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David Swasey
coqstdpp
Commits
034bdcad
Commit
034bdcad
authored
Nov 09, 2017
by
Johannes Kloos
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Renamed subset_difference_in and moved to collection
parent
01f94618
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2 changed files
with
5 additions
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5 deletions
+5
5
theories/collections.v
theories/collections.v
+3
0
theories/infinite.v
theories/infinite.v
+2
5
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theories/collections.v
View file @
034bdcad
...
...
@@ 620,6 +620,9 @@ Section collection.
Proof
.
set_solver
.
Qed
.
Lemma
difference_disjoint
X
Y
:
X
##
Y
→
X
∖
Y
≡
X
.
Proof
.
set_solver
.
Qed
.
Lemma
subset_difference_elem_of
{
x
:
A
}
{
s
:
C
}
(
inx
:
x
∈
s
)
:
s
∖
{[
x
]}
⊂
s
.
Proof
.
set_solver
.
Qed
.
Lemma
difference_mono
X1
X2
Y1
Y2
:
X1
⊆
X2
→
Y2
⊆
Y1
→
X1
∖
Y1
⊆
X2
∖
Y2
.
...
...
theories/infinite.v
View file @
034bdcad
...
...
@@ 27,13 +27,10 @@ Qed.
Section
Fresh
.
Context
`
{
FinCollection
A
C
}
`
{
Infinite
A
,
!
RelDecision
(@
elem_of
A
C
_
)}.
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
)

left
H
=>
rec
_
(
subset_difference_
elem_of
H
)
(
S
n
)

right
_
=>
cand
end
.
Lemma
fresh_generic_body_proper
s
(
f
g
:
∀
y
,
y
⊂
s
→
nat
→
A
)
:
...
...
@@ 64,7 +61,7 @@ Section Fresh.
induction
s
as
[
s
IH
]
using
(
well_founded_ind
collection_wf
)
;
intro
.
setoid_rewrite
fresh_generic_fixpoint_unfold
;
unfold
fresh_generic_body
.
destruct
decide
as
[
case

case
]
;
eauto
with
omega
.
destruct
(
IH
_
(
subset_difference_
in
case
)
(
S
n
))
as
[
m
[
mbound
[
eqfix
[
notin
inbelow
]]]].
destruct
(
IH
_
(
subset_difference_
elem_of
case
)
(
S
n
))
as
[
m
[
mbound
[
eqfix
[
notin
inbelow
]]]].
exists
m
;
repeat
split
;
auto
with
omega
.

rewrite
not_elem_of_difference
,
elem_of_singleton
in
notin
.
destruct
notin
as
[??%
inject_injective
]
;
auto
with
omega
.
...
...
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