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Vedant Chavda
PROSA  Formally Proven Schedulability Analysis
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7dc8adcc
Commit
7dc8adcc
authored
Feb 10, 2020
by
Marco Maida
🌱
Committed by
Björn Brandenburg
Feb 10, 2020
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util/bigcat.v
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7dc8adcc
Require
Export
prosa
.
util
.
tactics
prosa
.
util
.
notation
.
From
mathcomp
Require
Import
ssreflect
ssrbool
eqtype
ssrnat
seq
fintype
bigop
.
(* Lemmas about the big concatenation operator. *)
(** In this section, we introduce useful lemmas about the concatenation operation performed
over an arbitrary range of sequences. *)
Section
BigCatLemmas
.
Lemma
mem_bigcat_nat
:
forall
(
T
:
eqType
)
x
m
n
j
(
f
:
_
>
list
T
),
m
<=
j
<
n
>
x
\
in
(
f
j
)
>
x
\
in
\
cat_
(
m
<=
i
<
n
)
(
f
i
).
Proof
.
intros
T
x
m
n
j
f
LE
IN
;
move
:
LE
=>
/
andP
[
LE
LE0
].
rewrite
>
big_cat_nat
with
(
n
:
=
j
)
;
simpl
;
[
by
ins

by
apply
ltnW
].
rewrite
mem_cat
;
apply
/
orP
;
right
.
destruct
n
;
first
by
rewrite
ltn0
in
LE0
.
rewrite
big_nat_recl
;
last
by
ins
.
by
rewrite
mem_cat
;
apply
/
orP
;
left
.
Qed
.
(** Consider any type supporting equality comparisons... *)
Variable
T
:
eqType
.
Lemma
mem_bigcat_nat_exists
:
forall
(
T
:
eqType
)
x
m
n
(
f
:
nat
>
list
T
),
x
\
in
\
cat_
(
m
<=
i
<
n
)
(
f
i
)
>
exists
i
,
x
\
in
(
f
i
)
/\
m
<=
i
<
n
.
Proof
.
intros
T
x
m
n
f
IN
.
induction
n
;
first
by
rewrite
big_geq
//
in
IN
.
destruct
(
leqP
m
n
)
;
last
by
rewrite
big_geq
?in_nil
//
ltnW
in
IN
.
rewrite
big_nat_recr
//
/=
mem_cat
in
IN
.
move
:
IN
=>
/
orP
[
HEAD

TAIL
].
{
apply
IHn
in
HEAD
;
destruct
HEAD
;
exists
x0
.
move
:
H
=>
[
H
/
andP
[
H0
H1
]].
split
;
first
by
done
.
by
apply
/
andP
;
split
;
[
by
done

by
apply
ltnW
].
}
{
exists
n
;
split
;
first
by
done
.
apply
/
andP
;
split
;
[
by
done

by
apply
ltnSn
].
}
Qed
.
(** ...and a function that, given an index, yields a sequence. *)
Variable
f
:
nat
>
list
T
.
Lemma
bigcat_nat_uniq
:
forall
(
T
:
eqType
)
n1
n2
(
F
:
nat
>
list
T
),
(
forall
i
,
uniq
(
F
i
))
>
(
forall
x
i1
i2
,
x
\
in
(
F
i1
)
>
x
\
in
(
F
i2
)
>
i1
=
i2
)
>
uniq
(
\
cat_
(
n1
<=
i
<
n2
)
(
F
i
)).
Proof
.
intros
T
n1
n2
f
SINGLE
UNIQ
.
case
(
leqP
n1
n2
)
=>
[
LE

GT
]
;
last
by
rewrite
big_geq
//
ltnW
.
rewrite
[
n2
](
addKn
n1
).
rewrite

addnBA
//
;
set
delta
:
=
n2

n1
.
induction
delta
;
first
by
rewrite
addn0
big_geq
.
rewrite
addnS
big_nat_recr
/=
;
last
by
apply
leq_addr
.
rewrite
cat_uniq
;
apply
/
andP
;
split
;
first
by
apply
IHdelta
.
apply
/
andP
;
split
;
last
by
apply
SINGLE
.
rewrite

all_predC
;
apply
/
allP
;
intros
x
INx
.
simpl
;
apply
/
negP
;
unfold
not
;
intro
BUG
.
apply
mem_bigcat_nat_exists
in
BUG
.
move
:
BUG
=>
[
i
[
IN
/
andP
[
_
LTi
]]].
apply
UNIQ
with
(
i1
:
=
i
)
in
INx
;
last
by
done
.
by
rewrite
INx
ltnn
in
LTi
.
Qed
.
(** In this section, we prove that the concatenation over sequences works as expected:
no element is lost during the concatenation, and no new element is introduced. *)
Section
BigCatElements
.
(** First, we show that the concatenation comprises all the elements of each sequence;
i.e. any element contained in one of the sequences will also be an element of the
result of the concatenation. *)
Lemma
mem_bigcat_nat
:
forall
x
m
n
j
,
m
<=
j
<
n
>
x
\
in
f
j
>
x
\
in
\
cat_
(
m
<=
i
<
n
)
(
f
i
).
Proof
.
intros
x
m
n
j
LE
IN
;
move
:
LE
=>
/
andP
[
LE
LE0
].
rewrite
>
big_cat_nat
with
(
n
:
=
j
)
;
simpl
;
[
by
ins

by
apply
ltnW
].
rewrite
mem_cat
;
apply
/
orP
;
right
.
destruct
n
;
first
by
rewrite
ltn0
in
LE0
.
rewrite
big_nat_recl
;
last
by
ins
.
by
rewrite
mem_cat
;
apply
/
orP
;
left
.
Qed
.
(** Conversely, we prove that any element belonging to a concatenation of sequences
must come from one of the sequences. *)
Lemma
mem_bigcat_nat_exists
:
forall
x
m
n
,
x
\
in
\
cat_
(
m
<=
i
<
n
)
(
f
i
)
>
exists
i
,
x
\
in
f
i
/\
m
<=
i
<
n
.
Proof
.
intros
x
m
n
IN
.
induction
n
;
first
by
rewrite
big_geq
//
in
IN
.
destruct
(
leqP
m
n
)
;
last
by
rewrite
big_geq
?in_nil
//
ltnW
in
IN
.
rewrite
big_nat_recr
//
/=
mem_cat
in
IN
.
move
:
IN
=>
/
orP
[
HEAD

TAIL
].
{
apply
IHn
in
HEAD
;
destruct
HEAD
;
exists
x0
.
move
:
H
=>
[
H
/
andP
[
H0
H1
]].
split
;
first
by
done
.
by
apply
/
andP
;
split
;
[
by
done

by
apply
ltnW
].
}
{
exists
n
;
split
;
first
by
done
.
apply
/
andP
;
split
;
[
by
done

by
apply
ltnSn
].
}
Qed
.
End
BigCatElements
.
(** In this section, we show how we can preserve uniqueness of the elements
(i.e. the absence of a duplicate) over a concatenation of sequences. *)
Section
BigCatDistinctElements
.
(** Assume that there are no duplicates in each of the possible
sequences to concatenate... *)
Hypothesis
H_uniq_seq
:
forall
i
,
uniq
(
f
i
).
(** ...and that there are no elements in common between the sequences. *)
Hypothesis
H_no_elements_in_common
:
forall
x
i1
i2
,
x
\
in
f
i1
>
x
\
in
f
i2
>
i1
=
i2
.
(** We prove that the concatenation will yield a sequence with unique elements. *)
Lemma
bigcat_nat_uniq
:
forall
n1
n2
,
uniq
(
\
cat_
(
n1
<=
i
<
n2
)
(
f
i
)).
Proof
.
intros
n1
n2
.
case
(
leqP
n1
n2
)
=>
[
LE

GT
]
;
last
by
rewrite
big_geq
//
ltnW
.
rewrite
[
n2
](
addKn
n1
).
rewrite

addnBA
//
;
set
delta
:
=
n2

n1
.
induction
delta
;
first
by
rewrite
addn0
big_geq
.
rewrite
addnS
big_nat_recr
/=
;
last
by
apply
leq_addr
.
rewrite
cat_uniq
;
apply
/
andP
;
split
;
first
by
apply
IHdelta
.
apply
/
andP
;
split
;
last
by
apply
H_uniq_seq
.
rewrite

all_predC
;
apply
/
allP
;
intros
x
INx
.
simpl
;
apply
/
negP
;
unfold
not
;
intro
BUG
.
apply
mem_bigcat_nat_exists
in
BUG
.
move
:
BUG
=>
[
i
[
IN
/
andP
[
_
LTi
]]].
apply
H_no_elements_in_common
with
(
i1
:
=
i
)
in
INx
;
last
by
done
.
by
rewrite
INx
ltnn
in
LTi
.
Qed
.
End
BigCatDistinctElements
.
End
BigCatLemmas
.
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