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Felipe Cerqueira
rtproofs
Commits
a5cc6b86
Commit
a5cc6b86
authored
Mar 31, 2016
by
Felipe Cerqueira
Browse files
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Browse Files
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Plain Diff
Cleanup Vbase tactics
parent
266c9014
Changes
18
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Showing
18 changed files
with
113 additions
and
447 deletions
+113
447
analysis/basic/bertogna_edf_comp.v
analysis/basic/bertogna_edf_comp.v
+1
3
analysis/basic/bertogna_fp_comp.v
analysis/basic/bertogna_fp_comp.v
+1
2
analysis/basic/interference_bound_edf.v
analysis/basic/interference_bound_edf.v
+7
9
analysis/basic/workload_bound.v
analysis/basic/workload_bound.v
+5
6
analysis/jitter/bertogna_edf_comp.v
analysis/jitter/bertogna_edf_comp.v
+1
3
analysis/jitter/bertogna_fp_comp.v
analysis/jitter/bertogna_fp_comp.v
+1
2
analysis/jitter/interference_bound_edf.v
analysis/jitter/interference_bound_edf.v
+6
6
analysis/jitter/workload_bound.v
analysis/jitter/workload_bound.v
+4
4
analysis/parallel/bertogna_edf_comp.v
analysis/parallel/bertogna_edf_comp.v
+1
3
analysis/parallel/interference_bound_edf.v
analysis/parallel/interference_bound_edf.v
+2
2
analysis/parallel/workload_bound.v
analysis/parallel/workload_bound.v
+3
3
implementation/basic/arrival_sequence.v
implementation/basic/arrival_sequence.v
+15
15
implementation/jitter/arrival_sequence.v
implementation/jitter/arrival_sequence.v
+15
15
model/basic/arrival_sequence.v
model/basic/arrival_sequence.v
+0
3
model/basic/interference.v
model/basic/interference.v
+2
2
model/jitter/interference.v
model/jitter/interference.v
+10
12
util/Vbase.v
util/Vbase.v
+35
352
util/lemmas.v
util/lemmas.v
+4
5
No files found.
analysis/basic/bertogna_edf_comp.v
View file @
a5cc6b86
...
...
@@ 238,8 +238,7 @@ Module ResponseTimeIterationEDF.
Proof
.
intros
l
;
unfold
all_le
;
rewrite
eq_refl
andTb
.
destruct
l
;
first
by
done
.
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
));
try
(
by
ins
).
by
ins
;
apply
leqnn
.
by
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
)).
Qed
.
(
*
...
and
transitive
.
*
)
...
...
@@ 507,7 +506,6 @@ Module ResponseTimeIterationEDF.
try
(
by
done
);
[
by
apply
all_le_reflexive

by
apply
all_le_transitive

by
apply
leqnSn

by
apply
bertogna_edf_comp_iteration_preserves_order

by
apply
bertogna_edf_comp_iteration_preserves_minimum
].
Qed
.
...
...
analysis/basic/bertogna_fp_comp.v
View file @
a5cc6b86
...
...
@@ 402,8 +402,7 @@ Module ResponseTimeIterationFP.
{
apply
leq_trans
with
(
n
:=
W
task_cost
task_period
i
R
x1
);
first
by
apply
geq_minl
.
exploit
(
VALID
i
);
[
by
rewrite
mem_rcons
in_cons
IN
orbT

ins
;
des
].
by
apply
W_monotonic
;
try
(
by
ins
);
[
by
apply
GE_COST

by
apply
leqnn
].
by
apply
W_monotonic
;
try
by
ins
;
apply
GE_COST
.
}
{
apply
leq_trans
with
(
n
:=
x1

task_cost
tsk
+
1
);
first
by
apply
geq_minr
.
...
...
analysis/basic/interference_bound_edf.v
View file @
a5cc6b86
...
...
@@ 497,9 +497,8 @@ Module InterferenceBoundEDF.
by
apply
job_interference_seq_le_service
.
}
unfold
service_during
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
R_k
);
try
(
by
done
);
last
by
apply
leqnn
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
R_k
);
try
(
by
done
).
rewrite
addn0
;
apply
extend_sum
;
first
by
apply
leqnn
.
by
rewrite
leq_add2l
;
apply
H_R_k_le_deadline
.
}
...
...
@@ 515,8 +514,8 @@ Module InterferenceBoundEDF.
by
apply
job_interference_seq_le_service
.
}
unfold
service_during
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
R_k
);
try
(
by
done
);
last
by
apply
leqnn
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
R_k
);
try
(
by
done
)
.
rewrite
addn0
.
apply
leq_trans
with
(
n
:=
(
\
sum_
(
t1
<=
t
<
a_fst
+
R_k
)
1
)
+
\
sum_
(
a_fst
+
R_k
<=
t
<
a_fst
+
D_k
)
1
).
...
...
@@ 800,9 +799,9 @@ Module InterferenceBoundEDF.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
interfering_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
(
*
Using
the
lemma
above
,
we
prove
that
the
ratio
n_k
is
at
least
the
number
of
...
...
@@ 875,8 +874,7 @@ Module InterferenceBoundEDF.
rename
H_many_jobs
into
NUM
,
H_at_least_two_jobs
into
SIZE
.
have
NK
:=
interference_bound_edf_n_k_covers_middle_jobs_plus_one
.
move:
NK
;
rewrite
leq_eqVlt
orbC
;
move
=>
/
orP
NK
;
des
;
first
by
rewrite
SIZE
ltnS
leqNgt
NK
in
NUM
.
by
move
:
NK
=>
/
eqP
NK
;
rewrite
NK
.
[
by
rewrite
SIZE
ltnS
leqNgt
NK
in
NUM

by
done
].
Qed
.
(
*
After
proving
the
bounds
of
the
middle
and
last
jobs
,
we
do
the
same
for
...
...
analysis/basic/workload_bound.v
View file @
a5cc6b86
...
...
@@ 78,7 +78,7 @@ Module WorkloadBound.
rewrite
leq_eqVlt
in
LTp
;
move
:
LTp
=>
/
orP
LTp
;
des
;
last
by
rewrite
ltnS
in
LTp
;
apply
(
leq_trans
H_period_positive
)
in
LTp
.
{
move:
LTp
=>
/
eqP
LTp
;
rewrite
LTp
2
!
modn1
2
!
divn1
.
rewrite
LTp
2
!
modn1
2
!
divn1
.
rewrite
leq_add2l
leq_mul2r
;
apply
/
orP
;
right
.
by
rewrite
leq_sub2r
// leq_add2l.
}
...
...
@@ 253,7 +253,7 @@ Module WorkloadBound.
Proof
.
intros
j_i
LTi
.
rewrite

workload_bound_job_in_same_sequence
mem_filter
in
LTi
;
des
.
repeat
split
;
[
by
apply
/
eqP


by
done
].
repeat
split
;
[
by
done


by
done
].
unfold
jobs_scheduled_between
in
*
;
rewrite
mem_undup
in
LTi0
.
apply
mem_bigcat_nat_exists
in
LTi0
;
des
.
rewrite
mem_scheduled_jobs_eq_scheduled
in
LTi0
.
...
...
@@ 432,8 +432,7 @@ Module WorkloadBound.
[

by
apply
workload_bound_response_time_of_first_job_inside_interval

by
apply
ltnW
].
rewrite
{
2
}
[
\
sum_
(
_
<=
_
<
_
)
_
]
addn0
/=
leq_add2l
leqn0
;
apply
/
eqP
.
apply
(
cumulative_service_after_job_rt_zero
job_cost
)
with
(
R
:=
R_tsk
);
try
(
by
done
);
last
by
apply
leqnn
.
apply
(
cumulative_service_after_job_rt_zero
job_cost
)
with
(
R
:=
R_tsk
);
try
(
by
done
).
apply
H_response_time_bound
;
last
by
done
.
exploit
workload_bound_all_jobs_from_tsk
.
by
apply
mem_nth
;
instantiate
(
1
:=
0
);
rewrite
H_at_least_two_jobs
.
...
...
@@ 544,9 +543,9 @@ Module WorkloadBound.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
scheduled_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
(
*
Prove
that
n_k
is
at
least
the
number
of
the
middle
jobs
*
)
...
...
analysis/jitter/bertogna_edf_comp.v
View file @
a5cc6b86
...
...
@@ 292,8 +292,7 @@ Module ResponseTimeIterationEDF.
Proof
.
intros
l
;
unfold
all_le
;
rewrite
eq_refl
andTb
.
destruct
l
;
first
by
done
.
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
));
try
(
by
ins
).
by
ins
;
apply
leqnn
.
by
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
)).
Qed
.
(
*
...
and
transitive
.
*
)
...
...
@@ 562,7 +561,6 @@ Module ResponseTimeIterationEDF.
try
(
by
done
);
[
by
apply
all_le_reflexive

by
apply
all_le_transitive

by
apply
leqnSn

by
apply
bertogna_edf_comp_iteration_preserves_order

by
apply
bertogna_edf_comp_iteration_preserves_minimum
].
Qed
.
...
...
analysis/jitter/bertogna_fp_comp.v
View file @
a5cc6b86
...
...
@@ 407,8 +407,7 @@ Module ResponseTimeIterationFP.
apply
leq_trans
with
(
n
:=
W_jitter
task_cost
task_period
task_jitter
i
R
x1
);
first
by
apply
geq_minl
.
exploit
(
VALID
i
);
first
by
rewrite
mem_rcons
in_cons
;
apply
/
orP
;
right
.
by
ins
;
des
;
apply
W_monotonic
;
try
(
by
ins
);
[
by
apply
GE_COST

by
apply
leqnn
].
by
ins
;
des
;
apply
W_monotonic
;
try
(
by
ins
);
apply
GE_COST
.
}
{
apply
leq_trans
with
(
n
:=
x1

task_cost
tsk
+
1
);
first
by
apply
geq_minr
.
...
...
analysis/jitter/interference_bound_edf.v
View file @
a5cc6b86
...
...
@@ 527,8 +527,8 @@ Module InterferenceBoundEDFJitter.
unfold
service_during
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
J_k
+
R_k
);
try
(
by
done
);
rewrite
?
addnA
//; last by apply leqnn
.
rewrite
addn0
;
apply
extend_sum
;
first
by
apply
leqnn
.
rewrite
?
addnA
//
.
rewrite
addn0
;
apply
extend_sum
;
first
by
done
.
rewrite

addnA
leq_add2l
.
by
apply
H_R_k_le_deadline
.
}
...
...
@@ 552,7 +552,7 @@ Module InterferenceBoundEDFJitter.
}
unfold
service_during
.
rewrite
>
cumulative_service_after_job_rt_zero
with
(
job_cost0
:=
job_cost
)
(
R
:=
J_k
+
R_k
);
try
(
by
done
);
rewrite
?
addnA
//; last by apply leqnn
.
rewrite
?
addnA
//
.
rewrite
addn0
.
apply
leq_trans
with
(
n
:=
(
\
sum_
(
a_i
<=
t
<
a_fst
+
J_k
+
R_k
)
1
)
+
\
sum_
(
a_fst
+
J_k
+
R_k
<=
t
<
a_fst
+
D_k
)
1
).
...
...
@@ 884,9 +884,9 @@ Module InterferenceBoundEDFJitter.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
interfering_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
(
*
Using
the
lemma
above
,
we
prove
that
the
ratio
n_k
is
at
least
the
number
of
...
...
@@ 964,7 +964,7 @@ Module InterferenceBoundEDFJitter.
have
NK
:=
interference_bound_edf_n_k_covers_middle_jobs_plus_one
.
move:
NK
;
rewrite
leq_eqVlt
orbC
;
move
=>
/
orP
NK
;
des
;
first
by
rewrite
SIZE
ltnS
leqNgt
NK
in
NUM
.
by
move
:
NK
=>
/
eqP
NK
;
rewrite
NK
.
by
rewrite
NK
.
Qed
.
(
*
After
proving
the
bounds
of
the
middle
and
last
jobs
,
we
do
the
same
for
...
...
analysis/jitter/workload_bound.v
View file @
a5cc6b86
...
...
@@ 79,7 +79,7 @@ Module WorkloadBoundJitter.
rewrite
leq_eqVlt
in
LTp
;
move
:
LTp
=>
/
orP
LTp
;
des
;
last
by
rewrite
ltnS
in
LTp
;
apply
(
leq_trans
H_period_positive
)
in
LTp
.
{
move:
LTp
=>
/
eqP
LTp
;
rewrite
LTp
2
!
modn1
2
!
divn1
.
rewrite
LTp
2
!
modn1
2
!
divn1
.
rewrite
leq_add2l
leq_mul2r
;
apply
/
orP
;
right
.
by
rewrite
leq_sub2r
// leq_add2l.
}
...
...
@@ 257,7 +257,7 @@ Module WorkloadBoundJitter.
Proof
.
intros
j_i
LTi
.
rewrite

workload_bound_job_in_same_sequence
mem_filter
in
LTi
;
des
.
repeat
split
;
[
by
apply
/
eqP


by
done
].
repeat
split
;
[
by
done


by
done
].
unfold
jobs_scheduled_between
in
*
;
rewrite
mem_undup
in
LTi0
.
apply
mem_bigcat_nat_exists
in
LTi0
;
des
.
rewrite
mem_scheduled_jobs_eq_scheduled
in
LTi0
.
...
...
@@ 562,9 +562,9 @@ Module WorkloadBoundJitter.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
scheduled_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
(
*
Prove
that
n_k
is
at
least
the
number
of
the
middle
jobs
*
)
...
...
analysis/parallel/bertogna_edf_comp.v
View file @
a5cc6b86
...
...
@@ 239,8 +239,7 @@ Module ResponseTimeIterationEDF.
Proof
.
intros
l
;
unfold
all_le
;
rewrite
eq_refl
andTb
.
destruct
l
;
first
by
done
.
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
));
try
(
by
ins
).
by
ins
;
apply
leqnn
.
by
apply
/
(
zipP
(
fun
x
y
=>
snd
x
<=
snd
y
)).
Qed
.
(
*
...
and
transitive
.
*
)
...
...
@@ 508,7 +507,6 @@ Module ResponseTimeIterationEDF.
try
(
by
done
);
[
by
apply
all_le_reflexive

by
apply
all_le_transitive

by
apply
leqnSn

by
apply
bertogna_edf_comp_iteration_preserves_order

by
apply
bertogna_edf_comp_iteration_preserves_minimum
].
Qed
.
...
...
analysis/parallel/interference_bound_edf.v
View file @
a5cc6b86
...
...
@@ 618,9 +618,9 @@ Module InterferenceBoundEDF.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
interfering_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
Lemma
interference_bound_edf_slack_le_delta
:
...
...
analysis/parallel/workload_bound.v
View file @
a5cc6b86
...
...
@@ 186,7 +186,7 @@ Module WorkloadBound.
Proof
.
intros
j_i
LTi
.
rewrite

workload_bound_job_in_same_sequence
mem_filter
in
LTi
;
des
.
repeat
split
;
[
by
apply
/
eqP


by
done
].
repeat
split
;
[
by
done


by
done
].
unfold
jobs_scheduled_between
in
*
;
rewrite
mem_undup
in
LTi0
.
apply
mem_bigcat_nat_exists
in
LTi0
;
des
.
rewrite
mem_scheduled_jobs_eq_scheduled
in
LTi0
.
...
...
@@ 396,9 +396,9 @@ Module WorkloadBound.
by
apply
ltn_trans
with
(
n
:=
(
size
sorted_jobs
).

1
);
destruct
sorted_jobs
;
ins
.
by
destruct
sorted_jobs
;
ins
.
by
rewrite
sort_uniq
/
scheduled_jobs
filter_uniq
// undup_uniq.
by
move
:
INnth
INnth0
=>
/
eqP
INnth
/
eqP
INnth0
;
rewrite
INnth
INnth0
.
by
rewrite
INnth
INnth0
.
}
by
rewrite
subh3
// addnC
; move: INnth => /eqP INnth; rewrite
INnth.
by
rewrite
subh3
// addnC INnth.
Qed
.
(
*
Next
,
we
prove
that
n_k
covers
every
scheduled
job
,
...
*
)
...
...
implementation/basic/arrival_sequence.v
View file @
a5cc6b86
...
...
@@ 75,25 +75,25 @@ Module ConcreteArrivalSequence.
unfold
arrives_at
,
arr_seq
,
periodic_arrival_sequence
in
*
.
rewrite
2
!
mem_pmap
in
ARR
ARR
'
.
move:
ARR
ARR
'
=>
/
mapP
[
tsk_j
INj
SOMEj
]
/
mapP
[
tsk_j
'
INj
'
SOMEj
'
].
unfold
add_job
in
*
;
desf
;
simpl
in
*
;
subst
.
clear
INj
'
.
move:
Heq
Heq0
=>
/
dvdnP
DIV
'
/
dvdnP
DIV
;
des
.
unfold
add_job
in
SOMEj
,
SOMEj
'
;
desf
;
simpl
in
*
;
move:
Heq0
Heq
=>
/
dvdnP
[
k
DIV
]
/
dvdnP
[
k
'
DIV
'
]
.
{
rewrite
DIV
DIV
'

mulSnr
.
rewrite
leq_eqVlt
in
LE
;
move
:
LE
=>
/
orP
[
/
eqP
EQ

LESS
].
{
move:
DIFF
=>
/
orP
[
/
eqP
CASE1

/
eqP
CASE2
].

by
exfalso
;
apply
CASE1
;
repeat
f_equal
.

by
rewrite
EQ
in
CASE2
.
exfalso
;
move
:
DIFF
=>
/
negP
DIFF
;
apply
DIFF
.
by
subst
;
rewrite
EQ
.
}
subst
;
rewrite
leq_mul2r
;
apply
/
orP
;
right
.
by
rewrite
ltn_mul2r
in
LESS
;
move
:
LESS
=>
/
andP
[
_
LT
].
}
rewrite
leq_mul2r
;
apply
/
orP
;
right
.
rewrite
ltn_neqAle
;
apply
/
andP
;
split
.
{
apply
/
eqP
;
red
;
intro
EQ
;
subst
.
by
rewrite
ltnn
in
LESS
.
assert
(
LT
:
arr
<
arr
'
).
by
rewrite
ltn_neqAle
;
apply
/
andP
.
clear
LE
DIFF
;
subst
tsk_j
'
arr
arr
'
.
rewrite
ltn_mul2r
in
LT
;
move
:
LT
=>
/
andP
[
_
LT
].
by
apply
leq_trans
with
(
n
:=
k
.
+
1
*
task_period
tsk_j
);
[
by
rewrite
mulSnr

by
rewrite
leq_mul2r
;
apply
/
orP
;
right
].
}
rewrite
leqNgt
;
apply
/
negP
;
red
;
intro
LT
.
rewrite
ltnNge
in
LESS
;
move
:
LESS
=>
/
negP
LESS
;
apply
LESS
.
by
subst
;
rewrite
leq_mul2r
;
apply
/
orP
;
right
;
apply
ltnW
.
Qed
.
(
*
...
and
if
the
task
set
has
no
duplicates
,
the
same
applies
to
...
...
implementation/jitter/arrival_sequence.v
View file @
a5cc6b86
...
...
@@ 75,25 +75,25 @@ Module ConcreteArrivalSequence.
unfold
arrives_at
,
arr_seq
,
periodic_arrival_sequence
in
*
.
rewrite
2
!
mem_pmap
in
ARR
ARR
'
.
move:
ARR
ARR
'
=>
/
mapP
[
tsk_j
INj
SOMEj
]
/
mapP
[
tsk_j
'
INj
'
SOMEj
'
].
unfold
add_job
in
*
;
desf
;
simpl
in
*
;
subst
.
clear
INj
'
.
move:
Heq
Heq0
=>
/
dvdnP
DIV
'
/
dvdnP
DIV
;
des
.
unfold
add_job
in
SOMEj
,
SOMEj
'
;
desf
;
simpl
in
*
;
move:
Heq0
Heq
=>
/
dvdnP
[
k
DIV
]
/
dvdnP
[
k
'
DIV
'
]
.
{
rewrite
DIV
DIV
'

mulSnr
.
rewrite
leq_eqVlt
in
LE
;
move
:
LE
=>
/
orP
[
/
eqP
EQ

LESS
].
{
move:
DIFF
=>
/
orP
[
/
eqP
CASE1

/
eqP
CASE2
].

by
exfalso
;
apply
CASE1
;
repeat
f_equal
.

by
rewrite
EQ
in
CASE2
.
exfalso
;
move
:
DIFF
=>
/
negP
DIFF
;
apply
DIFF
.
by
subst
;
rewrite
EQ
.
}
subst
;
rewrite
leq_mul2r
;
apply
/
orP
;
right
.
by
rewrite
ltn_mul2r
in
LESS
;
move
:
LESS
=>
/
andP
[
_
LT
].
}
rewrite
leq_mul2r
;
apply
/
orP
;
right
.
rewrite
ltn_neqAle
;
apply
/
andP
;
split
.
{
apply
/
eqP
;
red
;
intro
EQ
;
subst
.
by
rewrite
ltnn
in
LESS
.
assert
(
LT
:
arr
<
arr
'
).
by
rewrite
ltn_neqAle
;
apply
/
andP
.
clear
LE
DIFF
;
subst
tsk_j
'
arr
arr
'
.
rewrite
ltn_mul2r
in
LT
;
move
:
LT
=>
/
andP
[
_
LT
].
by
apply
leq_trans
with
(
n
:=
k
.
+
1
*
task_period
tsk_j
);
[
by
rewrite
mulSnr

by
rewrite
leq_mul2r
;
apply
/
orP
;
right
].
}
rewrite
leqNgt
;
apply
/
negP
;
red
;
intro
LT
.
rewrite
ltnNge
in
LESS
;
move
:
LESS
=>
/
negP
LESS
;
apply
LESS
.
by
subst
;
rewrite
leq_mul2r
;
apply
/
orP
;
right
;
apply
ltnW
.
Qed
.
(
*
...
and
if
the
task
set
has
no
duplicates
,
the
same
applies
to
...
...
model/basic/arrival_sequence.v
View file @
a5cc6b86
...
...
@@ 100,9 +100,6 @@ Module ArrivalSequence.
if
(
j
\
in
arr_seq
t
)
is
true
then
Some
(
Build_JobIn
arr_seq
j
t
_
)
else
None
.
Next
Obligation
.
by
done
.
Qed
.
(
*
Now
we
define
the
list
of
JobIn
that
arrive
at
time
t
as
the
partial
map
of
is_JobIn
.
*
)
...
...
model/basic/interference.v
View file @
a5cc6b86
...
...
@@ 244,7 +244,7 @@ Module Interference.
rewrite
big_mkcond
(
bigD1_seq
j0
)
/=
;
last
by
rewrite
undup_uniq
.
{
rewrite

addn1
addnC
;
apply
leq_add
;
last
by
done
.
rewrite
EQtsk0
eq_refl
BACK
andTb
.
rewrite
EQtsk0
BACK
andTb
.
apply
eq_leq
;
symmetry
;
apply
/
eqP
;
rewrite
eqb1
.
unfold
scheduled
,
scheduled_on
.
by
apply
/
exists_inP
;
exists
x
;
[
by
done

by
rewrite
SCHED
].
...
...
@@ 252,7 +252,7 @@ Module Interference.
{
unfold
jobs_scheduled_between
.
rewrite
mem_undup
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
done
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
auto
.
unfold
jobs_scheduled_at
.
apply
mem_bigcat_ord
with
(
j
:=
x
);
first
by
apply
ltn_ord
.
by
unfold
make_sequence
;
rewrite
SCHED
mem_seq1
eq_refl
.
...
...
model/jitter/interference.v
View file @
a5cc6b86
...
...
@@ 175,26 +175,24 @@ Module Interference.
}
rewrite
big_mkcond
(
bigD1_seq
j0
)
/=
;
last
by
rewrite
undup_uniq
.
{
rewrite
EQtsk0
eq_refl
BACK
SCHEDULED
andbT
big_mkcond
.
rewrite
EQtsk0
BACK
SCHEDULED
andbT
big_mkcond
.
rewrite
(
eq_bigr
(
fun
x
=>
0
));
first
by
rewrite
big_const_seq
iter_addn
mul0n
addn0
addn0
.
intros
j1
_
;
desf
;
[
rewrite
andTb

by
done
]
.
intros
j1
_
;
desf
;
try
by
done
.
apply
/
eqP
;
rewrite
eqb0
;
apply
/
negP
;
unfold
not
;
intro
SCHEDULED
'
.
exploit
(
H_no_intratask_parallelism
j0
j1
t
);
try
(
by
done
).
by
move
:
Heq0
=>
/
eqP
EQtsk
;
rewrite
EQtsk
.
by
intros
EQj
;
rewrite
EQj
eq_refl
in
Heq
.
exploit
(
H_no_intratask_parallelism
j0
j1
t
);
try
by
eauto
.
}
{
rewrite
mem_undup
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
done
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
auto
.
apply
mem_bigcat_ord
with
(
j
:=
x
);
first
by
apply
ltn_ord
.
by
rewrite
SCHED
mem_seq1
eq_refl
.
by
rewrite
SCHED
mem_seq1
.
}
}
{
rewrite
big_mkcond
(
eq_bigr
(
fun
x
=>
0
));
first
by
rewrite
big_const_seq
iter_addn
mul0n
addn0
.
intros
i
_
;
desf
.
intros
i
_
;
desf
;
rewrite
// ?BACK ?andFb //
.
unfold
task_is_scheduled
in
BACK
.
apply
negbT
in
BACK
;
rewrite
negb_exists
in
BACK
.
move:
BACK
=>
/
forallP
BACK
.
...
...
@@ 206,7 +204,7 @@ Module Interference.
unfold
schedules_job_of_tsk
in
BACK
;
unfold
scheduled_on
.
destruct
(
sched
x
t
)
eqn
:
SCHED
;
last
by
ins
.
apply
/
negP
;
unfold
not
;
move
=>
/
eqP
BUG
;
inversion
BUG
;
subst
.
by
move
:
Heq
=>
/
eqP
Heq
;
rewrite
Heq
eq_refl
in
BACK
.
by
move
:
BACK
=>
/
negP
BACK
;
apply
BACK
.
}
by
rewrite
NOTSCHED
andbF
.
}
...
...
@@ 235,7 +233,7 @@ Module Interference.
rewrite
big_mkcond
(
bigD1_seq
j0
)
/=
;
last
by
rewrite
undup_uniq
.
{
rewrite

addn1
addnC
;
apply
leq_add
;
last
by
done
.
rewrite
EQtsk0
eq_refl
BACK
andTb
.
rewrite
EQtsk0
BACK
andTb
.
apply
eq_leq
;
symmetry
;
apply
/
eqP
;
rewrite
eqb1
.
unfold
scheduled
,
scheduled_on
.
by
apply
/
exists_inP
;
exists
x
;
[
by
done

by
rewrite
SCHED
].
...
...
@@ 243,10 +241,10 @@ Module Interference.
{
unfold
jobs_scheduled_between
.
rewrite
mem_undup
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
done
.
apply
mem_bigcat_nat
with
(
j
:=
t
);
first
by
auto
.
unfold
jobs_scheduled_at
.
apply
mem_bigcat_ord
with
(
j
:=
x
);
first
by
apply
ltn_ord
.
by
unfold
make_sequence
;
rewrite
SCHED
mem_seq1
eq_refl
.
by
unfold
make_sequence
;
rewrite
SCHED
mem_seq1
.
}
Qed
.
...
...
util/Vbase.v
View file @
a5cc6b86
This diff is collapsed.
Click to expand it.
util/lemmas.v
View file @
a5cc6b86
...
...
@@ 312,8 +312,8 @@ Section BigCatLemmas.
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
.
apply
IHn
in
HEAD
;
destruct
HEAD
as
[
x0
IN
]
.
by
eexists
(
widen_ord
_
x0
);
apply
IN
.
}
{
by
exists
ord_max
;
desf
.
...
...
@@ 924,14 +924,13 @@ Section UniqList.
apply
eq_trans
with
(
y
:=
[
seq
x
<
l

index
x
l
<
i
]).
{
apply
eq_in_filter
;
red
;
intros
x
IN
.
desf
;
move
:
Heq
=>
/
eqP
SUBST
;
subst
.
by
simpl
in
UNIQ
;
rewrite
IN
andFb
in
UNIQ
.
desf
;
subst
;
last
by
done
.
by
simpl
in
*
;
rewrite
IN
andFb
in
UNIQ
.
}
simpl
in
*
;
desf
.
rewrite
/=
ltnS
in
LT
.
rewrite
leq_eqVlt
in
LT
;
desf
.
{
move:
LT
=>
/
eqP
LT
;
subst
.
rewrite
take_size
.
apply
eq_trans
with
(
y
:=
filter
predT
l
);
last
by
rewrite
filter_predT
.
by
apply
eq_in_filter
;
red
;
ins
;
rewrite
index_mem
.
...
...
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