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stdpp
Commits
e409571d
Commit
e409571d
authored
Apr 22, 2015
by
Robbert Krebbers
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Restore axiomatic semantics.
parent
cc4ff176
Changes
5
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5 changed files
with
36 additions
and
43 deletions
+36
43
theories/ars.v
theories/ars.v
+7
34
theories/collections.v
theories/collections.v
+12
9
theories/fin_map_dom.v
theories/fin_map_dom.v
+6
0
theories/fin_maps.v
theories/fin_maps.v
+7
0
theories/option.v
theories/option.v
+4
0
No files found.
theories/ars.v
View file @
e409571d
...
...
@@ 64,8 +64,6 @@ Section rtc.
Proof
.
exact
rtc_transitive
.
Qed
.
Lemma
rtc_once
x
y
:
R
x
y
→
rtc
R
x
y
.
Proof
.
eauto
.
Qed
.
Instance
rtc_once_subrel
:
subrelation
R
(
rtc
R
).
Proof
.
exact
@
rtc_once
.
Qed
.
Lemma
rtc_r
x
y
z
:
rtc
R
x
y
→
R
y
z
→
rtc
R
x
z
.
Proof
.
intros
.
etransitivity
;
eauto
.
Qed
.
Lemma
rtc_inv
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
R
x
y
∧
rtc
R
y
z
.
...
...
@@ 156,8 +154,6 @@ Section rtc.
Proof
.
intros
Hxy
Hyz
.
revert
x
Hxy
.
induction
Hyz
;
eauto
using
tc_r
.
Qed
.
Lemma
tc_rtc
x
y
:
tc
R
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
.
Qed
.
Instance
tc_once_subrel
:
subrelation
(
tc
R
)
(
rtc
R
).
Proof
.
exact
@
tc_rtc
.
Qed
.
Lemma
all_loop_red
x
:
all_loop
R
x
→
red
R
x
.
Proof
.
destruct
1
;
auto
.
Qed
.
...
...
@@ 174,44 +170,21 @@ Section rtc.
Qed
.
End
rtc
.
(* Avoid too eager type class resolution *)
Hint
Extern
5
(
subrelation
_
(
rtc
_
))
=>
eapply
@
rtc_once_subrel
:
typeclass_instances
.
Hint
Extern
5
(
subrelation
_
(
tc
_
))
=>
eapply
@
tc_once_subrel
:
typeclass_instances
.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
ars
.
Hint
Resolve
rtc_once
rtc_r
tc_r
rtc_transitive
tc_rtc_l
tc_rtc_r
tc_rtc
bsteps_once
bsteps_r
bsteps_refl
bsteps_trans
:
ars
.
(** * Theorems on sub relations *)
Section
subrel
.
Context
{
A
}
(
R1
R2
:
relation
A
)
(
Hsub
:
subrelation
R1
R2
).
Lemma
red_subrel
x
:
red
R1
x
→
red
R2
x
.
Proof
.
intros
[
y
?].
exists
y
.
by
apply
Hsub
.
Qed
.
Lemma
nf_subrel
x
:
nf
R2
x
→
nf
R1
x
.
Proof
.
intros
H1
H2
.
destruct
H1
.
by
apply
red_subrel
.
Qed
.
Instance
rtc_subrel
:
subrelation
(
rtc
R1
)
(
rtc
R2
).
Proof
.
induction
1
;
[
left

eright
]
;
eauto
;
by
apply
Hsub
.
Qed
.
Instance
nsteps_subrel
:
subrelation
(
nsteps
R1
n
)
(
nsteps
R2
n
).
Proof
.
induction
1
;
[
left

eright
]
;
eauto
;
by
apply
Hsub
.
Qed
.
Instance
bsteps_subrel
:
subrelation
(
bsteps
R1
n
)
(
bsteps
R2
n
).
Proof
.
induction
1
;
[
left

eright
]
;
eauto
;
by
apply
Hsub
.
Qed
.
Instance
tc_subrel
:
subrelation
(
tc
R1
)
(
tc
R2
).
Proof
.
induction
1
;
[
left

eright
]
;
eauto
;
by
apply
Hsub
.
Qed
.
Context
{
A
}
(
R1
R2
:
relation
A
).
Notation
subrel
:
=
(
∀
x
y
,
R1
x
y
→
R2
x
y
).
Lemma
red_subrel
x
:
subrel
→
red
R1
x
→
red
R2
x
.
Proof
.
intros
?
[
y
?]
;
eauto
.
Qed
.
Lemma
nf_subrel
x
:
subrel
→
nf
R2
x
→
nf
R1
x
.
Proof
.
intros
?
H1
H2
;
destruct
H1
;
by
apply
red_subrel
.
Qed
.
End
subrel
.
Hint
Extern
5
(
subrelation
(
rtc
_
)
(
rtc
_
))
=>
eapply
@
rtc_subrel
:
typeclass_instances
.
Hint
Extern
5
(
subrelation
(
nsteps
_
)
(
nsteps
_
))
=>
eapply
@
nsteps_subrel
:
typeclass_instances
.
Hint
Extern
5
(
subrelation
(
bsteps
_
)
(
bsteps
_
))
=>
eapply
@
bsteps_subrel
:
typeclass_instances
.
Hint
Extern
5
(
subrelation
(
tc
_
)
(
tc
_
))
=>
eapply
@
tc_subrel
:
typeclass_instances
.
(** * Theorems on well founded relations *)
Notation
wf
:
=
well_founded
.
Section
wf
.
...
...
theories/collections.v
View file @
e409571d
...
...
@@ 138,28 +138,31 @@ Tactic Notation "decompose_elem_of" hyp(H) :=

_
∈
∅
=>
apply
elem_of_empty
in
H
;
destruct
H

?x
∈
{[
?y
]}
=>
apply
elem_of_singleton
in
H
;
try
first
[
subst
y

subst
x
]

?x
∉
{[
?y
]}
=>
apply
not_elem_of_singleton
in
H

_
∈
_
∪
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_union
in
H
;
destruct
H
as
[
H1

H2
]
;
[
go
H1

go
H2
]
apply
elem_of_union
in
H
;
destruct
H
as
[
H

H
]
;
[
go
H

go
H
]

_
∉
_
∪
_
=>
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
not_elem_of_union
in
H
;
destruct
H
as
[
H1
H2
]
;
go
H1
;
go
H2

_
∈
_
∩
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_intersection
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_intersection
in
H
;
destruct
H
as
[
H1
H2
]
;
go
H1
;
go
H2

_
∈
_
∖
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_difference
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_difference
in
H
;
destruct
H
as
[
H1
H2
]
;
go
H1
;
go
H2

?x
∈
_
<$>
_
=>
let
H1
:
=
fresh
in
apply
elem_of_fmap
in
H
;
destruct
H
as
[?
[?
H1
]]
;
try
(
subst
x
)
;
go
H1
apply
elem_of_fmap
in
H
;
destruct
H
as
[?
[?
H
]]
;
try
(
subst
x
)
;
go
H

_
∈
_
≫
=
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_bind
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_bind
in
H
;
destruct
H
as
[?
[
H1
H2
]]
;
go
H1
;
go
H2

?x
∈
mret
?y
=>
apply
elem_of_ret
in
H
;
try
first
[
subst
y

subst
x
]

_
∈
mjoin
_
≫
=
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_join
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_join
in
H
;
destruct
H
as
[?
[
H1
H2
]]
;
go
H1
;
go
H2

_
∈
guard
_;
_
=>
let
H1
:
=
fresh
in
let
H2
:
=
fresh
in
apply
elem_of_guard
in
H
;
let
H1
:
=
fresh
H
in
let
H2
:
=
fresh
H
in
apply
elem_of_guard
in
H
;
destruct
H
as
[
H1
H2
]
;
go
H2

_
∈
of_option
_
=>
apply
elem_of_of_option
in
H

_
=>
idtac
...
...
theories/fin_map_dom.v
View file @
e409571d
...
...
@@ 105,4 +105,10 @@ Proof.
unfold
is_Some
.
setoid_rewrite
lookup_difference_Some
.
destruct
(
m2
!!
i
)
;
naive_solver
.
Qed
.
Lemma
dom_fmap
{
A
B
}
(
f
:
A
→
B
)
m
:
dom
D
(
f
<$>
m
)
≡
dom
D
m
.
Proof
.
apply
elem_of_equiv
.
intros
i
.
rewrite
!
elem_of_dom
,
lookup_fmap
,
<!
not_eq_None_Some
.
destruct
(
m
!!
i
)
;
naive_solver
.
Qed
.
End
fin_map_dom
.
theories/fin_maps.v
View file @
e409571d
...
...
@@ 450,6 +450,13 @@ Lemma fmap_empty {A B} (f : A → B) : f <$> ∅ = ∅.
Proof
.
apply
map_empty
;
intros
i
.
by
rewrite
lookup_fmap
,
lookup_empty
.
Qed
.
Lemma
omap_empty
{
A
B
}
(
f
:
A
→
option
B
)
:
omap
f
∅
=
∅
.
Proof
.
apply
map_empty
;
intros
i
.
by
rewrite
lookup_omap
,
lookup_empty
.
Qed
.
Lemma
omap_singleton
{
A
B
}
(
f
:
A
→
option
B
)
i
x
y
:
f
x
=
Some
y
→
omap
f
{[
i
,
x
]}
=
{[
i
,
y
]}.
Proof
.
intros
;
apply
map_eq
;
intros
j
;
destruct
(
decide
(
i
=
j
))
as
[>].
*
by
rewrite
lookup_omap
,
!
lookup_singleton
.
*
by
rewrite
lookup_omap
,
!
lookup_singleton_ne
.
Qed
.
(** ** Properties of conversion to lists *)
Lemma
map_to_list_unique
{
A
}
(
m
:
M
A
)
i
x
y
:
...
...
theories/option.v
View file @
e409571d
...
...
@@ 261,6 +261,10 @@ Tactic Notation "simpl_option_monad" "by" tactic3(tac) :=

option
?A
=>
let
Hx
:
=
fresh
in
assert_Some_None
A
o
Hx
;
rewrite
Hx
;
clear
Hx
end

H
:
context
[
decide
_
]

_
=>
rewrite
decide_True
in
H
by
tac

H
:
context
[
decide
_
]

_
=>
rewrite
decide_False
in
H
by
tac

H
:
context
[
mguard
_
_
]

_
=>
rewrite
option_guard_False
in
H
by
tac

H
:
context
[
mguard
_
_
]

_
=>
rewrite
option_guard_True
in
H
by
tac

_
=>
rewrite
decide_True
by
tac

_
=>
rewrite
decide_False
by
tac

_
=>
rewrite
option_guard_True
by
tac
...
...
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