<s>
The	O
Isabelle	B-Application
automated	B-Application
theorem	I-Application
prover	I-Application
is	O
a	O
higher-order	B-Algorithm
logic	I-Algorithm
(	O
HOL	O
)	O
theorem	O
prover	O
,	O
written	O
in	O
Standard	B-Language
ML	I-Language
and	O
Scala	B-Application
.	O
</s>
<s>
Isabelle	B-Application
is	O
available	O
inside	O
a	O
flexible	O
system	O
framework	O
allowing	O
for	O
logically	O
safe	O
extensions	O
,	O
which	O
comprise	O
both	O
theories	O
as	O
well	O
as	O
implementations	O
for	O
code-generation	O
,	O
documentation	O
,	O
and	O
specific	O
support	O
for	O
a	O
variety	O
of	O
formal	B-Architecture
methods	I-Architecture
.	O
</s>
<s>
It	O
can	O
be	O
seen	O
as	O
an	O
IDE	B-Application
for	O
formal	B-Architecture
methods	I-Architecture
.	O
</s>
<s>
Isabelle	B-Application
was	O
named	O
by	O
Lawrence	O
Paulson	O
after	O
Gérard	O
Huet	O
's	O
daughter	O
.	O
</s>
<s>
The	O
Isabelle	B-Application
theorem	I-Application
prover	I-Application
is	O
free	B-Application
software	I-Application
,	O
released	O
under	O
the	O
revised	O
BSD	B-Operating_System
license	I-Operating_System
.	O
</s>
<s>
Isabelle	B-Application
is	O
generic	O
:	O
it	O
provides	O
a	O
meta-logic	O
(	O
a	O
weak	O
type	O
theory	O
)	O
,	O
which	O
is	O
used	O
to	O
encode	O
object	O
logics	O
like	O
first-order	O
logic	O
(	O
FOL	O
)	O
,	O
higher-order	B-Algorithm
logic	I-Algorithm
(	O
HOL	O
)	O
or	O
Zermelo	O
–	O
Fraenkel	O
set	O
theory	O
(	O
ZFC	O
)	O
.	O
</s>
<s>
The	O
most	O
widely	O
used	O
object	O
logic	O
is	O
Isabelle/HOL	B-Application
,	O
although	O
significant	O
set	O
theory	O
developments	O
were	O
completed	O
in	O
Isabelle/ZF	O
.	O
</s>
<s>
Isabelle	B-Application
's	O
main	O
proof	O
method	O
is	O
a	O
higher-order	O
version	O
of	O
resolution	O
,	O
based	O
on	O
higher-order	O
unification	B-Algorithm
.	O
</s>
<s>
Though	O
interactive	O
,	O
Isabelle	B-Application
features	O
efficient	O
automatic	O
reasoning	O
tools	O
,	O
such	O
as	O
a	O
term	O
rewriting	O
engine	O
and	O
a	O
tableaux	O
prover	O
,	O
various	O
decision	O
procedures	O
,	O
and	O
,	O
through	O
the	O
Sledgehammer	O
proof-automation	O
interface	O
,	O
external	O
satisfiability	B-Application
modulo	I-Application
theories	I-Application
(	O
SMT	O
)	O
solvers	O
(	O
including	O
CVC4	O
)	O
and	O
resolution-based	O
automated	B-Application
theorem	I-Application
provers	I-Application
(	O
ATPs	O
)	O
,	O
including	O
E	B-Language
and	O
SPASS	B-Application
(	O
the	O
Metis	O
proof	O
method	O
reconstructs	O
resolution	O
proofs	O
generated	O
by	O
these	O
ATPs	O
)	O
.	O
</s>
<s>
Isabelle	B-Application
features	O
locales	O
which	O
are	O
modules	O
that	O
structure	O
large	O
proofs	O
.	O
</s>
<s>
Isar	O
(	O
"	O
intelligible	O
semi-automated	O
reasoning	O
"	O
)	O
is	O
Isabelle	B-Application
's	O
formal	O
proof	O
language	O
.	O
</s>
<s>
It	O
is	O
inspired	O
by	O
the	O
Mizar	B-Language
system	I-Language
.	O
</s>
<s>
Isabelle	B-Application
allows	O
proofs	O
to	O
be	O
written	O
in	O
two	O
different	O
styles	O
,	O
the	O
procedural	B-Application
and	O
the	O
declarative	B-Language
.	O
</s>
<s>
Procedural	B-Application
proofs	O
specify	O
a	O
series	O
of	O
tactics	O
(	O
theorem	B-Application
proving	I-Application
functions/procedures	O
)	O
to	O
apply	O
;	O
while	O
reflecting	O
the	O
procedure	O
that	O
a	O
human	O
mathematician	O
might	O
apply	O
to	O
proving	O
a	O
result	O
,	O
they	O
are	O
typically	O
hard	O
to	O
read	O
as	O
they	O
do	O
not	O
describe	O
the	O
outcome	O
of	O
these	O
steps	O
.	O
</s>
<s>
Declarative	B-Language
proofs	O
(	O
supported	O
by	O
Isabelle	B-Application
's	O
proof	O
language	O
,	O
Isar	O
)	O
,	O
on	O
the	O
other	O
hand	O
,	O
specify	O
the	O
actual	O
mathematical	O
operations	O
to	O
be	O
performed	O
,	O
and	O
are	O
therefore	O
more	O
easily	O
read	O
and	O
checked	O
by	O
humans	O
.	O
</s>
<s>
The	O
procedural	B-Application
style	O
has	O
been	O
deprecated	O
in	O
recent	O
versions	O
of	O
Isabelle	B-Application
.	O
</s>
<s>
For	O
example	O
,	O
a	O
declarative	B-Language
proof	O
by	O
contradiction	O
in	O
Isar	O
that	O
the	O
square	O
root	O
of	O
two	O
is	O
not	O
rational	O
can	O
be	O
written	O
as	O
follows	O
.	O
</s>
<s>
Isabelle	B-Application
has	O
been	O
used	O
to	O
aid	O
formal	B-Architecture
methods	I-Architecture
for	O
the	O
specification	O
,	O
development	O
and	O
verification	O
of	O
software	O
and	O
hardware	O
systems	O
.	O
</s>
<s>
Isabelle	B-Application
has	O
been	O
used	O
to	O
formalize	O
numerous	O
theorems	O
from	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
like	O
Gödel	O
's	O
completeness	O
theorem	O
,	O
Gödel	O
's	O
theorem	O
about	O
the	O
consistency	O
of	O
the	O
axiom	O
of	O
choice	O
,	O
the	O
prime	O
number	O
theorem	O
,	O
correctness	O
of	O
security	B-Protocol
protocols	I-Protocol
,	O
and	O
properties	O
of	O
programming	B-Application
language	I-Application
semantics	I-Application
.	O
</s>
<s>
Many	O
of	O
the	O
formal	O
proofs	O
are	O
,	O
as	O
mentioned	O
,	O
maintained	O
in	O
the	B-Application
Archive	I-Application
of	I-Application
Formal	I-Application
Proofs	I-Application
,	O
which	O
contains	O
(	O
as	O
of	O
2019	O
)	O
at	O
least	O
500	O
articles	O
with	O
over	O
2	O
million	O
lines	O
of	O
proof	O
in	O
total	O
.	O
</s>
<s>
In	O
2009	O
,	O
the	O
L4.verified	O
project	O
at	O
NICTA	O
produced	O
the	O
first	O
formal	O
proof	O
of	O
functional	O
correctness	O
of	O
a	O
general-purpose	O
operating	O
system	O
kernel	O
:	O
the	O
seL4	O
(	O
secure	O
embedded	O
L4	B-Operating_System
)	O
microkernel	B-Operating_System
.	O
</s>
<s>
The	O
proof	O
is	O
constructed	O
and	O
checked	O
in	O
Isabelle/HOL	B-Application
and	O
comprises	O
over	O
200,000	O
lines	O
of	O
proof	O
script	O
to	O
verify	O
7,500	O
lines	O
of	O
C	B-Language
.	O
The	O
verification	O
covers	O
code	O
,	O
design	O
,	O
and	O
implementation	O
,	O
and	O
the	O
main	O
theorem	O
states	O
that	O
the	O
C	B-Language
code	O
correctly	O
implements	O
the	O
formal	O
specification	O
of	O
the	O
kernel	O
.	O
</s>
<s>
The	O
proof	O
uncovered	O
144	O
bugs	O
in	O
an	O
early	O
version	O
of	O
the	O
C	B-Language
code	O
of	O
the	O
seL4	O
kernel	O
,	O
and	O
about	O
150	O
issues	O
in	O
each	O
of	O
design	O
and	O
specification	O
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
programming	O
language	O
Lightweight	B-Language
Java	I-Language
was	O
proven	O
type-sound	B-Language
in	O
Isabelle	B-Application
.	O
</s>
<s>
Larry	O
Paulson	O
keeps	O
a	O
list	O
of	O
research	O
projects	O
that	O
use	O
Isabelle	B-Application
.	O
</s>
