The representation of Boolean algebras
in the spotlight of a proof checker
Rodica Ceterchi, E. G. Omodeo, Alexandru I. Tomescu
Dip. Matematica e Geoscienze — DMI
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
1/22
The representation of Boolean algebras1
in the spotlight of a proof checker
Rodica Ceterchi, E. G. Omodeo, Alexandru I. Tomescu
Dip. Matematica e Geoscienze — DMI
1
Work partially funded by INdAM/GNCS 2013, by Academy of Finland grant
250345 (CoECGR), and by FRA-UniTS PUMA
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
1/22
( Ideal ) itinerary of this talk
i Stone’s representation of Boolean algebras
ii The proof assistant Ref
iii Our Stone-related proof-verification experiment
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
2/22
‘Boolean algebras have an almost embarrassingly rich structure.’
[Hal62, p. 10]
A Boolean algebra is. . .
0
a distributive and complemented lattice
1
a ring with 1, meeting the special laws
X +X
X · X
One often sets:
X
X tY
X 6Y
etc.
Eugenio G. Omodeo
= 0
= X
=Def
X +1
=Def X · Y + X + Y
↔Def
X tY =Y
Proof-checking Stone’s representation th’m
3/22
‘Boolean algebras have an almost embarrassingly rich structure.’
[Hal62, p. 10]
A Boolean algebra is. . .
0
a distributive and complemented lattice
1
a ring with 1, meeting the special laws
X +X
X · X
One often sets:
X
X tY
X 6Y
etc.
Eugenio G. Omodeo
= 0
= X
=Def
X +1
=Def X · Y + X + Y
↔Def
X tY =Y
Proof-checking Stone’s representation th’m
3/22
‘Boolean algebras have an almost embarrassingly rich structure.’
[Hal62, p. 10]
A Boolean algebra is. . .
0
a distributive and complemented lattice
1
a ring with 1, meeting the special laws
X +X
X · X
One often sets:
X
X tY
X 6Y
etc.
Eugenio G. Omodeo
= 0
= X
=Def
X +1
=Def X · Y + X + Y
↔Def
X tY =Y
Proof-checking Stone’s representation th’m
3/22
‘Boolean algebras have an almost embarrassingly rich structure.’
[Hal62, p. 10]
A Boolean algebra is. . .
0
a distributive and complemented lattice
1
a ring with 1, meeting the special laws
X +X
X · X
One often sets:
X
X tY
X 6Y
etc.
Eugenio G. Omodeo
= 0
= X
=Def
X +1
=Def X · Y + X + Y
↔Def
X tY =Y
Proof-checking Stone’s representation th’m
3/22
Digression: The Robbins / McCune axioms
X tY
= Y tX
X t (Y t Z ) = (X t Y ) t Z
X tY tX tY
=
X
Cf. [McC97]
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
4/22
How does one plainly ascertain Boolean laws ?
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
5/22
Fields of sets
One calls field of sets any family B which
is closed under the operations of
intersection ( · ) and symmetric difference ( + ) ;
S
has B among its members, viz.,
owns a maximum w.r.t. set inclusion ;
differs from {∅} .
This clearly is an instance of a Boolean algebra. How general? A
renowned theorem by M. H. Stone [Sto36] gives us the answer:
Every Boolean algebra is isomorphic to a field of sets.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
6/22
Fields of sets
One calls field of sets any family B which
is closed under the operations of
intersection ( · ) and symmetric difference ( + ) ;
S
has B among its members, viz.,
owns a maximum w.r.t. set inclusion ;
differs from {∅} .
This clearly is an instance of a Boolean algebra. How general? A
renowned theorem by M. H. Stone [Sto36] gives us the answer:
Every Boolean algebra is isomorphic to a field of sets.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
6/22
Fields of sets
One calls field of sets any family B which
is closed under the operations of
intersection ( · ) and symmetric difference ( + ) ;
S
has B among its members, viz.,
owns a maximum w.r.t. set inclusion ;
differs from {∅} .
This clearly is an instance of a Boolean algebra. How general? A
renowned theorem by M. H. Stone [Sto36] gives us the answer:
Every Boolean algebra is isomorphic to a field of sets.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
6/22
Fields of sets
One calls field of sets any family B which
is closed under the operations of
intersection ( · ) and symmetric difference ( + ) ;
S
has B among its members, viz.,
owns a maximum w.r.t. set inclusion ;
differs from {∅} .
This clearly is an instance of a Boolean algebra. How general? A
renowned theorem by M. H. Stone [Sto36] gives us the answer:
Every Boolean algebra is isomorphic to a field of sets.
( Marshall Stone
1903–1989 )
Eugenio G. Omodeo
( Alfred Tarski
1902–1983 )
Proof-checking Stone’s representation th’m
6/22
‘A cardinal principle of modern mathematical research
may be stated as a maxim: “One must always topologize”.’
One calls Stone space any topological space (X , τ) such that
1
2
3
for every p ∈ X and q ∈ X \ {p} there exist P, Q ∈ τ such that
p ∈P , q ∈Q , P ∩Q =∅ ;
S
every O ⊆ τ s.t. O = X has a finite subfamily F ⊆ O such
S
that F = X ;
there is a β ⊆ τ ∩ { X \ Q : Q ∈ τ } such that
S
τ = { Y : Y ⊆ β} .
Every Boolean algebra is isomorphic to the field
τ ∩ {X \ Q : Q ∈ τ}
of all clopen sets of a Stone space (X , τ).
Cf. [Sto37], [Sto38, p. 814]
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
7/22
‘A cardinal principle of modern mathematical research
may be stated as a maxim: “One must always topologize”.’
One calls Stone space any topological space (X , τ) such that
1
2
3
for every p ∈ X and q ∈ X \ {p} there exist P, Q ∈ τ such that
p ∈P , q ∈Q , P ∩Q =∅ ;
S
every O ⊆ τ s.t. O = X has a finite subfamily F ⊆ O such
S
that F = X ;
there is a β ⊆ τ ∩ { X \ Q : Q ∈ τ } such that
S
τ = { Y : Y ⊆ β} .
Every Boolean algebra is isomorphic to the field
τ ∩ {X \ Q : Q ∈ τ}
of all clopen sets of a Stone space (X , τ).
Cf. [Sto37], [Sto38, p. 814]
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
7/22
ÆtnaNova aka Ref eree:
Cf. [SCO11]
( On-line worksheet )
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
8/22
Interaction with our proof-verifier
Eugenio G. Omodeo
( Input )
Proof-checking Stone’s representation th’m
9/22
Interaction with our proof-verifier( Output )
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
10/22
Interaction with our proof-verifier( Output )
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
10/22
3 basic constituents of a scenario ( examples )
( no ∈-recursion here! )
Definition:
-- After the celebrated paper Sur les ensembles fini ( Tarski, 1924 )
Def Fin: [Finitude]
Finite(F) ↔Def
h∀g ∈ P(P(F))\{∅}, ∃m | g ∩ P(m) = {m}i
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
11/22
3 basic constituents of a scenario ( examples )
( no ∈-recursion here! )
Definition:
-- After the celebrated paper Sur les ensembles fini ( Tarski, 1924 )
Def Fin: [Finitude]
Finite(F) ↔Def
h∀g ∈ P(P(F))\{∅}, ∃m | g ∩ P(m) = {m}i
Theorem and Proof:
( Monotonicity of finitude )
Thm fin0 . Y ⊇ X & Finite(Y) → Finite(X). Proof:
Suppose_not(y0 , x0 ) =⇒ y0 ⊇ x0 & Finite(y0 ) & ¬Finite(x0 )
hy0 , x0 i,→T pow1 =⇒ Py0 ⊇ Px0
Use_def(Finite) =⇒ Stat1 : ¬h∀g ∈ P(Px0 )\{∅}, ∃m |
g ∩ Pm = {m}i & h∀g 0 ∈ P(Py0 )\{∅}, ∃m | g 0 ∩ Pm = {m}i
hPy0 , Px0 i,→T pow1 =⇒ P(Py0 ) ⊇ P(Px0 )
hg0 , g0 i,→Stat1(Stat1?) =⇒ ¬h∃m | g0 ∩ Pm = {m}i &
h∃m | g0 ∩ Pm = {m}i
Discharge =⇒ Qed
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
11/22
4th , major constituent of a scenario (example)
A construct for proof reuse
Theory finite_image (s0 , g(X ))
Finite(s0 )
End finite_image
Enter_theory finite_image
..
..
..
..
.
.
.
.
Enter_theory Set_theory
Within a scenario, the discourse can momentarily digress into a
‘Theory’ that enforces certain local assumptions.
At the end of the digression, the upper theory will be re-entered.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
12/22
4th , major constituent of a scenario (example)
A construct for proof reuse
Theory finite_image (s0 , g(X ))
Finite(s0 )
End finite_image
Enter_theory finite_image
..
..
..
..
.
.
.
.
Enter_theory Set_theory
Within a scenario, the discourse can momentarily digress into a
‘Theory’ that enforces certain local assumptions.
At the end of the digression, the upper theory will be re-entered.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
12/22
4th , major constituent of a scenario (example)
A construct for proof reuse
Theory finite_image (s0 , g(X ))
Finite(s0 )
=⇒ fΘ
Finite { g(x) : x ∈ s0 }
fΘ ⊆ s0 & h ∀ t ⊆ fΘ | g(t) = g(s0 ) ↔ t = fΘ i
End finite_image
As an outcome of the digression, the Theory will be able to
instantiate new theorems
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
12/22
4th , major constituent of a scenario (example)
A construct for proof reuse
Theory finite_image (s0 , g(X ))
Finite(s0 )
=⇒ fΘ
Finite { g(x) : x ∈ s0 }
fΘ ⊆ s0 & h ∀ t ⊆ fΘ | g(t) = g(s0 ) ↔ t = fΘ i
End finite_image
As an outcome of the digression, the Theory will be able to
instantiate new theorems: possibly involving new symbols,
whose definition it encapsulates.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
12/22
Our experiment, in digits
The script-file containing our verified formal derivation of Stone’s
results ( with many asides ) from ‘first principles’:
comprises 42 definitions;
proves 210 theorems,
organized in 19 Theory s
Its processing takes ca. 25 seconds.
http://www2.units.it/eomodeo/StoneReprScenario.html
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
13/22
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.
An envisaged continuation of the present work will be in the
direction of MV-algebras
(cf. [Mun86])
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
14/22
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.
An envisaged continuation of the present work will be in the
direction of MV-algebras
(cf. [Mun86])
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
14/22
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.
An envisaged continuation of the present work will be in the
direction of MV-algebras
(cf. [Mun86])
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
14/22
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.
An envisaged continuation of the present work will be in the
direction of MV-algebras
(cf. [Mun86])
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
14/22
Thank you for your attention!
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
15/22
Bibliografic references
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
16/22
Paul R. Halmos. Algebraic Logic.
AMS Chelsea Publishing, Providence, Rhode Island, 1962.
W. W. McCune. Solution of the Robbins problem.
J. of Automated Reasoning, 19(3):263–276, 1997.
D. Mundici. Interpretation of AF C ∗ -algebras in Łukasiewicz
sentential calculus.
J. Funct. Anal., 65:15–63, 1986.
J.T. Schwartz, D. Cantone, and E.G. Omodeo.
Computational Logic and Set Theory - Applying Formalized
Logic to Analysis.
Springer, 2011.
Marshall H. Stone.
The theory of representations for Boolean algebras.
Transactions of the American Mathematical Society,
40:37–111, 1936.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
16/22
Marshall H. Stone. Applications
of the theory of Boolean rings to general topology.
Transactions of the American Mathematical Society,
41:375–481, 1937.
Marshall H. Stone.
The representation of Boolean algebras.
Bulletin of the American Mathematical Society, 44(Part
1):807–816, 1938.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
17/22
Excerpt 1 from scenario on Stone’s theorems
Theory pord dd, Le(U, V)
h∀x, y | {x, y} ⊆ dd → Le(x, y) & Le(y, x) ↔ x = y i
h∀x, y, z | {x, y, z} ⊆ dd → Le(x, y) & Le(y, z) → Le(x, z)i
=⇒ (poIsoΘ )
poIsoΘ = {[x, {v ∈ dd | Le(v, x)}] : x ∈ dd}
h∀x | x ∈ dd → Le(x, x) & poIsoΘ x = {v ∈ dd | Le(v, x)}i
h∀x, y | {x, y} ⊆ dd → Le(x, y) ↔ poIsoΘ x ⊆ poIsoΘ y i
1–1(poIsoΘ ) & dom(poIsoΘ ) = dd
End pord
Figure: Interface of a representation Theory about partial orderings
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
17/22
Excerpt 2 from scenario on Stone’s theorems
Theory booleanRing(bb, ·, ÷)
bb6=∅
h∀x, y | {x, y} ⊆ bb → x · y ∈ bb i
h∀x, y | {x, y} ⊆ bb → x ÷ y ∈ bb i
h∀x, y, z | {x, y, z} ⊆ bb → x · (y · z) = (x · y) · z i
h∀x, y, z | {x, y, z} ⊆ bb → x ÷ (y ÷ z) = (x ÷ y) ÷ z i
h∀x, y, z | {x, y, z} ⊆ bb → (x ÷ y) · z = z · y ÷ z · x i
h∀x, y | {x, y} ⊆ bb → x ÷ x = y ÷ y i
h∀x, y | {x, y} ⊆ bb → x ÷ (y ÷ x) = y i
h∀x | x ∈ bb → x · x = x i
=⇒ (zzΘ )
zzΘ = arb (bb) ÷ arb (bb)
h∀x | (x ∈ bb → x ÷ x = zzΘ & x ÷ zzΘ = x & zzΘ ÷ x = x) & zzΘ ∈ bb i
h∀x, y | x, y ∈ bb → x ÷ y = y ÷ x i
h∀x, y | x, y ∈ bb → x · y = y · x i
h∀x | x ∈ bb → zzΘ · x = zzΘ i
h∀u, v | {u, v} ⊆ bb & u · v = u & v · u = v → u = v i
End booleanRing
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
18/22
Excerpt 3 from scenario on Stone’s theorems
Our next Theory, presupposing the definition of symmetric
difference, shows that rings of sets match the assumptions of the
Theory booleanRing:
Theory protoBoolean(dd)
S
∅6= dd
h∀x, y | {x, y} ⊆ dd → x ∩ y ∈ dd i
h∀x, y | {x, y} ⊆ dd → x4y ∈ dd i
=⇒
dd6=∅
h∀x ∈ dd, y ∈ dd, z ∈ dd | x ∩ (y ∩ z) = (x ∩ y) ∩ z i
h∀x ∈ dd, y ∈ dd, z ∈ dd | x4(y4z) = (x4y)4z i
h∀x ∈ dd, y ∈ dd, z ∈ dd | (x4y) ∩ z = z ∩ y4z ∩ x i
h∀x ∈ dd, y ∈ dd | x4x = y4y i
h∀x ∈ dd, y ∈ dd | x4(y4x) = y i
h∀x ∈ dd | x ∩ x = x i
End protoBoolean
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
19/22
Excerpt 4 from scenario on Stone’s theorems
Two claims, proved inside the background Theory, namely
Set_theory, and presupposing 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 . W6=∅ & {X, Y} ⊆ PW → {X ∩ Y, X4Y} ⊆ PW & (PW)6=∅.
Thm . W6=∅ & D = {s ⊆ W | Finite(s) ∨ Finite(W\s)} & {X, Y} ⊆ D →
S
{X ∩ Y, X4Y} ⊆ D & D6=∅.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
20/22
Excerpt 5 from scenario on Stone’s theorems
Another Theory, akin to the preceding one, introduces a slightly
more specific algebraic variety than the one treated by
protoBoolean:
Theory archeoBoolean(dd)
S
∅6= dd
h∀x, y, z | {x, y} ⊆ dd & z ⊆ x ∪ y → z ∈ dd i
=⇒
h∀x, y | {x, y} ⊆ dd → x ∩ y ∈ dd i
h∀x, y | {x, y} ⊆ dd → x4y ∈ dd i
dd6=∅
End archeoBoolean
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
21/22
Excerpt 6 from scenario on Stone’s theorems
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 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 ∈ D & h∃z ⊆ W | z ∈
/ 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 ( see above ) and sets which are
not finite more than suffices for that purpose.
Eugenio G. Omodeo
Proof-checking Stone’s representation th’m
22/22