Skip to content
GitLab
Menu
Projects
Groups
Snippets
Loading...
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Menu
Open sidebar
George Pirlea
Iris
Commits
6dbe0c27
Commit
6dbe0c27
authored
Jul 22, 2016
by
Robbert Krebbers
Browse files
Move sorting stuff to separate file.
parent
f2c246c6
Changes
4
Show whitespace changes
Inline
Sidebyside
_CoqProject
View file @
6dbe0c27
...
...
@@ 36,6 +36,7 @@ prelude/list.v
prelude/error.v
prelude/functions.v
prelude/hlist.v
prelude/sorting.v
algebra/cmra.v
algebra/cmra_big_op.v
algebra/cmra_tactics.v
...
...
prelude/collections.v
View file @
6dbe0c27
...
...
@@ 3,7 +3,7 @@
(** This file collects definitions and theorems on collections. Most
importantly, it implements some tactics to automatically solve goals involving
collections. *)
From
iris
.
prelude
Require
Export
base
tactics
orders
.
From
iris
.
prelude
Require
Export
orders
list
.
Instance
collection_equiv
`
{
ElemOf
A
C
}
:
Equiv
C
:
=
λ
X
Y
,
∀
x
,
x
∈
X
↔
x
∈
Y
.
...
...
@@ 811,8 +811,7 @@ Section fresh.
Proof
.
induction
1
;
by
constructor
.
Qed
.
Lemma
Forall_fresh_elem_of
X
xs
x
:
Forall_fresh
X
xs
→
x
∈
xs
→
x
∉
X
.
Proof
.
intros
HX
;
revert
x
;
rewrite
<
Forall_forall
.
by
induction
HX
;
constructor
.
intros
HX
;
revert
x
;
rewrite
<
Forall_forall
.
by
induction
HX
;
constructor
.
Qed
.
Lemma
Forall_fresh_alt
X
xs
:
Forall_fresh
X
xs
↔
NoDup
xs
∧
∀
x
,
x
∈
xs
→
x
∉
X
.
...
...
prelude/fin_maps.v
View file @
6dbe0c27
...
...
@@ 5,7 +5,7 @@ finite maps and collects some theory on it. Most importantly, it proves useful
induction principles for finite maps and implements the tactic
[simplify_map_eq] to simplify goals involving finite maps. *)
From
Coq
Require
Import
Permutation
.
From
iris
.
prelude
Require
Export
relations
vector
orders
.
From
iris
.
prelude
Require
Export
relations
orders
vector
.
(** * Axiomatization of finite maps *)
(** We require Leibniz equality to be extensional on finite maps. This of
...
...
prelude/orders.v
View file @
6dbe0c27
(* Copyright (c) 20122015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects common properties of preorders and semi lattices. This
theory will mainly be used for the theory on collections and finite maps. *)
From
Coq
Require
Export
Sorted
.
From
iris
.
prelude
Require
Export
tactics
list
.
(** * Arbitrary pre, parial and total orders *)
(** Properties about arbitrary pre, partial, and total orders. We do not use
the relation [⊆] because we often have multiple orders on the same structure *)
From
iris
.
prelude
Require
Export
tactics
.
Section
orders
.
Context
{
A
}
{
R
:
relation
A
}.
Implicit
Types
X
Y
:
A
.
...
...
@@ 104,203 +100,3 @@ Ltac simplify_order := repeat
assert
(
R
x
z
)
by
(
by
trans
y
)
end
end
.
(** * Sorting *)
(** Merge sort. Adapted from the implementation of Hugo Herbelin in the Coq
standard library, but without using the module system. *)
Section
merge_sort
.
Context
{
A
}
(
R
:
relation
A
)
`
{
∀
x
y
,
Decision
(
R
x
y
)}.
Fixpoint
list_merge
(
l1
:
list
A
)
:
list
A
→
list
A
:
=
fix
list_merge_aux
l2
:
=
match
l1
,
l2
with

[],
_
=>
l2

_
,
[]
=>
l1

x1
::
l1
,
x2
::
l2
=>
if
decide_rel
R
x1
x2
then
x1
::
list_merge
l1
(
x2
::
l2
)
else
x2
::
list_merge_aux
l2
end
.
Global
Arguments
list_merge
!
_
!
_
/.
Local
Notation
stack
:
=
(
list
(
option
(
list
A
))).
Fixpoint
merge_list_to_stack
(
st
:
stack
)
(
l
:
list
A
)
:
stack
:
=
match
st
with

[]
=>
[
Some
l
]

None
::
st
=>
Some
l
::
st

Some
l'
::
st
=>
None
::
merge_list_to_stack
st
(
list_merge
l'
l
)
end
.
Fixpoint
merge_stack
(
st
:
stack
)
:
list
A
:
=
match
st
with

[]
=>
[]

None
::
st
=>
merge_stack
st

Some
l
::
st
=>
list_merge
l
(
merge_stack
st
)
end
.
Fixpoint
merge_sort_aux
(
st
:
stack
)
(
l
:
list
A
)
:
list
A
:
=
match
l
with

[]
=>
merge_stack
st

x
::
l
=>
merge_sort_aux
(
merge_list_to_stack
st
[
x
])
l
end
.
Definition
merge_sort
:
list
A
→
list
A
:
=
merge_sort_aux
[].
End
merge_sort
.
(** ** Properties of the [Sorted] and [StronglySorted] predicate *)
Section
sorted
.
Context
{
A
}
(
R
:
relation
A
).
Lemma
Sorted_StronglySorted
`
{!
Transitive
R
}
l
:
Sorted
R
l
→
StronglySorted
R
l
.
Proof
.
by
apply
Sorted
.
Sorted_StronglySorted
.
Qed
.
Lemma
StronglySorted_unique
`
{!
AntiSymm
(=)
R
}
l1
l2
:
StronglySorted
R
l1
→
StronglySorted
R
l2
→
l1
≡
ₚ
l2
→
l1
=
l2
.
Proof
.
intros
Hl1
;
revert
l2
.
induction
Hl1
as
[
x1
l1
?
IH
Hx1
]
;
intros
l2
Hl2
E
.
{
symmetry
.
by
apply
Permutation_nil
.
}
destruct
Hl2
as
[
x2
l2
?
Hx2
].
{
by
apply
Permutation_nil
in
E
.
}
assert
(
x1
=
x2
)
;
subst
.
{
rewrite
Forall_forall
in
Hx1
,
Hx2
.
assert
(
x2
∈
x1
::
l1
)
as
Hx2'
by
(
by
rewrite
E
;
left
).
assert
(
x1
∈
x2
::
l2
)
as
Hx1'
by
(
by
rewrite
<
E
;
left
).
inversion
Hx1'
;
inversion
Hx2'
;
simplify_eq
;
auto
.
}
f_equal
.
by
apply
IH
,
(
inj
(
x2
::
)).
Qed
.
Lemma
Sorted_unique
`
{!
Transitive
R
,
!
AntiSymm
(=)
R
}
l1
l2
:
Sorted
R
l1
→
Sorted
R
l2
→
l1
≡
ₚ
l2
→
l1
=
l2
.
Proof
.
auto
using
StronglySorted_unique
,
Sorted_StronglySorted
.
Qed
.
Global
Instance
HdRel_dec
x
`
{
∀
y
,
Decision
(
R
x
y
)}
l
:
Decision
(
HdRel
R
x
l
).
Proof
.
refine
match
l
with

[]
=>
left
_

y
::
l
=>
cast_if
(
decide
(
R
x
y
))
end
;
abstract
first
[
by
constructor

by
inversion
1
].
Defined
.
Global
Instance
Sorted_dec
`
{
∀
x
y
,
Decision
(
R
x
y
)}
:
∀
l
,
Decision
(
Sorted
R
l
).
Proof
.
refine
(
fix
go
l
:
=
match
l
return
Decision
(
Sorted
R
l
)
with

[]
=>
left
_

x
::
l
=>
cast_if_and
(
decide
(
HdRel
R
x
l
))
(
go
l
)
end
)
;
clear
go
;
abstract
first
[
by
constructor

by
inversion
1
].
Defined
.
Global
Instance
StronglySorted_dec
`
{
∀
x
y
,
Decision
(
R
x
y
)}
:
∀
l
,
Decision
(
StronglySorted
R
l
).
Proof
.
refine
(
fix
go
l
:
=
match
l
return
Decision
(
StronglySorted
R
l
)
with

[]
=>
left
_

x
::
l
=>
cast_if_and
(
decide
(
Forall
(
R
x
)
l
))
(
go
l
)
end
)
;
clear
go
;
abstract
first
[
by
constructor

by
inversion
1
].
Defined
.
Context
{
B
}
(
f
:
A
→
B
).
Lemma
HdRel_fmap
(
R1
:
relation
A
)
(
R2
:
relation
B
)
x
l
:
(
∀
y
,
R1
x
y
→
R2
(
f
x
)
(
f
y
))
→
HdRel
R1
x
l
→
HdRel
R2
(
f
x
)
(
f
<$>
l
).
Proof
.
destruct
2
;
constructor
;
auto
.
Qed
.
Lemma
Sorted_fmap
(
R1
:
relation
A
)
(
R2
:
relation
B
)
l
:
(
∀
x
y
,
R1
x
y
→
R2
(
f
x
)
(
f
y
))
→
Sorted
R1
l
→
Sorted
R2
(
f
<$>
l
).
Proof
.
induction
2
;
simpl
;
constructor
;
eauto
using
HdRel_fmap
.
Qed
.
Lemma
StronglySorted_fmap
(
R1
:
relation
A
)
(
R2
:
relation
B
)
l
:
(
∀
x
y
,
R1
x
y
→
R2
(
f
x
)
(
f
y
))
→
StronglySorted
R1
l
→
StronglySorted
R2
(
f
<$>
l
).
Proof
.
induction
2
;
csimpl
;
constructor
;
rewrite
?Forall_fmap
;
eauto
using
Forall_impl
.
Qed
.
End
sorted
.
(** ** Correctness of merge sort *)
Section
merge_sort_correct
.
Context
{
A
}
(
R
:
relation
A
)
`
{
∀
x
y
,
Decision
(
R
x
y
)}
`
{!
Total
R
}.
Lemma
list_merge_cons
x1
x2
l1
l2
:
list_merge
R
(
x1
::
l1
)
(
x2
::
l2
)
=
if
decide
(
R
x1
x2
)
then
x1
::
list_merge
R
l1
(
x2
::
l2
)
else
x2
::
list_merge
R
(
x1
::
l1
)
l2
.
Proof
.
done
.
Qed
.
Lemma
HdRel_list_merge
x
l1
l2
:
HdRel
R
x
l1
→
HdRel
R
x
l2
→
HdRel
R
x
(
list_merge
R
l1
l2
).
Proof
.
destruct
1
as
[
x1
l1
IH1
],
1
as
[
x2
l2
IH2
]
;
rewrite
?list_merge_cons
;
simpl
;
repeat
case_decide
;
auto
.
Qed
.
Lemma
Sorted_list_merge
l1
l2
:
Sorted
R
l1
→
Sorted
R
l2
→
Sorted
R
(
list_merge
R
l1
l2
).
Proof
.
intros
Hl1
.
revert
l2
.
induction
Hl1
as
[
x1
l1
IH1
]
;
induction
1
as
[
x2
l2
IH2
]
;
rewrite
?list_merge_cons
;
simpl
;
repeat
case_decide
;
constructor
;
eauto
using
HdRel_list_merge
,
HdRel_cons
,
total_not
.
Qed
.
Lemma
merge_Permutation
l1
l2
:
list_merge
R
l1
l2
≡
ₚ
l1
++
l2
.
Proof
.
revert
l2
.
induction
l1
as
[
x1
l1
IH1
]
;
intros
l2
;
induction
l2
as
[
x2
l2
IH2
]
;
rewrite
?list_merge_cons
;
simpl
;
repeat
case_decide
;
auto
.

by
rewrite
(
right_id_L
[]
(++)).

by
rewrite
IH2
,
Permutation_middle
.
Qed
.
Local
Notation
stack
:
=
(
list
(
option
(
list
A
))).
Inductive
merge_stack_Sorted
:
stack
→
Prop
:
=

merge_stack_Sorted_nil
:
merge_stack_Sorted
[]

merge_stack_Sorted_cons_None
st
:
merge_stack_Sorted
st
→
merge_stack_Sorted
(
None
::
st
)

merge_stack_Sorted_cons_Some
l
st
:
Sorted
R
l
→
merge_stack_Sorted
st
→
merge_stack_Sorted
(
Some
l
::
st
).
Fixpoint
merge_stack_flatten
(
st
:
stack
)
:
list
A
:
=
match
st
with

[]
=>
[]

None
::
st
=>
merge_stack_flatten
st

Some
l
::
st
=>
l
++
merge_stack_flatten
st
end
.
Lemma
Sorted_merge_list_to_stack
st
l
:
merge_stack_Sorted
st
→
Sorted
R
l
→
merge_stack_Sorted
(
merge_list_to_stack
R
st
l
).
Proof
.
intros
Hst
.
revert
l
.
induction
Hst
;
repeat
constructor
;
naive_solver
auto
using
Sorted_list_merge
.
Qed
.
Lemma
merge_list_to_stack_Permutation
st
l
:
merge_stack_flatten
(
merge_list_to_stack
R
st
l
)
≡
ₚ
l
++
merge_stack_flatten
st
.
Proof
.
revert
l
.
induction
st
as
[[
l'
]
st
IH
]
;
intros
l
;
simpl
;
auto
.
by
rewrite
IH
,
merge_Permutation
,
(
assoc_L
_
),
(
comm
(++)
l
).
Qed
.
Lemma
Sorted_merge_stack
st
:
merge_stack_Sorted
st
→
Sorted
R
(
merge_stack
R
st
).
Proof
.
induction
1
;
simpl
;
auto
using
Sorted_list_merge
.
Qed
.
Lemma
merge_stack_Permutation
st
:
merge_stack
R
st
≡
ₚ
merge_stack_flatten
st
.
Proof
.
induction
st
as
[[]
?
IH
]
;
intros
;
simpl
;
auto
.
by
rewrite
merge_Permutation
,
IH
.
Qed
.
Lemma
Sorted_merge_sort_aux
st
l
:
merge_stack_Sorted
st
→
Sorted
R
(
merge_sort_aux
R
st
l
).
Proof
.
revert
st
.
induction
l
;
simpl
;
auto
using
Sorted_merge_stack
,
Sorted_merge_list_to_stack
.
Qed
.
Lemma
merge_sort_aux_Permutation
st
l
:
merge_sort_aux
R
st
l
≡
ₚ
merge_stack_flatten
st
++
l
.
Proof
.
revert
st
.
induction
l
as
[??
IH
]
;
simpl
;
intros
.

by
rewrite
(
right_id_L
[]
(++)),
merge_stack_Permutation
.

rewrite
IH
,
merge_list_to_stack_Permutation
;
simpl
.
by
rewrite
Permutation_middle
.
Qed
.
Lemma
Sorted_merge_sort
l
:
Sorted
R
(
merge_sort
R
l
).
Proof
.
apply
Sorted_merge_sort_aux
.
by
constructor
.
Qed
.
Lemma
merge_sort_Permutation
l
:
merge_sort
R
l
≡
ₚ
l
.
Proof
.
unfold
merge_sort
.
by
rewrite
merge_sort_aux_Permutation
.
Qed
.
Lemma
StronglySorted_merge_sort
`
{!
Transitive
R
}
l
:
StronglySorted
R
(
merge_sort
R
l
).
Proof
.
auto
using
Sorted_StronglySorted
,
Sorted_merge_sort
.
Qed
.
End
merge_sort_correct
.
Write
Preview
Markdown
is supported
0%
Try again
or
attach a new file
.
Attach a file
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment