<s>
Coq	B-Application
is	O
an	O
interactive	O
theorem	O
prover	O
first	O
released	O
in	O
1989	O
.	O
</s>
<s>
It	O
allows	O
for	O
expressing	O
mathematical	O
assertions	O
,	O
mechanically	O
checks	O
proofs	O
of	O
these	O
assertions	O
,	O
helps	O
find	O
formal	O
proofs	O
,	O
and	O
extracts	O
a	O
certified	O
program	O
from	O
the	O
constructive	O
proof	O
of	O
its	O
formal	B-Application
specification	I-Application
.	O
</s>
<s>
Coq	B-Application
works	O
within	O
the	O
theory	O
of	O
the	O
calculus	B-Application
of	I-Application
inductive	I-Application
constructions	I-Application
,	O
a	O
derivative	O
of	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
.	O
</s>
<s>
Coq	B-Application
is	O
not	O
an	O
automated	B-Application
theorem	I-Application
prover	I-Application
but	O
includes	O
automatic	B-Application
theorem	I-Application
proving	I-Application
tactics	O
(	O
procedures	O
)	O
and	O
various	O
decision	O
procedures	O
.	O
</s>
<s>
The	O
Association	O
for	O
Computing	O
Machinery	O
awarded	O
Thierry	O
Coquand	O
,	O
Gérard	O
Huet	O
,	O
Christine	O
Paulin-Mohring	O
,	O
Bruno	O
Barras	O
,	O
Jean-Christophe	O
Filliâtre	O
,	O
Hugo	O
Herbelin	O
,	O
Chetan	O
Murthy	O
,	O
Yves	O
Bertot	O
,	O
and	O
Pierre	O
Castéran	O
with	O
the	O
2013	O
ACM	O
Software	O
System	O
Award	O
for	O
Coq	B-Application
.	O
</s>
<s>
The	O
name	O
"	O
Coq	B-Application
"	O
is	O
a	O
wordplay	O
on	O
the	O
name	O
of	O
Thierry	O
Coquand	O
,	O
Calculus	B-Application
of	I-Application
Constructions	I-Application
or	O
"	O
CoC	O
"	O
and	O
follows	O
the	O
French	O
computer	O
science	O
tradition	O
of	O
naming	O
software	O
after	O
animals	O
(	O
coq	B-Application
in	O
French	O
meaning	O
rooster	O
)	O
.	O
</s>
<s>
When	O
viewed	O
as	O
a	O
programming	O
language	O
,	O
Coq	B-Application
implements	O
a	O
dependently	O
typed	O
functional	B-Language
programming	I-Language
language	I-Language
;	O
when	O
viewed	O
as	O
a	O
logical	O
system	O
,	O
it	O
implements	O
a	O
higher-order	B-Algorithm
type	O
theory	O
.	O
</s>
<s>
The	O
development	O
of	O
Coq	B-Application
has	O
been	O
supported	O
since	O
1984	O
by	O
INRIA	O
,	O
now	O
in	O
collaboration	O
with	O
École	O
Polytechnique	O
,	O
University	O
of	O
Paris-Sud	O
,	O
Paris	O
Diderot	O
University	O
,	O
and	O
CNRS	O
.	O
</s>
<s>
The	O
development	O
of	O
Coq	B-Application
was	O
initiated	O
by	O
Gérard	O
Huet	O
and	O
Thierry	O
Coquand	O
,	O
and	O
more	O
than	O
40	O
people	O
,	O
mainly	O
researchers	O
,	O
have	O
contributed	O
features	O
to	O
the	O
core	O
system	O
since	O
its	O
inception	O
.	O
</s>
<s>
Coq	B-Application
is	O
mainly	O
implemented	O
in	O
OCaml	B-Language
with	O
a	O
bit	O
of	O
C	B-Language
.	O
The	O
core	O
system	O
can	O
be	O
extended	O
by	O
way	O
of	O
a	O
plug-in	B-Application
mechanism	O
.	O
</s>
<s>
Up	O
until	O
1991	O
,	O
Coquand	O
was	O
implementing	O
a	O
language	O
called	O
the	O
Calculus	B-Application
of	I-Application
Constructions	I-Application
and	O
it	O
was	O
simply	O
called	O
CoC	O
at	O
this	O
time	O
.	O
</s>
<s>
In	O
1991	O
,	O
a	O
new	O
implementation	O
based	O
on	O
the	O
extended	O
Calculus	B-Application
of	I-Application
Inductive	I-Application
Constructions	I-Application
was	O
started	O
and	O
the	O
name	O
was	O
changed	O
from	O
CoC	O
to	O
Coq	B-Application
in	O
an	O
indirect	O
reference	O
to	O
Coquand	O
,	O
who	O
developed	O
the	O
Calculus	B-Application
of	I-Application
Constructions	I-Application
along	O
with	O
Gérard	O
Huet	O
and	O
contributed	O
to	O
the	O
Calculus	B-Application
of	I-Application
Inductive	I-Application
Constructions	I-Application
with	O
Christine	O
Paulin-Mohring	O
.	O
</s>
<s>
Coq	B-Application
provides	O
a	O
specification	B-Application
language	O
called	O
Gallina	O
(	O
"	O
hen	O
"	O
in	O
Latin	O
,	O
Spanish	O
,	O
Italian	O
and	O
Catalan	O
)	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
a	O
proof	O
of	O
commutativity	O
of	O
addition	O
on	O
natural	O
numbers	O
in	O
Coq	B-Application
:	O
</s>
<s>
stands	O
for	O
mathematical	B-Algorithm
induction	I-Algorithm
,	O
for	O
substitution	O
of	O
equals	O
,	O
and	O
for	O
taking	O
the	O
same	O
function	O
on	O
both	O
sides	O
of	O
the	O
equality	O
.	O
</s>
<s>
Georges	O
Gonthier	O
of	O
Microsoft	O
Research	O
in	O
Cambridge	O
,	O
England	O
and	O
Benjamin	O
Werner	O
of	O
INRIA	O
used	O
Coq	B-Application
to	O
create	O
a	O
surveyable	O
proof	O
of	O
the	O
four	O
color	O
theorem	O
,	O
which	O
was	O
completed	O
in	O
2002	O
.	O
</s>
<s>
Their	O
work	O
led	O
to	O
the	O
development	O
of	O
the	O
SSReflect	O
(	O
"	O
Small	O
Scale	O
Reflection	O
"	O
)	O
package	O
,	O
which	O
was	O
a	O
significant	O
extension	O
to	O
Coq	B-Application
.	O
</s>
<s>
Despite	O
its	O
name	O
,	O
most	O
of	O
the	O
features	O
added	O
to	O
Coq	B-Application
by	O
SSReflect	O
are	O
general-purpose	O
features	O
and	O
are	O
not	O
limited	O
to	O
the	O
computational	O
reflection	O
style	O
of	O
proof	O
.	O
</s>
<s>
SSReflect	O
1.11	O
is	O
freely	O
available	O
,	O
dual-licensed	O
under	O
the	O
open	O
source	O
CeCILL-B	B-License
or	O
CeCILL-2.0	O
license	O
,	O
and	O
compatible	O
with	O
Coq	B-Application
8.11	O
.	O
</s>
<s>
CompCert	B-Application
:	O
an	O
optimizing	O
compiler	O
for	O
almost	O
all	O
of	O
the	O
C	B-Language
programming	I-Language
language	I-Language
which	O
is	O
largely	O
programmed	O
and	O
proven	O
correct	O
in	O
Coq	B-Application
.	O
</s>
<s>
Disjoint-set	O
data	O
structure	O
:	O
correctness	O
proof	O
in	O
Coq	B-Application
was	O
published	O
in	O
2007	O
.	O
</s>
<s>
Feit	O
–	O
Thompson	O
theorem	O
:	O
formal	O
proof	O
using	O
Coq	B-Application
was	O
completed	O
in	O
September	O
2012	O
.	O
</s>
