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Why3
why3
Commits
5459fb63
Commit
5459fb63
authored
Oct 03, 2011
by
Guillaume Melquiond
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Add realization of real.Abs.
parent
ff115c6a
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realizations/coq/real/Abs.v
realizations/coq/real/Abs.v
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realizations/coq/real/Abs.v
0 → 100644
View file @
5459fb63
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
Rbasic_fun
.
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/theories"
.
*
)
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/modules"
.
*
)
Require
real
.
Real
.
Definition
abs
:
R
>
R
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Rabs
.
Defined
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Lemma
abs_def
:
forall
(
x
:
R
),
((
0
%
R
<=
x
)
%
R
>
((
abs
x
)
=
x
))
/
\
((
~
(
0
%
R
<=
x
)
%
R
)
>
((
abs
x
)
=
(

x
)
%
R
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
split
;
intros
H
.
apply
Rabs_right
.
now
apply
Rle_ge
.
apply
Rabs_left
.
now
apply
Rnot_le_lt
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Lemma
Abs_le
:
forall
(
x
:
R
)
(
y
:
R
),
((
abs
x
)
<=
y
)
%
R
<>
(((

y
)
%
R
<=
x
)
%
R
/
\
(
x
<=
y
)
%
R
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
.
unfold
abs
,
Rabs
.
case
Rcase_abs
;
intros
H
;
(
split
;
[
intros
H0
;
split

intros
(
H0
,
H1
)]).
rewrite
<
(
Ropp_involutive
x
).
now
apply
Ropp_le_contravar
.
apply
Rlt_le
.
apply
Rlt_le_trans
with
(
1
:=
H
).
apply
Rle_trans
with
(
2
:=
H0
).
rewrite
<
Ropp_0
.
apply
Ropp_le_contravar
.
now
apply
Rlt_le
.
rewrite
<
(
Ropp_involutive
y
).
now
apply
Ropp_le_contravar
.
apply
Rge_le
in
H
.
apply
Rle_trans
with
(
2
:=
H
).
apply
Rle_trans
with
(
Ropp
x
).
now
apply
Ropp_le_contravar
.
rewrite
<
Ropp_0
.
now
apply
Ropp_le_contravar
.
exact
H0
.
exact
H1
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Lemma
Abs_pos
:
forall
(
x
:
R
),
(
0
%
R
<=
(
abs
x
))
%
R
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Rabs_pos
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
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