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ON STREAMS THAT ARE FINITELY RED
MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
Department of Informatics, University of Bergen
e-mail address: bezem@ii.uib.no
Institute of Cybernetics at Tallinn University of Technology
e-mail address: keiko@cs.ioc.ee
Institute of Cybernetics at Tallinn University of Technology
e-mail address: tarmo@cs.ioc.ee
Abstract. Mixing induction and coinduction, we study alternative definitions of streams
being finitely red. We organize our definitions into a hierarchy including also some well-
known alternatives in intuitionistic analysis. The hierarchy collapses classically, but is
intuitionistically of strictly decreasing strength. We characterize the differences in strength
in a precise way by weak instances of the Law of Excluded Middle.
1. Introduction
Finiteness is a concept that seems as intuitive as it is fundamental in all of mathematics.
At the same time finiteness is notoriously difficult to capture axiomatically. First, due to
compactness, finiteness is not first-order definable. Second, in ZF set theory, there exist
several different approximations (as ZF is a first-order theory). Tarski [4] is still a very
readable introduction to different definitions of finiteness in set theory without the axioms
of infinity and choice. These include the definitions by Dedekind, by Kuratowski and the
inductive definition by Tarski himself: a set Ais finite if it is in the inductively defined subset
of the powerset of Acontaining the empty set and closed under adding one element of A.
These different approximations of finiteness are only equivalent if one assumes additional
axioms. And all this already in the realm of classical mathematics.
It will therefore not come as a surprise that in intuitionistic mathematics the situation is
even more complicated. In this paper, we will study several classically equivalent definitions
of binary sequences that are almost always zero, that is, there are at most finitely many
positions where the sequence is one. From the constructive point of view, one has at least
the following main variants.
(1) ∃n. ∀m≥n. (s m) = 0. This definition expresses that all finitely many mfor
which (s m) = 1 occur in sbefore some position n. By the decidability of =, they
can all be looked up and counted. This is clearly the strongest definition giving
1998 ACM Subject Classification: F.4.1.
Key words and phrases: Type theory, constructive mathematics, (co)induction, finiteness.
LOGICAL METHODS
IN COMPUTER SCIENCE DOI:10.2168/LMCS-???
c
Marc Bezem, Keiko Nakata, and Tarmo Uustalu
Creative Commons
1
2 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
all information. By the decidability of =, this definition is also intuitionistically
equivalent to ∃n. ¬¬∀m≥n. (s m) = 0, in spite of the double negation prefixing
the universal quantifier.
(2) ∃n. ∀m. #{k≤m|(s k)=1}< n. This definition is weaker than the first one. It
only states that there is an upper bound to the number of ones in the sequence, but
does not provide information on where to find them.
(3) ¬(∀n. ¬¬∃m≥n. (s m) = 1). This definition is equivalent to ¬¬(1). Note that (3)
is stable since it is negative, and therefore does not imply (2). Surprisingly, (3) is
also equivalent to ¬¬(2). The reason is that (1) and (2) are classically equivalent,
do not contain disjunction, and have only existential quantification as the main
connective of the formula. Therefore their respective double negation translations
¬¬(1) and ¬¬(2) are constructively equivalent, so also equally weak.
(4) ¬(∀n. ∃m≥n. (s m) = 1). This definition expresses that the set of positions where
the sequence is one is not infinite. It does not give a clue where to find the ones or
how many ones there are. Definition (4) is the weakest of all: It negates a strong,
positive statement allowing the construction of an infinite subsequence of ones in s.
The variants are listed in decreasing constructive strength. Variants (1) and (2) are
positive and therefore strictly stronger than the negative variants (3) and (4). Reversing the
implications above requires some form of classical logic. For instance we know that (4) =⇒
(2) is not constructively valid. We use the occasion to introduce an argument employed
more rigorously later in this paper. Let sbe an arbitrary binary function. Construct s0
starting from n= 0 by taking (s0n) = 0 as long as (s n) = 0. There is no constructive
way to find out whether (s n) is always 0 or not, but if (s n) = 1 for the first time, we take
(s0k) = 1 for n <=k <= 2nand (s0k) = 0 for k > 2n. One easily verifies (4) for s0. Now,
if (2) would hold for s0we would be able to decide whether sis constant 0 or not. For if
there are at most nones occurring in s0, the first one would occur not later than at n, and
this can constructively be tested. In other words, (4) =⇒(2) implies an instance of the
excluded middle which is not constructively valid.
The paper sets out an expedition to the concept of finiteness from the constructive
point of view, with strong assumptions on the set whose finiteness we study. Namely,
(1) The set is enclosed in another set with decidable equality and carved out by a
predicate (whether a bit-valued function returns 1).
(2) This predicate is decidable.
(3) The enclosing set is traversible.
In one word, therefore, we could summarize our setting as ”searchable”. As we will see in
the paper, even in a searchable setting, there are at least six different notions of ”finiteness”.
The remainder of the paper is structured as follows. In the next section, we set up a
basis for our development in the paper. Section 3 introduces a spectrum of definitions for
sequences being finitely one. In Section 4, we study relative strength of these definitions
from the constructive point of view. In Section 5, we relate our analysis to that of finiteness
of sets in Bishop’s set theory due to Coquand and Spiwack [2]. We also corroborate a
conjecture posed by them. We conclude in Section 6.
For methodological uniformity, we prefer to define all datatypes inductively (rules de-
noted by a single line) or coinductively (rules denoted by a double line).
ON STREAMS THAT ARE FINITELY RED 3
2. Two views of infinite sequences
We may look at binary infinite sequences in two ways. We may view them as bit-valued
functions on natural numbers or, which will amount to the same, as streams of bits, i.e.,
as elements of a coinductive type. Correspondingly, we will use two different languages
to speak about them: arithmetic (as is traditional in logic) for bit-valued functions and
the language of inductive and coinductive predicates (as is more customary in functional,
in particular, dependently typed, programming) for bitstreams. As a warming-up, in this
section, we connect the two views, setting up a basis for our development along the way.
For this paper to have some color, we take a bit to be one of the two colors, red and
black:
R:color B:color
In the function-view, an infinite sequence is therefore a function f:nat →color mapping
natural numbers (positions in the sequence) to colors. Our intended notion of equality of
these functions is the extensional function equality defined by
∀n. f n =f0n
f≡f0
In the stream-view, an infinite sequence is a stream s:str where the stream type is defined
coinductively by the following rule:
c:color s:str
c s :str
Two streams sand s0are equal for us, if they are bisimilar. This notion of equality is
defined coinductively by the rule
s∼s0
c s ∼c s0
The two types are isomorphic. Indeed we can define two functions s2f :str →nat →color
and f2s : (nat →color)→str mediating between the two types. The function s2f is defined
by (structural) recursion by
s2f (c s) 0 = c
s2f (c s) (n+ 1) = s2f s n
while the function f2s is defined by (guarded-by-constructors) corecursion by
f2s f= (f0) (f2s (λn. f (n+ 1)))
We have that ∀f, s. f ≡s2f s⇔f2s f∼s. The ⇒direction is proved by coinduction, the
⇐direction by induction. From this fact it follows immediately that ∀f. f ≡s2f (f2s f)
and ∀s. f2s (s2f s)∼s, i.e., that the function and stream types are isomorphic, as well
as that ∀f, f 0. f ≡f0⇒f2s f∼f2s f0and ∀s, s0. s ∼s0⇒f2s s≡f2s s0, i.e., that the
conversion functions appropriately preserve equality. (In general, we have to ensure that
all functions and predicates we define on bit-valued functions and bitstreams respect our
notions of equality for them, i.e., extensional function equality and bisimilarity.1)
1The correspondence between extensional function equality and bisimilarity shows that bisimilarity is the
one and only reasonable notion of “extensional stream equality”.
4 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
Properties of binary infinite sequences can now be defined and analyzed in either one of
the two equivalent views. For the stream-view, it is convenient to introduce some operations
and predicates as primitives in our language for streams. We define
(c s)@0 = c(c s)@(n+ 1) = s@n
s|0=s(c s)|(n+1) =s|n
so that s@ndenotes the color at the position nin sand s|ndenotes the suffix of sat n.
We also define
red (R s)black (B s)
X s
FX s
FX s
FX(c s)
X(c s)GX s
GX(c s)
Here, Fand Gare the “sometime in the future” (“finally”) and “always in the future”
(“globally”) modalities of linear-time temporal logic. They are stream predicates parame-
terized over stream predicates.2Induction and coinduction give us simple proofs of basic
facts such as the equivalence
∀s. G(λt. ¬X t)s⇔ ¬F X s
and the implication (converse does not hold)
∀s. F(λt. ¬X t)s⇒ ¬G X s
Importantly, we can also prove that
∀s. FX s ⇔ ∃n. X (s|n)
∀s. GX s ⇔ ∀n. X (s|n)
noticing that ∀s, n. s2f (s|n) = λm. s2f s(n+m).
Both modalities are expressible in the function-view, but the definitions are (perhaps)
less elegant, as they involve explicit arithmetical manipulation of positions:
∀f. F(λs. Y (s2f s)) (f2s f)⇔ ∃n. Y (λm. f (n+m))
∀f. G(λs. Y (s2f s)) (f2s f)⇔ ∀n. Y (λm. f (n+m))
In particular,
∀s. Fred s⇔ ∃n. s2f s n =R
∀s. Gblack s⇔ ∀n. s2f s n =B
Accordingly, we have
∀s. (¬G black s⇒ F red s)⇔(¬(∀n. s2f s n =B)⇒ ∃n. s2f s n =R)
and hence
[∀s. ¬G black s⇒ F red s]⇔[∀f. ¬(∀n. f n =B)⇒ ∃n. f n =R)]
We now have arrived at two equivalent formulations of Markov’s Principle (MP). Markov’s
Principle is an important principle that is neither valid nor inconsistent constructively, but
only classically valid. It is computationally meaningful, however, being realizable by search.
In the function-view (the right-hand side), which is how it is traditionally presented,
Markov’s Principle is the statement that
∀f. ¬(∀n. f n =B)⇒ ∃n. f n =R
2There is no need to see them as “first-class” predicate transformers, as there is no real impredicativity
involved: the argument of Fis constantly Xin the definition of F, and the same is true of the definition of
G.
ON STREAMS THAT ARE FINITELY RED 5
(or, equivalently, as ∀n. ¬A⇔ ¬∃n. A, the statement ∀f. ¬¬(∃n. f n =R)⇒ ∃n. f n =R.)
The computational interpretation is the natural one: if it cannot be that all positions
in a given infinite sequence are black, then we find a red position by exhaustively checking
all positions in the natural order 0,1,2. . .. (Cf. computability theory: this is minimization,
not primitive recursion.)
In the stream-view (the left-hand side), Markov’s Principle is
∀s. ¬G black s⇒ F red s
stating that if a stream sis not all black, then it is eventually red. But, in a certain sense,
it is more than just any equivalent statement to the function-view counterpart. It is a
concise formulation of Markov’s Principle based on the stream view of infinite sequences
and canonical inductive and coinductive predicates on streams. We would therefore like
to think that, for computer scientists, it should be natural to take namely this statement
rather than the traditional arithmetical version as the definition of Markov’s Principle.
This applies to another important classical axiom of the Lesser Principle of Omniscience
which is meaningful as a special case of the Law of Excluded Middle.
The Lesser Principle of Omniscience (LPO) is the assertion of the statement
∀f. (∀n. f n =B)∨(∃n. f n =R)
that, in the light of what we already learned, is equivalent to
∀s. Gblack s∨ F red s
Again, the latter statement is perhaps more basic for a computer scientist than the for-
mer: it states that any stream is either all black or eventually red (which is constructively
impossible).
As we have constructively A∨B⇒(¬A⇒B), but not vice versa, LPO implies Markov,
but not the other way around.
3. Some notions of “finitely red”
With these preparations done, we can now proceed to possible mathematizations of the
informal property of a given infinite sequence (function for stream s) being “finitely red”.
We consider six variations. They are all equivalent classically. In Section 4, we will study
their relative strength from the constructive point of view.
3.1. Eventually All Black. The simplest mathematization is: “from some position on,
the sequence is all black”.
In the function view, this is stated as
∃n. ∀m≥n.f m =B
while the stream-view statement is at least as simple, namely, the stream is “finally” “glob-
ally” black:
F(Gblack)s
The two statements are equivalent.
∀s. F (Gblack )s⇔ ∃n. ∀m≥n. s2f s m =B
6 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
3.2. Boundedly Red. Next comes: “the number of red positions in the sequence is
bounded”.
In the function view, this is stated as
∃n. ∀m. #{k≤m|f k =R}< n
so that for a fixed n,fis red fewer than nup to the m-th position for any m.
The formation of the stream view is similar. We first define a predicate lens, which
states that sis fewer than nred, coinductively by
le(n+1) s
le(n+1) (B s)
lens
le(n+1) (R s)
Then the stream-view is simply:
∃n. lens
Again, the two statements are equivalent
∀s, n. lens⇔ ∀m. #{k≤m|s2f s k =R}< n
3.3. Almost Always Black. The third definition amounts to the least fixed point of a
weak until operator in linear-time temporal logic. It is also found in the thesis of C. Raf-
falli [5]. We formulate it in the stream view. The weak until operator, WX, is parameterized
over any predicate Xon streams and defined coinductively by
WXs
WX(B s)
X s
WX(R s)
so that WXsholds if, whenever the first occurrence of red in sis encountered, Xholds
on the suffix after the occurrence. Classically it is equivalent to that sis either all black
or it is eventually red and Xholds on the suffix after the first occurrence of red (which is
guaranteed to exist as sis eventually red). Our definition of WXavoids upfront decisions
of LPO, i.e., whether a stream is all black or eventually red.
We then take the least fixed point of WX. Define µWinductively in terms of WXby
the (Park-style) rule:
WµWs
µWs
As WXis monotone on X, the above definition makes sense. For the purpose of proof, in
particular to avoid explicitly invoking monotonicity of the underlying predicate transformer
WX, it is however convenient to use Mendler-style rule
(∀s. X s ⇒µWs)WXs
µWs
The Park-style rule is derivable from the Mendler-style rule. As WXis monotone on X, we
can also recover the natural inversion principle for µW.
The statement µWsdoes not give a clue as to where to find the red positions in sor
how many of them there are. Nonetheless it refutes that the stream is infinitely often red
(to be formulated below). Therefore µWsexpresses that sis almost always black, and in
the remainder of the paper we phrase µWas almost always black.
The function view corresponding to µWcould be given by the second-order encoding of
induction and coinduction, which is inevitably more verbose and therefore omitted. Instead,
ON STREAMS THAT ARE FINITELY RED 7
in the following subsections, we will take a closer look at WXand µW, giving alternative
characterizations of streams that are almost always black.
3.4. Streamless Red Positions. The fourth definition is inspired by [2]. It states that
the set of red positions in the sequence is streamless. A set Ais streamless if every stream
over Ahas a duplicate. As equality on Ais decidable for us, this is equivalent to saying
that a set Ais streamless if any duplicate-free colist over Ais finite.
For any set A, we define duplicate-free colists over Acoinductively by
hi :colist A
x:A ` :colist (A\ {x})
x ` :colist A
We define finiteness of colists inductively by
hi ↓
`↓
x ` ↓
For any sequence, namely function for stream s, let Rf(resp. Rs) denote the set of red
positions in f(resp. s). Formally, n∈ Rf(resp. n∈ Rs) if f n =R(resp. s@n=R).
Then, the forth definition of streams being finitely red is stated in the stream view as
∀`:colist Rs. ` ↓
or, trivially equivalently in the function view, as
∀`:colist Rf. ` ↓
3.5. Not Not Eventually All Black. In this paper, we are mainly interested in positive
variations. However, two negative variations appear natural to consider for us. One of them
is the double negation of the first definition of eventually all black.
Our fifth definition is stated in the function view as,
¬¬∃n. ∀m≥n.f m =B
or in the stream view as
¬¬F (Gblack)s
which is equivalent to
¬G (¬G black)s
The last formulation, ¬G (¬G black)s, turns out handy in proofs and we will use either of
them interchangeably.
8 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
3.6. Not Infinitely Often Red. The last definition of streams being finitely red is given
by streams not being infinitely often red. So we first look at definitions of streams being
infinitely often red, which admit less variety of definitions.
A well-known definition is given by streams that are “globally” “finally” red, or
G(Fred)s
This definition is dual to that of eventually-all-black streams, i.e., F(Gblack )s. The modal-
ities Gand Fare flipped, so are the colors red and black. The function view of this is stated
as
∀n. ∀m≥n. f m =R
The function and stream views are equivalent
∀s. G(Fred)s⇔ ∀n. ∃m≥n. s2f s m =R
Similarly, we obtain a definition of streams being infinitely often red, by dualizing the
definitions of WXand µW, yielding
UXs
UX(B s)
X s
UX(R s)
UνUs
νUs
The (strong) until operator UXis dual to the weak until operator WX: The statement UXs
says that the suffix of safter the first occurrence of red must satisfy Xand the occurrence
must exist. Then νUtakes the greatest fixed point of UX, whereas µWis the least fixed
point of WX.
Interestingly, νUis equivalent to G(Fred)
∀s. νUs⇔ G (Fred)s
As we will see in Section 4, µWand F(Gblack) are not equivalent constructively. (Col-
lapsing the two amounts to LPO.)
We conclude this section with the weakest definition in our spectrum of streams being
finitely red. Namely,
¬G (Fred)s
or in its equivalent function view
¬(∀n. ∃m≥n. f m =R)
3.7. Accessibility. In this section, we characterize streams that are almost always black
in terms of accessibility of (decidable) relations on natural numbers induced by streams.
We define accessibility of a binary relation on a set Uby
∀m. n m⇒accm
accn
For any stream s, we define a decidable relation son natural numbers by taking
nsmto mean that mis the position following the first red position from nonward
(including n). Formally,
n≤`∀k. n ≤k < ` ⇒s@k=B s@`=R ` + 1 = m
nsm
An equivalent inductive definition is:
0(R s)1
0sm
0(B s)m+ 1
nsm
n+ 1 (c s)m+ 1
ON STREAMS THAT ARE FINITELY RED 9
The intuition is that nsmshould hold if and only if, whenever WXs|nis true, then this
is justified by X s|m. (This means that sis deterministic, but not functional.) This is
what the next lemma proves.
Lemma 3.1. ∀s, n. WXs|n⇔(∀m. n sm⇒X s|m)
Proof. (⇒): We prove ∀s, n, m. n sm⇒ WXs|n⇒X s|mby induction on the proof of
nsm.
The case of s=R s0,n= 0 and m= 1: From the assumption WXs|0, i.e., WXs, we
directly learn that X s0, i.e., X s|1.
The case of s=B s0,n= 0 and m=m0+ 1 and 0 s0m0: The assumption WXs|0, i.e.,
WXs, assures us that WXs0, and by the induction hypothesis we have WXs0|0⇒X s0|m0.
Hence X s0|m0, i.e., X s|m.
The case of s=c s0,n=n0+ 1 and m=m0+ 1 and n0s0m0: The assumption WXs|n
amounts to WXs0|n0. By the induction hypothesis, WXs0|n0⇒X s0|m0, we get that X s0|m0,
i.e., X s|m.
(⇐): We prove ∀s, n. (∀m. n sm⇒X s|m)⇒ WXs|nby coinduction and by case
analysis on n. In case n= 0, we further perform case analysis on the head color of s.3
The case of n= 0 and s=R s0: we know that 0 s1. Hence the assumption
∀m. 0sm⇒X s|mgives us that X s|1, i.e., X s0, from where it follows that WXs, i.e.,
WXs|0.
The case of n= 0 and s=B s0: We know that, if 0 smfor any m, then m=m0+ 1
for some m0and 0 s0m0. Hence the assumption ∀m. 0sm⇒X s|mgives us that
∀m0.0s0m0⇒X s0|m0. By the coinduction hypothesis, it follows that WXs0|0, i.e.,
WXs0, from where we learn WXs, i.e., WXs|0.
The case of n=n0+1 and s=c s0: We observe that nsmif n0s0m0and m=m0+1.
Therefore the assumption ∀m. n sm⇒X s|mgives us that ∀m0. n0s0m0⇒X s0|m0. By
the coinduction hypothesis, we get that WXs0|n0which is the same as WXs|n.
It is noteworthy that this lemma, instantiated at n= 0, gives us a possible arithmetical
definition of the weak until operator WXthat avoids impredicativity (quantification over
predicates). Indeed, it suggests that we could have defined:
WXs⇔ ∀`. (∀k < `. s@k=B)∧s@`=R⇒X s|`+1
To compare, the impredicative definition is:
WXs⇔ ∃Y. (∀s0. Y (R s0)⇒X s0)∧(∀s0. Y (B s0)⇒Y s0)∧Y s
Further, we have that, for any stream s,sis almost always black, µWs, if and only if
0 is accessible with respect to s. The claim follows from the following lemma.
Lemma 3.2. ∀s, n. µWs|n⇔accsn.
Proof. (⇒): We prove ∀s, n. µWs|n⇒accsnby induction on the proof of µWs|n.4
From this proof, we have that, for some stream predicate X,∀s0. X s0⇒µWs0and WXs|n.
By the induction hypothesis, the former gives us ∀m. X s|m⇒accsmwhile, by the
previous lemma, the latter gives ∀m. n sm⇒X s|m. Putting the two together, we get
∀m. n sm⇒accsm, hence accsn.
3To be fully precise, we prove ∀s0.(∃s, n. s0=s|n∧ ∀m. n sm⇒X s|m)⇒ WXs0by coinduction.
4To be fully precise, we prove ∀s0. µWs0⇒(∀s, n. s0=s|n⇒accsn) by induction on the proof of
µWs0. In further proofs we will use these generalizations of coinduction and induction without comments.
10 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
(⇐): By induction on the proof of accsn. We have ∀m. n sm⇒accsmand by
the induction hypothesis, ∀m. n sm⇒µWs|m. The previous lemma therefore gives us
WµWs|n, hence µWs|n, as required.
Corollary 3.1. ∀s. µWs⇔accs0.
We can in fact rephrase the fourth variant (streams for which the sets of red positions
are streamless) and sixth variant (streams that are not infinitely often red) in terms of s,
as we will do now.
3.7.1. Strong Normalization. Streams whose red positions form streamless sets correspond
to streams sfor which sis strongly normalizing at 0.
For any set Uand any relation on U, we define (descending) chains in coinductively
by x0:U
hi :chainx0
x0x1`:chainx1
x1`:chainx0
so that x1x2...xnhi :chainx0means that x0x1x2. . . xn. Note that a chain in
may be infinite.
We define finiteness of chains inductively by
hi ↓
`↓
x ` ↓
We use the same notation for finiteness of colists and chains.
A binary relation on a set Uis strongly normalizing at x:U, SN sx, if any -chain
starting at xis finite, or ∀`:chainx. ` ↓.
For any stream s,sis strongly normalizing at 0 if and only if Rsis streamless.
Lemma 3.3. ∀s. SN s0⇔ Rsis streamless.
Proof. (⇒): We first notice that SN s0 if and only if SN +
s0, where +
sis the transitive
closure of s. Define a function f:nat →colist nat →colist nat by recursion by
f n hi =hi
f n (m `) = m ` if n<m
f n (m `) = f n ` if n≥m
The computation of f n ` is terminating as `is duplicate-free. (So, f n ` is welldefined.)
Moreover, define a function g:colist nat →colist nat by corecursion by
ghi =hi g(n `)=(n+ 1) (g(f n `))
We have that, for any duplicate-free colist `over Rs,`is finite if and only if g ` is finite,
and moreover g ` is a chain in +
sstarting at 0.
Now, for any given duplicate-free colist `:colist Rs, by our assumption, g ` is finite,
which implies `is finite, as required.
(⇐): Define a function f:colist (nat \ {0})→colist nat by corecursion by
fhi =hi f(n `) = (n−1) (f `)
so that f ` shifts the elements in `by subtracting one.
For any given `:chains0, f ` is a duplicate-free colist over Rs, therefore f ` is finite
by our assumption. By construction of f,`is finite, which completes the proof.
ON STREAMS THAT ARE FINITELY RED 11
3.7.2. Antifoundedness. Streams that are infinitely often red correspond to streams sfor
which sis antifounded.
We define antifoundedness of binary relation on a set Ucoinductively by
nmdivm
divn
so that divnmeans that there is an infinite descending chain in starting from n.
Firstly we rephrase the strong until operator, UX, which, unlike the weak until operator
WX, requires Xto hold at some point.
Lemma 3.4. ∀s, n. UXs|n⇔(∃m. n sm∧X s|m).
Proof. (⇒): By induction on the proof of UXs|n.
The case of s=R s0and n= 0: We have that 0 ≺s1 and X s0and can choose m= 1.
The case of s=B s0and n= 0: We have that UXs0, the induction hypothesis gives
us that there is an m0such that 0 sm0∧X s0|m0. But then we also have that ns
m0+ 1 ∧X s|m0+1, so the desired result is witnessed by m=m0+ 1.
The case of s=c s0and n=n0+ 1: The assumption UXs|namounts to UXs0|n0.
By the induction hypothesis, there is an m0such that n0sm0∧X s0|m0. But then also
nsm0+ 1 ∧X s|m0+1, so we can choose m=m0+ 1.
(⇐): We prove ∀s, n, m. n sm∧X s|m⇒ UXs|nby induction on the proof of nsm.
The case of s=R s0,n= 0 and m= 1: The assumption X s|1, i.e., X s0, implies UXs,
i.e., UXs|0.
The case of s=B s0,n= 0, m=m0+ 1 and 0 s0m0: The assumption X s|mamounts
to X s0|m0. By the induction hypothesis, we have that UXs0|0, from where UXs|0follows in
turn.
The case of s=c s0,n=n0+ 1, m=m0+ 1 and n0s0m0: The assumption X s|m
amounts to X s0|m0. By the induction hypothesis, it holds that UXs0|n0, which is the same
as UXs|n.
Then we have that, for any stream s,sis infinitely often red, νUs, if and only if 0 is
antifounded with respect to s. The claim follows from the following lemma.
Lemma 3.5. ∀s, n. ν Us|n⇔divsn.
Proof. (⇒): By coinduction. From the assumption νUs|n, we have that, for some stream
predicate X,∀s0. X s0⇒νUs0and UXs|n. The former and the coinduction hypothesis
together give us that, ∀m0. X s|m0⇒div sm0. From the latter and the previous lemma,
it follows that there exists an msuch that nsmand X s|m. Hence divsmand we can
also conclude that divsn.
(⇐): By coinduction. From the assumption divsn, we have that there exists some m
such that nsmand divsm. By the coinduction hypothesis, we have νUsm. By the
previous lemma it follows now that UνUsnwhereby we also learn that νUsn.
Corollary 3.2. ∀s. ν Us⇔divs0.
12 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
3.8. Classical fixed point. It turns out that the weak until operator WXreaches the
fixed point by ω-iteration only classically. In fact, we have a stronger result: closure at ω
is equivalent to LPO.
Define: Fns
Fωs
where F0=False and Fn+1 =WFn, so that Fωis WXiterated ωtimes.
Lemma 3.6. (∀s. WFωs⇒Fωs)⇔(∀s. Fred s∨ G black s).
Proof. (⇒): Define f:nat →str →str and g:nat →str by corecursion
f n (B s) = B(f(n+ 1) s)f n (R s) = g n
g(n+ 1) = R(g n)g0 = B∞
where B∞denotes a stream of black, defined by corecursion by B∞=B B∞. The compu-
tation of f0slooks for the first occurrence of red in s, while keeping track of the number
of black it has seen so far in the second argument. On encountering the first red (if exists),
it invokes g, passing nas argument. The stream that g n produces is red up to the n-th
position, followed by an all black stream. The trick is to record the position of the first
occurrence of red in sin terms of the number of red in f0s. If sdoes not contain red, then
f0sdoes not either. This way, if we know the bound on the number of red in f0s, then
we know the bound on the depth of the first occurrence of red in s. We prove ∀n. F n+1 (g n)
by induction on n, then ∀n, s. WFω(f n s) by coinduction. We deduce ∀s. F ω(f0s) by
our assumption, therefore ∀s. ∃n. F n(f0s) by definition. For any s, given Fn(f0s) for
some n, however, it suffices to examine the initial (n+ 1)-segment of sto know whether s
contains red or not, enabling us to decide whether Fred sor Gblack sholds.
(⇐): For any given s, suppose WFωs. By our assumption, we have either Gblack s
or Fred s. In the case of Gblack s, we immediately have F1s, therefore Fωs. In the case
of Fred s, let nbe the position of the first occurrence of red in s, which is guaranteed to
exist by Fred s. From WFωs, we deduce Fωs|n+1, i.e., Fms|n+1 for some m, which yields
Fm+1 s, therefore Fωsas required.
In fact, Fnis equivalent to len. Namely we have that, ∀n, s. F ns⇔le ns. It is an
open question whether there is a constructive closure ordinal.
4. Analysis of the spectrum
In this section, we analyze our spectrum of streams being finitely red. We have presented
six variants:
(a) Eventually all black
(b) Boundedly red
(c) Almost always black
(d) Streamless red positions
(e) Double negation of eventually all black
(f) Negation of infinitely often red
We have a clear view on relative strength between positive variations. For negative ones,
open questions remain. The overall picture is given in Section 6.
ON STREAMS THAT ARE FINITELY RED 13
We start from downward implications. The six variations above are listed in decreasing
order of constructive strength, except that we do not know whether (d) implies (e): we only
know that (c) implies (d) and (e), both of which imply (f) (lemmata 4.3, 4.4, 4.5 and 4.6)
and that (e) ⇒(d) amounts to Markov’s Principle (Lemma 4.9).
If a stream is eventually all black, then it is boundedly red.
Lemma 4.1. ∀s. F(Gblack )s⇒ ∃n. le ns.
Proof. By induction on the proof of F(Gblack )s.
If a stream is boundedly red, then it is almost always black.
Lemma 4.2. ∀n, s. le ns⇒µWs.
Proof. By induction on n. The case of n= 0 is immediate. The case of n=n0+ 1: We
prove that, ∀s. len0+1 s⇒ WµWsby coinduction and case analysis on the head color of s.
The case of s@0 = Bfollows from the coinduction hypothesis. The case of s@0 = Rfollows
from the main induction hypothesis.
If a stream is almost always black, then the set of its red positions is streamless.
Lemma 4.3. ∀s. µWs⇒ Rsis streamless.
Proof. The claim follows from Corollary 3.1 and Lemma 3.3, since accessibility implies
strong normalization.
If a stream sis almost always black, then it is not the case that sis not eventually all
black.
Lemma 4.4. ∀s. µWs⇒ ¬G (¬G black)s.
Proof. We prove a slightly stronger statement, ∀s. (∀n. ¬G black s|n)⇒ ∀n. µWs|n⇒False,
from which the claim follows. For a given s, we assume ∀n. ¬G black s|n. We prove
∀n. µWs|n⇒False by induction on the proof of µWs|n. We are given as induction hypoth-
esis that, ∀n. X s|n⇒False. We have to prove False, given WXs|n. From our assumption,
however, it suffices to prove Gblack s|n. We do so by proving ∀n. WXs|n⇒ G black s|nby
coinduction using the main induction hypothesis.
If the set of red positions of a stream sis streamless, then sis not infinitely often red.
Lemma 4.5. ∀s. Rsis streamless ⇒ ¬νUs.
Proof. The claim follows from Lemma 3.3 and Corollary 3.2, since strong normalization
contradicts antifoundedness.
If it is not the case that a stream sis not eventually all black, then sis not infinitely
often red.
Lemma 4.6. ∀s. ¬G (¬G black )s⇒ ¬νUs.
Proof. Noticing ∀s. νUs⇔ G (Fred)s, the claim follows by contraposition from a tautology
∀s. G(Fred)s⇒ G (¬G black )s.
14 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
We now proceed to study strength of upward implications, which are technically more
interesting than downward implications. We know that differences between the first three
positive variants amount to LPO (Lemma 4.7 and 4.8). Moreover, (e) =⇒(d) amounts to
Markov’s Principle (Lemma 4.9) and (f) =⇒(e) to an instance of Double Negation Shift
for a Σ0
1-formula (Lemma 4.10). As immediate corollaries from Section 3.7, we have that
(d) =⇒(c) is equivalent to that SN of sat 0 implies accessibility of 0 with respect to s
(Corollary 4.1) and that (f) =⇒(d) is equivalent to that non-antifoundedness of 0 with
respect to simplies SN of sat 0 (Corollary 4.3).
Lemma 4.7. (∀n, s. le ns⇒ F (Gblack )s)⇔(∀s. Fred s∨ G black s).
Proof. (⇒): Define f:str →str by corecursion
f(B s) = B(f s)f(R s) = R B∞
so that f s contains (exactly) one red if and only if scontains at least one red. We have
that, ∀s. le2(f s), proved by coinduction and case analysis on the head color of f s. By
our assumption, we have that, ∀s. F(Gblack ) (f s). The proof of F(Gblack ) (f s) tells us
whether f s contains red or not, deciding whether sis eventually red, Fred sor all black,
Gblack s, as required.
(⇐): We prove that, ∀n, s. le ns⇒ F (Gblack )sby induction on n, assuming ∀s. Fred s∨
Gblack s. The case of n= 0 is immediate. The case of n=n0+ 1: Suppose len0+1 s. By
our assumption, we have either Fred sor Gblack s. The latter case immediately yields
F(Gblack)s. For the former case, we prove ∀s. Fred s→len0+1 s→ F (Gblack )sby
induction on Fred sand case analysis on the head color of s, using the main induction
hypothesis.
Lemma 4.8. (∀s. µWs⇒ ∃n. lens)⇔(∀s. Fred s∨ G black s).
Proof. (⇒): We prove ∀s. µWs⇒ ∃n. lensby induction on the proof of µWs, assuming
∀s. Fred s∨G black s. We first prove an auxiliary lemma: ∀s0.Fred s0⇒µWs0⇒ ∃n. lens0
by induction on the proof of Fred s0and case analysis on the head color of s0. The case of
s0@0 = Bfollows from the induction hypothesis. The case of s0@0 = Rfollows from the
main induction hypothesis.
Now suppose µWs. We have to prove that there exists some nsuch that lens. By
our assumption, we have either Gblack sor Fred s. The former case follows immediate by
coinduction by taking n= 1. The latter case is closed by the auxiliary lemma just proved.
(⇐): We prove that, ∀s. WFωs⇒Fωs, assuming ∀s. µWs⇒ ∃n. le ns, where Fω
was defined in Section 3.8. Then the case follows from Lemma 3.6. Suppose WFωs. By
Lemma 4.2 and the monotonicity of WXon X, we have WµWs, which yields µWsby
definition. From our assumption and the equivalence between Fnand len, we conclude
Fωs, as required.
The following claim is corollary from Corollary 3.1 and Lemma 3.3.
Corollary 4.1. (∀s. Rsis streamless ⇒µWs)⇔(∀s. SN s0⇒accs0)
Lemma 4.9. (∀s. ¬G (¬G black )s⇒ Rsis streamless) ⇔(∀s. ¬G black s⇒ F red s)
Proof. (⇒): Define a function f:str →str by corecursion by
f(R s) = B∞f(B s) = R(f s)
ON STREAMS THAT ARE FINITELY RED 15
so that f s is red until the first occurrence of red in sis encountered, from where f s becomes
all black.
For any given s, we assume ¬G black s. We have to prove Fred s. Firstly, we prove
¬G (¬G black) (f s). It suffices to prove ∀s. G(¬G black) (f s)⇒ G black s. We do so by
coinduction and case analysis on the head color of s. The case of s@0 = R: This is impossible
as we then have Gblack (f s), contradicting the assumption G(¬G black) (f s). The case of
s@0 = B: From the assumption G(¬G black) (f s), it follows that, G(¬G black) (f s)|1. By
the coinduction hypothesis, we obtain Gblack s|1, hence Gblack s.
Applying our assumption, ∀s. ¬G (¬G black )s⇒ Rsis streamless, to ¬G (¬G black ) (f s)
yields that (f s)is strongly normalizing at 0 by Lemma 3.3. Below we prove Fred s, as-
suming SN (f s)0, which completes the proof.
Define a function g:nat →colist nat by recursion by
g n = (n+ 1) (g(n+ 1)) if (f s)@n=R
g n =hi if (f s)@n=B
As g0 is a chain in (f s)starting at 0, i.e., g0 : chain(f s)0, by our assumption g0 is finite.
By construction of g, we have (f s)@n=B, where nis the length of g0. (As g0 is finite, its
length is welldefined.) By construction of f, we now have s@n=R, which yields Fred s,
as required.
(⇐): For any given s, we assume ¬G (¬G black)s. We have to prove, for any given
`:chains0, `is finite.
Define a function f:colist nat →str by corecursion by
fhi =R∞f(n `0) = B(f `0)
By definition of f, we have that, ∀`0:chains0.Gblack (f `0)⇒ G (¬G black )s, proved
by coinduction. Hence from the assumption ¬G (¬G black)s, we are entitled to conclude
¬(Gblack) (f `). By Markov’s Principle it follows that, Fred (f `). However this means
that `↓, which completes the proof.
Lemma 4.10. (∀s. ¬G (Fred)s⇒ ¬G (¬G black)s)⇔(∀s. G(¬¬F red)s⇒ ¬¬G (Fred)s).
Proof. For any given s, we have
¬¬G (¬¬F red)s⇔ ¬¬¬F (¬F red)s⇔ ¬F (¬F red )s⇔ G (¬¬F red)s
Now the claim follows by taking contrapositions of the respective assumptions, noticing
∀s. ¬G black s⇔ ¬¬F red sand the above equivalence.
The corollary below follows from lemmata 4.5, 4.9 and 4.10.
Corollary 4.2. (∀s. ¬νUs⇒ Rsis streamless) ⇔(∀s. ¬G black s⇒ F red s)
The following claim is corollary from Corollary 3.2 and Lemma 3.3.
Corollary 4.3. (∀s. ¬νUs⇒ Rsis streamless) ⇔(∀s. ¬div 00⇒ ∀`:chains0. ` ↓).
16 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
5. Related work: finiteness of sets of red positions
In [2], Coquand and Spiwack introduce four notions of finiteness of sets in Bishop’s set
theory [3]. For understanding some of their arguments, for example, on page 222, the 9-th
line from below, we had to assume that equality is decidable. Under this assumption their
results may be rendered as follows:
(i) Set Ais enumerated if it is given by a list.
(ii) Set Ais of bounded size if there exists a bound such that any list over Acontains
duplicates whenever its length exceeds the bound.
(iii) Set Ais noetherian if the root of the tree of duplicate-free lists over Ais accessible
(cf. Section 3.7).
(iv) Set Ais streamless if every stream over Ahas a duplicate.
These four notions are classically equivalent but of decreasing constructive strength. The
authors conjecture that there exists a set which is streamless but not noetherian (unless
classical logic is assumed). Their hierarchy of finiteness matches pleasantly with our hier-
archy of positive variations of streams being finitely red, if we look at sets of red positions
in our streams. An important difference is that Coquand and Spiwack consider sets that
may not be decidable, whereas we work with decidable sets of natural numbers. Moreover
natural numbers are traversible. As a result, our hierarchy becomes tighter than theirs,
allowing us to capture differences in strength of our hierarchy in terms of weak instances of
the Law of Excluded Middle.
In this section, we rephrase our hierarchy in terms of Coquand and Spiwack’s. Their
streamless sets directly correspond to our streams sfor which the set of red positions, Rs,
is streamless. We will therefore only consider (i) - (iii). Furthermore, we will corrobate
their conjecture by showing that there is a unary predicate which is streamless in recursive
analysis, but not noetherian. Our results in this section assume decidable equality.
Enumerated sets. A set Ais enumerated, enum A, if all its elements can be listed, or
∀x:A. false
enum A
x:Aenum A\ {x}
enum A
Note that a proof of enum Ais essentially an exhaustive duplicate-free list of elements of
A.
It is easy to see that a stream sis eventually all black if and only if the set of red
positions in sis enumerated.
Lemma 5.1. ∀s. F(Gblack )s⇔enum Rs.
Proof. (⇒): Given F(Gblack)s, we can construct a list of the red positions in s, from
which enum Rsfollows.
(⇐): Given enum Rs, we know the position of the last occurrence of red in s, which
yields F(Gblack)s.
Size-bounded sets. A set Ais of bounded size if there exists a natural number nsuch that
any duplicate-free list over Ais of length less than n. Specifically, we say Ais size-bounded
by nif any duplicate-free list over Ais of length of less than n. Formally,
∀x:A. boundednA\ {x}
boundedn+1 A
ON STREAMS THAT ARE FINITELY RED 17
Lemma 5.2. ∀n, s. le ns⇔bounded nRs.
Proof. For any decidable set Aof natural numbers, we define a stream sAby
sA@k=Rwhen k∈A
sA@k=Botherwise
so that sAis red exactly at the positions in A.
(⇒): By induction on n. The case of n= 0 is immediate. The case of n=n0+ 1:
We are given as induction hypothesis that, ∀s. len0s⇒boundedn0Rs. We have to prove
boundedn0+1 Rs, given len0+1 s. Suppose x∈ Rs. It suffices to prove boundedn0Rs\ {x}.
From len0+1 s, we deduce len0sRs\{x}. By induction hypothesis, we obtain boundedn0Rs\
{x}, as required.
(⇐): We prove ∀n, A. boundednA⇒le nsAby induction on n, from which the case
follows. The case of n= 0 is immediate. The case of n=n0+ 1: We are given as induction
hypothesis that, ∀A. boundedn0A⇒len0sA. We have to prove ∀A. boundedn0+1 A⇒
len0+1 sA. We do so by coinduction and case analysis on the head color of sA. The case
of sA@0 = B: We have boundedn0+1 RsA|1. We close the case by coinduction hypothesis.
The case of sA@0 = R: We have boundedn0RsA|1. We close the case by the main induction
hypothesis.
Noetherian sets. A set Ais Noetherian, Noet A, if, for all x∈A,A\{x}is Noetherian.
Formally,
∀n∈A. Noet A\{n}
Noet A
Then, a stream sis almost always black, µWs, if and only if the set of red positions in
sis Noetherian. To prove this, it is convenient to reformulate Noetherianness by removing
the elements up to n(including n):
∀n∈A. Noet0A\{0, . . . , n}
Noet0A
The two definitions are equivalent.
Lemma 5.3. ∀A. Noet A⇔Noet0A.
Proof. (⇒): We prove that, ∀A. Noet A⇒ ∀n∈A. Noet0A\{0, . . . , n}by induction
on the proof of Noet A. We are given as induction hypothesis that, ∀n∈A. ∀m∈
A\{n}.Noet0A\{n}\{0, . . . , m}. We have to prove that, ∀n∈A. Noet0A\{0, . . . , n}, which
follows from the induction hypothesis and by case analysis on whether there is m<nsuch
that m∈A.
(⇐): We prove by induction on the proof of Noet0A. We are given as induction
hypothesis that, ∀n∈A. Noet A\{0, . . . , n}. We have to prove ∀n∈A. Noet A\{n}, which
follows from an auxiliary lemma, ∀n, A. Noet A⇒Noet (A∪ {n}), proved by induction.
18 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
Given a set Aof natural numbers, we define a relation Aon natural numbers such
that nAmif m=`+ 1 with `being the least natural number such that n≤`and `∈A.
Formally, n≤`∀k. n ≤k < ` ⇒k6∈ A ` ∈A ` + 1 = m
nAm
Note that, for any stream s,sis equivalent to Rsby definition. So our task is to prove
equivalence of Abeing Noetherian and accessibility of 0 with respect to A.
For a relation over a set A,∗denotes the reflexive and transitive closure of and
+denotes the transitive closure.
Lemma 5.4. ∀ .(∀n. accn⇔ ∀n. acc+n).
Proof. (⇒): We prove a slightly stronger statement, ∀n. accn⇒ ∀m. n ∗m⇒acc+m
by induction on the proof of accn, from which the claim follows.
(⇐): By induction on the proof of acc+n.
Lemma 5.5. ∀A. Noet0A⇔accA0.
Proof. (⇒): By induction on the proof of Noet0A. We are given as induction hypothesis
that, ∀n∈A. accA\{0,...,n}0. We have to prove ∀n. 0An⇒accAn, which follows from
the induction hypothesis and by observing that, ∀A, n. accA\{0,...,n−1}0⇒accAn.
(⇐): We prove that, ∀A, n. acc+
An⇒Noet0A\{0, . . . , n−1}by induction on the proof
of acc+
An. Then the case follows from lemma 5.4. We are given as induction hypothesis
that, ∀m. n +
Am⇒Noet0A\{0, . . . , m −1}. We have to prove Noet0A\{0, . . . , n −1},
which follows from the induction hypothesis and by case analysis on whether n∈Aor
not.
Combining lemmata 3.2, 5.3 and 5.5, we obtain:
Corollary 5.1. ∀s. µWs⇔Noet Rs.
5.1. Analysis of the conjecture of Coquand and Spiwack. In this subsection we show
that there is a unary predicate that, interpreted in recursive analysis, is streamless but not
noetherian. Since noetherian implies streamless, we conclude that noetherian is strictly
stronger than streamless, at least in recursive analysis. Notations in this subsection comply
with traditional recursive analysis rather than with type theory.
We need some prerequisites from recursion theory. By {i}nwe denote the partial
recursive function with Kleene index iapplied to n. We define H(i) to be the halting
predicate, that is, H(i) is equivalent to ∃k T (i, i, k), with Kleene’s T-predicate. Another
familiar notation for H(i) we use is {i}i↓(convergence), whereas {i}i↑expresses the
opposite (divergence). We denote binary lists as in [2] by [b0;...;bn−1], and assume them to
be encoded as natural numbers in some obvious, recursive way.
We call a binary list [b0;...;bn−1] a partial solution to the halting problem if H(i)⇐⇒
bi= 1 for all i<n. Let AH(l) abbreviate that lis a partial solution to the halting problem.
Note that the predicate AHis not decidable. Distinct elements of AHhave different lengths.
We view a stream over AHsimply as a (total) function on the natural numbers whose values
satisfy AH.
We first show that any recursive stream over AHhas duplicates. Classically this would
follow from the unsolvability of the halting problem. Constructively, we first define a partial
ON STREAMS THAT ARE FINITELY RED 19
recursive function ψas follows. Computing ψ(x, y) starts by computing {x}0,...,{x}(y+1).
If there is a duplicate in this sequence, we put ψ(x, y) = y+1. Otherwise, let [b0;...;bn−1] be
a binary list of maximal length among the binary lists encoded by {x}0,...,{x}(y+ 1). If
n≤ywe let the computation of ψ(x, y) diverge. If y < n, we look at byand diagonalize over
H: we let the computation of ψ(x, y) converge if by= 0, and diverge otherwise. Clearly, ψ
is a partial recursive function. By the S-m-n Theorem there exists a total recursive function
fsuch that {f(x)}y=ψ(x, y). Now, if sis a stream in AHwith Kleene index i, then s
has duplicates among its first f(i) + 2 values. Since this is a decidable property, we may
prove it by contradiction. Assume shas no duplicates among its first f(i) + 2 values. Then
{i}0,...,{i}(f(i) + 1) are distinct numbers satisfying AH. This means that the longest
binary list l= [b0;...;bn−1] encoded by these values has length n>f(i). Both bf(i)= 0
and bf(i)= 1 conflict with AH(l) because of the diagonalization in the definition of ψ.
Contradiction, and hence the stream shas duplicates.
Coquand and Spiwack define noetherian by first defining inductively NAas the smallest
set containing all lists lover Ahaving duplicates or satisfying a:: l∈Nfor all a∈A. Then
Ais noetherian if the empty list is in NA. To see that our AHis not noetherian, define the
following recursive set S:
S={[l0;...;ln−1]|li=ljfor some 0 ≤i < j < n}
We stress that Scontains all lists of natural numbers containing duplicates, including lists of
elements of AHencoded as natural numbers. Let ll be a list over AHand assume l:: ll ∈S
for all l∈AH. It is decidable whether ll has duplicates or not. If so, we have ll ∈S. If
not, let [b0;...;bn−1] be the longest list in ll. Define li= [b0;...;bn−1;i] for i= 0,1. Then
li:: ll not in Sand hence li/∈AHfor i= 0,1. By the definition of AHthis would mean
¬(H(n)⇐⇒ i= 1) for i= 0,1, which is absurd. Hence we have ll ∈Sand Ssatisfies
both closure conditions defining NAH. Since Sdoes clearly not contain the empty list, it
follows that AHis not noetherian.
The above construction yields an unprovability result if one formalizes the notions of
[2] in a theory like HAω, see [6, Ch. 9]. (A very subtle point here is the inductive definition
of NA. Since the second-order quantification in HAωis weak, NAbecomes slightly larger
than in type theory, but this makes our result that AHis not noetherian just stronger.) It
is possible to completely eliminate recursion theory from our argument by abstracting from
Hand proving ¬¬∀n(H(n)∨ ¬H(n)) from the assumption that AHis streamless implies
that AHis noetherian. This will be elaborated by Coquand in a forthcoming paper and
will settle the conjecture.
6. Conclusion
The following diagram summarizes our current understanding of the constructive in-
terrelations between the various notions of finiteness. Implications that are annotated have
not been proved constructively; the annotations explain which principle is sufficient and, in
20 MARC BEZEM, KEIKO NAKATA, AND TARMO UUSTALU
some cases, necessary to prove the implication.
F(Gblack)s
∃n. lens
⇔LPO
V^
µWs
⇔LPO
W_
"*
SN s0
"*
⇐BI
V^
¬ G (¬ G black)s
⇔MP
W_
¬G (Fred)s
⇐Σ0
1−DNS
V^
We do not know whether the implication SN s0⇒ ¬G (¬G black )sholds. The implica-
tion SN s0⇒µWscan be proved by a very weak instance of Bar Induction (BI, see [6,
Ch. 4]), as observed in [2]. Since Rsis decidable, non-constructivity of SN s0⇒µWs
is even more difficult to prove than the conjecture by Coquand and Spiwack.
Acknowledgements. We would like to thank Thierry Coquand, Arnaud Spiwack and Nils
Anders Danielsson for a fruitful discussion in a late stage of the preparation of this paper.
K. Nakata and T. Uustalu’s research was supported by the European Regional De-
velopment Fund (ERDF) through the Estonian Centre of Excellence in Computer Science
(EXCS). M. Bezem’s visit to Estonia in Feb. 2011 was supported by the same project.
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