=Paper=
{{Paper
|id=Vol-1195/long19
|storemode=property
|title=The Representation of Boolean Algebras in the Spotlight of a Proof Checker
|pdfUrl=https://ceur-ws.org/Vol-1195/long19.pdf
|volume=Vol-1195
|dblpUrl=https://dblp.org/rec/conf/cilc/CeterchiOT14
}}
==The Representation of Boolean Algebras in the Spotlight of a Proof Checker==
https://ceur-ws.org/Vol-1195/long19.pdf
The representation of Boolean algebras in the
spotlight of a proof checker?
Rodica Ceterchi1 , Eugenio G. Omodeo2 , Alexandru I. Tomescu3
1
Facultatea de Matematică s, i Informatică, Universitatea din Bucures, ti
email: rceterchi@gmail.com
2
Dipartimento di Matematica e Geoscienze, Università di Trieste,
Via Valerio 12/1, I-34127 – Trieste, Italy
email: eomodeo@units.it
3
Helsinki Institute for Information Technology HIIT,
Department of Computer Science, University of Helsinki,
P.O. 68 (Gustaf Hällströmin katu 2b), FI-00014 – Helsinki, Finland
email: tomescu@cs.helsinki.fi
Abstract. We report on a proof-checked version of Stone’s result on
the representability of Boolean algebras via the clopen sets of a totally
disconnected compact Hausdor↵ space. Our experiment is based on a
proof verifier based on set theory, whose usability can in its turn benefit
from fully formalized proofs of representation theorems akin to the one
discussed in this note.
Key words: Theory-based automated reasoning; proof checking; Referee
aka ÆtnaNova; Boolean rings; Boolean algebras; Stone spaces.
“Boolean algebras have an almost embarrassingly rich structure.” [7, p. 10]
Introduction
This paper reports on a formally verified proof of Stone’s celebrated results
[20,21,22] on the representability of Boolean algebras as fields of sets, carried
out along the lines found in [4, pp. 41–43] by means of Jacob T. Schwartz’s
proof-checker Referee, aka ÆtnaNova, to be simply called ‘Ref’ for brevity in the
ongoing. The proof of Zorn’s lemma, formalized much earlier (cf. [19, Chapter 7])
and previously exploited in the compactness proof for classical propositional logic
[16], played again a crucial role in our present scenario.
A website reporting on our experiment is at http://www2.units.it/eomodeo/
StoneReprScenario.html. In its final form, the script-file supporting it and
leading from first principles to the topological version of Stone’s result, com-
prises 42 definitions and proves 210 theorems altogether, organized in 19 The-
ory s, including the background Theory Set theory. Its processing takes about
25 seconds.
?
Work partially supported by the INdAM/GNCS 2013 project “Specifica e verifica di
algoritmi tramite strumenti basati sulla teoria degli insiemi” and by the Academy
of Finland under grant 250345 (CoECGR)
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
1 Boolean algebras and rings
A ring is said to be Boolean when every one of its elements is self-inverse w.r.t.
addition and idempotent w.r.t. multiplication:
X +X = 0,
X · X =X.
The former of these laws makes it superfluous to postulate the commutativity
of addition (for, it implies it); the latter implies the commutativity of multipli-
cation. When endowed with multiplicative identity, 1, a Boolean ring is called a
Boolean algebra. (To avoid trivialities, we will require that 06=1).
An aspect of the richness of Boolean rings is that the relation
X 6 Y $Def X · Y = X
is a partial order, in which every pair X, Y of elements has greatest common
lower bound X u Y = X · Y and least common upper bound X t Y = X · Y +
X +Y . In this ordering 0 acts as the minimum and—when present—1 acts as the
maximum. Historically, Boolean algebras were first studied as lattices endowed
with peculiar properties (namely, distributivity and complementedness).4 The
salient operations, from this viewpoint, were u, t, ; the algebraic kinship with
the rings of numbers remained unnoticed for quite a while (cf.[9, p. 208]).
From the ring-based point of view—the one which will prevail in these pages—
the complementation operation turns out to be: X =Def 1 + X.
X 2 B ! X 2 B & X = X & 1B = 0 B & 0B = 1 B
{X, Y } ✓ B ! X + X = 1B & Y · X + Y · X = Y & (Y · X) · (Y · X) = 0B
{X, Y } ✓ B ! X + Y = X · Y + X · Y
X 2 B ! X 6= X & (X 2 / {0B , 1B } ! X 2 B \ {0B , 1B })
{X, Y } ✓ B ! (X · Y = 1B ! X = 1B & Y = 1B )
{U, V, X, Y } ✓ B ! U · X + V · Y = (U · X + V · Y ) · X · Y
Fig. 1. A few derived Boolean laws
4
From the lattice-based point of view, the simplest available characterization of the
structure ‘Boolean algebra’ is the one proposed by Herbert Robbins, ca. 1933 (cfr.
[12,10]). Ignoring the construct u, which can be introduced by way of shortening
notation, the Robbins laws are:
X t Y = Y t X, X t (Y t Z) = (X t Y ) t Z, X t Y t X t Y = X.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
2 Fields of sets
Starting with a non-void set S, let us construct the following families of sets:5
– P(S) =Def { x : x ✓ S } , the family of all subsets of S;
– F(S) =Def { x ✓ S | |x| 2 N } , the family of all finite subsets of S;
– B(S) =Def F(S) [ { S \ x : x 2 F(S) } , the family formed by those subsets
of S each of which is either finite or has a finite complement (relative to S).
If S is finite, P(S) = F(S) = B(S) holds; otherwise, we get three families.
To get Boolean rings out of these, it suffices to define:
X · Y =Def X \ Y (intersection),
X + Y =Def (X [ Y ) \ (X \ Y ) (symmetric di↵erence).
One readily sees that P(S) and B(S) thus become Boolean algebras, whose
additive identity, 0, and multiplicative identity, 1, are ; and S; as for F(S), it
lacks 1 when S is infinite.
By generalizing the case of P(S) and B(S), one calls field of sets any
family B which
– is closed
S under the operations of intersection and symmetric di↵erence;
– has B among its members, viz., owns a maximum w.r.t. set inclusion;
– di↵ers from {;}.
This clearly is an instance of a Boolean algebra. How general? A renowned
theorem by Marshall H. Stone [20] gives us the answer:
Every Boolean algebra is isomorphic to a field of sets.
This field is not always of the form P(S) (this holds only for particular Boolean
algebras, among which the ones whose underlying domain is finite6 ): in fact it
is trivial that the field B(N), whose cardinality equals the one of N, cannot be
isomorphic to P(S) for any S.7
3 Stone spaces
We define a base of a topological space to be a pair (X , ) such that:
(i) X =
6 ;; (ii) S ✓ P(X ); (iii) enjoys, w.r.t. dyadic intersection, \, and to
monadic union, , the following closure properties:
5
We designate by |X| the cardinality of a set X and by N the set {0, 1, 2, . . . } of all
natural numbers (which is, in its turn, a cardinal number—the first infinite one).
6
More generally, the Boolean algebras which are isomorphic to P(S) for some S are
the ones which are completely distributive; [2, pp. 221–222] credits this result to
Alfred Tarski.
7
Indeed, |P(S)| 2 N (i.e., |P(S)| is smaller than the cardinal number N) when |S| 2 N;
whereas |P(S)| exceeds N—because |P(S)| exceeds |S|—when S is infinite.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
S
1) = X, S
2) (A 2 & B 2 ) ! (A \ B) ✓ { c 2 | c ✓ (A \ B) }.
The topological space generated by such a base is, by definition, the pair
(X , ⌧ ) where S
⌧ = { a : a ✓ }.
A topological space hence is a pair (X , ⌧ ) such that: (i) X 6= ;; (ii)S⌧ ✓ P(X ); (iii)
⌧ enjoys, w.r.t. dyadic intersection, \, and to monadic union, , the following
closure properties:
– X 2 ⌧, S
– A, B 2 ⌧ ! (A
S \ B) ✓ { c 2 ⌧ | c ✓ (A \ B) },
– A 2 P(⌧ ) ! A 2 ⌧ ,
T follows that A, B 2 ⌧ ! A \ B 2 ⌧ and, more
whence it plainly T generally, that
A 2 F(⌧ ) ! A 2 ⌧ , under the proviso that the intersection ; equals X .
The open and the closed sets of such a space are, respectively, the members
of ⌧ and their complements X \ A (with A 2 ⌧ ). A subset of X which is open
and closed is called a clopen set: examples are ; and X . It is apparent that
clopen sets form a field of sets.
A topological space is said to be
a Hausdorff space if: for every pair of distinct members p, q 2 X , there exist
disjoint open sets P, Q ✓ X such that p 2 P , q 2 Q;8
compact if: every family of open sets whose union is X includes a finite sub-
family whose union is X .
One calls Stone space any topological space which, in addition to enjoying the
two properties just stated, also
– owns a base—namely a ✓ ⌧ such that every open set is the union of a
subfamily of —entirely formed by clopen sets.
By relying on these concepts, one enhances the claim of Stone’s theorem as
follows (cf. [21,22]):
Every Boolean algebra is isomorphic to a field of sets consisting of the
clopen sets of a Stone space.
In sight of the proof of this theorem, it is worthwhile to recall another way
of stating the compactness property, dual to the one proposed above:
A topological space (X , ⌧ ) is compact if and
T only if every (non-void)
family C of closed sets whose intersection C is void has T some finite
subfamily whose intersection is void : F ✓ C, |F| 2 N \ {0}, F = ;.
8
This definition implies that in a Hausdor↵ space every singleton subset of X is closed.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
4 Boolean ideals and their maximal enlargements
The study of an abstract algebraic structure forcibly leads one to investigate
the associated homomorphisms. In the Boolean case at hand, to move reso-
lutely in the direction that best suits our purposes, we will just consider those
homomorphisms which translate the operations of a Boolean algebra (the homo-
morphism’s domain) into the set-operations \, 4 of intersection and symmetric
di↵erence. The images of such a homomorphism will hence form a field of sets;
particularly worth of consideration, among the homomorphisms of interest, are
the ones whose values form the special field 2 = ;, {;} , \, 4, {;}, ; .
Let us refer by B = (B, ·, +, 1B , 0B ) to a Boolean algebra which, tacitly, will
act as domain of the homomorphisms that will enter into play. A homomorphism
of B into 2 is fully characterized by either one of the two subsets of the underlying
domain B which are counter-images, respectively, of 0 = ; and of 1 = {;}. Not all
subsets of B can play the role of counter-images of the minimum via a Boolean
homomorphism; let us hence figure out which conditions a set must meet in
order to qualify for such a role. In investigations of this nature, algebraists tend
to focus on the counter-images of the minimum, the so-called ideals, or ‘kernels’;
logicians, on the opposite, tend to focus on the counter-images of the maximum,
the so-called filters, or ‘shells’. We will conform to the algebraic habit; moreover,
since we must concentrate mainly on homomorphisms into 2, we will tribute
special attention to ideals which are maximal w.r.t. to set inclusion.
Definition 1. An ideal is a subset of the underlying domain B which is closed
with respect to addition, as well as to multiplication of its elements by elements
of B, and which is not one of the (exceedingly trivial) sets ;, {0B }, B.
Three theorems about ideals play a crucial role in the proof of Stone’s results:
a) Every element x of B which is neither 1B nor 0B belongs to at least one
ideal: the least such ideal, named a principal ideal, is simply formed by
the multiples of x. It hence follows, save in the case when B = {0B , 1B }, that
there is at least one ideal.
b) To each ideal I and each x 6= 1B not belonging to I, there corresponds an
ideal J ◆ I one of whose elements is x: this is {a · x + y : a 2 B, y 2 I}.
c) Every ideal I is included in an ideal which is maximal w.r.t. ✓.
Checking the first two of these is very plain; the third can be proved by means
of Zorn’s lemma, after observing that every chain of ideals is closed w.r.t. union.
5 1st Stone’s representation theorem
Let us recall first the algebraic version of Stone’s theorem:
Theorem 1 (Stone’s algebraic representation). Every Boolean algebra
B = (B, ·, +, 1B , 0B )
is isomorphic to a field H of sets whose underlying domain is included in P(H),
where H is the set of all homomorphisms from B into 2.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
Proof. Associate with each x 2 B the set x e of those homomorphisms in H which
send x to {;}; thus, clearly, 0f f
B = ; and 1B = H. Moreover x g·y = x e \ ye
holds: in fact, when h is a homomorphism, h(x · y) = h(x) \ h(y) equals {;} if
and only if h(x) = h(y) = {;}, i.e. i↵ h 2 x e \ ye. By an analogous argument,
h(x + y) = h(x) 4 h(y) and x] +y = x e 4 ye. Take H to be the image-set of the
function x 7! x e.
In order to see the injectivity of this function, consider the di↵erence e0 =
x0 + x0 · x1 between two elements x0 , x1 2 B such that x0 · x1 6= x0 (whence
e0 6= 0B ). We will show that there is an h 2 H sending e0 to 1B ; accordingly,
since 1B = h(e0 ) = h(x0 + x0 · x1 ) = h(x0 ) 4 h(x0 ) \ h(x1 ) , we will have
h(x0 ) = {;}, h(x1 ) = ;, and therefore h 2 x e0 \ x
e1 as desired. We readily get the
sought h if B = {0B , 1B }; otherwise we pick a y0 2 B \ {0B , 1B }, choosing y0 = e0
if e0 6= 1B . This y0 belongs to a principal ideal and hence to a maximal ideal M ,
and it is plain that the opposite h = 1 M of the characteristic function of M ,
manifestly a homomorphism from B to 2, does to our case. a
Call set-representation of the algebra B the field of sets just built. We
will see next that this H generates a very peculiar topology on H.
6 2nd Stone’s representation theorem
We are now ready for the topological version of Stone’s theorem:
Theorem 2 (Stone’s topological representation). The field of sets by which
we have represented a Boolean algebra B is the base—as well as the family of all
clopen sets—of a topology on the set H of all homomorphisms from B into 2.
Once endowed with such a topology, H turns out to be a Stone space.
Proof. To see that H (constructed as in the preceding proof) S is the base of a
topology onT H, we can directly check that H ✓ P(H) and H=H= 6 ; hold,
and that F belongs to H for every finite non-void subset F of H. Indeed, H
is the image-set of an injective function x 7! xe from B into
S P(H), where B is at
least doubleton; hence, readily, H = 6 ;, H ✓ P(H), and H ✓ H hold. The last
inclusion is in fact an equality, because H, which is 1fB , belongs to H. Then we
get that H is closed under intersection through the remark, made above, that
e \ ye = xg
x · y.
Knowing, at this point, that H qualifies as the base for a topology on H,
let us notice that all sets in H are clopen in the topology ⌧ generated by it: in
fact, since H \ xe = H4x e = 1fB 4x e = 1^B + x, the complement of a set in the
base belongs to the base in its turn. This remark readily gives us that (H, ⌧ ) is
a Hausdor↵ space. In fact, when f, g 2 H di↵er, there is an x 2 B such that
f (x) = 1 $ g(x) 6= 1; thus, since H \ x e we can find sets u, v 2 ⌧ such that
e = x,
f 2 u, g 2 v, and u \ v = ;, by also insisting that {u, v} = {x,e x
e }.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
It remains to be shown that the space is compact, which will also yield that
every clopen set of ⌧ belongs to H.9 One easily sees that the closed sets in ⌧
are: H and all intersections of non-void subsets of {ex : x 2 B}; compactness
will
T hence readily follow if we manage to show that whenever ; 6= OT✓ H and
O = ; hold, there is an F ✓ O such that |F| T 2 N \ {0} and F = ;.
Equivalently, assuming that ; 6= O ✓ H andTthat F = 6 ; holds for every finite
non-void subset F of O, we will show that O = 6 ;.
Notice that the subset
n \ o
B0 =Def x : F ✓ O | |F| 2 N \ {0} & x e= F [ {1B }
of B meets the conditions:
1) x · y 2 B0 for all x, y 2 B0 ,
2) 0B 2/ B0 6✓ {1B },
implying that { a · x : a 2 B , x 2 B0 } is an ideal of B. By enlarging this into a
maximal ideal, we get the kernel of a homomorphism h sending all complements
T
of el’ts of B0 to ;, hence sending all elements of B0 to {;}. Thus, h 2 O. a
7 Formalization of Stone’s representation theorems in Ref
This section o↵ers glimpses of our formal development of Stone’s result on the
representability of Boolean algebras via the clopen sets of a totally disconnected,
compact Hausdor↵ space. In carrying out this task, we relied on a proof checker:
Ref. Our experiment culminated in a rather elaborate series of mathematical
claims, shown not in this section, but in the appendix.
Organization and rationale of our automated proof assistant are extensively
discussed in [19]; but so far, due to the short time elapsed from its implementa-
tion, Ref is not widely known. Hence, putting aside our report on our experiment
for a moment, we devote a quick subsection to introducing Ref itself.
7.1 Brief presentation of the proof-checking framework
Ref is a proof assistant for mathematics based on a variant of Zemelo-Fraenkel
set theory, which is “hardwired” in Ref’s proof-checking abilities. From the user
Ref receives script files, called scenarios, consisting of successive definitions, the-
orems, and auxiliary commands, which it either certifies as constituting a valid
sequence or rejects as defective. In the case of rejection, the verifier attempts
to pinpoint the troublesome locations within a scenario, so that errors can be
9 S S
To derive from compactness that O 2 H holds when O ✓ HSand O isSclosed,
S argueTas follows. Assuming, w.l.o.g., that ; =
we 6 O, since ; = ( O) \ (H \ O) =
( O) \ {H \ u : u 2 O} is the intersectionS of a family
T of closed sets, we can pick
an
S F such
T that ; 6
= F ✓ O, |F | 2 N, and ( O) \ {H \ u : u 2 F} = ;. Thus
O = {H \ u : u 2 F } 2 H, because H is closed relative to complementation and
to finite intersection.
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
located and repaired. Step timings are produced even for correct proofs, to help
the user in spotting places where appropriate modifications could speed up proof
processing.
The bulk of the text normally submitted to the verifier consists of theorems
and proofs. Some theorems (and their proofs) are enclosed within so-called The-
orys, whose external conclusions are justified by these internal theorems. This
lets scenarios be subdivided into modules, which increases the readability and
supports proof reuse.
Many theorems are not enclosed within a user-defined Theory; when this
happens, they belong on their own right to the underlying “big” Theory identified
by the name Set theory.
Of all relationships treated within Set theory, the most fundamental is mem-
bership, 2, which is supposed to be well founded in the sense that no infinite
sequence x0 , x1 , x2 , . . . of sets can satisfy xi+1 2 xi for every i. The well founded-
ness of 2 is witnessed by the built-in arbitrary selection operator, arb, meeting
the conditions arb(x) 2 x and arb(x) \ x = ; for every set x other than the null
set ; (about which the equality arb(;) = ; is assumed).
At their simplest definitions are merely abbreviations which concentrate at-
tention on interesting constructs by assigning them names which shorten their
syntactic form (an example of this kind is the definition of finitude that we will
soon meet). Beyond this simple level, Ref o↵ers a primitive scheme of 2-recursive
definition legitimatized by the well foundedness of 2 and illustrated, e.g., by the
following specification:
Def : [Join singletons] filum(X) =Def {X} [ arb filum(y) : y 2 X | X = {y}
(This collects together into filum(X) all elements of the finitely many singletons
“spreading”—in a quite definite sense—from {X}.)
This example also shows the availability, in the formal language of Ref, of
perspicuous set-formers such as filum(y) : y 2 X | X = {y} . Just in order to
see a few other set-formers
S at work, consider the following shorthand definitions:
Y =Def { z : x 2 Y , z 2 x} ,
edges(V, E) =Def {x, y} : x 2SV, y S 2 V | x 6= y \ E ,
Connected(E) $Def {b : b ✓ E | b \ (E \ b) = ;} ✓ {;, E} .
A very small Ref scenario, consisting of one definition (involving yet another
set-former) and two theorems with their proofs, is shown in Fig. 2. As one sees
there, each inference step in a Ref proof has two components, separated by the
‘)’ sign: on the right of which a logical statement (sometimes hidden behind one
of the keywords Auto, Qed) appears; while, on the left, there is a justification
of the statement, namely an indication of which inference method enables its
derivation from the preceding part of the proof.
Last but not least, Ref supports proof reuse through a costruct named The-
ory, essentially a second-order form of Skolemization. Fig. 3 shows the interface
of a specific Theory, named finite image, which has two input parameters: s0 , a
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
Def P: [Family of all subsets of a given set] PS =Def {x : x ✓ S}
Thm pow0 : [No set equals its own powerset] (X ◆ Y $ Y 2 PX) & X 6= PX. Proof:
Suppose not(x0 , y0 ) ) Auto
Use def(Px0 ) ) Auto
Suppose ) x0 = Px0
ELEM ) Stat0 : x0 2 / {y : y ✓ x0 }
hx0 i,!Stat0 ) false; Discharge ) Auto
EQUAL ) Stat1 : x0 ◆ y0 6= y0 2 {y : y ✓ x0 }
Suppose ) Stat2 : y0 2 {y : y ✓ x0 }
hy1 i,!Stat2(Stat1?) ) false; Discharge ) Stat3 : y0 2
/ {y : y ✓ x0 }
hy0 i,!Stat3(Stat1?) ) false; Discharge ) Qed
Thm pow1 : [Monotonicity of powerset] S ◆ X ! PX [ {;, X} ✓ PS. Proof:
Suppose not(s0 , x0 ) ) Auto
Set monot ) {x : x ✓ x0 } ✓ {x : x ✓ s0 }
Use def(P) ) Stat1 : ; 2 / {x : x ✓ s0 } _ x0 2
/ {x : x ✓ s0 }
h;, x0 i,!Stat1 ) false; Discharge ) Qed
Thm pow2 : [Powerset of null set and of singletons] P; = {;} &
P {X} = {;, {X}} . Proof:
Suppose not(x0 ) ) Auto
Suppose ) P; 6= {;}
h;, ;i,!T pow1 ) Stat0 : P; 6✓ {;}
hy0 i,!Stat0(Stat0?) ) Stat1 : y0 2 P; & y0 2/ {;}
h;, y0 i,!T pow0 (Stat1?) ) false; Discharge ) P {x0 } =6 {;, {x0 }}
h {x0 } , {x0 } i,!T pow1 ) Stat2 : P {x0 } 6✓ {;, {x0 }}
hy1 i,!Stat2 ) Stat3 : y1 2 P {x0 } & y1 2/ {;, {x0 }}
h {x0 } , y1 i,!T pow0 (Stat3?) ) false; Discharge ) Qed
Fig. 2. Tiny sample of a Ref scenario
finite set, and g, a global function. Inside finite image, from the assumed finite-
ness of s0 the user has derived that { g(x) : x 2 s0 } is a finite set; moreover, (s)he
has defined the output parameter f⇥ so as to insure that f⇥ be an ✓ -minimal
subset of s0 such that f⇥ and s0 are sent by g to the same image.
Theory finite image (s0 , g(X))
Finite(s0 )
) f⇥
Finite { g(x) : x 2 s0 }
f⇥ ✓ s0 & h 8 t ✓ f⇥ | g(t) = g(s0 ) $ t = f⇥ i
End finite image
Fig. 3. Interface of a Theory specifying an induction principle
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R. Ceterchi et al. The representation of Boolean algebras in the spotlight of a proof checker
7.2 Excerpts from our proof scenario of Stone’s theorems
As a warm-up exercise, we developed with the assistance of Ref the Theory
pord displayed in Fig. 4, showing that every partially ordered set is isomorphic
to a family of sets partially ordered by inclusion. The assumptions of pord state
that Le must be a partial ordering of dd. By sending each element x of dd to the
set consisting of those elements of dd which are smaller than or equal to x, we get
an order monomorphism between (dd, Le) and (P(dd), ✓): whose name ‘poIso⇥ ’,
as indicated by the subscript ⇥, is specified—along with its definition—inside
the Theory.
Theory pord dd, Le(U, V)
h8x, y | {x, y} ✓ dd ! Le(x, y) & Le(y, x) $ x = y i
h8x, y, z | {x, y, z} ✓ dd ! Le(x, y) & Le(y, z) ! Le(x, z)i
) (poIso⇥ )
poIso⇥ = {[x, {v 2 dd | Le(v, x)}] : x 2 dd}
h8x | x 2 dd ! Le(x, x) & poIso⇥ x = {v 2 dd | Le(v, x)} i
h8x, y | {x, y} ✓ dd ! Le(x, y) $ poIso⇥ x ✓ poIso⇥ y i
1–1(poIso⇥ ) & domain(poIso⇥ ) = dd
End pord
Fig. 4. Interface of a representation Theory about partial orderings
Our next step consisted in developing a theory of Boolean rings:
Theory booleanRing(bb, ·, ÷)
bb 6= ;
h8x, y | {x, y} ✓ bb ! x · y 2 bb i
h8x, y | {x, y} ✓ bb ! x ÷ y 2 bb i
h8x, y, z | {x, y, z} ✓ bb ! x · (y · z) = (x · y) · z i
h8x, y, z | {x, y, z} ✓ bb ! x ÷ (y ÷ z) = (x ÷ y) ÷ z i
h8x, y, z | {x, y, z} ✓ bb ! (x ÷ y) · z = z · y ÷ z · x i
h8x, y | {x, y} ✓ bb ! x ÷ x = y ÷ y i
h8x, y | {x, y} ✓ bb ! x ÷ (y ÷ x) = y i
h8x | x 2 bb ! x · x = x i
) (zz⇥ )
zz⇥ = arb(bb) ÷ arb(bb)
h8x | (x 2 bb ! x ÷ x = zz⇥ & x ÷ zz⇥ = x & zz⇥ ÷ x = x) & zz⇥ 2 bb i
h8x, y | x, y 2 bb ! x ÷ y = y ÷ x i
h8x, y | x, y 2 bb ! x · y = y · x i
h8x | x 2 bb ! zz⇥ · x = zz⇥ i
h8u, v | {u, v} ✓ bb & u · v = u & v · u = v ! u = v i
End booleanRing
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Here zz⇥ designates the additive identity. Notice, among the internally derived
claims, the commutativity laws.
Due to its entirely algebraic character, this Theory could have been de-
veloped somewhat more easily with an autonomous theorem prover oriented to
the treatment of equality: as announced in [6, Sec. 3], we plan to implement
interfaces between Ref and outer automated proof assistants.
Our next Theory, presupposing the definition of symmetric di↵erence, shows
that rings of sets match the assumptions of the Theory booleanRing:
TheorySprotoBoolean(dd)
; 6= dd
h8x, y | {x, y} ✓ dd ! x \ y 2 dd i
h8x, y | {x, y} ✓ dd ! x 4 y 2 dd i
)
dd 6= ;
h8x 2 dd, y 2 dd, z 2 dd | x \ (y \ z) = (x \ y) \ z i
h8x 2 dd, y 2 dd, z 2 dd | x 4 (y 4 z) = (x 4 y) 4 z i
h8x 2 dd, y 2 dd, z 2 dd | (x 4 y) \ z = z \ y 4 z \ x i
h8x 2 dd, y 2 dd | x 4 x = y 4 y i
h8x 2 dd, y 2 dd | x 4 (y 4 x) = y i
h8x 2 dd | x \ x = x i
End protoBoolean
Two claims, proved inside the background Theory, namely Set theory, and pre-
supposing the definition of P, show that the family of all subsets, and the
one of all finite and cofinite subsets, of a non-void set constitute instances of
protoBoolean:
S
Thm . W 6= ; & {X, Y} ✓ PW ! {X \ Y, X 4 Y} ✓ PW & (PW) 6= ; ,
Thm . W 6= ; & D = {s ✓ W | Finite(s) _ Finite(W\s)} & {X, Y}S✓ D !
{X \ Y, X 4 Y} ✓ D & D 6= ;.
Another Theory, akin to the preceding one, introduces a slightly more spe-
cific algebraic variety than the one treated by protoBoolean:
TheorySarcheoBoolean(dd)
; 6= dd
h8x, y, z | {x, y} ✓ dd & z ✓ x [ y ! z 2 dd i
)
h8x, y | {x, y} ✓ dd ! x \ y 2 dd i
h8x, y | {x, y} ✓ dd ! x 4 y 2 dd i
dd 6= ;
End archeoBoolean
After switching back to the background Set theory level, one proves that
there are fields of sets which are instances of protoBoolean but are not instances
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of archeoBoolean. Indeed, the collection of all finite and cofinite subsets of an
infinite set is not closed with respect to inclusion.
Thm . ¬Finite(W) & D = {s ✓ W | Finite(s) _ Finite(W\s)} !
W 2 D & h9z ✓ W | z 2 / D i.
Surprisingly enough, it is unnecessary to resort to a theory of cardinals of any
sophistication in order to get the result just cited: the distinction between finite
sets and sets which are not finite more than suffices for that purpose, where the
following definition applies:
Def : [Finitude] Finite(F) $Def h8g 2 P(PF)\ {;} , 9m | g \ Pm = {m} i
Last but not least, we developed the Theory booleanAlgebra whose interface
is shown in the Appendix.
Conclusions and future work
Proof-verification can highly benefit from representation theorems of the kind
illustrated by Stone’s results on Boolean algebras. On the human side, such
results disclose new insights by shedding light on a discipline from unusual angles;
on the technological side, they enable the transfer of proof methods from one
realm of mathematics to another.
Examples of this can be found in various recent proofs concerning connected
claw-free graphs:10 thanks to a convenient choice on how to represent those
graphs, Milanič and Tomescu [13] proved with relative ease two classical propo-
sitions, namely that any such graph owns a near-perfect matching and has a
Hamiltonian cycle in its square; a proof of the somewhat deeper theorem [8]
that all connected claw-free graphs have a vertex-pancyclic square was also at-
tained cheaply through the same representation [23]. Specifically, the facilitation
stems from transferring those results to the special class of the membership di-
graphs, whose set of vertices is a hereditarily finite set and whose arcs precisely
reflect the membership relation between vertices. Under this change of perspec-
tive, a fully formal reconstruction of the first two results became a↵ordable and,
once carried out, was certified correct with the Ref proof-checker [15,17,18].
This motivated us in undertaking the formal development, with Ref, of proofs
of various representation theorems (see also [1]). An envisaged continuation of
the present work will be in the direction of MV-algebras (cf. [14,5,3,11]).
10
A graph is said to be claw-free if no induced subgraph of its is isomorphic to the
graph, called the claw : K1,3 = {w, x, y, z}, {y, x}, {y, z}, {y, w} .
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A The main theory in our scenario on Boolean algebras
Theory booleanAlgebra(B, ·, ÷, 1B )
1B 2 B
1B 6= 1B ÷ 1B
h8x, y | {x, y} ✓ B ! x · y 2 Bi
h8x, y | {x, y} ✓ B ! x ÷ y 2 Bi
h8x, y, z | {x, y, z} ✓ B ! x · (y · z) = (x · y) · z i
h8x, y, z | {x, y, z} ✓ B ! x ÷ (y ÷ z) = (x ÷ y) ÷ z i
h8x, y, z | {x, y, z} ✓ B ! (x ÷ y) · z = z · y ÷ z · x i
h8x, y | {x, y} ✓ B ! x ÷ x = y ÷ y i
h8x, y | {x, y} ✓ B ! x ÷ (y ÷ x) = y i
h8x | x 2 B ! x · x = x i
h8x | x 2 B ! 1B · x = x i
) (0⇥ , ( )⇥ , Ideal⇥ , BooHom⇥ , H⇥ , '⇥ )
0⇥ = arb(B) ÷ arb(B)
h8x | (x 2 B ! x ÷ x = 0⇥ & x ÷ 0⇥ = x & 0⇥ ÷ x = x) & 0⇥ 2 Bi
h8x, y | {x, y} ✓ B ! x ÷ y = y ÷ x i
h8x, y | {x, y} ✓ B ! x · y = y · x i
h8u, v | {u, v} ✓ B & u · v = u & v · u = v ! u = v i
h8x | ⇣x 2 B ! (x)⇥ = 1B ÷ x⇣i ⌘ ⌘
h8x | x 2 B ! (x)⇥ 2 B & (x)⇥ = x & 1B ⇥ = 0⇥ & 0⇥ ⇥ = 1B i
⇥
h8x, y | {x, y} ✓ B ! (x)⇥ ÷ x = 1B & y · x ÷ y · (x)⇥ = y & y · x · y · (x)⇥ = 0⇥ i
h8x, y | {x, y} ✓ B ! (x ÷ y)⇥ = x · y ÷ (x)⇥ · (y)⇥ i
h8x | x 2 B ! (x)⇥ 6= x & x 2/ {0⇥ , 1B } ! (x)⇥ 2 B\ {0⇥ , 1B } i
h8u, v | {u, v} ✓ B & u · v = 1B ! u = 1B & v = 1B i
h8u, v, x, y | {u, v, x, y} ✓ B ! u · (x)⇥ ÷ v · (y)⇥ = u · (x)⇥ ÷ v · (y)⇥ · (x · y)⇥ i
h8i | Ideal⇥ (i) $ {x ÷ y : x 2 i, y 2 i} ✓ i & {x · y : x 2 B, y 2 i} ✓ i & i ✓ B\ {1B } & i 6✓ {0⇥ } i
h8i, x, y | Ideal⇥ (i) & {x, y} ✓ i ! x ÷ y 2 i i
h8i | Ideal⇥ (i) ! 0⇥ 2 i & x 2 i & y 2 B ! x · y, y · x 2 i & (x)⇥ 2/ i & 1B 2/ i i
h8i | Ideal⇥ (i) ! h9m | i ✓ m & h8j | Ideal⇥ (j) & m ✓ j $ j = m iii
h8b | b ✓ B\ {0⇥ } & {x · y : x 2 b, y 2 b} ✓ b & b 6✓ {1B } ! Ideal⇥ {a · (x)⇥ : a 2 B, x 2 b}
h8x | x 2 B\ {0⇥ , 1B } ! Ideal⇥ ({a · x : a 2 B}) & x 2 {a · x S: a 2 B} i
h8h | BooHom⇥ (h) $ Svm(h) & domain(h) = B & h 1B = range(h) & h 1B 6= h 0⇥ &
h8x 2 B, y 2 B | h (x · y) = h x \ h y & h (x ÷ y) = h x 4 h y ii
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H⇥ = {h ✓ B ⇥ 2 | BooHom⇥ (h)}
h8h | h 2 H⇥ ! h 0⇥ = ; & h 1B = 1i
h8h | h 2 H⇥ & {x, y} ✓ B & h (x ÷ x · y) = 1 & h y = ; ! h x = 1i
h8i, x | Ideal⇥ (i) & x 2 B & (x)⇥ 2/ i ! h9j | Ideal⇥ (j) & i [ {x} ✓ jii
h8x, m | x 2/ m & x 2 B & h8j | Ideal⇥ (j) & m ✓ j $ j = m i ! (x)⇥ 2 m i
h8m | h8j | Ideal⇥ (j) & m ✓ j $ j = m i ! {[x, if x 2 m then ; else 1 fi] : x 2 B} 2 H⇥ i
B ✓ {0⇥ , 1B } ! {[0⇥ , ;] , [1B , 1]} 2 H⇥
h8x 2 B\ {0⇥ } | {h 2 H⇥ | h x = 1} = 6 ;i & H⇥ 6= ;
'⇥ = {[b, {h 2 H⇥ | h b = 1}] : b 2 B}
h8x 2 B | '⇥ x = {h 2 H⇥ | h x = 1} i
h8x, y | {x, y} ✓ B ! '⇥ (x · y) = '⇥ x \ '⇥ y & '⇥ (x ÷ y) = '⇥ x 4 '⇥ y i
h8x, y |S{x, y} ✓ B & x · y 6= x ! '⇥ x 6= '⇥ yi
H⇥ = range('⇥ ) & '⇥ 0⇥ = ; & '⇥ 1B 6= '⇥ 0⇥ & '⇥ 1B = H⇥
h8x 2 B | '⇥ (x)⇥ = H⇥ \'⇥ x i
1–1('⇥ ) & domain('⇥ ) = B
BooHom⇥ ('⇥ )
range('⇥ ) ✓ {x : x ✓ H⇥ } & ; 2 range('⇥ ) & H⇥ 2 range('⇥ )
h8u 2 range('⇥ ) | H⇥ \u 2 range('⇥ )i
h8f, g | {f, g} ✓ H⇥ & f 6= g ! h9u 2 range('
T ⇥ ), v 2 range('⇥ ) | f 2 u & g 2 v & u \ v = ;ii
h8f ✓ range('⇥ ) | f 6= ; & Finite(f)
T ! f 2 range(' ⇥ )i T
h8k ✓ range('⇥ ) | k 6= ; & k = ; ! h9f ✓ k | f 6= ; & Finite(f) ! f = ;ii
End booleanAlgebra
301