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834c66e0
Commit
834c66e0
authored
Sep 30, 2016
by
Robbert Krebbers
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Extensionality (for Leibniz equality) of big ops.
parent
c3c31e88
Changes
2
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2 changed files
with
22 additions
and
17 deletions
+22
-17
algebra/cmra_big_op.v
algebra/cmra_big_op.v
+17
-9
algebra/upred_big_op.v
algebra/upred_big_op.v
+5
-8
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algebra/cmra_big_op.v
View file @
834c66e0
...
...
@@ -138,6 +138,10 @@ Section list.
(
∀
k
y
,
l
!!
k
=
Some
y
→
f
k
y
≼
g
k
y
)
→
([
⋅
list
]
k
↦
y
∈
l
,
f
k
y
)
≼
[
⋅
list
]
k
↦
y
∈
l
,
g
k
y
.
Proof
.
apply
big_opL_forall
;
apply
_
.
Qed
.
Lemma
big_opL_ext
f
g
l
:
(
∀
k
y
,
l
!!
k
=
Some
y
→
f
k
y
=
g
k
y
)
→
([
⋅
list
]
k
↦
y
∈
l
,
f
k
y
)
=
[
⋅
list
]
k
↦
y
∈
l
,
g
k
y
.
Proof
.
apply
big_opL_forall
;
apply
_
.
Qed
.
Lemma
big_opL_proper
f
g
l
:
(
∀
k
y
,
l
!!
k
=
Some
y
→
f
k
y
≡
g
k
y
)
→
([
⋅
list
]
k
↦
y
∈
l
,
f
k
y
)
≡
([
⋅
list
]
k
↦
y
∈
l
,
g
k
y
).
...
...
@@ -207,6 +211,10 @@ Section gmap.
-
by
apply
big_op_contains
,
fmap_contains
,
map_to_list_contains
.
-
apply
big_opM_forall
;
apply
_
||
auto
.
Qed
.
Lemma
big_opM_ext
f
g
m
:
(
∀
k
x
,
m
!!
k
=
Some
x
→
f
k
x
=
g
k
x
)
→
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
=
([
⋅
map
]
k
↦
x
∈
m
,
g
k
x
).
Proof
.
apply
big_opM_forall
;
apply
_
.
Qed
.
Lemma
big_opM_proper
f
g
m
:
(
∀
k
x
,
m
!!
k
=
Some
x
→
f
k
x
≡
g
k
x
)
→
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
≡
([
⋅
map
]
k
↦
x
∈
m
,
g
k
x
).
...
...
@@ -314,14 +322,14 @@ Section gset.
-
by
apply
big_op_contains
,
fmap_contains
,
elements_contains
.
-
apply
big_opS_forall
;
apply
_
||
auto
.
Qed
.
Lemma
big_opS_
proper
f
g
X
Y
:
X
≡
Y
→
(
∀
x
,
x
∈
X
→
x
∈
Y
→
f
x
≡
g
x
)
→
([
⋅
set
]
x
∈
X
,
f
x
)
≡
([
⋅
set
]
x
∈
Y
,
g
x
).
Proof
.
intros
HX
Hf
.
trans
([
⋅
set
]
x
∈
Y
,
f
x
).
-
apply
big_op_permutation
.
by
rewrite
HX
.
-
apply
big_opS_forall
;
try
apply
_
||
set_solver
.
Qed
.
Lemma
big_opS_
ext
f
g
X
:
(
∀
x
,
x
∈
X
→
f
x
=
g
x
)
→
([
⋅
set
]
x
∈
X
,
f
x
)
=
([
⋅
set
]
x
∈
X
,
g
x
).
Proof
.
apply
big_opS_forall
;
apply
_
.
Qed
.
Lemma
big_opS_proper
f
g
X
:
(
∀
x
,
x
∈
X
→
f
x
≡
g
x
)
→
([
⋅
set
]
x
∈
X
,
f
x
)
≡
([
⋅
set
]
x
∈
X
,
g
x
)
.
Proof
.
apply
big_opS_forall
;
apply
_
.
Qed
.
Lemma
big_opS_ne
X
n
:
Proper
(
pointwise_relation
_
(
dist
n
)
==>
dist
n
)
(
big_opS
(
M
:
=
M
)
X
).
...
...
@@ -345,7 +353,7 @@ Section gset.
≡
(
f
x
b
⋅
[
⋅
set
]
y
∈
X
,
f
y
(
h
y
)).
Proof
.
intros
.
rewrite
big_opS_insert
//
fn_lookup_insert
.
apply
cmra_op_proper'
,
big_opS_proper
;
auto
=>
y
?
?.
apply
cmra_op_proper'
,
big_opS_proper
;
auto
=>
y
?.
by
rewrite
fn_lookup_insert_ne
;
last
set_solver
.
Qed
.
Lemma
big_opS_fn_insert'
f
X
x
P
:
...
...
algebra/upred_big_op.v
View file @
834c66e0
...
...
@@ -125,7 +125,6 @@ Section list.
(
∀
k
y
,
l
!!
k
=
Some
y
→
Φ
k
y
⊢
Ψ
k
y
)
→
([
★
list
]
k
↦
y
∈
l
,
Φ
k
y
)
⊢
[
★
list
]
k
↦
y
∈
l
,
Ψ
k
y
.
Proof
.
apply
big_opL_forall
;
apply
_
.
Qed
.
Lemma
big_sepL_proper
Φ
Ψ
l
:
(
∀
k
y
,
l
!!
k
=
Some
y
→
Φ
k
y
⊣
⊢
Ψ
k
y
)
→
([
★
list
]
k
↦
y
∈
l
,
Φ
k
y
)
⊣
⊢
([
★
list
]
k
↦
y
∈
l
,
Ψ
k
y
).
...
...
@@ -219,7 +218,6 @@ Section gmap.
by
apply
fmap_contains
,
map_to_list_contains
.
-
apply
big_opM_forall
;
apply
_
||
auto
.
Qed
.
Lemma
big_sepM_proper
Φ
Ψ
m
:
(
∀
k
x
,
m
!!
k
=
Some
x
→
Φ
k
x
⊣
⊢
Ψ
k
x
)
→
([
★
map
]
k
↦
x
∈
m
,
Φ
k
x
)
⊣
⊢
([
★
map
]
k
↦
x
∈
m
,
Ψ
k
x
).
...
...
@@ -344,16 +342,15 @@ Section gset.
by
apply
fmap_contains
,
elements_contains
.
-
apply
big_opS_forall
;
apply
_
||
auto
.
Qed
.
Lemma
big_sepS_proper
Φ
Ψ
X
:
(
∀
x
,
x
∈
X
→
Φ
x
⊣
⊢
Ψ
x
)
→
([
★
set
]
x
∈
X
,
Φ
x
)
⊣
⊢
([
★
set
]
x
∈
X
,
Ψ
x
).
Proof
.
apply
:
big_opS_proper
.
Qed
.
Lemma
big_sepS_mono'
X
:
Global
Instance
big_sepS_mono'
X
:
Proper
(
pointwise_relation
_
(
⊢
)
==>
(
⊢
))
(
big_opS
(
M
:
=
uPredUR
M
)
X
).
Proof
.
intros
f
g
Hf
.
apply
big_opS_forall
;
apply
_
||
intros
;
apply
Hf
.
Qed
.
Lemma
big_sepS_proper
Φ
Ψ
X
Y
:
X
≡
Y
→
(
∀
x
,
x
∈
X
→
x
∈
Y
→
Φ
x
⊣
⊢
Ψ
x
)
→
([
★
set
]
x
∈
X
,
Φ
x
)
⊣
⊢
([
★
set
]
x
∈
Y
,
Ψ
x
).
Proof
.
apply
:
big_opS_proper
.
Qed
.
Lemma
big_sepS_empty
Φ
:
([
★
set
]
x
∈
∅
,
Φ
x
)
⊣
⊢
True
.
Proof
.
by
rewrite
big_opS_empty
.
Qed
.
...
...
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