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Marco Maida
PROSA  Formally Proven Schedulability Analysis
Commits
cc47990f
Commit
cc47990f
authored
Jan 13, 2016
by
Felipe Cerqueira
Browse files
Add lemmas about big_cat_ord
parent
afb151ae
Changes
1
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Inline
Sidebyside
util_lemmas.v
View file @
cc47990f
...
...
@@ 26,6 +26,13 @@ Reserved Notation "\cat_ ( m <= i < n ) F"
Notation
"\cat_ ( m <= i < n ) F"
:
=
(
\
big
[
cat
/[
::
]]
_
(
m
<=
i
<
n
)
F
%
N
)
:
nat_scope
.
Reserved
Notation
"\cat_ ( m <= i < n  P ) F"
(
at
level
41
,
F
at
level
41
,
P
at
level
41
,
i
,
m
,
n
at
level
50
,
format
"'[' \cat_ ( m <= i < n  P ) '/ ' F ']'"
).
Notation
"\cat_ ( m <= i < n  P ) F"
:
=
(
\
big
[
cat
/[
::
]]
_
(
m
<=
i
<
n

P
)
F
%
N
)
:
nat_scope
.
Reserved
Notation
"\sum_ ( ( m , n ) < r ) F"
(
at
level
41
,
F
at
level
41
,
m
,
n
at
level
50
,
format
"'[' \sum_ ( ( m , n ) < r ) '/ ' F ']'"
).
...
...
@@ 68,6 +75,13 @@ Reserved Notation "\cat_ ( i < n ) F"
Notation
"\cat_ ( i < n ) F"
:
=
(
\
big
[
cat
/[
::
]]
_
(
i
<
n
)
F
%
N
)
:
nat_scope
.
Reserved
Notation
"\cat_ ( i < n  P ) F"
(
at
level
41
,
F
at
level
41
,
i
,
n
at
level
50
,
format
"'[' \cat_ ( i < n  P ) '/ ' F ']'"
).
Notation
"\cat_ ( i < n  P ) F"
:
=
(
\
big
[
cat
/[
::
]]
_
(
i
<
n

P
)
F
%
N
)
:
nat_scope
.
Reserved
Notation
"x \In A"
(
at
level
70
,
format
"'[hv' x '/ ' \In A ']'"
,
no
associativity
).
...
...
@@ 147,6 +161,59 @@ Proof.
[
by
apply
/
andP
;
split

by
rewrite
eq_fun_ord_to_nat
].
Qed
.
Lemma
mem_bigcat_ord_exists
:
forall
(
T
:
eqType
)
x
n
(
f
:
'
I_n
>
list
T
),
x
\
in
\
cat_
(
i
<
n
)
(
f
i
)
>
exists
i
,
x
\
in
(
f
i
).
Proof
.
intros
T
x
n
f
IN
.
induction
n
;
first
by
rewrite
big_ord0
in_nil
in
IN
.
{
rewrite
big_ord_recr
/=
mem_cat
in
IN
.
move
:
IN
=>
/
orP
[
HEAD

TAIL
].
{
apply
IHn
in
HEAD
;
destruct
HEAD
.
by
eexists
(
widen_ord
_
x0
)
;
desf
.
}
{
by
exists
ord_max
;
desf
.
}
}
Qed
.
Lemma
bigcat_ord_uniq
:
forall
(
T
:
eqType
)
n
(
f
:
'
I_n
>
list
T
),
(
forall
i
,
uniq
(
f
i
))
>
(
forall
x
i1
i2
,
x
\
in
(
f
i1
)
>
x
\
in
(
f
i2
)
>
i1
=
i2
)
>
uniq
(
\
cat_
(
i
<
n
)
(
f
i
)).
Proof
.
intros
T
n
f
SINGLE
UNIQ
.
induction
n
;
first
by
rewrite
big_ord0
.
{
rewrite
big_ord_recr
cat_uniq
;
apply
/
andP
;
split
.
{
apply
IHn
;
first
by
done
.
intros
x
i1
i2
IN1
IN2
.
exploit
(
UNIQ
x
)
;
[
by
apply
IN1

by
apply
IN2

intro
EQ
;
inversion
EQ
].
by
apply
ord_inj
.
}
apply
/
andP
;
split
;
last
by
apply
SINGLE
.
{
rewrite

all_predC
;
apply
/
allP
;
intros
x
INx
.
simpl
;
apply
/
negP
;
unfold
not
;
intro
BUG
.
rewrite

big_ord_narrow
in
BUG
.
rewrite
big_mkcond
/=
in
BUG
.
have
EX
:
=
mem_bigcat_ord_exists
T
x
n
.+
1
_
.
apply
EX
in
BUG
;
clear
EX
;
desf
.
apply
UNIQ
with
(
i1
:
=
ord_max
)
in
BUG
;
last
by
done
.
by
desf
;
unfold
ord_max
in
*
;
rewrite
/=
ltnn
in
Heq
.
}
}
Qed
.
Lemma
addnb
(
b1
b2
:
bool
)
:
(
b1
+
b2
)
!=
0
=
b1

b2
.
Proof
.
by
destruct
b1
,
b2
;
...
...
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