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David Swasey
coqstdpp
Commits
b9117014
Commit
b9117014
authored
Aug 04, 2014
by
Robbert Krebbers
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Add monoid operation on option.
parent
71214d32
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theories/option.v
theories/option.v
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theories/option.v
View file @
b9117014
...
...
@@ 138,6 +138,46 @@ Qed.
Lemma
bind_with_Some
{
A
}
(
x
:
option
A
)
:
x
≫
=
Some
=
x
.
Proof
.
by
destruct
x
.
Qed
.
(** * Union, intersection and difference *)
Instance
option_union_with
{
A
}
:
UnionWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with

Some
a
,
Some
b
=>
f
a
b

Some
a
,
None
=>
Some
a

None
,
Some
b
=>
Some
b

None
,
None
=>
None
end
.
Instance
option_intersection_with
{
A
}
:
IntersectionWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with
Some
a
,
Some
b
=>
f
a
b

_
,
_
=>
None
end
.
Instance
option_difference_with
{
A
}
:
DifferenceWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with

Some
a
,
Some
b
=>
f
a
b

Some
a
,
None
=>
Some
a

None
,
_
=>
None
end
.
Instance
option_union
{
A
}
:
Union
(
option
A
)
:
=
union_with
(
λ
x
_
,
Some
x
).
Lemma
option_union_Some
{
A
}
(
x
y
:
option
A
)
z
:
x
∪
y
=
Some
z
→
x
=
Some
z
∨
y
=
Some
z
.
Proof
.
destruct
x
,
y
;
intros
;
simplify_equality
;
auto
.
Qed
.
Section
option_union_intersection_difference
.
Context
{
A
}
(
f
:
A
→
A
→
option
A
).
Global
Instance
:
LeftId
(=)
None
(
union_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
RightId
(=)
None
(
union_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
Commutative
(=)
f
→
Commutative
(=)
(
union_with
f
).
Proof
.
by
intros
?
[?]
[?]
;
compute
;
rewrite
1
?(
commutative
f
).
Qed
.
Global
Instance
:
LeftAbsorb
(=)
None
(
intersection_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
RightAbsorb
(=)
None
(
intersection_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
Commutative
(=)
f
→
Commutative
(=)
(
intersection_with
f
).
Proof
.
by
intros
?
[?]
[?]
;
compute
;
rewrite
1
?(
commutative
f
).
Qed
.
Global
Instance
:
RightId
(=)
None
(
difference_with
f
).
Proof
.
by
intros
[?].
Qed
.
End
option_union_intersection_difference
.
(** * Tactics *)
Tactic
Notation
"case_option_guard"
"as"
ident
(
Hx
)
:
=
match
goal
with

H
:
context
C
[@
mguard
option
_
?P
?dec
_
?x
]

_
=>
...
...
@@ 199,6 +239,7 @@ Tactic Notation "simplify_option_equality" "by" tactic3(tac) :=
repeat
match
goal
with

_
=>
progress
simplify_equality'

_
=>
progress
simpl_option_monad
by
tac

H
:
_
∪
_
=
Some
_

_
=>
apply
option_union_Some
in
H
;
destruct
H

H
:
mbind
(
M
:
=
option
)
?f
?o
=
?x

_
=>
match
o
with
Some
_
=>
fail
1

None
=>
fail
1

_
=>
idtac
end
;
match
x
with
Some
_
=>
idtac

None
=>
idtac

_
=>
fail
1
end
;
...
...
@@ 223,38 +264,3 @@ Tactic Notation "simplify_option_equality" "by" tactic3(tac) :=

_
=>
progress
case_option_guard
end
.
Tactic
Notation
"simplify_option_equality"
:
=
simplify_option_equality
by
eauto
.
(** * Union, intersection and difference *)
Instance
option_union_with
{
A
}
:
UnionWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with

Some
a
,
Some
b
=>
f
a
b

Some
a
,
None
=>
Some
a

None
,
Some
b
=>
Some
b

None
,
None
=>
None
end
.
Instance
option_intersection_with
{
A
}
:
IntersectionWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with
Some
a
,
Some
b
=>
f
a
b

_
,
_
=>
None
end
.
Instance
option_difference_with
{
A
}
:
DifferenceWith
A
(
option
A
)
:
=
λ
f
x
y
,
match
x
,
y
with

Some
a
,
Some
b
=>
f
a
b

Some
a
,
None
=>
Some
a

None
,
_
=>
None
end
.
Section
option_union_intersection_difference
.
Context
{
A
}
(
f
:
A
→
A
→
option
A
).
Global
Instance
:
LeftId
(=)
None
(
union_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
RightId
(=)
None
(
union_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
Commutative
(=)
f
→
Commutative
(=)
(
union_with
f
).
Proof
.
by
intros
?
[?]
[?]
;
compute
;
rewrite
1
?(
commutative
f
).
Qed
.
Global
Instance
:
LeftAbsorb
(=)
None
(
intersection_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
RightAbsorb
(=)
None
(
intersection_with
f
).
Proof
.
by
intros
[?].
Qed
.
Global
Instance
:
Commutative
(=)
f
→
Commutative
(=)
(
intersection_with
f
).
Proof
.
by
intros
?
[?]
[?]
;
compute
;
rewrite
1
?(
commutative
f
).
Qed
.
Global
Instance
:
RightId
(=)
None
(
difference_with
f
).
Proof
.
by
intros
[?].
Qed
.
End
option_union_intersection_difference
.
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