If $A \in {\mathcal A} _n$, then $\exists x_n.A$ is equivalent to some $B \in {\mathcal A} _{n-1}$.

We may now finish the proof of Theorem 2.

7 Main Theorem

Let ${\mathcal R}$, ${\mathcal D}$ as in Sect. 5. From the properties of the partial bijections in ${\mathcal R}$ and from the quantifier elimination result (Sect. 6) we derive our main result.

Theorem 3

(Counterexample to Brotherston-Simpson Conjecture). Let H be the formula defined in Definition 1. Then H has a proof in $\mathtt{CLKID}^\omega (\varSigma _ N , \varPhi _ N )$, and no proof in $\mathtt{LKID} (\varSigma _ N , \varPhi _ N ) + (0,{ s})$-axioms.

Proof. The proof in $\mathtt{CLKID}^\omega$ is shown in Theorem 1. The non-provability in $\mathtt{LKID}$ is shown as follows. Any atomic formulas in ${\mathcal M}$ is in ${\mathcal R}$. By definition, ${\mathcal R}$ contains all finite bijections and is closed under restriction to any set $D \in {\mathcal D}$. Thus, by Lemma 6, ${\mathcal R}$ satisfies all hypothesis of Theorem 2. We deduce that all definable sets of ${\mathcal M}$ are in ${\mathcal D}$. By Lemma 3 point 4, all sets in ${\mathcal D}$ have a dyadic measure, and by Lemma 2 satisfy the induction schema. According to Definition 2.10 of [6], this is a sufficient condition for ${\mathcal M}$ being an Henkin model of $\mathtt{LKID} (\varSigma _ N , \varPhi _ N )$. ${\mathcal M}$ satisfies the (0, S)-axioms by construction. ${\mathcal M}$ falsifies H by Lemma 1.    $\Box$

8 Non-conservativity of Martin-Löf’s Inductive Definition System

This section shows non-conservativity of $\mathtt{LKID}$ with respect to additional inductive predicates, by giving a counterexample.

We assume the inductive predicate $\le$ and the production rules for it:

We call the set of these production rules $\varPhi _{\le }$. Let 0-axiom be $\forall x \in N . \ { s}x \not = 0$. In $\mathtt{LKID} (\varSigma _N+\{\le \},\varPhi _N+\varPhi _{\le })$, we can show any number $\le 0$ is only 0.

Lemma 11

The proof is in [1].

The next theorem shows 2-Hydra is provable in $\mathtt{LKID}$ with $\le$.

Theorem 4

is provable in $\mathtt{LKID} (\varSigma _N+\{\le \},\varPhi _N+\varPhi _{\le })$.

We may show by induction on n. The proof is given in [1] in case.

In the standard model, the truth of formula does not change when we extend the model with inductive predicates that do not appear in the formula. On the other hand, this is not the case for provability in Martin-Löf’s inductive definition system $\mathtt{LKID}$. Namely, a system may change the provability of a formula even when we add inductive predicates that do not appear in the formula. Namely, for a given system, the system with additional inductive predicates may not be conservative over the original system. Theorems 3 and 4 give such an example: the sequent ${ 0\text{-axiom } } \vdash H$ is in the language of $\mathtt{LKID}$ but it is not provable in $\mathtt{LKID}$, while it is provable in $\mathtt{LKID}$ extended with $\le$.

9 Conclusion

We proved in Theorem 3 that $\mathtt{CLKID}^\omega$, the formal system of cyclic proofs ([6]) proves strictly more that $\mathtt{LKID}$, Martin-Löf formal system of inductive definitions with classical logic. This settles an open question given in [6]. Our proof also shows that if we add more inductive predicates to $\mathtt{LKID}$ we may obtain a non-conservative extension (Theorem 4).