Skip to content
GitLab
Projects
Groups
Snippets
Help
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in
Toggle navigation
F
flocq
Project overview
Project overview
Details
Activity
Releases
Repository
Repository
Files
Commits
Branches
Tags
Contributors
Graph
Compare
Issues
1
Issues
1
List
Boards
Labels
Service Desk
Milestones
Merge Requests
1
Merge Requests
1
CI / CD
CI / CD
Pipelines
Jobs
Schedules
Operations
Operations
Incidents
Environments
Packages & Registries
Packages & Registries
Container Registry
Analytics
Analytics
CI / CD
Repository
Value Stream
Wiki
Wiki
Snippets
Snippets
Members
Members
Collapse sidebar
Close sidebar
Activity
Graph
Create a new issue
Jobs
Commits
Issue Boards
Open sidebar
Flocq
flocq
Commits
bfb097f9
Commit
bfb097f9
authored
Nov 08, 2013
by
BOLDO Sylvie
Browse files
Options
Browse Files
Download
Email Patches
Plain Diff
WIP
parent
dc1a2788
Changes
1
Hide whitespace changes
Inline
Side-by-side
Showing
1 changed file
with
55 additions
and
28 deletions
+55
-28
src/Appli/Fappli_Muller.v
src/Appli/Fappli_Muller.v
+55
-28
No files found.
src/Appli/Fappli_Muller.v
View file @
bfb097f9
...
...
@@ -1147,34 +1147,31 @@ rewrite bpow_opp, bpow_1, H; reflexivity.
Qed
.
Lemma
Zceil_lt_0
:
forall
x
,
0
<
x
->
(
0
<
Zceil
x
)
%
Z
.
Admitted
.
Let
k
:=
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
(
2
+
bpow
(
1
-
prec
)))
-
1
)
%
Z
.
Let
k
:=
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
2
)
-
1
)
%
Z
.
Lemma
kpos
:
(
0
<=
k
)
%
Z
.
assert
(
0
<
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
(
2
+
bpow
(
1
-
prec
))
).
assert
(
0
<
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
2
).
apply
Fourier_util
.
Rlt_mult_inv_pos
.
apply
Rmult_lt_0_compat
.
assumption
.
apply
bpow_gt_0
.
rewrite
Rplus_comm
,
<-
Rplus_assoc
;
apply
Rle_lt_0_plus_1
.
apply
Rlt_le
,
Rle_lt_0_plus_1
,
bpow_ge_0
.
apply
Zceil_lt_0
in
H
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
assert
(
0
<
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
2
))
%
Z
.
apply
lt_Z2R
;
simpl
(
Z2R
0
).
apply
Rlt_le_trans
with
(
1
:=
H
).
apply
Zceil_ub
.
unfold
k
;
omega
.
Qed
.
Lemma
kLe
:
(
k
<
radix_val
beta
)
%
Z
.
cut
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
(
2
+
bpow
(
1
-
prec
))
)
<=
beta
)
%
Z
.
cut
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
2
)
<=
beta
)
%
Z
.
unfold
k
;
omega
.
apply
Zceil_glb
.
apply
Rle_trans
with
(
bpow
1
/
1
).
unfold
Rdiv
;
apply
Rmult_le_compat
.
apply
Rmult_le_pos
;
try
apply
bpow_ge_0
;
now
left
.
apply
Rlt_le
,
Rinv_0_lt_compat
.
rewrite
Rplus_comm
,
<-
Rplus_assoc
;
apply
Rle_lt_0_plus_1
.
apply
Rlt_le
,
Rle_lt_0_plus_1
,
bpow_ge_0
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
apply
Rle_trans
with
(
bpow
(
ln_beta
beta
x
)
*
bpow
(
1
-
ln_beta
beta
x
)).
apply
Rmult_le_compat_r
.
apply
bpow_ge_0
.
...
...
@@ -1185,21 +1182,19 @@ apply bpow_le; omega.
apply
Rinv_le
.
exact
Rlt_0_1
.
apply
Rplus_le_reg_l
with
(
-
1
);
ring_simplify
.
apply
Rlt_le
,
Rl
e_lt_0_plus_1
,
bpow_ge_0
.
apply
Rlt_le
,
Rl
t_0_1
.
rewrite
bpow_1
;
right
;
field
.
Qed
.
Lemma
kLe2
:
(
k
<=
Zceil
(
Z2R
(
radix_val
beta
)
/
2
)
-
1
)
%
Z
.
cut
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
(
2
+
bpow
(
1
-
prec
))
)
cut
(
Zceil
(
x
*
bpow
(
1
-
ln_beta
beta
x
)
/
2
)
<=
Zceil
(
Z2R
(
radix_val
beta
)
/
2
))
%
Z
.
unfold
k
;
omega
.
apply
Zceil_glb
.
apply
Rle_trans
with
(
bpow
1
/
2
).
unfold
Rdiv
;
apply
Rmult_le_compat
.
apply
Rmult_le_pos
;
try
apply
bpow_ge_0
;
now
left
.
unfold
Rdiv
;
apply
Rmult_le_compat_r
.
apply
Rlt_le
,
Rinv_0_lt_compat
.
rewrite
Rplus_comm
,
<-
Rplus_assoc
;
apply
Rle_lt_0_plus_1
.
apply
Rlt_le
,
Rle_lt_0_plus_1
,
bpow_ge_0
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
apply
Rle_trans
with
(
bpow
(
ln_beta
beta
x
)
*
bpow
(
1
-
ln_beta
beta
x
)).
apply
Rmult_le_compat_r
.
apply
bpow_ge_0
.
...
...
@@ -1207,10 +1202,6 @@ apply Rle_trans with (1:=RRle_abs _).
left
;
apply
bpow_ln_beta_gt
.
rewrite
<-
bpow_plus
.
apply
bpow_le
;
omega
.
apply
Rinv_le
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
apply
Rplus_le_reg_l
with
(
-
2
);
ring_simplify
.
apply
bpow_ge_0
.
rewrite
bpow_1
.
apply
Zceil_ub
.
Qed
.
...
...
@@ -1241,27 +1232,63 @@ apply Rplus_lt_reg_r with ((Z2R k + / 2) * (Z2R k + / 2) * bpow (ln_beta beta x
apply
Rlt_le_trans
with
((
2
*
Z2R
k
+
1
)
*
x
).
2
:
right
;
ring
.
apply
Rle_lt_trans
with
(((
Z2R
(
Zceil
(
Z2R
(
radix_val
beta
)
/
2
))
-/
2
)
*
(
Z2R
(
Zceil
(
Z2R
(
radix_val
beta
)
/
2
))
-/
2
))
*
bpow
(
ln_beta
beta
x
-
prec
)
+
/
2
*
bpow
(
ln_beta
beta
x
)).
(
/
4
*
bpow
(
2
+
(
ln_beta
beta
x
-
prec
))
+
/
2
*
bpow
(
ln_beta
beta
x
)).
apply
Rplus_le_compat_r
.
rewrite
bpow_plus
,
<-
Rmult_assoc
.
apply
Rmult_le_compat_r
.
apply
bpow_ge_0
.
assert
(
0
<=
Z2R
k
+
/
2
).
replace
0
with
(
Z2R
0
+
0
)
by
(
simpl
;
ring
);
apply
Rplus_le_compat
.
apply
Z2R_le
,
kpos
.
apply
Rlt_le
,
Rinv_0_lt_compat
,
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
assert
(
Z2R
k
+
/
2
<=
Z2R
(
Zceil
(
Z2R
beta
/
2
))
-
/
2
).
assert
(
Z2R
k
+
/
2
<=
Z2R
beta
/
2
).
apply
Rle_trans
with
((
Z2R
(
Zceil
(
Z2R
beta
/
2
)
-
1
))
+
/
2
).
apply
Rplus_le_compat_r
.
apply
Z2R_le
,
kLe2
.
rewrite
Z2R_minus
;
simpl
.
right
;
field
.
generalize
(
beta
);
intros
n
.
case
(
Zeven_odd_dec
n
);
intros
V
.
apply
Zeven_ex_iff
in
V
;
destruct
V
as
(
m
,
Hm
).
rewrite
Hm
,
Z2R_mult
.
replace
(
Z2R
2
*
Z2R
m
/
2
)
with
(
Z2R
m
).
rewrite
Zceil_Z2R
.
apply
Rplus_le_reg_l
with
(
-
Z2R
m
+
/
2
).
field_simplify
.
unfold
Rdiv
;
apply
Rmult_le_compat_r
.
apply
Rlt_le
,
Rinv_0_lt_compat
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
apply
Rlt_le
,
Rlt_0_1
.
simpl
;
field
.
apply
Zodd_ex_iff
in
V
;
destruct
V
as
(
m
,
Hm
).
rewrite
Hm
,
Z2R_plus
,
Z2R_mult
.
replace
((
Z2R
2
*
Z2R
m
+
Z2R
1
)
/
2
)
with
(
Z2R
m
+/
2
).
replace
(
Zceil
(
Z2R
m
+
/
2
))
with
(
m
+
1
)
%
Z
.
rewrite
Z2R_plus
;
simpl
;
right
;
field
.
apply
sym_eq
,
Zceil_imp
.
split
.
ring_simplify
(
m
+
1
-
1
)
%
Z
.
apply
Rplus_lt_reg_r
with
(
-
Z2R
m
).
ring_simplify
.
apply
Rinv_0_lt_compat
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
rewrite
Z2R_plus
;
apply
Rplus_le_compat_l
.
apply
Rplus_le_reg_l
with
(
-/
2
).
simpl
;
field_simplify
.
unfold
Rdiv
;
apply
Rmult_le_compat_r
.
apply
Rlt_le
,
Rinv_0_lt_compat
.
apply
Rle_lt_0_plus_1
,
Rlt_le
,
Rlt_0_1
.
apply
Rlt_le
,
Rlt_0_1
.
simpl
;
field
.
apply
Rle_trans
with
((
Z2R
beta
/
2
)
*
(
Z2R
beta
/
2
)).
now
apply
Rmult_le_compat
.
simpl
;
unfold
Z
.
pow_pos
;
simpl
.
rewrite
Zmult_1_r
,
Z2R_mult
.
right
;
field
.
unfold
k
.
destruct
(
ln_beta
beta
x
)
as
(
e
,
He
).
simpl
(
ln_beta_val
beta
x
(
Build_ln_beta_prop
beta
x
e
He
))
in
*
.
apply
Rle_lt_trans
with
(
bpow
(
e
-
1
)
*
(
bpow
(
1
-
prec
)
*
Rsqr
(
Z2R
(
Zceil
(
Z2R
beta
/
2
))
-
/
2
)
+
(
Z2R
beta
)
/
2
)).
rewrite
Rmult_plus_distr_l
.
apply
Rle_lt_trans
with
(
bpow
(
e
-
1
)
*
(
/
4
*
bpow
(
3
-
prec
)
+
(
Z2R
beta
)
/
2
)).
rewrite
(
Rmult_plus_distr_l
(
bpow
(
e
-
1
)))
.
apply
Rplus_le_compat
.
rewrite
(
Rmult_comm
_
(
bpow
(
e
-
prec
))).
rewrite
<-
(
Rmult_assoc
(
bpow
(
e
-
1
))).
...
...
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment