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409e0c1b
Commit
409e0c1b
authored
May 26, 2016
by
Robbert Krebbers
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More inversion properties for equiv/dist on option.
parent
184838d3
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+22
-4
algebra/cofe.v
algebra/cofe.v
+13
-0
prelude/option.v
prelude/option.v
+9
-4
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algebra/cofe.v
View file @
409e0c1b
...
...
@@ -577,6 +577,19 @@ Section option.
Proof
.
inversion_clear
1
;
constructor
.
Qed
.
Global
Instance
Some_timeless
x
:
Timeless
x
→
Timeless
(
Some
x
).
Proof
.
by
intros
?
;
inversion_clear
1
;
constructor
;
apply
timeless
.
Qed
.
Lemma
dist_None
n
mx
:
mx
≡
{
n
}
≡
None
↔
mx
=
None
.
Proof
.
split
;
[
by
inversion_clear
1
|
by
intros
->].
Qed
.
Lemma
dist_Some_inv_l
n
mx
my
x
:
mx
≡
{
n
}
≡
my
→
mx
=
Some
x
→
∃
y
,
my
=
Some
y
∧
x
≡
{
n
}
≡
y
.
Proof
.
destruct
1
;
naive_solver
.
Qed
.
Lemma
dist_Some_inv_r
n
mx
my
y
:
mx
≡
{
n
}
≡
my
→
my
=
Some
y
→
∃
x
,
mx
=
Some
x
∧
x
≡
{
n
}
≡
y
.
Proof
.
destruct
1
;
naive_solver
.
Qed
.
Lemma
dist_Some_inv_l'
n
my
x
:
Some
x
≡
{
n
}
≡
my
→
∃
x'
,
Some
x'
=
my
∧
x
≡
{
n
}
≡
x'
.
Proof
.
intros
?%(
dist_Some_inv_l
_
_
_
x
)
;
naive_solver
.
Qed
.
Lemma
dist_Some_inv_r'
n
mx
y
:
mx
≡
{
n
}
≡
Some
y
→
∃
y'
,
mx
=
Some
y'
∧
y
≡
{
n
}
≡
y'
.
Proof
.
intros
?%(
dist_Some_inv_r
_
_
_
y
)
;
naive_solver
.
Qed
.
End
option
.
Typeclasses
Opaque
option_dist
.
...
...
prelude/option.v
View file @
409e0c1b
...
...
@@ -110,6 +110,7 @@ Instance option_equiv `{Equiv A} : Equiv (option A) := option_Forall2 (≡).
Section
setoids
.
Context
`
{
Equiv
A
}
`
{!
Equivalence
((
≡
)
:
relation
A
)}.
Implicit
Types
mx
my
:
option
A
.
Lemma
equiv_option_Forall2
mx
my
:
mx
≡
my
↔
option_Forall2
(
≡
)
mx
my
.
Proof
.
done
.
Qed
.
...
...
@@ -121,14 +122,18 @@ Section setoids.
Global
Instance
option_leibniz
`
{!
LeibnizEquiv
A
}
:
LeibnizEquiv
(
option
A
).
Proof
.
intros
x
y
;
destruct
1
;
fold_leibniz
;
congruence
.
Qed
.
Lemma
equiv_None
(
mx
:
option
A
)
:
mx
≡
None
↔
mx
=
None
.
Lemma
equiv_None
mx
:
mx
≡
None
↔
mx
=
None
.
Proof
.
split
;
[
by
inversion_clear
1
|
by
intros
->].
Qed
.
Lemma
equiv_Some_inv_l
(
mx
my
:
option
A
)
x
:
Lemma
equiv_Some_inv_l
mx
my
x
:
mx
≡
my
→
mx
=
Some
x
→
∃
y
,
my
=
Some
y
∧
x
≡
y
.
Proof
.
destruct
1
;
naive_solver
.
Qed
.
Lemma
equiv_Some_inv_r
(
mx
my
:
option
A
)
y
:
mx
≡
my
→
m
x
=
Some
y
→
∃
x
,
mx
=
Some
x
∧
x
≡
y
.
Lemma
equiv_Some_inv_r
mx
my
y
:
mx
≡
my
→
m
y
=
Some
y
→
∃
x
,
mx
=
Some
x
∧
x
≡
y
.
Proof
.
destruct
1
;
naive_solver
.
Qed
.
Lemma
equiv_Some_inv_l'
my
x
:
Some
x
≡
my
→
∃
x'
,
Some
x'
=
my
∧
x
≡
x'
.
Proof
.
intros
?%(
equiv_Some_inv_l
_
_
x
)
;
naive_solver
.
Qed
.
Lemma
equiv_Some_inv_r'
mx
y
:
mx
≡
Some
y
→
∃
y'
,
mx
=
Some
y'
∧
y
≡
y'
.
Proof
.
intros
?%(
equiv_Some_inv_r
_
_
y
)
;
naive_solver
.
Qed
.
Global
Instance
is_Some_proper
:
Proper
((
≡
)
==>
iff
)
(@
is_Some
A
).
Proof
.
inversion_clear
1
;
split
;
eauto
.
Qed
.
...
...
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