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Iris
stdpp
Commits
5446fba3
Commit
5446fba3
authored
Jun 11, 2012
by
Robbert Krebbers
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+2212
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theories/base.v
theories/base.v
+293
0
theories/collections.v
theories/collections.v
+192
0
theories/decidable.v
theories/decidable.v
+62
0
theories/fin_collections.v
theories/fin_collections.v
+182
0
theories/fin_maps.v
theories/fin_maps.v
+218
0
theories/list.v
theories/list.v
+341
0
theories/listset.v
theories/listset.v
+100
0
theories/monads.v
theories/monads.v
+19
0
theories/nmap.v
theories/nmap.v
+52
0
theories/numbers.v
theories/numbers.v
+58
0
theories/option.v
theories/option.v
+163
0
theories/orders.v
theories/orders.v
+101
0
theories/pmap.v
theories/pmap.v
+299
0
theories/prelude.v
theories/prelude.v
+11
0
theories/subset.v
theories/subset.v
+14
0
theories/trs.v
theories/trs.v
+107
0
No files found.
theories/base.v
0 → 100644
View file @
5446fba3
Global
Generalizable
All
Variables
.
Global
Set
Automatic
Coercions
Import
.
Require
Export
Morphisms
RelationClasses
List
Bool
Utf8
Program
Setoid
NArith
.
Arguments
existT
{
_
_
}
_
_
.
Arguments
existT2
{
_
_
_
}
_
_
_
.
Definition
proj1_T2
{
A
}
{
P
Q
:
A
→
Type
}
(
x
:
sigT2
P
Q
)
:
A
:
=
match
x
with
existT2
a
_
_
=>
a
end
.
Definition
proj2_T2
{
A
}
{
P
Q
:
A
→
Type
}
(
x
:
sigT2
P
Q
)
:
P
(
proj1_T2
x
)
:
=
match
x
with
existT2
_
p
_
=>
p
end
.
Definition
proj3_T2
{
A
}
{
P
Q
:
A
→
Type
}
(
x
:
sigT2
P
Q
)
:
Q
(
proj1_T2
x
)
:
=
match
x
with
existT2
_
_
q
=>
q
end
.
(* Common notations *)
Delimit
Scope
C_scope
with
C
.
Global
Open
Scope
C_scope
.
Notation
"(=)"
:
=
eq
(
only
parsing
)
:
C_scope
.
Notation
"( x =)"
:
=
(
eq
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(= x )"
:
=
(
λ
y
,
eq
y
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(≠)"
:
=
(
λ
x
y
,
x
≠
y
)
(
only
parsing
)
:
C_scope
.
Notation
"( x ≠)"
:
=
(
λ
y
,
x
≠
y
)
(
only
parsing
)
:
C_scope
.
Notation
"(≠ x )"
:
=
(
λ
y
,
y
≠
x
)
(
only
parsing
)
:
C_scope
.
Hint
Extern
0
(
?x
=
?x
)
=>
reflexivity
.
Notation
"(→)"
:
=
(
λ
x
y
,
x
→
y
)
:
C_scope
.
Notation
"( T →)"
:
=
(
λ
y
,
T
→
y
)
:
C_scope
.
Notation
"(→ T )"
:
=
(
λ
y
,
y
→
T
)
:
C_scope
.
Notation
"t $ r"
:
=
(
t
r
)
(
at
level
65
,
right
associativity
,
only
parsing
)
:
C_scope
.
Infix
"∘"
:
=
compose
:
C_scope
.
Notation
"(∘)"
:
=
compose
(
only
parsing
)
:
C_scope
.
Notation
"( f ∘)"
:
=
(
compose
f
)
(
only
parsing
)
:
C_scope
.
Notation
"(∘ f )"
:
=
(
λ
g
,
compose
g
f
)
(
only
parsing
)
:
C_scope
.
Notation
"x ↾ p"
:
=
(
exist
_
x
p
)
(
at
level
20
)
:
C_scope
.
Notation
"` x"
:
=
(
proj1_sig
x
)
:
C_scope
.
(* Provable propositions *)
Class
PropHolds
(
P
:
Prop
)
:
=
prop_holds
:
P
.
(* Decidable propositions *)
Class
Decision
(
P
:
Prop
)
:
=
decide
:
{
P
}
+
{
¬
P
}.
Arguments
decide
_
{
_
}.
(* Common relations & operations *)
Class
Equiv
A
:
=
equiv
:
relation
A
.
Infix
"≡"
:
=
equiv
(
at
level
70
,
no
associativity
)
:
C_scope
.
Notation
"(≡)"
:
=
equiv
(
only
parsing
)
:
C_scope
.
Notation
"( x ≡)"
:
=
(
equiv
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(≡ x )"
:
=
(
λ
y
,
y
≡
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(≢)"
:
=
(
λ
x
y
,
¬
x
≡
y
)
(
only
parsing
)
:
C_scope
.
Notation
"x ≢ y"
:
=
(
¬
x
≡
y
)
(
at
level
70
,
no
associativity
)
:
C_scope
.
Notation
"( x ≢)"
:
=
(
λ
y
,
x
≢
y
)
(
only
parsing
)
:
C_scope
.
Notation
"(≢ x )"
:
=
(
λ
y
,
y
≢
x
)
(
only
parsing
)
:
C_scope
.
Instance
equiv_default_relation
`
{
Equiv
A
}
:
DefaultRelation
(
≡
)

3
.
Hint
Extern
0
(
?x
≡
?x
)
=>
reflexivity
.
Class
Empty
A
:
=
empty
:
A
.
Notation
"∅"
:
=
empty
:
C_scope
.
Class
Union
A
:
=
union
:
A
→
A
→
A
.
Infix
"∪"
:
=
union
(
at
level
50
,
left
associativity
)
:
C_scope
.
Notation
"(∪)"
:
=
union
(
only
parsing
)
:
C_scope
.
Notation
"( x ∪)"
:
=
(
union
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(∪ x )"
:
=
(
λ
y
,
union
y
x
)
(
only
parsing
)
:
C_scope
.
Class
Intersection
A
:
=
intersection
:
A
→
A
→
A
.
Infix
"∩"
:
=
intersection
(
at
level
40
)
:
C_scope
.
Notation
"(∩)"
:
=
intersection
(
only
parsing
)
:
C_scope
.
Notation
"( x ∩)"
:
=
(
intersection
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(∩ x )"
:
=
(
λ
y
,
intersection
y
x
)
(
only
parsing
)
:
C_scope
.
Class
Difference
A
:
=
difference
:
A
→
A
→
A
.
Infix
"∖"
:
=
difference
(
at
level
40
)
:
C_scope
.
Notation
"(∖)"
:
=
difference
(
only
parsing
)
:
C_scope
.
Notation
"( x ∖)"
:
=
(
difference
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(∖ x )"
:
=
(
λ
y
,
difference
y
x
)
(
only
parsing
)
:
C_scope
.
Class
SubsetEq
A
:
=
subseteq
:
A
→
A
→
Prop
.
Infix
"⊆"
:
=
subseteq
(
at
level
70
)
:
C_scope
.
Notation
"(⊆)"
:
=
subseteq
(
only
parsing
)
:
C_scope
.
Notation
"( X ⊆ )"
:
=
(
subseteq
X
)
(
only
parsing
)
:
C_scope
.
Notation
"( ⊆ X )"
:
=
(
λ
Y
,
subseteq
Y
X
)
(
only
parsing
)
:
C_scope
.
Notation
"X ⊈ Y"
:
=
(
¬
X
⊆
Y
)
(
at
level
70
)
:
C_scope
.
Notation
"(⊈)"
:
=
(
λ
X
Y
,
X
⊈
Y
)
(
only
parsing
)
:
C_scope
.
Notation
"( X ⊈ )"
:
=
(
λ
Y
,
X
⊈
Y
)
(
only
parsing
)
:
C_scope
.
Notation
"( ⊈ X )"
:
=
(
λ
Y
,
Y
⊈
X
)
(
only
parsing
)
:
C_scope
.
Hint
Extern
0
(
?x
⊆
?x
)
=>
reflexivity
.
Class
Singleton
A
B
:
=
singleton
:
A
→
B
.
Notation
"{{ x }}"
:
=
(
singleton
x
)
:
C_scope
.
Notation
"{{ x ; y ; .. ; z }}"
:
=
(
union
..
(
union
(
singleton
x
)
(
singleton
y
))
..
(
singleton
z
))
:
C_scope
.
Class
ElemOf
A
B
:
=
elem_of
:
A
→
B
→
Prop
.
Infix
"∈"
:
=
elem_of
(
at
level
70
)
:
C_scope
.
Notation
"(∈)"
:
=
elem_of
(
only
parsing
)
:
C_scope
.
Notation
"( x ∈)"
:
=
(
elem_of
x
)
(
only
parsing
)
:
C_scope
.
Notation
"(∈ X )"
:
=
(
λ
x
,
elem_of
x
X
)
(
only
parsing
)
:
C_scope
.
Notation
"x ∉ X"
:
=
(
¬
x
∈
X
)
(
at
level
80
)
:
C_scope
.
Notation
"(∉)"
:
=
(
λ
x
X
,
x
∉
X
)
(
only
parsing
)
:
C_scope
.
Notation
"( x ∉)"
:
=
(
λ
X
,
x
∉
X
)
(
only
parsing
)
:
C_scope
.
Notation
"(∉ X )"
:
=
(
λ
x
,
x
∉
X
)
(
only
parsing
)
:
C_scope
.
Class
UnionWith
M
:
=
union_with
:
∀
{
A
},
(
A
→
A
→
A
)
→
M
A
→
M
A
→
M
A
.
Class
IntersectWith
M
:
=
intersect_with
:
∀
{
A
},
(
A
→
A
→
A
)
→
M
A
→
M
A
→
M
A
.
(* Common properties *)
Class
Injective
{
A
B
}
R
S
(
f
:
A
→
B
)
:
=
injective
:
∀
x
y
:
A
,
S
(
f
x
)
(
f
y
)
→
R
x
y
.
Class
Idempotent
{
A
}
R
(
f
:
A
→
A
→
A
)
:
=
idempotent
:
∀
x
,
R
(
f
x
x
)
x
.
Class
Commutative
{
A
B
}
R
(
f
:
B
→
B
→
A
)
:
=
commutative
:
∀
x
y
,
R
(
f
x
y
)
(
f
y
x
).
Class
LeftId
{
A
}
R
(
i
:
A
)
(
f
:
A
→
A
→
A
)
:
=
left_id
:
∀
x
,
R
(
f
i
x
)
x
.
Class
RightId
{
A
}
R
(
i
:
A
)
(
f
:
A
→
A
→
A
)
:
=
right_id
:
∀
x
,
R
(
f
x
i
)
x
.
Class
Associative
{
A
}
R
(
f
:
A
→
A
→
A
)
:
=
associative
:
∀
x
y
z
,
R
(
f
x
(
f
y
z
))
(
f
(
f
x
y
)
z
).
Arguments
injective
{
_
_
_
_
}
_
{
_
}
_
_
_
.
Arguments
idempotent
{
_
_
}
_
{
_
}
_
.
Arguments
commutative
{
_
_
_
}
_
{
_
}
_
_
.
Arguments
left_id
{
_
_
}
_
_
{
_
}
_
.
Arguments
right_id
{
_
_
}
_
_
{
_
}
_
.
Arguments
associative
{
_
_
}
_
{
_
}
_
_
_
.
(* Monadic operations *)
Section
monad_ops
.
Context
(
M
:
Type
→
Type
).
Class
MRet
:
=
mret
:
∀
{
A
},
A
→
M
A
.
Class
MBind
:
=
mbind
:
∀
{
A
B
},
(
A
→
M
B
)
→
M
A
→
M
B
.
Class
MJoin
:
=
mjoin
:
∀
{
A
},
M
(
M
A
)
→
M
A
.
Class
FMap
:
=
fmap
:
∀
{
A
B
},
(
A
→
B
)
→
M
A
→
M
B
.
End
monad_ops
.
Arguments
mret
{
M
MRet
A
}
_
.
Arguments
mbind
{
M
MBind
A
B
}
_
_
.
Arguments
mjoin
{
M
MJoin
A
}
_
.
Arguments
fmap
{
M
FMap
A
B
}
_
_
.
Notation
"m ≫= f"
:
=
(
mbind
f
m
)
(
at
level
60
,
right
associativity
)
:
C_scope
.
Notation
"x ← y ; z"
:
=
(
y
≫
=
(
λ
x
:
_
,
z
))
(
at
level
65
,
next
at
level
35
,
right
associativity
)
:
C_scope
.
Infix
"<$>"
:
=
fmap
(
at
level
65
,
right
associativity
,
only
parsing
)
:
C_scope
.
(* Ordered structures *)
Class
BoundedPreOrder
A
`
{
Empty
A
}
`
{
SubsetEq
A
}
:
=
{
bounded_preorder
:
>>
PreOrder
(
⊆
)
;
subseteq_empty
x
:
∅
⊆
x
}.
(* Note: no equality to avoid the need for setoids. We define equality in a generic way. *)
Class
BoundedJoinSemiLattice
A
`
{
Empty
A
}
`
{
SubsetEq
A
}
`
{
Union
A
}
:
=
{
jsl_preorder
:
>>
BoundedPreOrder
A
;
subseteq_union_l
x
y
:
x
⊆
x
∪
y
;
subseteq_union_r
x
y
:
y
⊆
x
∪
y
;
union_least
x
y
z
:
x
⊆
z
→
y
⊆
z
→
x
∪
y
⊆
z
}.
Class
MeetSemiLattice
A
`
{
Empty
A
}
`
{
SubsetEq
A
}
`
{
Intersection
A
}
:
=
{
msl_preorder
:
>>
BoundedPreOrder
A
;
subseteq_intersection_l
x
y
:
x
∩
y
⊆
x
;
subseteq_intersection_r
x
y
:
x
∩
y
⊆
y
;
intersection_greatest
x
y
z
:
z
⊆
x
→
z
⊆
y
→
z
⊆
x
∩
y
}.
(* Containers *)
Class
Size
C
:
=
size
:
C
→
nat
.
Class
Map
A
C
:
=
map
:
(
A
→
A
)
→
(
C
→
C
).
Class
Collection
A
C
`
{
ElemOf
A
C
}
`
{
Empty
C
}
`
{
Union
C
}
`
{
Intersection
C
}
`
{
Difference
C
}
`
{
Singleton
A
C
}
`
{
Map
A
C
}
:
=
{
elem_of_empty
(
x
:
A
)
:
x
∉
∅
;
elem_of_singleton
(
x
y
:
A
)
:
x
∈
{{
y
}}
↔
x
=
y
;
elem_of_union
X
Y
(
x
:
A
)
:
x
∈
X
∪
Y
↔
x
∈
X
∨
x
∈
Y
;
elem_of_intersection
X
Y
(
x
:
A
)
:
x
∈
X
∩
Y
↔
x
∈
X
∧
x
∈
Y
;
elem_of_difference
X
Y
(
x
:
A
)
:
x
∈
X
∖
Y
↔
x
∈
X
∧
x
∉
Y
;
elem_of_map
f
X
(
x
:
A
)
:
x
∈
map
f
X
↔
∃
y
,
x
=
f
y
∧
y
∈
X
}.
Class
Elements
A
C
:
=
elements
:
C
→
list
A
.
Class
FinCollection
A
C
`
{
Empty
C
}
`
{
Union
C
}
`
{
Intersection
C
}
`
{
Difference
C
}
`
{
Singleton
A
C
}
`
{
ElemOf
A
C
}
`
{
Map
A
C
}
`
{
Elements
A
C
}
:
=
{
fin_collection
:
>>
Collection
A
C
;
elements_spec
X
x
:
x
∈
X
↔
In
x
(
elements
X
)
;
elements_nodup
X
:
NoDup
(
elements
X
)
}.
Class
Fresh
A
C
:
=
fresh
:
C
→
A
.
Class
FreshSpec
A
C
`
{!
Fresh
A
C
}
`
{!
ElemOf
A
C
}
:
=
{
fresh_proper
X
Y
:
(
∀
x
,
x
∈
X
↔
x
∈
Y
)
→
fresh
X
=
fresh
Y
;
is_fresh
(
X
:
C
)
:
fresh
X
∉
X
}.
(* Maps *)
Class
Lookup
K
M
:
=
lookup
:
∀
{
A
},
K
→
M
A
→
option
A
.
Notation
"m !! i"
:
=
(
lookup
i
m
)
(
at
level
20
)
:
C_scope
.
Notation
"(!!)"
:
=
lookup
(
only
parsing
)
:
C_scope
.
Notation
"( m !!)"
:
=
(
λ
i
,
lookup
i
m
)
(
only
parsing
)
:
C_scope
.
Notation
"(!! i )"
:
=
(
lookup
i
)
(
only
parsing
)
:
C_scope
.
Class
PartialAlter
K
M
:
=
partial_alter
:
∀
{
A
},
(
option
A
→
option
A
)
→
K
→
M
A
→
M
A
.
Class
Alter
K
M
:
=
alter
:
∀
{
A
},
(
A
→
A
)
→
K
→
M
A
→
M
A
.
Class
Dom
K
M
:
=
dom
:
∀
C
`
{
Empty
C
}
`
{
Union
C
}
`
{
Singleton
K
C
},
M
→
C
.
Class
Merge
M
:
=
merge
:
∀
{
A
},
(
option
A
→
option
A
→
option
A
)
→
M
A
→
M
A
→
M
A
.
Class
Insert
K
M
:
=
insert
:
∀
{
A
},
K
→
A
→
M
A
→
M
A
.
Notation
"<[ k := a ]>"
:
=
(
insert
k
a
)
(
at
level
5
,
right
associativity
)
:
C_scope
.
Class
Delete
K
M
:
=
delete
:
K
→
M
→
M
.
(* Misc *)
Instance
pointwise_reflexive
{
A
}
`
{
R
:
relation
B
}
:
Reflexive
R
→
Reflexive
(
pointwise_relation
A
R
)

9
.
Proof
.
firstorder
.
Qed
.
Instance
pointwise_symmetric
{
A
}
`
{
R
:
relation
B
}
:
Symmetric
R
→
Symmetric
(
pointwise_relation
A
R
)

9
.
Proof
.
firstorder
.
Qed
.
Instance
pointwise_transitive
{
A
}
`
{
R
:
relation
B
}
:
Transitive
R
→
Transitive
(
pointwise_relation
A
R
)

9
.
Proof
.
firstorder
.
Qed
.
Definition
fst_map
{
A
A'
B
}
(
f
:
A
→
A'
)
(
p
:
A
*
B
)
:
A'
*
B
:
=
(
f
(
fst
p
),
snd
p
).
Definition
snd_map
{
A
B
B'
}
(
f
:
B
→
B'
)
(
p
:
A
*
B
)
:
A
*
B'
:
=
(
fst
p
,
f
(
snd
p
)).
Definition
prod_relation
{
A
B
}
(
R1
:
relation
A
)
(
R2
:
relation
B
)
:
relation
(
A
*
B
)
:
=
λ
x
y
,
R1
(
fst
x
)
(
fst
y
)
∧
R2
(
snd
x
)
(
snd
y
).
Section
prod_relation
.
Context
`
{
R1
:
relation
A
}
`
{
R2
:
relation
B
}.
Global
Instance
:
Reflexive
R1
→
Reflexive
R2
→
Reflexive
(
prod_relation
R1
R2
).
Proof
.
firstorder
eauto
.
Qed
.
Global
Instance
:
Symmetric
R1
→
Symmetric
R2
→
Symmetric
(
prod_relation
R1
R2
).
Proof
.
firstorder
eauto
.
Qed
.
Global
Instance
:
Transitive
R1
→
Transitive
R2
→
Transitive
(
prod_relation
R1
R2
).
Proof
.
firstorder
eauto
.
Qed
.
Global
Instance
:
Equivalence
R1
→
Equivalence
R2
→
Equivalence
(
prod_relation
R1
R2
).
Proof
.
split
;
apply
_
.
Qed
.
Global
Instance
:
Proper
(
R1
==>
R2
==>
prod_relation
R1
R2
)
pair
.
Proof
.
firstorder
eauto
.
Qed
.
Global
Instance
:
Proper
(
prod_relation
R1
R2
==>
R1
)
fst
.
Proof
.
firstorder
eauto
.
Qed
.
Global
Instance
:
Proper
(
prod_relation
R1
R2
==>
R2
)
snd
.
Proof
.
firstorder
eauto
.
Qed
.
End
prod_relation
.
Definition
lift_relation
{
A
B
}
(
R
:
relation
A
)
(
f
:
B
→
A
)
:
relation
B
:
=
λ
x
y
,
R
(
f
x
)
(
f
y
).
Definition
lift_relation_equivalence
{
A
B
}
(
R
:
relation
A
)
(
f
:
B
→
A
)
:
Equivalence
R
→
Equivalence
(
lift_relation
R
f
).
Proof
.
unfold
lift_relation
.
firstorder
.
Qed
.
Hint
Extern
0
(
Equivalence
(
lift_relation
_
_
))
=>
eapply
@
lift_relation_equivalence
:
typeclass_instances
.
Instance
:
∀
A
B
(
x
:
B
),
Commutative
(=)
(
λ
_
_
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
:
∀
A
(
x
:
A
),
Associative
(=)
(
λ
_
_
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
:
∀
A
,
Associative
(=)
(
λ
x
_
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
:
∀
A
,
Associative
(=)
(
λ
_
x
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
:
∀
A
,
Idempotent
(=)
(
λ
x
_
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
:
∀
A
,
Idempotent
(=)
(
λ
_
x
:
A
,
x
).
Proof
.
easy
.
Qed
.
Instance
left_id_propholds
{
A
}
(
R
:
relation
A
)
i
f
:
LeftId
R
i
f
→
∀
x
,
PropHolds
(
R
(
f
i
x
)
x
).
Proof
.
easy
.
Qed
.
Instance
right_id_propholds
{
A
}
(
R
:
relation
A
)
i
f
:
RightId
R
i
f
→
∀
x
,
PropHolds
(
R
(
f
x
i
)
x
).
Proof
.
easy
.
Qed
.
Instance
idem_propholds
{
A
}
(
R
:
relation
A
)
f
:
Idempotent
R
f
→
∀
x
,
PropHolds
(
R
(
f
x
x
)
x
).
Proof
.
easy
.
Qed
.
Ltac
simplify_eqs
:
=
repeat
match
goal
with


_
=>
progress
subst


_
=
_
=>
reflexivity

H
:
_
≠
_

_
=>
now
destruct
H

H
:
_
=
_
→
False

_
=>
now
destruct
H

H
:
_
=
_

_
=>
discriminate
H

H
:
_
=
_

?G
=>
change
(
id
G
)
;
injection
H
;
clear
H
;
intros
;
unfold
id
at
1

H
:
?f
_
=
?f
_

_
=>
apply
(
injective
f
)
in
H

H
:
?f
_
?x
=
?f
_
?x

_
=>
apply
(
injective
(
λ
y
,
f
y
x
))
in
H
end
.
Hint
Extern
0
(
PropHolds
_
)
=>
assumption
:
typeclass_instances
.
Instance
:
Proper
(
iff
==>
iff
)
PropHolds
.
Proof
.
now
repeat
intro
.
Qed
.
Ltac
solve_propholds
:
=
match
goal
with

[

PropHolds
(
?P
)
]
=>
apply
_

[

?P
]
=>
change
(
PropHolds
P
)
;
apply
_
end
.
Tactic
Notation
"remember"
constr
(
t
)
"as"
"("
ident
(
x
)
","
ident
(
E
)
")"
:
=
remember
t
as
x
;
match
goal
with

E'
:
x
=
_

_
=>
rename
E'
into
E
end
.
theories/collections.v
0 → 100644
View file @
5446fba3
Require
Export
base
orders
.
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_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
.
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
.
Proof
.
firstorder
.
Qed
.
Lemma
elem_of_subseteq
X
Y
:
X
⊆
Y
↔
∀
x
,
x
∈
X
→
x
∈
Y
.
Proof
.
easy
.
Qed
.
Lemma
elem_of_equiv
X
Y
:
X
≡
Y
↔
∀
x
,
x
∈
X
↔
x
∈
Y
.
Proof
.
firstorder
.
Qed
.
Lemma
elem_of_equiv_alt
X
Y
:
X
≡
Y
↔
(
∀
x
,
x
∈
X
→
x
∈
Y
)
∧
(
∀
x
,
x
∈
Y
→
x
∈
X
).
Proof
.
firstorder
.
Qed
.
Global
Instance
:
Proper
((=)
==>
(
≡
)
==>
iff
)
(
∈
).
Proof
.
intros
???.
subst
.
firstorder
.
Qed
.
Lemma
empty_ne_singleton
x
:
∅
≢
{{
x
}}.
Proof
.
intros
[
_
E
].
destruct
(
elem_of_empty
x
).
apply
E
.
now
apply
elem_of_singleton
.
Qed
.
End
collection
.
Section
cmap
.
Context
`
{
Collection
A
C
}.
Lemma
elem_of_map_1
(
f
:
A
→
A
)
(
X
:
C
)
(
x
:
A
)
:
x
∈
X
→
f
x
∈
map
f
X
.
Proof
.
intros
.
apply
(
elem_of_map
_
).
eauto
.
Qed
.
Lemma
elem_of_map_1_alt
(
f
:
A
→
A
)
(
X
:
C
)
(
x
:
A
)
y
:
x
∈
X
→
y
=
f
x
→
y
∈
map
f
X
.
Proof
.
intros
.
apply
(
elem_of_map
_
).
eauto
.
Qed
.
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
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_singleton
in
H

H
:
context
[
_
∈
_
∪
_
]

_
=>
setoid_rewrite
elem_of_union
in
H

H
:
context
[
_
∈
_
∩
_
]

_
=>
setoid_rewrite
elem_of_intersection
in
H

H
:
context
[
_
∈
_
∖
_
]

_
=>
setoid_rewrite
elem_of_difference
in
H

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_singleton


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


context
[
_
∈
_
∩
_
]
=>
setoid_rewrite
elem_of_intersection


context
[
_
∈
_
∖
_
]
=>
setoid_rewrite
elem_of_difference


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 *)
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
[
_
∧
_
]

_
=>
edestruct
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
).
Section
no_dup
.
Context
`
{
Collection
A
B
}
(
R
:
relation
A
)
`
{!
Equivalence
R
}.
Definition
elem_of_upto
(
x
:
A
)
(
X
:
B
)
:
=
∃
y
,
y
∈
X
∧
R
x
y
.
Definition
no_dup
(
X
:
B
)
:
=
∀
x
y
,
x
∈
X
→
y
∈
X
→
R
x
y
→
x
=
y
.
Global
Instance
:
Proper
((
≡
)
==>
iff
)
(
elem_of_upto
x
).
Proof
.
firstorder
.
Qed
.
Global
Instance
:
Proper
(
R
==>
(
≡
)
==>
iff
)
elem_of_upto
.
Proof
.
intros
??
E1
??
E2
.
split
;
intros
[
z
[??]]
;
exists
z
.
rewrite
<
E1
,
<
E2
;
intuition
.
rewrite
E1
,
E2
;
intuition
.
Qed
.
Global
Instance
:
Proper
((
≡
)
==>
iff
)
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
.
esimplify_elem_of
.
Qed
.
Lemma
elem_of_upto_empty
x
:
¬
elem_of_upto
x
∅
.
Proof
.
unfold
elem_of_upto
.
esimplify_elem_of
.
Qed
.
Lemma
elem_of_upto_singleton
x
y
:
elem_of_upto
x
{{
y
}}
↔
R
x
y
.
Proof
.
unfold
elem_of_upto
.
esimplify_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
.
esimplify_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
.
esimplify_elem_of
.
Qed
.
Lemma
no_dup_empty
:
no_dup
∅
.
Proof
.
unfold
no_dup
.
simplify_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
.
esimplify_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
;
simplify_elem_of
.
Qed
.
Lemma
no_dup_inv_union_l
X
Y
:
no_dup
(
X
∪
Y
)
→
no_dup
X
.
Proof
.
unfold
no_dup
.
simplify_elem_of
.
Qed
.
Lemma
no_dup_inv_union_r
X
Y
:
no_dup
(
X
∪
Y
)
→
no_dup
Y
.
Proof
.
unfold
no_dup
.
simplify_elem_of
.
Qed
.
End
no_dup
.
Section
quantifiers
.
Context
`
{
Collection
A
B
}
(
P
:
A
→
Prop
).
Definition
cforall
X
:
=
∀
x
,
x
∈
X
→
P
x
.
Definition
cexists
X
:
=
∃
x
,
x
∈
X
∧
P
x
.
Lemma
cforall_empty
:
cforall
∅
.
Proof
.
unfold
cforall
.
simplify_elem_of
.
Qed
.
Lemma
cforall_singleton
x
:
cforall
{{
x
}}
↔
P
x
.
Proof
.
unfold
cforall
.
simplify_elem_of
.
Qed
.
Lemma
cforall_union
X
Y
:
cforall
X
→
cforall
Y
→
cforall
(
X
∪
Y
).
Proof
.
unfold
cforall
.
simplify_elem_of
.
Qed
.
Lemma
cforall_union_inv_1
X
Y
:
cforall
(
X
∪
Y
)
→
cforall
X
.
Proof
.
unfold
cforall
.
simplify_elem_of
.
Qed
.
Lemma
cforall_union_inv_2
X
Y
:
cforall
(
X
∪
Y
)
→
cforall
Y
.
Proof
.
unfold
cforall
.
simplify_elem_of
.
Qed
.
Lemma
cexists_empty
:
¬
cexists
∅
.
Proof
.
unfold
cexists
.
esimplify_elem_of
.
Qed
.
Lemma
cexists_singleton
x
:
cexists
{{
x
}}
↔
P
x
.
Proof
.
unfold
cexists
.
esimplify_elem_of
.
Qed
.
Lemma
cexists_union_1
X
Y
:
cexists
X
→
cexists
(
X
∪
Y
).
Proof
.
unfold
cexists
.
esimplify_elem_of
.
Qed
.
Lemma
cexists_union_2
X
Y
:
cexists
Y
→
cexists
(
X
∪
Y
).
Proof
.
unfold
cexists
.
esimplify_elem_of
.
Qed
.
Lemma
cexists_union_inv
X
Y
:
cexists
(
X
∪
Y
)
→
cexists
X
∨
cexists
Y
.
Proof
.
unfold
cexists
.
esimplify_elem_of
.
Qed
.
End
quantifiers
.
Lemma
cforall_weak
`
{
Collection
A
B
}
(
P
Q
:
A
→
Prop
)
(
Hweak
:
∀
x
,
P
x
→
Q
x
)
X
:
cforall
P
X
→
cforall
Q
X
.
Proof
.
firstorder
.
Qed
.
Lemma
cexists_weak
`
{
Collection
A
B
}
(
P
Q
:
A
→
Prop
)
(
Hweak
:
∀
x
,
P
x
→
Q
x
)
X
:
cexists
P
X
→
cexists
Q
X
.
Proof
.
firstorder
.
Qed
.
theories/decidable.v
0 → 100644
View file @
5446fba3
Require
Export
base
.
Definition
decide_rel
{
A
B
}
(
R
:
A
→
B
→
Prop
)
{
dec
:
∀
x
y
,
Decision
(
R
x
y
)}
(
x
:
A
)
(
y
:
B
)
:
Decision
(
R
x
y
)
:
=
dec
x
y
.
Ltac
case_decide
:
=
match
goal
with

H
:
context
[@
decide
?P
?dec
]

_
=>
case
(@
decide
P
dec
)
in
*

H
:
context
[@
decide_rel
_
_
?R
?x
?y
?dec
]

_
=>
case
(@
decide_rel
_
_
R
x
y
dec
)
in
*


context
[@
decide
?P
?dec
]
=>
case
(@
decide
P
dec
)
in
*


context
[@
decide_rel
_
_
?R
?x
?y
?dec
]
=>
case
(@
decide_rel
_
_
R
x
y
dec
)
in
*
end
.
Ltac
solve_trivial_decision
:
=
match
goal
with

[

Decision
(
?P
)
]
=>
apply
_

[

sumbool
?P
(
¬
?P
)
]
=>
change
(
Decision
P
)
;
</