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c189f3d6
Commit
c189f3d6
authored
Feb 14, 2016
by
Robbert Krebbers
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Move uPred big_op stuff to separate file.
parent
bdfb180a
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3
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-61
_CoqProject
_CoqProject
+1
-0
algebra/upred.v
algebra/upred.v
+0
-61
algebra/upred_big_op.v
algebra/upred_big_op.v
+69
-0
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_CoqProject
View file @
c189f3d6
...
...
@@ -50,6 +50,7 @@ algebra/excl.v
algebra/iprod.v
algebra/functor.v
algebra/upred.v
algebra/upred_big_op.v
program_logic/model.v
program_logic/adequacy.v
program_logic/hoare_lifting.v
...
...
algebra/upred.v
View file @
c189f3d6
...
...
@@ -219,29 +219,15 @@ Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope.
Definition
uPred_iff
{
M
}
(
P
Q
:
uPred
M
)
:
uPred
M
:
=
((
P
→
Q
)
∧
(
Q
→
P
))%
I
.
Infix
"↔"
:
=
uPred_iff
:
uPred_scope
.
Fixpoint
uPred_big_and
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
match
Ps
with
[]
=>
True
|
P
::
Ps
=>
P
∧
uPred_big_and
Ps
end
%
I
.
Instance
:
Params
(@
uPred_big_and
)
1
.
Notation
"'Π∧' Ps"
:
=
(
uPred_big_and
Ps
)
(
at
level
20
)
:
uPred_scope
.
Fixpoint
uPred_big_sep
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
match
Ps
with
[]
=>
True
|
P
::
Ps
=>
P
★
uPred_big_sep
Ps
end
%
I
.
Instance
:
Params
(@
uPred_big_sep
)
1
.
Notation
"'Π★' Ps"
:
=
(
uPred_big_sep
Ps
)
(
at
level
20
)
:
uPred_scope
.
Class
TimelessP
{
M
}
(
P
:
uPred
M
)
:
=
timelessP
:
▷
P
⊑
(
P
∨
▷
False
).
Arguments
timelessP
{
_
}
_
{
_
}
_
_
_
_
.
Class
AlwaysStable
{
M
}
(
P
:
uPred
M
)
:
=
always_stable
:
P
⊑
□
P
.
Arguments
always_stable
{
_
}
_
{
_
}
_
_
_
_
.
Class
AlwaysStableL
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
always_stableL
:
Forall
AlwaysStable
Ps
.
Arguments
always_stableL
{
_
}
_
{
_
}.
Module
uPred
.
Section
uPred_logic
.
Context
{
M
:
cmraT
}.
Implicit
Types
φ
:
Prop
.
Implicit
Types
P
Q
:
uPred
M
.
Implicit
Types
Ps
Qs
:
list
(
uPred
M
).
Implicit
Types
A
:
Type
.
Notation
"P ⊑ Q"
:
=
(@
uPred_entails
M
P
%
I
Q
%
I
).
(* Force implicit argument M *)
Arguments
uPred_holds
{
_
}
!
_
_
_
/.
...
...
@@ -849,36 +835,6 @@ Proof. done. Qed.
Lemma
ownM_invalid
(
a
:
M
)
:
¬
✓
{
0
}
a
→
uPred_ownM
a
⊑
False
.
Proof
.
by
intros
;
rewrite
ownM_valid
valid_elim
.
Qed
.
(* Big ops *)
Global
Instance
big_and_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
uPred_big_and
M
).
Proof
.
by
induction
1
as
[|
P
Q
Ps
Qs
HPQ
?
IH
]
;
rewrite
/=
?HPQ
?IH
.
Qed
.
Global
Instance
big_sep_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
uPred_big_sep
M
).
Proof
.
by
induction
1
as
[|
P
Q
Ps
Qs
HPQ
?
IH
]
;
rewrite
/=
?HPQ
?IH
.
Qed
.
Global
Instance
big_and_perm
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
uPred_big_and
M
).
Proof
.
induction
1
as
[|
P
Ps
Qs
?
IH
|
P
Q
Ps
|]
;
simpl
;
auto
.
*
by
rewrite
IH
.
*
by
rewrite
!
assoc
(
comm
_
P
).
*
etransitivity
;
eauto
.
Qed
.
Global
Instance
big_sep_perm
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
uPred_big_sep
M
).
Proof
.
induction
1
as
[|
P
Ps
Qs
?
IH
|
P
Q
Ps
|]
;
simpl
;
auto
.
*
by
rewrite
IH
.
*
by
rewrite
!
assoc
(
comm
_
P
).
*
etransitivity
;
eauto
.
Qed
.
Lemma
big_and_app
Ps
Qs
:
(
Π
∧
(
Ps
++
Qs
))%
I
≡
(
Π
∧
Ps
∧
Π
∧
Qs
)%
I
.
Proof
.
by
induction
Ps
as
[|??
IH
]
;
rewrite
/=
?left_id
-
?assoc
?IH
.
Qed
.
Lemma
big_sep_app
Ps
Qs
:
(
Π★
(
Ps
++
Qs
))%
I
≡
(
Π★
Ps
★
Π★
Qs
)%
I
.
Proof
.
by
induction
Ps
as
[|??
IH
]
;
rewrite
/=
?left_id
-
?assoc
?IH
.
Qed
.
Lemma
big_sep_and
Ps
:
(
Π★
Ps
)
⊑
(
Π
∧
Ps
).
Proof
.
by
induction
Ps
as
[|
P
Ps
IH
]
;
simpl
;
auto
.
Qed
.
Lemma
big_and_elem_of
Ps
P
:
P
∈
Ps
→
(
Π
∧
Ps
)
⊑
P
.
Proof
.
induction
1
;
simpl
;
auto
.
Qed
.
Lemma
big_sep_elem_of
Ps
P
:
P
∈
Ps
→
(
Π★
Ps
)
⊑
P
.
Proof
.
induction
1
;
simpl
;
auto
.
Qed
.
(* Timeless *)
Lemma
timelessP_spec
P
:
TimelessP
P
↔
∀
x
n
,
✓
{
n
}
x
→
P
0
x
→
P
n
x
.
Proof
.
...
...
@@ -967,23 +923,6 @@ Global Instance default_always_stable {A} P (Q : A → uPred M) (mx : option A)
AS
P
→
(
∀
x
,
AS
(
Q
x
))
→
AS
(
default
P
mx
Q
).
Proof
.
destruct
mx
;
apply
_
.
Qed
.
(* Always stable for lists *)
Local
Notation
ASL
:
=
AlwaysStableL
.
Global
Instance
big_and_always_stable
Ps
:
ASL
Ps
→
AS
(
Π
∧
Ps
).
Proof
.
induction
1
;
apply
_
.
Qed
.
Global
Instance
big_sep_always_stable
Ps
:
ASL
Ps
→
AS
(
Π★
Ps
).
Proof
.
induction
1
;
apply
_
.
Qed
.
Global
Instance
nil_always_stable
:
ASL
(@
nil
(
uPred
M
)).
Proof
.
constructor
.
Qed
.
Global
Instance
cons_always_stable
P
Ps
:
AS
P
→
ASL
Ps
→
ASL
(
P
::
Ps
).
Proof
.
by
constructor
.
Qed
.
Global
Instance
app_always_stable
Ps
Ps'
:
ASL
Ps
→
ASL
Ps'
→
ASL
(
Ps
++
Ps'
).
Proof
.
apply
Forall_app_2
.
Qed
.
Global
Instance
zip_with_always_stable
{
A
B
}
(
f
:
A
→
B
→
uPred
M
)
xs
ys
:
(
∀
x
y
,
AS
(
f
x
y
))
→
ASL
(
zip_with
f
xs
ys
).
Proof
.
unfold
ASL
=>
?
;
revert
ys
;
induction
xs
=>
-[|??]
;
constructor
;
auto
.
Qed
.
(* Derived lemmas for always stable *)
Lemma
always_always
P
`
{!
AlwaysStable
P
}
:
(
□
P
)%
I
≡
P
.
Proof
.
apply
(
anti_symm
(
⊑
))
;
auto
using
always_elim
.
Qed
.
...
...
algebra/upred_big_op.v
0 → 100644
View file @
c189f3d6
From
algebra
Require
Export
upred
.
Fixpoint
uPred_big_and
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
match
Ps
with
[]
=>
True
|
P
::
Ps
=>
P
∧
uPred_big_and
Ps
end
%
I
.
Instance
:
Params
(@
uPred_big_and
)
1
.
Notation
"'Π∧' Ps"
:
=
(
uPred_big_and
Ps
)
(
at
level
20
)
:
uPred_scope
.
Fixpoint
uPred_big_sep
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
match
Ps
with
[]
=>
True
|
P
::
Ps
=>
P
★
uPred_big_sep
Ps
end
%
I
.
Instance
:
Params
(@
uPred_big_sep
)
1
.
Notation
"'Π★' Ps"
:
=
(
uPred_big_sep
Ps
)
(
at
level
20
)
:
uPred_scope
.
Class
AlwaysStableL
{
M
}
(
Ps
:
list
(
uPred
M
))
:
=
always_stableL
:
Forall
AlwaysStable
Ps
.
Arguments
always_stableL
{
_
}
_
{
_
}.
Section
big_op
.
Context
{
M
:
cmraT
}.
Implicit
Types
P
Q
:
uPred
M
.
Implicit
Types
Ps
Qs
:
list
(
uPred
M
).
Implicit
Types
A
:
Type
.
(* Big ops *)
Global
Instance
big_and_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
uPred_big_and
M
).
Proof
.
by
induction
1
as
[|
P
Q
Ps
Qs
HPQ
?
IH
]
;
rewrite
/=
?HPQ
?IH
.
Qed
.
Global
Instance
big_sep_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
uPred_big_sep
M
).
Proof
.
by
induction
1
as
[|
P
Q
Ps
Qs
HPQ
?
IH
]
;
rewrite
/=
?HPQ
?IH
.
Qed
.
Global
Instance
big_and_perm
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
uPred_big_and
M
).
Proof
.
induction
1
as
[|
P
Ps
Qs
?
IH
|
P
Q
Ps
|]
;
simpl
;
auto
.
*
by
rewrite
IH
.
*
by
rewrite
!
assoc
(
comm
_
P
).
*
etransitivity
;
eauto
.
Qed
.
Global
Instance
big_sep_perm
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
uPred_big_sep
M
).
Proof
.
induction
1
as
[|
P
Ps
Qs
?
IH
|
P
Q
Ps
|]
;
simpl
;
auto
.
*
by
rewrite
IH
.
*
by
rewrite
!
assoc
(
comm
_
P
).
*
etransitivity
;
eauto
.
Qed
.
Lemma
big_and_app
Ps
Qs
:
(
Π
∧
(
Ps
++
Qs
))%
I
≡
(
Π
∧
Ps
∧
Π
∧
Qs
)%
I
.
Proof
.
by
induction
Ps
as
[|??
IH
]
;
rewrite
/=
?left_id
-
?assoc
?IH
.
Qed
.
Lemma
big_sep_app
Ps
Qs
:
(
Π★
(
Ps
++
Qs
))%
I
≡
(
Π★
Ps
★
Π★
Qs
)%
I
.
Proof
.
by
induction
Ps
as
[|??
IH
]
;
rewrite
/=
?left_id
-
?assoc
?IH
.
Qed
.
Lemma
big_sep_and
Ps
:
(
Π★
Ps
)
⊑
(
Π
∧
Ps
).
Proof
.
by
induction
Ps
as
[|
P
Ps
IH
]
;
simpl
;
auto
with
I
.
Qed
.
Lemma
big_and_elem_of
Ps
P
:
P
∈
Ps
→
(
Π
∧
Ps
)
⊑
P
.
Proof
.
induction
1
;
simpl
;
auto
with
I
.
Qed
.
Lemma
big_sep_elem_of
Ps
P
:
P
∈
Ps
→
(
Π★
Ps
)
⊑
P
.
Proof
.
induction
1
;
simpl
;
auto
with
I
.
Qed
.
(* Always stable *)
Local
Notation
AS
:
=
AlwaysStable
.
Local
Notation
ASL
:
=
AlwaysStableL
.
Global
Instance
big_and_always_stable
Ps
:
ASL
Ps
→
AS
(
Π
∧
Ps
).
Proof
.
induction
1
;
apply
_
.
Qed
.
Global
Instance
big_sep_always_stable
Ps
:
ASL
Ps
→
AS
(
Π★
Ps
).
Proof
.
induction
1
;
apply
_
.
Qed
.
Global
Instance
nil_always_stable
:
ASL
(@
nil
(
uPred
M
)).
Proof
.
constructor
.
Qed
.
Global
Instance
cons_always_stable
P
Ps
:
AS
P
→
ASL
Ps
→
ASL
(
P
::
Ps
).
Proof
.
by
constructor
.
Qed
.
Global
Instance
app_always_stable
Ps
Ps'
:
ASL
Ps
→
ASL
Ps'
→
ASL
(
Ps
++
Ps'
).
Proof
.
apply
Forall_app_2
.
Qed
.
Global
Instance
zip_with_always_stable
{
A
B
}
(
f
:
A
→
B
→
uPred
M
)
xs
ys
:
(
∀
x
y
,
AS
(
f
x
y
))
→
ASL
(
zip_with
f
xs
ys
).
Proof
.
unfold
ASL
=>
?
;
revert
ys
;
induction
xs
=>
-[|??]
;
constructor
;
auto
.
Qed
.
End
big_op
.
\ No newline at end of file
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