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Flocq
flocq
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471def1b
Commit
471def1b
authored
Jul 24, 2013
by
BOLDO Sylvie
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parent
0dd873d3
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src/Appli/Fappli_rnd_odd.v
src/Appli/Fappli_rnd_odd.v
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src/Appli/Fappli_rnd_odd.v
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471def1b
...
...
@@ -369,18 +369,50 @@ Hypothesis fexpe_fexp: forall e, (fexpe e <= fexp e -2)%Z. (* ??? *)
Lemma
generic_format_F2R_2
:
forall
c
,
forall
(
x
:
R
)
(
f
:
float
beta
),
x
=
F2R
f
->
((
x
<>
0
)
%
R
->
F2R
f
=
x
->
((
x
<>
0
)
%
R
->
(
canonic_exp
beta
c
x
<=
Fexp
f
)
%
Z
)
->
generic_format
beta
c
x
.
Proof
.
intros
c
x
f
H1
H2
.
rewrite
H1
;
destruct
f
as
(
m
,
e
).
rewrite
<-
H1
;
destruct
f
as
(
m
,
e
).
apply
generic_format_F2R
.
simpl
in
*
;
intros
H3
.
rewrite
<-
H1
;
apply
H2
.
rewrite
H1
;
apply
H2
.
intros
Y
;
apply
H3
.
apply
F2R_eq_0_reg
with
beta
e
.
now
rewrite
<-
H1
.
now
rewrite
H1
.
Qed
.
Lemma
exists_even_fexp_lt
:
forall
(
c
:
Z
->
Z
),
forall
(
x
:
R
),
(
exists
f
:
float
beta
,
F2R
f
=
x
/
\
(
c
(
ln_beta
beta
x
)
<
Fexp
f
)
%
Z
)
->
exists
f
:
float
beta
,
F2R
f
=
x
/
\
canonic
beta
c
f
/
\
Zeven
(
Fnum
f
)
=
true
.
Proof
with
auto
with
typeclass_instances
.
intros
c
x
(
g
,(
Hg1
,
Hg2
)).
exists
(
Float
beta
(
Fnum
g
*
Z
.
pow
(
radix_val
beta
)
(
Fexp
g
-
c
(
ln_beta
beta
x
)))
(
c
(
ln_beta
beta
x
))).
assert
(
F2R
(
Float
beta
(
Fnum
g
*
Z
.
pow
(
radix_val
beta
)
(
Fexp
g
-
c
(
ln_beta
beta
x
)))
(
c
(
ln_beta
beta
x
)))
=
x
).
unfold
F2R
;
simpl
.
rewrite
Z2R_mult
,
Z2R_Zpower
.
rewrite
Rmult_assoc
,
<-
bpow_plus
.
rewrite
<-
Hg1
;
unfold
F2R
.
apply
f_equal
,
f_equal
.
ring
.
omega
.
split
;
trivial
.
split
.
unfold
canonic
,
canonic_exp
.
now
rewrite
H
.
simpl
.
rewrite
Zeven_mult
.
rewrite
Zeven_Zpower
.
rewrite
Even_beta
.
apply
Bool
.
orb_true_intro
.
now
right
.
omega
.
Qed
.
...
...
@@ -570,21 +602,46 @@ Qed.
Lemma
Zm
:
exists
g
:
float
beta
,
m
=
F2R
g
/
\
canonic
beta
fexp
g
/
\
Zeven
(
Fnum
g
)
=
true
.
exists
g
:
float
beta
,
F2R
g
=
m
/
\
canonic
beta
fexpe
g
/
\
Zeven
(
Fnum
g
)
=
true
.
Proof
with
auto
with
typeclass_instances
.
destruct
m_eq
as
(
g
,(
Hg1
,
Hg2
)).
apply
generic_format_F2R_2
with
g
.
apply
exists_even_fexp_lt
.
exists
g
;
split
;
trivial
.
rewrite
Hg2
.
rewrite
ln_beta_m
.
rewrite
<-
Fexp_d
.
rewrite
Cd
.
unfold
canonic_exp
.
generalize
(
fexpe_fexp
(
ln_beta
beta
(
F2R
d
))).
omega
.
Qed
.
Theorem
r
nd_opp
:
forall
x
,
Theorem
r
ound_odd_prop
:
round
beta
fexp
ZnearestE
(
round
beta
fexpe
Zrnd_odd
x
)
=
round
beta
fexp
ZnearestE
x
.
Proof
with
auto
with
typeclass_instances
.
intros
x
.
case
(
Rle_or_lt
x
m
);
intros
Y
;[
destruct
Y
|
idtac
].
(
*
.
*
)
apply
trans_eq
with
(
F2R
d
).
apply
round_N_DN_betw
with
(
F2R
u
)...
needs:
Rnd_DN_pt
(
generic_format
beta
fexp
)
(
round
beta
fexpe
Zrnd_odd
x
)
(
F2R
d
)
Rnd_UP_pt
(
generic_format
beta
fexp
)
(
round
beta
fexpe
Zrnd_odd
x
)
(
F2R
u
)
TOTO
.
apply
sym_eq
,
round_N_DN_betw
with
(
F2R
u
)...
split
.
apply
Hd
.
exact
H
.
(
*
.
*
)
Rle_or_lt
x
m
round_N_UP_betw
...
...
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