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POTTIER Francois
mpri2.4public
Commits
c0c83a49
Commit
c0c83a49
authored
Oct 20, 2017
by
POTTIER Francois
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Add a demo of equational reasoning in Coq.
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DemoEqReasoning.v
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Require
Import
List
.
Section
Demo
.
(
*

*
)
Variables
A
B
:
Type
.
Variable
p
:
B
>
bool
.
Variable
f
:
A
>
B
.
(
*
The
composition
of
[
filter
]
and
[
map
]
can
be
computed
by
the
specialized
function
[
filter_map
]
.
*
)
Fixpoint
filter_map
xs
:=
match
xs
with

nil
=>
nil

cons
x
xs
=>
let
y
:=
f
x
in
if
p
y
then
y
::
filter_map
xs
else
filter_map
xs
end
.
Lemma
filter_map_spec
:
forall
xs
,
filter
p
(
map
f
xs
)
=
filter_map
xs
.
Proof
.
induction
xs
as
[

x
xs
]
;
simpl
.
{
reflexivity
.
}
{
rewrite
IHxs
.
reflexivity
.
}
Qed
.
(
*

*
)
(
*
[
filter
]
and
[
map
]
commute
in
a
certain
sense
.
*
)
Variable
q
:
A
>
bool
.
Lemma
filter_map_commute
:
(
forall
x
,
p
(
f
x
)
=
q
x
)
>
forall
xs
,
filter
p
(
map
f
xs
)
=
map
f
(
filter
q
xs
)
.
Proof
.
intros
h
.
induction
xs
as
[

x
xs
]
;
simpl
;
intros
.
(
*
Case
:
[
nil
]
.
*
)
{
reflexivity
.
}
(
*
Case
:
[
x
::
xs
]
.
*
)
{
rewrite
h
.
rewrite
IHxs
.
(
*
Case
analysis
:
[
q
x
]
is
either
true
or
false
.
In
either
case
,
the
result
is
immediate
.
*
)
destruct
(
q
x
)
;
reflexivity
.
}
Qed
.
(
*
In
a
slightly
stronger
version
of
the
lemma
,
the
equality
[
p
(
f
x
)
=
q
x
]
needs
to
be
proved
only
under
the
hypothesis
that
[
x
]
is
an
element
of
the
list
[
xs
]
.
*
)
Lemma
filter_map_commute_stronger
:
forall
xs
,
(
forall
x
,
In
x
xs
>
p
(
f
x
)
=
q
x
)
>
filter
p
(
map
f
xs
)
=
map
f
(
filter
q
xs
)
.
Proof
.
induction
xs
as
[

x
xs
]
;
simpl
;
intro
h
.
{
reflexivity
.
}
{
(
*
The
proof
is
the
same
as
above
,
except
the
two
rewriting
steps
have
side
conditions
,
which
are
immediately
proved
by
[
eauto
]
.
*
)
rewrite
h
by
eauto
.
rewrite
IHxs
by
eauto
.
destruct
(
q
x
)
;
reflexivity
.
}
Qed
.
End
Demo
.
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