Skip to content
GitLab
Projects
Groups
Snippets
Help
Loading...
Help
What's new
7
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Open sidebar
David Swasey
coqstdpp
Commits
9d0d6825
Commit
9d0d6825
authored
Feb 22, 2013
by
Robbert Krebbers
Browse files
Options
Browse Files
Download
Email Patches
Plain Diff
Update prelude. Reorganize list functions.
parent
415a4f1c
Changes
8
Expand all
Hide whitespace changes
Inline
Sidebyside
Showing
8 changed files
with
1045 additions
and
868 deletions
+1045
868
theories/base.v
theories/base.v
+19
2
theories/list.v
theories/list.v
+894
799
theories/mapset.v
theories/mapset.v
+8
3
theories/nmap.v
theories/nmap.v
+9
3
theories/option.v
theories/option.v
+56
1
theories/orders.v
theories/orders.v
+7
7
theories/pmap.v
theories/pmap.v
+1
1
theories/tactics.v
theories/tactics.v
+51
52
No files found.
theories/base.v
View file @
9d0d6825
...
...
@@ 266,6 +266,22 @@ Class Filter A B :=
(* Arguments filter {_ _ _} _ {_} !_ / : simpl nomatch. *)
(** We define variants of the relations [(≡)] and [(⊆)] that are indexed by
an environment. *)
Class
EquivEnv
A
B
:
=
equiv_env
:
A
→
relation
B
.
Notation
"X ≡@{ E } Y"
:
=
(
equiv_env
E
X
Y
)
(
at
level
70
,
format
"X ≡@{ E } Y"
)
:
C_scope
.
Notation
"(≡@{ E } )"
:
=
(
equiv_env
E
)
(
E
at
level
1
,
only
parsing
)
:
C_scope
.
Instance
:
Params
(@
equiv_env
)
4
.
Class
SubsetEqEnv
A
B
:
=
subseteq_env
:
A
→
relation
B
.
Notation
"X ⊆@{ E } Y"
:
=
(
subseteq_env
E
X
Y
)
(
at
level
70
,
format
"X ⊆@{ E } Y"
)
:
C_scope
.
Notation
"(⊆@{ E } )"
:
=
(
subseteq_env
E
)
(
E
at
level
1
,
only
parsing
)
:
C_scope
.
Instance
:
Params
(@
subseteq_env
)
4
.
(** ** Monadic operations *)
(** We define operational type classes for the monadic operations bind, join
and fmap. These type classes are defined in a nonstandard way by taking the
...
...
@@ 282,7 +298,7 @@ in the appropriate way, and so that it can be used for mutual recursion
(the mapped function [f] is not part of the fixpoint) as well. This is a hack,
and should be replaced by something more appropriate in future versions. *)
(* We use these type classes merely for convenient overloading of notations and
(*
*
We use these type classes merely for convenient overloading of notations and
do not formalize any theory on monads (we do not even define a class with the
monad laws). *)
Class
MRet
(
M
:
Type
→
Type
)
:
=
mret
:
∀
{
A
},
A
→
M
A
.
...
...
@@ 316,11 +332,12 @@ Class MGuard (M : Type → Type) :=
mguard
:
∀
P
{
dec
:
Decision
P
}
{
A
},
M
A
→
M
A
.
Notation
"'guard' P ; o"
:
=
(
mguard
P
o
)
(
at
level
65
,
only
parsing
,
next
at
level
35
,
right
associativity
)
:
C_scope
.
Arguments
mguard
_
_
_
!
_
_
!
_
/
:
simpl
nomatch
.
(** ** Operations on maps *)
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
The function lookup [m !! k] should yield the element at key [k] in [m]. *)
The function look
up [m !! k] should yield the element at key [k] in [m]. *)
Class
Lookup
(
K
A
M
:
Type
)
:
=
lookup
:
K
→
M
→
option
A
.
Instance
:
Params
(@
lookup
)
4
.
...
...
theories/list.v
View file @
9d0d6825
This diff is collapsed.
Click to expand it.
theories/mapset.v
View file @
9d0d6825
...
...
@@ 47,9 +47,14 @@ Proof.
apply
option_eq
.
intros
[].
by
apply
E
.
Qed
.
Global
Instance
mapset_eq_dec
`
{
∀
m1
m2
:
M
unit
,
Decision
(
m1
=
m2
)}
:
∀
X1
X2
:
mapset
M
,
Decision
(
X1
=
X2
)

1
.
Proof
.
solve_decision
.
Defined
.
Global
Instance
mapset_eq_dec
`
{
∀
m1
m2
:
M
unit
,
Decision
(
m1
=
m2
)}
(
X1
X2
:
mapset
M
)
:
Decision
(
X1
=
X2
)

1
.
Proof
.
refine
match
X1
,
X2
with

Mapset
m1
,
Mapset
m2
=>
cast_if
(
decide
(
m1
=
m2
))
end
;
abstract
congruence
.
Defined
.
Global
Instance
mapset_elem_of_dec
x
(
X
:
mapset
M
)
:
Decision
(
x
∈
X
)

1
.
Proof
.
solve_decision
.
Defined
.
...
...
theories/nmap.v
View file @
9d0d6825
...
...
@@ 12,9 +12,15 @@ Arguments Nmap_0 {_} _.
Arguments
Nmap_pos
{
_
}
_
.
Arguments
NMap
{
_
}
_
_
.
Instance
Pmap_dec
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
:
∀
x
y
:
Nmap
A
,
Decision
(
x
=
y
).
Proof
.
solve_decision
.
Defined
.
Instance
Nmap_eq_dec
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
t1
t2
:
Nmap
A
)
:
Decision
(
t1
=
t2
).
Proof
.
refine
match
t1
,
t2
with

NMap
x
t1
,
NMap
y
t2
=>
cast_if_and
(
decide
(
x
=
y
))
(
decide
(
t1
=
t2
))
end
;
abstract
congruence
.
Defined
.
Instance
Nempty
{
A
}
:
Empty
(
Nmap
A
)
:
=
NMap
None
∅
.
Instance
Nlookup
{
A
}
:
Lookup
N
A
(
Nmap
A
)
:
=
λ
i
t
,
...
...
theories/option.v
View file @
9d0d6825
...
...
@@ 122,6 +122,14 @@ Instance option_fmap: FMap option := @option_map.
Instance
option_guard
:
MGuard
option
:
=
λ
P
dec
A
x
,
if
dec
then
x
else
None
.
Definition
mapM
`
{!
MBind
M
}
`
{!
MRet
M
}
{
A
B
}
(
f
:
A
→
M
B
)
:
list
A
→
M
(
list
B
)
:
=
fix
go
l
:
=
match
l
with

[]
=>
mret
[]

x
::
l
=>
y
←
f
x
;
k
←
go
l
;
mret
(
y
::
k
)
end
.
Lemma
fmap_is_Some
{
A
B
}
(
f
:
A
→
B
)
(
x
:
option
A
)
:
is_Some
(
f
<$>
x
)
↔
is_Some
x
.
Proof
.
split
;
inversion
1
.
by
destruct
x
.
done
.
Qed
.
...
...
@@ 138,6 +146,51 @@ Proof. by destruct x. Qed.
Lemma
option_bind_assoc
{
A
B
C
}
(
f
:
A
→
option
B
)
(
g
:
B
→
option
C
)
(
x
:
option
A
)
:
(
x
≫
=
f
)
≫
=
g
=
x
≫
=
(
mbind
g
∘
f
).
Proof
.
by
destruct
x
;
simpl
.
Qed
.
Lemma
option_bind_ext
{
A
B
}
(
f
g
:
A
→
option
B
)
x
y
:
(
∀
a
,
f
a
=
g
a
)
→
x
=
y
→
x
≫
=
f
=
y
≫
=
g
.
Proof
.
intros
.
destruct
x
,
y
;
simplify_equality
;
simpl
;
auto
.
Qed
.
Lemma
option_bind_ext_fun
{
A
B
}
(
f
g
:
A
→
option
B
)
x
:
(
∀
a
,
f
a
=
g
a
)
→
x
≫
=
f
=
x
≫
=
g
.
Proof
.
intros
.
by
apply
option_bind_ext
.
Qed
.
Section
mapM
.
Context
{
A
B
:
Type
}
(
f
:
A
→
option
B
).
Lemma
mapM_ext
(
g
:
A
→
option
B
)
l
:
(
∀
x
,
f
x
=
g
x
)
→
mapM
f
l
=
mapM
g
l
.
Proof
.
intros
Hfg
.
by
induction
l
;
simpl
;
rewrite
?Hfg
,
?IHl
.
Qed
.
Lemma
Forall2_mapM_ext
(
g
:
A
→
option
B
)
l
k
:
Forall2
(
λ
x
y
,
f
x
=
g
y
)
l
k
→
mapM
f
l
=
mapM
g
k
.
Proof
.
induction
1
as
[????
Hfg
?
IH
]
;
simpl
.
done
.
by
rewrite
Hfg
,
IH
.
Qed
.
Lemma
Forall_mapM_ext
(
g
:
A
→
option
B
)
l
:
Forall
(
λ
x
,
f
x
=
g
x
)
l
→
mapM
f
l
=
mapM
g
l
.
Proof
.
induction
1
as
[??
Hfg
?
IH
]
;
simpl
.
done
.
by
rewrite
Hfg
,
IH
.
Qed
.
Lemma
mapM_Some_1
l
k
:
mapM
f
l
=
Some
k
→
Forall2
(
λ
x
y
,
f
x
=
Some
y
)
l
k
.
Proof
.
revert
k
.
induction
l
as
[
x
l
]
;
intros
[
y
k
]
;
simpl
;
try
done
.
*
destruct
(
f
x
)
;
simpl
;
[
discriminate
].
by
destruct
(
mapM
f
l
).
*
destruct
(
f
x
)
eqn
:
?
;
simpl
;
[
discriminate
].
destruct
(
mapM
f
l
)
;
intros
;
simplify_equality
.
constructor
;
auto
.
Qed
.
Lemma
mapM_Some_2
l
k
:
Forall2
(
λ
x
y
,
f
x
=
Some
y
)
l
k
→
mapM
f
l
=
Some
k
.
Proof
.
induction
1
as
[????
Hf
?
IH
]
;
simpl
;
[
done
].
rewrite
Hf
.
simpl
.
by
rewrite
IH
.
Qed
.
Lemma
mapM_Some
l
k
:
mapM
f
l
=
Some
k
↔
Forall2
(
λ
x
y
,
f
x
=
Some
y
)
l
k
.
Proof
.
split
;
auto
using
mapM_Some_1
,
mapM_Some_2
.
Qed
.
End
mapM
.
Tactic
Notation
"simplify_option_equality"
"by"
tactic3
(
tac
)
:
=
repeat
match
goal
with
...
...
@@ 222,11 +275,13 @@ Tactic Notation "simplify_option_equality" "by" tactic3(tac) := repeat
assert
(
y
=
x
)
by
congruence
;
clear
H2

H1
:
?o
=
Some
?x
,
H2
:
?o
=
None

_
=>
congruence

H
:
mapM
_
_
=
Some
_

_
=>
apply
mapM_Some
in
H
end
.
Tactic
Notation
"simplify_option_equality"
:
=
simplify_option_equality
by
eauto
.
Hint
Extern
100
=>
simplify_option_equality
:
simplify_option_equality
.
Hint
Extern
800
=>
progress
simplify_option_equality
:
simplify_option_equality
.
(** * Union, intersection and difference *)
Instance
option_union_with
{
A
}
:
UnionWith
A
(
option
A
)
:
=
λ
f
x
y
,
...
...
theories/orders.v
View file @
9d0d6825
...
...
@@ 209,7 +209,7 @@ Section bounded_join_sl.
by
transitivity
(
X
∪
Y
)
;
[
auto

rewrite
E
].
*
intros
[
E1
E2
].
by
rewrite
E1
,
E2
,
(
left_id
_
_
).
Qed
.
Lemma
empty_
list_
union
Xs
:
⋃
Xs
≡
∅
↔
Forall
(
≡
∅
)
Xs
.
Lemma
empty_union
_list
Xs
:
⋃
Xs
≡
∅
↔
Forall
(
≡
∅
)
Xs
.
Proof
.
split
.
*
induction
Xs
;
simpl
;
rewrite
?empty_union
;
intuition
.
...
...
@@ 248,8 +248,8 @@ Section bounded_join_sl.
Lemma
empty_union_L
X
Y
:
X
∪
Y
=
∅
↔
X
=
∅
∧
Y
=
∅
.
Proof
.
unfold_leibniz
.
apply
empty_union
.
Qed
.
Lemma
empty_
list_
union_L
Xs
:
⋃
Xs
=
∅
↔
Forall
(=
∅
)
Xs
.
Proof
.
unfold_leibniz
.
apply
empty_
list_
union
.
Qed
.
Lemma
empty_union
_list
_L
Xs
:
⋃
Xs
=
∅
↔
Forall
(=
∅
)
Xs
.
Proof
.
unfold_leibniz
.
apply
empty_union
_list
.
Qed
.
End
leibniz
.
Section
dec
.
...
...
@@ 257,14 +257,14 @@ Section bounded_join_sl.
Lemma
non_empty_union
X
Y
:
X
∪
Y
≢
∅
→
X
≢
∅
∨
Y
≢
∅
.
Proof
.
rewrite
empty_union
.
destruct
(
decide
(
X
≡
∅
))
;
intuition
.
Qed
.
Lemma
non_empty_
list_
union
Xs
:
⋃
Xs
≢
∅
→
Exists
(
≢
∅
)
Xs
.
Proof
.
rewrite
empty_
list_
union
.
apply
(
not_Forall_Exists
_
).
Qed
.
Lemma
non_empty_union
_list
Xs
:
⋃
Xs
≢
∅
→
Exists
(
≢
∅
)
Xs
.
Proof
.
rewrite
empty_union
_list
.
apply
(
not_Forall_Exists
_
).
Qed
.
Context
`
{!
LeibnizEquiv
A
}.
Lemma
non_empty_union_L
X
Y
:
X
∪
Y
≠
∅
→
X
≠
∅
∨
Y
≠
∅
.
Proof
.
unfold_leibniz
.
apply
non_empty_union
.
Qed
.
Lemma
non_empty_
list_
union_L
Xs
:
⋃
Xs
≠
∅
→
Exists
(
≠
∅
)
Xs
.
Proof
.
unfold_leibniz
.
apply
non_empty_
list_
union
.
Qed
.
Lemma
non_empty_union
_list
_L
Xs
:
⋃
Xs
≠
∅
→
Exists
(
≠
∅
)
Xs
.
Proof
.
unfold_leibniz
.
apply
non_empty_union
_list
.
Qed
.
End
dec
.
End
bounded_join_sl
.
...
...
theories/pmap.v
View file @
9d0d6825
...
...
@@ 69,7 +69,7 @@ thereby obtain a data type that ensures canonicity. *)
Definition
Pmap
A
:
=
dsig
(@
Pmap_wf
A
).
(** * Operations on the data structure *)
Global
Instance
Pmap_dec
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
t1
t2
:
Pmap
A
)
:
Global
Instance
Pmap_
eq_
dec
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
(
t1
t2
:
Pmap
A
)
:
Decision
(
t1
=
t2
)
:
=
match
Pmap_raw_eq_dec
(
`
t1
)
(
`
t2
)
with

left
H
=>
left
(
proj2
(
dsig_eq
_
t1
t2
)
H
)
...
...
theories/tactics.v
View file @
9d0d6825
...
...
@@ 55,14 +55,10 @@ Ltac case_match :=


context
[
match
?x
with
_
=>
_
end
]
=>
destruct
x
eqn
:
?
end
.
(** The tactic [assert T unless tac_fail by tac_success] is used to assert
[T] only if it is not provable by [tac_fail]. This is useful to build other
tactics where only propositions that are not trivially provable are being
asserted. *)
Tactic
Notation
"assert"
constr
(
T
)
"unless"
tactic3
(
tac_fail
)
"by"
tactic3
(
tac_success
)
:
=
first
[
assert
T
by
tac_fail
;
fail
1

assert
T
by
tac_success
].
(** The tactic [unless T by tac_fail] succeeds if [T] is not provable by
the tactic [tac_fail]. *)
Tactic
Notation
"unless"
constr
(
T
)
"by"
tactic3
(
tac_fail
)
:
=
first
[
assert
T
by
tac_fail
;
fail
1

idtac
].
(** The tactic [repeat_on_hyps tac] repeatedly applies [tac] in unspecified
order on all hypotheses until it cannot be applied to any hypothesis anymore. *)
...
...
@@ 255,21 +251,33 @@ Tactic Notation "feed" "destruct" constr(H) "as" simple_intropattern(IP) :=
feed
(
fun
p
=>
let
H'
:
=
fresh
in
pose
proof
p
as
H'
;
destruct
H'
as
IP
)
H
.
(** Coq's [firstorder] tactic fails or loops on rather small goals already. In
particular, on those generated by the tactic [unfold_elem_ofs]
to solve
propositions on collections. The [naive_solver] tactic implements an
adhoc
and incomplete [firstorder]like solver using Ltac's backtracking
mechanism.
The tactic suffers from the following limitations:
particular, on those generated by the tactic [unfold_elem_ofs]
which is used
to solve
propositions on collections. The [naive_solver] tactic implements an
adhoc
and incomplete [firstorder]like solver using Ltac's backtracking
mechanism.
The tactic suffers from the following limitations:
 It might leave unresolved evars as Ltac provides no way to detect that.
 To avoid the tactic
going into pointless loops, it just does not allow a
universally quantified hypothesis to be used more than once
.
 To avoid the tactic
becoming too slow, we allow a universally quantified
hypothesis to be instantiated only once during each search path
.
 It does not perform backtracking on instantiation of universally quantified
assumptions.
We use a counter to make the search breath first. Breath first search ensures
that a minimal number of hypotheses is instantiated, and thus reduced the
posibility that an evar remains unresolved.
Despite these limitations, it works much better than Coq's [firstorder] tactic
for the purposes of this development. This tactic either fails or proves the
goal. *)
Lemma
forall_and_distr
(
A
:
Type
)
(
P
Q
:
A
→
Prop
)
:
(
∀
x
,
P
x
∧
Q
x
)
↔
(
∀
x
,
P
x
)
∧
(
∀
x
,
Q
x
).
Proof
.
firstorder
.
Qed
.
Tactic
Notation
"naive_solver"
tactic
(
tac
)
:
=
unfold
iff
,
not
in
*
;
repeat
match
goal
with

H
:
context
[
∀
_
,
_
∧
_
]

_
=>
repeat
setoid_rewrite
forall_and_distr
in
H
;
revert
H
end
;
let
rec
go
n
:
=
repeat
match
goal
with
(**i intros *)
...
...
@@ 278,52 +286,43 @@ Tactic Notation "naive_solver" tactic(tac) :=

H
:
False

_
=>
destruct
H

H
:
_
∧
_

_
=>
destruct
H

H
:
∃
_
,
_

_
=>
destruct
H

H
:
?P
→
?Q
,
H2
:
?Q

_
=>
specialize
(
H
H2
)
(**i simplify and solve equalities *)


_
=>
progress
simpl
in
*


_
=>
progress
simplify_equality
(**i solve the goal *)


_
=>
solve
[
eassumption

symmetry
;
eassumption

apply
not_symmetry
;
eassumption

reflexivity
]


_
=>
solve
[
eassumption

symmetry
;
eassumption

apply
not_symmetry
;
eassumption

reflexivity
]
(**i operations that generate more subgoals *)


_
∧
_
=>
split

H
:
_
∨
_

_
=>
destruct
H
(**i solve the goal using the user supplied tactic *)


_
=>
solve
[
tac
]
end
;
(**i use recursion to enable backtracking on the following clauses. We use
a counter to minimize the number of instantiations, and thus to reduce the
number of potentially unresolved meta variables. *)
first
[
iter
(
fun
n'
=>
match
goal
with
(**i instantiations of assumptions *)

H
:
_
→
_

_
=>
is_non_dependent
H
;
eapply
H
;
clear
H
;
go
n'

H
:
_
→
_

_
=>
is_non_dependent
H
;
(**i create subgoals for all premises *)
efeed
H
using
(
fun
p
=>
match
type
of
p
with

_
∧
_
=>
let
H'
:
=
fresh
in
pose
proof
p
as
H'
;
destruct
H'

∃
_
,
_
=>
let
H'
:
=
fresh
in
pose
proof
p
as
H'
;
destruct
H'

_
∨
_
=>
let
H'
:
=
fresh
in
pose
proof
p
as
H'
;
destruct
H'

False
=>
let
H'
:
=
fresh
in
pose
proof
p
as
H'
;
destruct
H'
end
)
by
(
clear
H
;
go
n'
)
;
(**i solve these subgoals, but clear [H] to avoid loops *)
clear
H
;
go
n
end
)
(
eval
compute
in
(
seq
0
n
))

match
goal
with
(**i instantiation of the conclusion *)


∃
x
,
_
=>
eexists
;
go
n


_
∨
_
=>
first
[
left
;
go
n

right
;
go
n
]
end
]
in
go
10
.
(**i use recursion to enable backtracking on the following clauses. *)
match
goal
with
(**i instantiation of the conclusion *)


∃
x
,
_
=>
eexists
;
go
n


_
∨
_
=>
first
[
left
;
go
n

right
;
go
n
]

_
=>
(**i instantiations of assumptions. *)
lazymatch
n
with

S
?n'
=>
(**i we give priority to assumptions that fit on the conclusion. *)
match
goal
with

H
:
_
→
_

_
=>
is_non_dependent
H
;
eapply
H
;
clear
H
;
go
n'

H
:
_
→
_

_
=>
is_non_dependent
H
;
try
(
eapply
H
;
fail
2
)
;
efeed
pose
proof
H
;
clear
H
;
go
n'
end
end
end
in
iter
(
fun
n'
=>
go
n'
)
(
eval
compute
in
(
seq
0
6
)).
Tactic
Notation
"naive_solver"
:
=
naive_solver
eauto
.
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