<s>
In	O
mathematical	O
logic	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
(	O
CoC	O
)	O
is	O
a	O
type	O
theory	O
created	O
by	O
Thierry	O
Coquand	O
.	O
</s>
<s>
For	O
this	O
second	O
reason	O
,	O
the	O
CoC	O
and	O
its	O
variants	O
have	O
been	O
the	O
basis	O
for	O
Coq	B-Application
and	O
other	O
proof	O
assistants	O
.	O
</s>
<s>
Some	O
of	O
its	O
variants	O
include	O
the	O
calculus	B-Application
of	I-Application
inductive	I-Application
constructions	I-Application
(	O
which	O
adds	O
inductive	O
types	O
)	O
,	O
the	O
calculus	O
of	O
(	O
co	O
)	O
inductive	O
constructions	O
(	O
which	O
adds	O
coinduction	B-Application
)	O
,	O
and	O
the	O
predicative	O
calculus	B-Application
of	I-Application
inductive	I-Application
constructions	I-Application
(	O
which	O
removes	O
some	O
impredicativity	O
)	O
.	O
</s>
<s>
It	O
is	O
well	O
known	O
for	O
being	O
at	O
the	O
top	O
of	O
Barendregt	O
's	O
lambda	B-Application
cube	I-Application
.	O
</s>
<s>
The	O
CoC	O
has	O
been	O
developed	O
alongside	O
the	O
Coq	B-Application
proof	I-Application
assistant	I-Application
.	O
</s>
<s>
As	O
features	O
were	O
added	O
(	O
or	O
possible	O
liabilities	O
removed	O
)	O
to	O
the	O
theory	O
,	O
they	O
became	O
available	O
in	O
Coq	B-Application
.	O
</s>
<s>
Variants	O
of	O
the	O
CoC	O
are	O
used	O
in	O
other	O
proof	O
assistants	O
,	O
such	O
as	O
Matita	B-Application
.	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
can	O
be	O
considered	O
an	O
extension	O
of	O
the	O
Curry	O
–	O
Howard	O
isomorphism	O
.	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
extends	O
this	O
isomorphism	O
to	O
proofs	O
in	O
the	O
full	O
intuitionistic	O
predicate	O
calculus	O
,	O
which	O
includes	O
proofs	O
of	O
quantified	O
statements	O
(	O
which	O
we	O
will	O
also	O
call	O
"	O
propositions	O
"	O
)	O
.	O
</s>
<s>
A	O
term	O
in	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
is	O
constructed	O
using	O
the	O
following	O
rules	O
:	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
has	O
five	O
kinds	O
of	O
objects	O
:	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
allows	O
proving	O
typing	O
judgments	O
:	O
</s>
<s>
The	O
valid	O
judgments	O
for	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
are	O
derivable	O
from	O
a	O
set	O
of	O
inference	O
rules	O
.	O
</s>
<s>
The	O
calculus	B-Application
of	I-Application
constructions	I-Application
has	O
very	O
few	O
basic	O
operators	O
:	O
the	O
only	O
logical	O
operator	O
for	O
forming	O
propositions	O
is	O
.	O
</s>
<s>
The	O
basic	O
data	O
types	O
used	O
in	O
computer	B-General_Concept
science	I-General_Concept
can	O
be	O
defined	O
within	O
the	O
calculus	B-Application
of	I-Application
constructions	I-Application
:	O
</s>
<s>
Note	O
that	O
Booleans	O
and	O
Naturals	O
are	O
defined	O
in	O
the	O
same	O
way	O
as	O
in	O
Church	B-Application
encoding	I-Application
.	O
</s>
