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Why3
why3
Commits
54f46f3b
Commit
54f46f3b
authored
Feb 25, 2012
by
Guillaume Melquiond
Browse files
Update some additional Coq realizations now that notations are supported.
parent
47fb5a4c
Changes
5
Hide whitespace changes
Inline
Side-by-side
Makefile.in
View file @
54f46f3b
...
...
@@ -979,7 +979,7 @@ clean::
rm
-f
$(COQVO)
$(COQVD)
$(
addsuffix
.glob,
$(COQLIBS_FILES)
)
update-coq
:
bin/why3
#
for f in
$(COQLIBS_INT_FILES)
;
do
bin/why3
--realize
-D
drivers/coq-realize.drv
-T
int.
$$
f
-o
lib/coq/int/
;
done
for
f
in
$(COQLIBS_INT_FILES)
;
do
bin/why3
--realize
-D
drivers/coq-realize.drv
-T
int.
$$
f
-o
lib/coq/int/
;
done
for
f
in
$(COQLIBS_REAL_FILES)
;
do
bin/why3
--realize
-D
drivers/coq-realize.drv
-T
real.
$$
f
-o
lib/coq/real/
;
done
#for f in
$(COQLIBS_FP_FILES)
;
do
bin/why3
--realize
-D
drivers/coq-realize.drv
-T
floating_point.
$$
f
-o
lib/coq/floating_point/
;
done
...
...
lib/coq/int/Abs.v
View file @
54f46f3b
...
...
@@ -2,19 +2,14 @@
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/theories"
.
*
)
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/modules"
.
*
)
Require
int
.
Int
.
(
*
Why3
goal
*
)
Notation
abs
:=
Zabs
(
only
parsing
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
abs_def
:
forall
(
x
:
Z
),
((
0
%
Z
<=
x
)
%
Z
->
((
abs
x
)
=
x
))
/
\
((
~
(
0
%
Z
<=
x
)
%
Z
)
->
((
abs
x
)
=
(
-
x
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
.
split
;
intros
H
.
now
apply
Zabs_eq
.
...
...
@@ -24,29 +19,18 @@ contradict H.
apply
Zlt_le_weak
.
now
apply
Zgt_lt
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
abs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
.
zify
.
omega
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Abs_pos
:
forall
(
x
:
Z
),
(
0
%
Z
<=
(
abs
x
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zabs_pos
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
lib/coq/int/ComputerDivision.v
View file @
54f46f3b
...
...
@@ -3,34 +3,25 @@
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
ZOdiv
.
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/theories"
.
*
)
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/modules"
.
*
)
Require
int
.
Int
.
Require
int
.
Abs
.
(
*
Why3
goal
*
)
Notation
div
:=
ZOdiv
(
only
parsing
).
(
*
Why3
goal
*
)
Notation
mod1
:=
ZOmod
(
only
parsing
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_mod
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
~
(
y
=
0
%
Z
))
->
(
x
=
((
y
*
(
div
x
y
))
%
Z
+
(
mod1
x
y
))
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
_.
apply
ZO_div_mod_eq
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_bound
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
0
%
Z
<=
(
div
x
y
))
%
Z
/
\
((
div
x
y
)
<=
x
)
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
).
split
.
apply
ZO_div_pos
with
(
1
:=
Hx
).
...
...
@@ -44,15 +35,10 @@ apply ZO_div_lt with (1 := H').
omega
.
now
rewrite
<-
H
'
,
ZOdiv_0_l
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_bound
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
~
(
y
=
0
%
Z
))
->
(((
-
(
Zabs
y
))
%
Z
<
(
mod1
x
y
))
%
Z
/
\
((
mod1
x
y
)
<
(
Zabs
y
))
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
Zy
.
destruct
(
Zle_or_lt
0
x
)
as
[
Hx
|
Hx
].
refine
((
fun
H
=>
conj
(
Zlt_le_trans
_
0
_
_
(
proj1
H
))
(
proj2
H
))
_
).
...
...
@@ -63,66 +49,41 @@ clear -Zy ; zify ; omega.
apply
ZOmod_lt_neg
with
(
2
:=
Zy
).
now
apply
Zlt_le_weak
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_sign_pos
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(
0
%
Z
<=
(
div
x
y
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
).
apply
ZO_div_pos
with
(
1
:=
Hx
).
now
apply
Zlt_le_weak
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_sign_neg
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
x
<=
0
%
Z
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
div
x
y
)
<=
0
%
Z
)
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
).
generalize
(
ZO_div_pos
(
-
x
)
y
).
rewrite
ZOdiv_opp_l
.
omega
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_sign_pos
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
~
(
y
=
0
%
Z
))
->
(
0
%
Z
<=
(
mod1
x
y
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Zy
).
now
apply
ZOmod_lt_pos
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_sign_neg
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
x
<=
0
%
Z
)
%
Z
/
\
~
(
y
=
0
%
Z
))
->
((
mod1
x
y
)
<=
0
%
Z
)
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Zy
).
now
apply
ZOmod_lt_neg
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Rounds_toward_zero
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
~
(
y
=
0
%
Z
))
->
((
Zabs
((
div
x
y
)
*
y
)
%
Z
)
<=
(
Zabs
x
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
Zy
.
rewrite
Zmult_comm
.
zify
.
...
...
@@ -130,57 +91,32 @@ generalize (ZO_mult_div_le x y).
generalize
(
ZO_mult_div_ge
x
y
).
omega
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_1
:
forall
(
x
:
Z
),
((
div
x
1
%
Z
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
ZOdiv_1_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_1
:
forall
(
x
:
Z
),
((
mod1
x
1
%
Z
)
=
0
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
ZOmod_1_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_inf
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
x
<
y
)
%
Z
)
->
((
div
x
y
)
=
0
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
ZOdiv_small
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_inf
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
x
<
y
)
%
Z
)
->
((
mod1
x
y
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
ZOmod_small
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Div_mult
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
0
%
Z
<
x
)
%
Z
/
\
((
0
%
Z
<=
y
)
%
Z
/
\
(
0
%
Z
<=
z
)
%
Z
))
->
((
div
((
x
*
y
)
%
Z
+
z
)
%
Z
x
)
=
(
y
+
(
div
z
x
))
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
(
Hx
&
Hy
&
Hz
).
rewrite
(
Zplus_comm
y
).
rewrite
<-
ZO_div_plus
.
...
...
@@ -192,15 +128,10 @@ now apply Zlt_le_weak.
intros
H
.
now
rewrite
H
in
Hx
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mod_mult
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
0
%
Z
<
x
)
%
Z
/
\
((
0
%
Z
<=
y
)
%
Z
/
\
(
0
%
Z
<=
z
)
%
Z
))
->
((
mod1
((
x
*
y
)
%
Z
+
z
)
%
Z
x
)
=
(
mod1
z
x
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
(
Hx
&
Hy
&
Hz
).
rewrite
Zplus_comm
,
Zmult_comm
.
apply
ZO_mod_plus
.
...
...
@@ -209,6 +140,5 @@ apply Zplus_le_0_compat with (1 := Hz).
apply
Zmult_le_0_compat
with
(
1
:=
Hy
).
now
apply
Zlt_le_weak
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
lib/coq/int/Int.v
View file @
54f46f3b
...
...
@@ -2,11 +2,11 @@
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/theories"
.
*
)
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/modules"
.
*
)
(
*
Why3
goal
*
)
Notation
infix_ls
:=
Zlt
(
only
parsing
).
(
*
Why3
assumption
*
)
Definition
infix_lseq
(
x
:
Z
)
(
y
:
Z
)
:
Prop
:=
(
infix_ls
x
y
)
\
/
(
x
=
y
).
Lemma
infix_lseq_Zle
:
...
...
@@ -17,156 +17,96 @@ apply iff_Symmetric.
apply
Zle_lt_or_eq_iff
.
Qed
.
(
*
Why3
goal
*
)
Notation
infix_pl
:=
Zplus
(
only
parsing
).
(
*
Why3
goal
*
)
Notation
prefix_mn
:=
Zopp
(
only
parsing
).
(
*
Why3
goal
*
)
Notation
infix_as
:=
Zmult
(
only
parsing
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Unit_def
:
forall
(
x
:
Z
),
((
infix_pl
x
0
%
Z
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zplus_0_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Assoc
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
infix_pl
(
infix_pl
x
y
)
z
)
=
(
infix_pl
x
(
infix_pl
y
z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
.
apply
sym_eq
.
apply
Zplus_assoc
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Inv_def
:
forall
(
x
:
Z
),
((
infix_pl
x
(
prefix_mn
x
))
=
0
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zplus_opp_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Comm
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
infix_pl
x
y
)
=
(
infix_pl
y
x
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zplus_comm
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Assoc1
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
infix_as
(
infix_as
x
y
)
z
)
=
(
infix_as
x
(
infix_as
y
z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
.
apply
sym_eq
.
apply
Zmult_assoc
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Mul_distr
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
infix_as
x
(
infix_pl
y
z
))
=
(
infix_pl
(
infix_as
x
y
)
(
infix_as
x
z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmult_plus_distr_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
assumption
*
)
Definition
infix_mn
(
x
:
Z
)
(
y
:
Z
)
:
Z
:=
(
infix_pl
x
(
prefix_mn
y
)).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Comm1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
infix_as
x
y
)
=
(
infix_as
y
x
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmult_comm
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Unitary
:
forall
(
x
:
Z
),
((
infix_as
1
%
Z
x
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmult_1_l
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
NonTrivialRing
:
~
(
0
%
Z
=
1
%
Z
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
discriminate
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Refl
:
forall
(
x
:
Z
),
(
infix_lseq
x
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
.
apply
infix_lseq_Zle
.
apply
Zle_refl
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Trans
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
(
infix_lseq
x
y
)
->
((
infix_lseq
y
z
)
->
(
infix_lseq
x
z
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
H1
H2
.
apply
infix_lseq_Zle
.
apply
Zle_trans
with
y
;
now
apply
infix_lseq_Zle
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Antisymm
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
infix_lseq
x
y
)
->
((
infix_lseq
y
x
)
->
(
x
=
y
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
H1
H2
.
apply
Zle_antisym
;
now
apply
infix_lseq_Zle
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Total
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
infix_lseq
x
y
)
\
/
(
infix_lseq
y
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
.
destruct
(
Zle_or_lt
x
y
)
as
[
H
|
H
].
left
.
...
...
@@ -175,34 +115,28 @@ right.
apply
infix_lseq_Zle
.
now
apply
Zlt_le_weak
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
ZeroLessOne
:
(
infix_lseq
0
%
Z
1
%
Z
).
now
left
.
Qed
.
(
*
Why3
goal
*
)
Lemma
CompatOrderAdd
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
(
infix_lseq
x
y
)
->
(
infix_lseq
(
infix_pl
x
z
)
(
infix_pl
y
z
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
H
.
apply
infix_lseq_Zle
.
apply
Zplus_le_compat_r
.
now
apply
infix_lseq_Zle
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
CompatOrderMult
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
(
infix_lseq
x
y
)
->
((
infix_lseq
0
%
Z
z
)
->
(
infix_lseq
(
infix_as
x
z
)
(
infix_as
y
z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
z
H1
H2
.
apply
infix_lseq_Zle
.
apply
Zmult_le_compat_r
;
now
apply
infix_lseq_Zle
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
lib/coq/int/MinMax.v
View file @
54f46f3b
...
...
@@ -2,33 +2,24 @@
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/theories"
.
*
)
(
*
Add
Rec
LoadPath
"/home/guillaume/bin/why3/share/why3/modules"
.
*
)
Require
int
.
Int
.
(
*
Why3
goal
*
)
Notation
min
:=
Zmin
(
only
parsing
).
(
*
Why3
goal
*
)
Notation
max
:=
Zmax
(
only
parsing
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Max_is_ge
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
x
<=
(
max
x
y
))
%
Z
/
\
(
y
<=
(
max
x
y
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
split
.
apply
Zle_max_l
.
apply
Zle_max_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Max_is_some
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
max
x
y
)
=
x
)
\
/
((
max
x
y
)
=
y
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
.
unfold
Zmax
.
case
Zcompare
.
...
...
@@ -36,27 +27,17 @@ now left.
now
right
.
now
left
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Min_is_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
min
x
y
)
<=
x
)
%
Z
/
\
((
min
x
y
)
<=
y
)
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
split
.
apply
Zle_min_l
.
apply
Zle_min_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Min_is_some
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
min
x
y
)
=
x
)
\
/
((
min
x
y
)
=
y
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
.
unfold
Zmin
.
case
Zcompare
.
...
...
@@ -64,68 +45,37 @@ now left.
now
left
.
now
right
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Max_x
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
y
<=
x
)
%
Z
->
((
max
x
y
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmax_l
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Max_y
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
x
<=
y
)
%
Z
->
((
max
x
y
)
=
y
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmax_r
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
(
*
Why3
goal
*
)
Lemma
Min_x
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
x
<=
y
)
%
Z
->
((
min
x
y
)
=
x
).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
exact
Zmin_l
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
