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Felipe Cerqueira
rtproofs
Commits
76263149
Commit
76263149
authored
Nov 03, 2016
by
Felipe Cerqueira
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Add more lemmas about zip
parent
af7806cd
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util/list.v
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76263149
...
...
@@ 228,6 +228,60 @@ Section Zip.
}
Qed
.
Lemma
unzip1_pair
:
forall
{
T1
T2
:
eqType
}
(
l
:
seq
T1
)
(
f
:
T1
>
T2
),
unzip1
[
seq
(
x
,
f
x
)

x
<
l
]
=
l
.
Proof
.
intros
T1
T2
.
induction
l
;
first
by
done
.
by
intros
f
;
simpl
;
f_equal
.
Qed
.
Lemma
unzip2_pair
:
forall
{
T1
T2
:
eqType
}
(
l
:
seq
T1
)
(
f
:
T1
>
T2
),
unzip2
[
seq
(
f
x
,
x
)

x
<
l
]
=
l
.
Proof
.
intros
T1
T2
.
induction
l
;
first
by
done
.
by
intros
f
;
simpl
;
f_equal
.
Qed
.
Lemma
eq_unzip1
:
forall
{
T1
T2
:
eqType
}
(
l1
l2
:
seq
(
T1
*
T2
))
x0
,
size
l1
=
size
l2
>
(
forall
i
,
i
<
size
l1
>
(
fst
(
nth
x0
l1
i
))
=
(
fst
(
nth
x0
l2
i
)))
>
unzip1
l1
=
unzip1
l2
.
Proof
.
intros
T1
T2
.
induction
l1
;
simpl
;
first
by
destruct
l2
.
intros
l2
x0
SIZE
ALL
.
destruct
l2
;
first
by
done
.
simpl
;
f_equal
;
first
by
feed
(
ALL
0
).
case:
SIZE
=>
SIZE
.
apply
IHl1
with
(
x0
:=
x0
);
first
by
done
.
intros
i
LTi
.
by
feed
(
ALL
i
.
+
1
);
first
by
rewrite

[
X
in
X
<
_
]
addn1

[
X
in
_
<
X
]
addn1
ltn_add2r
.
Qed
.
Lemma
eq_unzip2
:
forall
{
T1
T2
:
eqType
}
(
l1
l2
:
seq
(
T1
*
T2
))
x0
,
size
l1
=
size
l2
>
(
forall
i
,
i
<
size
l1
>
(
snd
(
nth
x0
l1
i
))
=
(
snd
(
nth
x0
l2
i
)))
>
unzip2
l1
=
unzip2
l2
.
Proof
.
intros
T1
T2
.
induction
l1
;
simpl
;
first
by
destruct
l2
.
intros
l2
x0
SIZE
ALL
.
destruct
l2
;
first
by
done
.
simpl
;
f_equal
;
first
by
feed
(
ALL
0
).
case:
SIZE
=>
SIZE
.
apply
IHl1
with
(
x0
:=
x0
);
first
by
done
.
intros
i
LTi
.
by
feed
(
ALL
i
.
+
1
);
first
by
rewrite

[
X
in
X
<
_
]
addn1

[
X
in
_
<
X
]
addn1
ltn_add2r
.
Qed
.
End
Zip
.
(
*
Restate
nth_error
function
from
Coq
library
.
*
)
...
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