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c2064b60
Commit
c2064b60
authored
Sep 30, 2008
by
Adam Chlipala
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Make exercises display properly in HTML
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Predicates.v
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src/Predicates.v
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c2064b60
...
...
@@ -866,12 +866,10 @@ Qed.
%
\
item
%
#
<
li
>
#
[(
True
\
/
False
)
/
\
(
False
\
/
True
)]#
</
li
>
#
%
\
item
%
#
<
li
>
#
[
P
->
~
~
P
]#
</
li
>
#
%
\
item
%
#
<
li
>
#
[
P
/
\
(
Q
\
/
R
)
->
(
P
/
\
Q
)
\
/
(
P
/
\
R
)]#
</
li
>
#
#
</
ol
>
</
li
>
#
%
\
end
{
enumerate
}%
*
)
#
</
ol
>
</
li
>
#
%
\
end
{
enumerate
}%
(
**
remove
printing
exists
*
)
(
**
%
\
item
%
#
<
li
>
#
Prove
the
following
tautology
of
first
-
order
logic
,
using
only
the
tactics
[
apply
]
,
[
assert
]
,
[
assumption
]
,
[
destruct
]
,
[
eapply
]
,
[
eassumption
]
,
and
[
exists
]
.
You
will
probably
find
[
assert
]
useful
for
stating
and
proving
an
intermediate
lemma
,
enabling
a
kind
of
"forward reasoning,"
in
contrast
to
the
"backward reasoning"
that
is
the
default
for
Coq
tactics
.
[
eassumption
]
is
a
version
of
[
assumption
]
that
will
do
matching
of
unification
variables
.
Let
some
variable
[
T
]
of
type
[
Set
]
be
the
set
of
individuals
.
[
x
]
is
a
constant
symbol
,
[
p
]
is
a
unary
predicate
symbol
,
[
q
]
is
a
binary
predicate
symbol
,
and
[
f
]
is
a
unary
function
symbol
.
**
)
(
**
printing
exists
$
\
exists
$
*
)
(
**
%
\
begin
{
enumerate
}%
#
<
ol
>
#
%
\
item
%
#
<
li
>
#
Prove
the
following
tautology
of
first
-
order
logic
,
using
only
the
tactics
[
apply
]
,
[
assert
]
,
[
assumption
]
,
[
destruct
]
,
[
eapply
]
,
[
eassumption
]
,
and
%
\
textit
{%
#
<
tt
>
#
exists
#
</
tt
>
#
%}%.
You
will
probably
find
[
assert
]
useful
for
stating
and
proving
an
intermediate
lemma
,
enabling
a
kind
of
"forward reasoning,"
in
contrast
to
the
"backward reasoning"
that
is
the
default
for
Coq
tactics
.
[
eassumption
]
is
a
version
of
[
assumption
]
that
will
do
matching
of
unification
variables
.
Let
some
variable
[
T
]
of
type
[
Set
]
be
the
set
of
individuals
.
[
x
]
is
a
constant
symbol
,
[
p
]
is
a
unary
predicate
symbol
,
[
q
]
is
a
binary
predicate
symbol
,
and
[
f
]
is
a
unary
function
symbol
.
%
\
begin
{
enumerate
}%
#
<
ol
>
#
%
\
item
%
#
<
li
>
#
[
p
x
->
(
forall
x
,
p
x
->
exists
y
,
q
x
y
)
->
(
forall
x
y
,
q
x
y
->
q
y
(
f
y
))
->
exists
z
,
q
z
(
f
z
)]#
</
li
>
#
#
</
ol
>
</
li
>
#
%
\
end
{
enumerate
}%
...
...
@@ -892,7 +890,6 @@ Qed.
%
\
item
%
#
<
li
>
#
Prove
that
any
expression
that
has
type
[
t
]
under
variable
typing
[
vt
]
evaluates
under
variable
assignment
[
va
]
to
some
value
that
also
has
type
[
t
]
in
[
vt
]
,
as
long
as
[
va
]
and
[
vt
]
agree
.
#
</
li
>
#
%
\
item
%
#
<
li
>
#
Prove
that
any
command
that
has
type
[
t
]
under
variable
typing
[
vt
]
evaluates
under
variable
assignment
[
va
]
to
some
value
that
also
has
type
[
t
]
in
[
vt
]
,
as
long
as
[
va
]
and
[
vt
]
agree
.
#
</
li
>
#
#
</
ol
>
</
li
>
#
%
\
end
{
enumerate
}%
A
few
hints
that
may
be
helpful
:
%
\
begin
{
enumerate
}%
#
<
ol
>
#
%
\
item
%
#
<
li
>
#
One
easy
way
of
defining
variable
assignments
and
typings
is
to
define
both
as
instances
of
a
polymorphic
map
type
.
The
map
type
at
parameter
[
T
]
can
be
defined
to
be
the
type
of
arbitrary
functions
from
variables
to
[
T
]
.
A
helpful
function
for
implementing
insertion
into
such
a
functional
map
is
[
eq_nat_dec
]
,
which
you
can
make
available
with
[
Require
Import
Arith
.
]
.
[
eq_nat_dec
]
has
a
dependent
type
that
tells
you
that
it
makes
accurate
decisions
on
whether
two
natural
numbers
are
equal
,
but
you
can
use
it
as
if
it
returned
a
boolean
,
e
.
g
.,
[
if
eq_nat_dec
n
m
then
E1
else
E2
]
.
#
</
li
>
#
...
...
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