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David Swasey
coq-stdpp
Commits
a8d02255
Commit
a8d02255
authored
Oct 31, 2017
by
Johannes Kloos
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Generic fresh element generator for infinite types.
This implements a simple linear search for fresh elements.
parent
04b92602
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1
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theories/fin_collections.v
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theories/fin_collections.v
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a8d02255
...
...
@@ -283,4 +283,89 @@ Proof.
refine
(
cast_if
(
decide
(
Exists
P
(
elements
X
))))
;
by
rewrite
set_Exists_elements
.
Defined
.
(** * Fresh elements *)
Section
Fresh
.
Context
`
{
Infinite
A
,
∀
(
x
:
A
)
(
s
:
C
),
Decision
(
x
∈
s
)}.
Lemma
subset_difference_in
{
x
:
A
}
{
s
:
C
}
(
inx
:
x
∈
s
)
:
s
∖
{[
x
]}
⊂
s
.
Proof
.
set_solver
.
Qed
.
Definition
fresh_generic_body
(
s
:
C
)
(
rec
:
∀
s'
,
s'
⊂
s
→
nat
→
A
)
(
n
:
nat
)
:
=
let
cand
:
=
inject
n
in
match
decide
(
cand
∈
s
)
with
|
left
H
=>
rec
_
(
subset_difference_in
H
)
(
S
n
)
|
right
_
=>
cand
end
.
Lemma
fresh_generic_body_proper
s
(
f
g
:
∀
y
,
y
⊂
s
→
nat
→
A
)
:
(
∀
y
Hy
Hy'
,
pointwise_relation
nat
eq
(
f
y
Hy
)
(
g
y
Hy'
))
→
pointwise_relation
nat
eq
(
fresh_generic_body
s
f
)
(
fresh_generic_body
s
g
).
Proof
.
intros
relfg
i
.
unfold
fresh_generic_body
.
destruct
decide
.
2
:
done
.
apply
relfg
.
Qed
.
Global
Instance
fresh_generic
:
Fresh
A
C
|
20
:
=
λ
s
,
Fix
collection_wf
(
const
(
nat
→
A
))
fresh_generic_body
s
0
.
Global
Instance
fresh_generic_spec
:
FreshSpec
A
C
.
Proof
.
assert
(
unfold
:
∀
s
n
,
Fix
collection_wf
(
const
(
nat
→
A
))
fresh_generic_body
s
n
=
fresh_generic_body
s
(
λ
s'
_
n
,
Fix
collection_wf
(
const
(
nat
→
A
))
fresh_generic_body
s'
n
)
n
).
{
intros
.
apply
(
Fix_unfold_rel
collection_wf
(
const
(
nat
→
A
))
(
const
(
pointwise_relation
nat
(=)))
fresh_generic_body
fresh_generic_body_proper
s
n
).
}
split
.
-
apply
_
.
-
unfold
fresh
,
fresh_generic
.
intro
X
.
generalize
0
as
n
.
induction
X
as
[
X
IH
]
using
(
well_founded_ind
collection_wf
).
intros
n
Y
eqXY
.
rewrite
(
unfold
X
n
),
(
unfold
Y
n
).
unfold
fresh_generic_body
at
1
3
.
destruct
decide
as
[
case
|
case
],
decide
as
[
case'
|
case'
].
2
,
3
:
specialize
(
eqXY
(
inject
n
))
;
tauto
.
2
:
done
.
erewrite
(
IH
(
X
∖
{[
inject
n
]})).
+
done
.
+
by
apply
subset_difference_in
.
+
by
intro
;
rewrite
!
elem_of_difference
,
eqXY
.
-
unfold
fresh
,
fresh_generic
.
intro
X
.
generalize
0
as
n
.
induction
X
as
[
X
IH
]
using
(
well_founded_ind
collection_wf
).
intro
.
rewrite
unfold
;
unfold
fresh_generic_body
at
1
.
destruct
decide
as
[
case
|
case
].
2
:
done
.
specialize
(
IH
_
(
subset_difference_in
case
)
(
S
n
)).
rewrite
not_elem_of_difference
in
IH
.
destruct
IH
as
[?|
contra
].
1
:
done
.
clear
case
.
assert
(
∀
Y
n
,
∃
m
,
Fix
collection_wf
(
const
(
nat
→
A
))
fresh_generic_body
Y
n
=
inject
m
∧
n
≤
m
)
as
candidates
.
{
clear
X
n
contra
.
induction
Y
as
[
Y
IH
]
using
(
well_founded_ind
collection_wf
).
intro
.
setoid_rewrite
unfold
;
unfold
fresh_generic_body
at
1
.
destruct
decide
as
[
case
|
case
].
{
destruct
(
IH
_
(
subset_difference_in
case
)
(
S
n
))
as
[
m
[
eqx
bound
]].
exists
m
;
split
;
auto
with
arith
.
}
{
exists
n
;
auto
.
}
}
revert
contra
.
destruct
(
candidates
(
X
∖
{[
inject
n
]})
(
S
n
))
as
[
m
[->
bound
]].
rewrite
elem_of_singleton
;
intro
eqinj
.
enough
(
n
=
m
)
by
lia
.
apply
inject_injective
;
by
rewrite
eqinj
.
Qed
.
End
Fresh
.
End
fin_collection
.
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