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stdpp
Commits
e79e91f7
Commit
e79e91f7
authored
Aug 29, 2012
by
Robbert Krebbers
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Add documentation, add license, simplify build process, some reorganization,
improve some definitions, simplify some proofs.
parent
fb763df2
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+1190
545
theories/ars.v
theories/ars.v
+159
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theories/base.v
theories/base.v
+187
130
theories/collections.v
theories/collections.v
+119
96
theories/decidable.v
theories/decidable.v
+75
42
theories/fin_collections.v
theories/fin_collections.v
+57
37
theories/fin_maps.v
theories/fin_maps.v
+100
53
theories/list.v
theories/list.v
+112
52
theories/listset.v
theories/listset.v
+30
13
theories/monads.v
theories/monads.v
+0
20
theories/nmap.v
theories/nmap.v
+8
2
theories/numbers.v
theories/numbers.v
+24
3
theories/option.v
theories/option.v
+81
76
theories/orders.v
theories/orders.v
+11
0
theories/pmap.v
theories/pmap.v
+42
16
theories/prelude.v
theories/prelude.v
+8
4
theories/subset.v
theories/subset.v
+6
1
theories/tactics.v
theories/tactics.v
+171
0
No files found.
theories/
t
rs.v
→
theories/
a
rs.v
View file @
e79e91f7
Require
Export
base
.
Definition
red
`
(
R
:
relation
A
)
(
x
:
A
)
:
=
∃
y
,
R
x
y
.
Definition
nf
`
(
R
:
relation
A
)
(
x
:
A
)
:
=
¬
red
R
x
.
(* The reflexive transitive closure
*)
Inductive
rtc
`
(
R
:
relation
A
)
:
relation
A
:
=

rtc_refl
x
:
rtc
R
x
x

rtc_l
x
y
z
:
R
x
y
→
rtc
R
y
z
→
rtc
R
x
z
.
(* A reduction of exactly n steps *)
Inductive
nsteps
`
(
R
:
relation
A
)
:
nat
→
relation
A
:
=

nsteps_O
x
:
nsteps
R
0
x
x

nsteps_l
n
x
y
z
:
R
x
y
→
nsteps
R
n
y
z
→
nsteps
R
(
S
n
)
x
z
.
(* A reduction whose length is bounded by n *)
Inductive
bsteps
`
(
R
:
relation
A
)
:
nat
→
relation
A
:
=

bsteps_refl
n
x
:
bsteps
R
n
x
x

bsteps_l
n
x
y
z
:
R
x
y
→
bsteps
R
n
y
z
→
bsteps
R
(
S
n
)
x
z
.
(* The transitive closure *)
Inductive
tc
`
(
R
:
relation
A
)
:
relation
A
:
=

tc_once
x
y
:
R
x
y
→
tc
R
x
y

tc_l
x
y
z
:
R
x
y
→
tc
R
y
z
→
tc
R
x
z
.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
trs
.
Arguments
rtc_l
{
_
_
_
_
_
}
_
_
.
Arguments
nsteps_l
{
_
_
_
_
_
_
}
_
_
.
Arguments
bsteps_refl
{
_
_
}
_
_
.
Arguments
bsteps_l
{
_
_
_
_
_
_
}
_
_
.
Arguments
tc_once
{
_
_
_
}
_
_
.
Arguments
tc_l
{
_
_
_
_
_
}
_
_
.
Ltac
generalize_rtc
H
:
=
match
type
of
H
with

rtc
?R
?x
?y
=>
let
Hx
:
=
fresh
in
let
Hy
:
=
fresh
in
let
Heqx
:
=
fresh
in
let
Heqy
:
=
fresh
in
remember
x
as
(
Hx
,
Heqx
)
;
remember
y
as
(
Hy
,
Heqy
)
;
revert
Heqx
Heqy
;
repeat
match
x
with

context
[
?z
]
=>
revert
z
end
;
repeat
match
y
with

context
[
?z
]
=>
revert
z
end
end
.
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on abstract rewriting systems
.
These are particularly useful as we define the operational semantics as a
small step semantics. This file defines a hint database [ars] containing
some theorems on abstract rewriting systems.
*)
Require
Export
tactics
base
.
(** * Definitions *)
Section
definitions
.
Context
`
(
R
:
relation
A
).
(** An element is reducible if a step is possible. *)
Definition
red
(
x
:
A
)
:
=
∃
y
,
R
x
y
.
(** An element is in normal form if no further steps are possible. *)
Definition
nf
(
x
:
A
)
:
=
¬
red
x
.
(** The reflexive transitive closure. *)
Inductive
rtc
:
relation
A
:
=

rtc_refl
x
:
rtc
x
x

rtc_l
x
y
z
:
R
x
y
→
rtc
y
z
→
rtc
x
z
.
(** Reductions of exactly [n] steps. *)
Inductive
nsteps
:
nat
→
relation
A
:
=

nsteps_O
x
:
nsteps
0
x
x

nsteps_l
n
x
y
z
:
R
x
y
→
nsteps
n
y
z
→
nsteps
(
S
n
)
x
z
.
(** Reduction of at most [n] steps. *)
Inductive
bsteps
:
nat
→
relation
A
:
=

bsteps_refl
n
x
:
bsteps
n
x
x

bsteps_l
n
x
y
z
:
R
x
y
→
bsteps
n
y
z
→
bsteps
(
S
n
)
x
z
.
(** The transitive closure. *)
Inductive
tc
:
relation
A
:
=

tc_once
x
y
:
R
x
y
→
tc
x
y

tc_l
x
y
z
:
R
x
y
→
tc
y
z
→
tc
x
z
.
(** An element [x] is looping if all paths starting at [x] are infinite. *)
CoInductive
looping
:
A
→
Prop
:
=

looping_do_step
x
:
red
x
→
(
∀
y
,
R
x
y
→
looping
y
)
→
looping
x
.
End
definitions
.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
ars
.
(** * General theorems *)
Section
rtc
.
Context
`
{
R
:
relation
A
}.
Global
Instance
:
Reflexive
(
rtc
R
).
Proof
rtc_refl
R
.
Global
Instance
rtc_trans
:
Transitive
(
rtc
R
).
Proof
.
red
;
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
rtc_once
{
x
y
}
:
R
x
y
→
rtc
R
x
y
.
Proof
.
eauto
with
t
rs
.
Qed
.
Proof
.
red
;
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
rtc_once
x
y
:
R
x
y
→
rtc
R
x
y
.
Proof
.
eauto
with
a
rs
.
Qed
.
Global
Instance
:
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
with
t
rs
.
Qed
.
Lemma
rtc_r
x
y
z
:
rtc
R
x
y
→
R
y
z
→
rtc
R
x
z
.
Proof
.
intros
.
etransitivity
;
eauto
with
a
rs
.
Qed
.
Lemma
rtc_inv
{
x
z
}
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
R
x
y
∧
rtc
R
y
z
.
Lemma
rtc_inv
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
R
x
y
∧
rtc
R
y
z
.
Proof
.
inversion_clear
1
;
eauto
.
Qed
.
Lemma
rtc_ind_r
(
P
:
A
→
A
→
Prop
)
Lemma
rtc_ind_r
(
P
:
A
→
A
→
Prop
)
(
Prefl
:
∀
x
,
P
x
x
)
(
Pstep
:
∀
x
y
z
,
rtc
R
x
y
→
R
y
z
→
P
x
y
→
P
x
z
)
:
∀
y
z
,
rtc
R
y
z
→
P
y
z
.
Proof
.
...
...
@@ 70,58 +70,76 @@ Section rtc.
induction
1
;
eauto
using
rtc_r
.
Qed
.
Lemma
rtc_inv_r
{
x
z
}
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
rtc
R
x
y
∧
R
y
z
.
Lemma
rtc_inv_r
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
rtc
R
x
y
∧
R
y
z
.
Proof
.
revert
x
z
.
apply
rtc_ind_r
;
eauto
.
Qed
.
Lemma
nsteps_once
{
x
y
}
:
R
x
y
→
nsteps
R
1
x
y
.
Proof
.
eauto
with
t
rs
.
Qed
.
Lemma
nsteps_trans
{
n
m
x
y
z
}
:
Lemma
nsteps_once
x
y
:
R
x
y
→
nsteps
R
1
x
y
.
Proof
.
eauto
with
a
rs
.
Qed
.
Lemma
nsteps_trans
n
m
x
y
z
:
nsteps
R
n
x
y
→
nsteps
R
m
y
z
→
nsteps
R
(
n
+
m
)
x
z
.
Proof
.
induction
1
;
simpl
;
eauto
with
t
rs
.
Qed
.
Lemma
nsteps_r
{
n
x
y
z
}
:
nsteps
R
n
x
y
→
R
y
z
→
nsteps
R
(
S
n
)
x
z
.
Proof
.
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
nsteps_rtc
{
n
x
y
}
:
nsteps
R
n
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
rtc_nsteps
{
x
y
}
:
rtc
R
x
y
→
∃
n
,
nsteps
R
n
x
y
.
Proof
.
induction
1
;
firstorder
eauto
with
t
rs
.
Qed
.
Lemma
bsteps_once
{
n
x
y
}
:
R
x
y
→
bsteps
R
(
S
n
)
x
y
.
Proof
.
eauto
with
t
rs
.
Qed
.
Lemma
bsteps_plus_r
{
n
m
x
y
}
:
Proof
.
induction
1
;
simpl
;
eauto
with
a
rs
.
Qed
.
Lemma
nsteps_r
n
x
y
z
:
nsteps
R
n
x
y
→
R
y
z
→
nsteps
R
(
S
n
)
x
z
.
Proof
.
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
nsteps_rtc
n
x
y
:
nsteps
R
n
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
rtc_nsteps
x
y
:
rtc
R
x
y
→
∃
n
,
nsteps
R
n
x
y
.
Proof
.
induction
1
;
firstorder
eauto
with
a
rs
.
Qed
.
Lemma
bsteps_once
n
x
y
:
R
x
y
→
bsteps
R
(
S
n
)
x
y
.
Proof
.
eauto
with
a
rs
.
Qed
.
Lemma
bsteps_plus_r
n
m
x
y
:
bsteps
R
n
x
y
→
bsteps
R
(
n
+
m
)
x
y
.
Proof
.
induction
1
;
simpl
;
eauto
with
t
rs
.
Qed
.
Lemma
bsteps_weaken
{
n
m
x
y
}
:
Proof
.
induction
1
;
simpl
;
eauto
with
a
rs
.
Qed
.
Lemma
bsteps_weaken
n
m
x
y
:
n
≤
m
→
bsteps
R
n
x
y
→
bsteps
R
m
x
y
.
Proof
.
intros
.
rewrite
(
Minus
.
le_plus_minus
n
m
)
;
auto
using
bsteps_plus_r
.
Qed
.
Lemma
bsteps_plus_l
{
n
m
x
y
}
:
Lemma
bsteps_plus_l
n
m
x
y
:
bsteps
R
n
x
y
→
bsteps
R
(
m
+
n
)
x
y
.
Proof
.
apply
bsteps_weaken
.
auto
with
arith
.
Qed
.
Lemma
bsteps_S
{
n
x
y
}
:
bsteps
R
n
x
y
→
bsteps
R
(
S
n
)
x
y
.
Lemma
bsteps_S
n
x
y
:
bsteps
R
n
x
y
→
bsteps
R
(
S
n
)
x
y
.
Proof
.
apply
bsteps_weaken
.
auto
with
arith
.
Qed
.
Lemma
bsteps_trans
{
n
m
x
y
z
}
:
Lemma
bsteps_trans
n
m
x
y
z
:
bsteps
R
n
x
y
→
bsteps
R
m
y
z
→
bsteps
R
(
n
+
m
)
x
z
.
Proof
.
induction
1
;
simpl
;
eauto
using
bsteps_plus_l
with
t
rs
.
Qed
.
Lemma
bsteps_r
{
n
x
y
z
}
:
bsteps
R
n
x
y
→
R
y
z
→
bsteps
R
(
S
n
)
x
z
.
Proof
.
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
bsteps_rtc
{
n
x
y
}
:
bsteps
R
n
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
rtc_bsteps
{
x
y
}
:
rtc
R
x
y
→
∃
n
,
bsteps
R
n
x
y
.
Proof
.
induction
1
.
exists
0
.
auto
with
trs
.
firstorder
eauto
with
t
rs
.
Qed
.
Proof
.
induction
1
;
simpl
;
eauto
using
bsteps_plus_l
with
a
rs
.
Qed
.
Lemma
bsteps_r
n
x
y
z
:
bsteps
R
n
x
y
→
R
y
z
→
bsteps
R
(
S
n
)
x
z
.
Proof
.
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
bsteps_rtc
n
x
y
:
bsteps
R
n
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
rtc_bsteps
x
y
:
rtc
R
x
y
→
∃
n
,
bsteps
R
n
x
y
.
Proof
.
induction
1
.
exists
0
.
auto
with
ars
.
firstorder
eauto
with
a
rs
.
Qed
.
Global
Instance
tc_trans
:
Transitive
(
tc
R
).
Proof
.
red
;
induction
1
;
eauto
with
t
rs
.
Qed
.
Lemma
tc_r
{
x
y
z
}
:
tc
R
x
y
→
R
y
z
→
tc
R
x
z
.
Proof
.
intros
.
etransitivity
;
eauto
with
t
rs
.
Qed
.
Lemma
tc_rtc
{
x
y
}
:
tc
R
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
t
rs
.
Qed
.
Proof
.
red
;
induction
1
;
eauto
with
a
rs
.
Qed
.
Lemma
tc_r
x
y
z
:
tc
R
x
y
→
R
y
z
→
tc
R
x
z
.
Proof
.
intros
.
etransitivity
;
eauto
with
a
rs
.
Qed
.
Lemma
tc_rtc
x
y
:
tc
R
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
with
a
rs
.
Qed
.
Global
Instance
:
subrelation
(
tc
R
)
(
rtc
R
).
Proof
.
exact
@
tc_rtc
.
Qed
.
Lemma
looping_red
x
:
looping
R
x
→
red
R
x
.
Proof
.
destruct
1
;
auto
.
Qed
.
Lemma
looping_step
x
y
:
looping
R
x
→
R
x
y
→
looping
R
y
.
Proof
.
destruct
1
;
auto
.
Qed
.
Lemma
looping_rtc
x
y
:
looping
R
x
→
rtc
R
x
y
→
looping
R
y
.
Proof
.
induction
2
;
eauto
using
looping_step
.
Qed
.
Lemma
looping_alt
x
:
looping
R
x
↔
∀
y
,
rtc
R
x
y
→
red
R
y
.
Proof
.
split
.
*
eauto
using
looping_red
,
looping_rtc
.
*
intros
H
.
cut
(
∀
z
,
rtc
R
x
z
→
looping
R
z
).
{
eauto
with
ars
.
}
cofix
FIX
.
constructor
;
eauto
using
rtc_r
with
ars
.
Qed
.
End
rtc
.
Hint
Resolve
rtc_once
rtc_r
tc_r
:
t
rs
.
Hint
Resolve
rtc_once
rtc_r
tc_r
:
a
rs
.
(** * Theorems on sub relations *)
Section
subrel
.
Context
{
A
}
(
R1
R2
:
relation
A
)
(
Hsub
:
subrelation
R1
R2
).
...
...
theories/base.v
View file @
e79e91f7
This diff is collapsed.
Click to expand it.
theories/collections.v
View file @
e79e91f7
Require
Export
base
orders
.
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
Require
Export
base
tactics
orders
.
(** * Theorems *)
Section
collection
.
Context
`
{
Collection
A
B
}.
Lemma
elem_of_empty_iff
x
:
x
∈
∅
↔
False
.
Proof
.
split
.
apply
elem_of_empty
.
easy
.
Qed
.
Lemma
elem_of_empty
x
:
x
∈
∅
↔
False
.
Proof
.
split
.
apply
not_elem_of_empty
.
easy
.
Qed
.
Lemma
elem_of_union_l
x
X
Y
:
x
∈
X
→
x
∈
X
∪
Y
.
Proof
.
intros
.
apply
elem_of_union
.
auto
.
Qed
.
Lemma
elem_of_union_r
x
X
Y
:
x
∈
Y
→
x
∈
X
∪
Y
.
Proof
.
intros
.
apply
elem_of_union
.
auto
.
Qed
.
Lemma
not_elem_of_singleton
x
y
:
x
∉
{[
y
]}
↔
x
≠
y
.
Proof
.
now
rewrite
elem_of_singleton
.
Qed
.
Lemma
not_elem_of_union
x
X
Y
:
x
∉
X
∪
Y
↔
x
∉
X
∧
x
∉
Y
.
Proof
.
rewrite
elem_of_union
.
tauto
.
Qed
.
Global
Instance
collection_subseteq
:
SubsetEq
B
:
=
λ
X
Y
,
∀
x
,
x
∈
X
→
x
∈
Y
.
Global
Instance
collection_subseteq
:
SubsetEq
B
:
=
λ
X
Y
,
∀
x
,
x
∈
X
→
x
∈
Y
.
Global
Instance
:
BoundedJoinSemiLattice
B
.
Proof
.
firstorder
.
Qed
.
Global
Instance
:
MeetSemiLattice
B
.
...
...
@@ 25,17 +35,20 @@ Section collection.
X
≡
Y
↔
(
∀
x
,
x
∈
X
→
x
∈
Y
)
∧
(
∀
x
,
x
∈
Y
→
x
∈
X
).
Proof
.
firstorder
.
Qed
.
Global
Instance
:
Proper
((=)
==>
(
≡
)
==>
iff
)
(
∈
).
Global
Instance
singleton_proper
:
Proper
((=)
==>
(
≡
))
singleton
.
Proof
.
repeat
intro
.
now
subst
.
Qed
.
Global
Instance
elem_of_proper
:
Proper
((=)
==>
(
≡
)
==>
iff
)
(
∈
).
Proof
.
intros
???.
subst
.
firstorder
.
Qed
.
Lemma
empty_ne_singleton
x
:
∅
≢
{[
x
]}.
Lemma
empty_ne_singleton
x
:
∅
≢
{[
x
]}.
Proof
.
intros
[
_
E
].
destruct
(
elem_of_empty
x
).
intros
[
_
E
].
apply
(
elem_of_empty
x
).
apply
E
.
now
apply
elem_of_singleton
.
Qed
.
Qed
.
End
collection
.
Section
cmap
.
(** * Theorems about map *)
Section
map
.
Context
`
{
Collection
A
C
}.
Lemma
elem_of_map_1
(
f
:
A
→
A
)
(
X
:
C
)
(
x
:
A
)
:
...
...
@@ 47,19 +60,18 @@ Section cmap.
Lemma
elem_of_map_2
(
f
:
A
→
A
)
(
X
:
C
)
(
x
:
A
)
:
x
∈
map
f
X
→
∃
y
,
x
=
f
y
∧
y
∈
X
.
Proof
.
intros
.
now
apply
(
elem_of_map
_
).
Qed
.
End
cmap
.
Definition
fresh_sig
`
{
FreshSpec
A
C
}
(
X
:
C
)
:
{
x
:
A

x
∉
X
}
:
=
exist
(
∉
X
)
(
fresh
X
)
(
is_fresh
X
).
Lemma
elem_of_fresh_iff
`
{
FreshSpec
A
C
}
(
X
:
C
)
:
fresh
X
∈
X
↔
False
.
Proof
.
split
.
apply
is_fresh
.
easy
.
Qed
.
Ltac
split_elem_ofs
:
=
repeat
End
map
.
(** * Tactics *)
(** The first pass consists of eliminating all occurrences of [(∪)], [(∩)],
[(∖)], [map], [∅], [{[_]}], [(≡)], and [(⊆)], by rewriting these into
logically equivalent propositions. For example we rewrite [A → x ∈ X ∪ ∅] into
[A → x ∈ X ∨ False]. *)
Ltac
unfold_elem_of
:
=
repeat
match
goal
with

H
:
context
[
_
⊆
_
]

_
=>
setoid_rewrite
elem_of_subseteq
in
H

H
:
context
[
_
≡
_
]

_
=>
setoid_rewrite
elem_of_equiv_alt
in
H

H
:
context
[
_
∈
∅
]

_
=>
setoid_rewrite
elem_of_empty
_iff
in
H

H
:
context
[
_
∈
∅
]

_
=>
setoid_rewrite
elem_of_empty
in
H

H
:
context
[
_
∈
{[
_
]}
]

_
=>
setoid_rewrite
elem_of_singleton
in
H

H
:
context
[
_
∈
_
∪
_
]

_
=>
setoid_rewrite
elem_of_union
in
H

H
:
context
[
_
∈
_
∩
_
]

_
=>
setoid_rewrite
elem_of_intersection
in
H
...
...
@@ 67,7 +79,7 @@ Ltac split_elem_ofs := repeat

H
:
context
[
_
∈
map
_
_
]

_
=>
setoid_rewrite
elem_of_map
in
H


context
[
_
⊆
_
]
=>
setoid_rewrite
elem_of_subseteq


context
[
_
≡
_
]
=>
setoid_rewrite
elem_of_equiv_alt


context
[
_
∈
∅
]
=>
setoid_rewrite
elem_of_empty
_iff


context
[
_
∈
∅
]
=>
setoid_rewrite
elem_of_empty


context
[
_
∈
{[
_
]}
]
=>
setoid_rewrite
elem_of_singleton


context
[
_
∈
_
∪
_
]
=>
setoid_rewrite
elem_of_union


context
[
_
∈
_
∩
_
]
=>
setoid_rewrite
elem_of_intersection
...
...
@@ 75,56 +87,49 @@ Ltac split_elem_ofs := repeat


context
[
_
∈
map
_
_
]
=>
setoid_rewrite
elem_of_map
end
.
Ltac
destruct_elem_ofs
:
=
repeat
match
goal
with

H
:
context
[
@
elem_of
(
_
*
_
)
_
_
?x
_
]

_
=>
is_var
x
;
destruct
x

H
:
context
[
@
elem_of
(
_
+
_
)
_
_
?x
_
]

_
=>
is_var
x
;
destruct
x
end
.
Tactic
Notation
"simplify_elem_of"
tactic
(
t
)
:
=
intros
;
(* due to bug #2790 *)
(** The tactic [solve_elem_of tac] composes the above tactic with [intuition].
For goals that do not involve [≡], [⊆], [map], or quantifiers this tactic is
generally powerful enough. This tactic either fails or proves the goal. *)
Tactic
Notation
"solve_elem_of"
tactic
(
tac
)
:
=
simpl
in
*
;
split_elem_ofs
;
destruct_elem_ofs
;
intuition
(
simplify_eqs
;
t
).
Tactic
Notation
"simplify_elem_of"
:
=
simplify_elem_of
auto
.
Ltac
naive_firstorder
t
:
=
match
goal
with
(* intros *)


∀
_
,
_
=>
intro
;
naive_firstorder
t
(* destructs without information loss *)

H
:
False

_
=>
destruct
H

H
:
?X
,
Hneg
:
¬
?X

_
=>
now
destruct
Hneg

H
:
_
∧
_

_
=>
destruct
H
;
naive_firstorder
t

H
:
∃
_
,
_

_
=>
destruct
H
;
naive_firstorder
t
(* simplification *)


_
=>
progress
(
simplify_eqs
;
simpl
in
*)
;
naive_firstorder
t
(* constructs *)


_
∧
_
=>
split
;
naive_firstorder
t
(* solve *)


_
=>
solve
[
t
]
(* dirty destructs *)

H
:
context
[
∃
_
,
_
]

_
=>
edestruct
H
;
clear
H
;
naive_firstorder
t

clear
H
;
naive_firstorder
t

H
:
context
[
_
∧
_
]

_
=>
destruct
H
;
clear
H
;
naive_firstorder
t

clear
H
;
naive_firstorder
t

H
:
context
[
_
∨
_
]

_
=>
edestruct
H
;
clear
H
;
naive_firstorder
t

clear
H
;
naive_firstorder
t
(* dirty constructs *)


∃
x
,
_
=>
eexists
;
naive_firstorder
t


_
∨
_
=>
left
;
naive_firstorder
t

right
;
naive_firstorder
t

H
:
_
→
False

_
=>
destruct
H
;
naive_firstorder
t
end
.
Tactic
Notation
"naive_firstorder"
tactic
(
t
)
:
=
unfold
iff
,
not
in
*
;
naive_firstorder
t
.
Tactic
Notation
"esimplify_elem_of"
tactic
(
t
)
:
=
(
simplify_elem_of
t
)
;
try
naive_firstorder
t
.
Tactic
Notation
"esimplify_elem_of"
:
=
esimplify_elem_of
(
eauto
5
).
unfold_elem_of
;
solve
[
intuition
(
simplify_equality
;
tac
)].
Tactic
Notation
"solve_elem_of"
:
=
solve_elem_of
auto
.
(** For goals with quantifiers we could use the above tactic but with
[firstorder] instead of [intuition] as finishing tactic. However, [firstorder]
fails or loops on very small goals generated by [solve_elem_of] already. We
use the [naive_solver] tactic as a substitute. This tactic either fails or
proves the goal. *)
Tactic
Notation
"esolve_elem_of"
tactic
(
tac
)
:
=
simpl
in
*
;
unfold_elem_of
;
naive_solver
tac
.
Tactic
Notation
"esolve_elem_of"
:
=
esolve_elem_of
eauto
.
(** Given an assumption [H : _ ∈ _], the tactic [destruct_elem_of H] will
recursively split [H] for [(∪)], [(∩)], [(∖)], [map], [∅], [{[_]}]. *)
Tactic
Notation
"destruct_elem_of"
hyp
(
H
)
:
=
let
rec
go
H
:
=
lazymatch
type
of
H
with

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

_
∈
{[
?l'
]}
=>
apply
elem_of_singleton
in
H
;
subst
l'

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

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

_
∈
map
_
_
=>
let
H1
:
=
fresh
in
apply
elem_of_map
in
H
;
destruct
H
as
[?[?
H1
]]
;
go
H1

_
=>
idtac
end
in
go
H
.
(** * Sets without duplicates up to an equivalence *)
Section
no_dup
.
Context
`
{
Collection
A
B
}
(
R
:
relation
A
)
`
{!
Equivalence
R
}.
...
...
@@ 143,33 +148,34 @@ Section no_dup.
Proof
.
firstorder
.
Qed
.
Lemma
elem_of_upto_elem_of
x
X
:
x
∈
X
→
elem_of_upto
x
X
.
Proof
.
unfold
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
elem_of_upto_empty
x
:
¬
elem_of_upto
x
∅
.
Proof
.
unfold
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
elem_of_upto_singleton
x
y
:
elem_of_upto
x
{[
y
]}
↔
R
x
y
.
Proof
.
unfold
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
elem_of_upto_union
X
Y
x
:
elem_of_upto
x
(
X
∪
Y
)
↔
elem_of_upto
x
X
∨
elem_of_upto
x
Y
.
Proof
.
unfold
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
not_elem_of_upto
x
X
:
¬
elem_of_upto
x
X
→
∀
y
,
y
∈
X
→
¬
R
x
y
.
Proof
.
unfold
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
no_dup_empty
:
no_dup
∅
.
Proof
.
unfold
no_dup
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
no_dup
.
s
olve
_elem_of
.
Qed
.
Lemma
no_dup_add
x
X
:
¬
elem_of_upto
x
X
→
no_dup
X
→
no_dup
({[
x
]}
∪
X
).
Proof
.
unfold
no_dup
,
elem_of_upto
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
no_dup
,
elem_of_upto
.
es
olve
_elem_of
.
Qed
.
Lemma
no_dup_inv_add
x
X
:
x
∉
X
→
no_dup
({[
x
]}
∪
X
)
→
¬
elem_of_upto
x
X
.
Proof
.
intros
Hin
Hnodup
[
y
[??]].
rewrite
(
Hnodup
x
y
)
in
Hin
;
s
implify
_elem_of
.
rewrite
(
Hnodup
x
y
)
in
Hin
;
s
olve
_elem_of
.
Qed
.
Lemma
no_dup_inv_union_l
X
Y
:
no_dup
(
X
∪
Y
)
→
no_dup
X
.
Proof
.
unfold
no_dup
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
no_dup
.
s
olve
_elem_of
.
Qed
.
Lemma
no_dup_inv_union_r
X
Y
:
no_dup
(
X
∪
Y
)
→
no_dup
Y
.
Proof
.
unfold
no_dup
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
no_dup
.
s
olve
_elem_of
.
Qed
.
End
no_dup
.
(** * Quantifiers *)
Section
quantifiers
.
Context
`
{
Collection
A
B
}
(
P
:
A
→
Prop
).
...
...
@@ 177,48 +183,65 @@ Section quantifiers.
Definition
cexists
X
:
=
∃
x
,
x
∈
X
∧
P
x
.
Lemma
cforall_empty
:
cforall
∅
.
Proof
.
unfold
cforall
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
cforall
.
s
olve
_elem_of
.
Qed
.
Lemma
cforall_singleton
x
:
cforall
{[
x
]}
↔
P
x
.
Proof
.
unfold
cforall
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
cforall
.
s
olve
_elem_of
.
Qed
.
Lemma
cforall_union
X
Y
:
cforall
X
→
cforall
Y
→
cforall
(
X
∪
Y
).
Proof
.
unfold
cforall
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
cforall
.
s
olve
_elem_of
.
Qed
.
Lemma
cforall_union_inv_1
X
Y
:
cforall
(
X
∪
Y
)
→
cforall
X
.
Proof
.
unfold
cforall
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
cforall
.
s
olve
_elem_of
.
Qed
.
Lemma
cforall_union_inv_2
X
Y
:
cforall
(
X
∪
Y
)
→
cforall
Y
.
Proof
.
unfold
cforall
.
s
implify
_elem_of
.
Qed
.
Proof
.
unfold
cforall
.
s
olve
_elem_of
.
Qed
.
Lemma
cexists_empty
:
¬
cexists
∅
.
Proof
.
unfold
cexists
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
cexists
.
es
olve
_elem_of
.
Qed
.
Lemma
cexists_singleton
x
:
cexists
{[
x
]}
↔
P
x
.
Proof
.
unfold
cexists
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
cexists
.
es
olve
_elem_of
.
Qed
.
Lemma
cexists_union_1
X
Y
:
cexists
X
→
cexists
(
X
∪
Y
).
Proof
.
unfold
cexists
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
cexists
.
es
olve
_elem_of
.
Qed
.
Lemma
cexists_union_2
X
Y
:
cexists
Y
→
cexists
(
X
∪
Y
).
Proof
.
unfold
cexists
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
cexists
.
es
olve
_elem_of
.
Qed
.
Lemma
cexists_union_inv
X
Y
:
cexists
(
X
∪
Y
)
→
cexists
X
∨
cexists
Y
.
Proof
.
unfold
cexists
.
es
implify
_elem_of
.
Qed
.
Proof
.
unfold
cexists
.
es
olve
_elem_of
.
Qed
.
End
quantifiers
.
Section
more_quantifiers
.
Context
`
{
Collection
A
B
}.
Lemma
cforall_weak
(
P
Q
:
A
→
Prop
)
(
Hweak
:
∀
x
,
P
x
→
Q
x
)
X
:
cforall
P
X
→
cforall
Q
X
.
Proof
.
firstord
er
.
Qed
.
Proof
.
unfold
cforall
.
naive_solv
er
.
Qed
.
Lemma
cexists_weak
(
P
Q
:
A
→
Prop
)
(
Hweak
:
∀
x
,
P
x
→
Q
x
)
X
:
cexists
P
X
→
cexists
Q
X
.
Proof
.
firstord
er
.
Qed
.
Proof
.
unfold
cexists
.
naive_solv
er
.
Qed
.
End
more_quantifiers
.
(** * Fresh elements *)
(** We collect some properties on the [fresh] operation. In particular we
generalize [fresh] to generate lists of fresh elements. *)
Section
fresh
.
Context
`
{
Collection
A
C
}
`
{
Fresh
A
C
}
`
{!
FreshSpec
A
C
}
.
Definition
fresh_sig
(
X
:
C
)
:
{
x
: