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e15c090e
Commit
e15c090e
authored
Mar 11, 2016
by
Robbert Krebbers
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Choice principle for finite types.
parent
8013e09a
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17
prelude/finite.v
prelude/finite.v
+49
1
prelude/vector.v
prelude/vector.v
+1
16
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prelude/finite.v
View file @
e15c090e
(* Copyright (c) 20122015, Robbert Krebbers. *)
(* Copyright (c) 20122015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(* This file is distributed under the terms of the BSD license. *)
From
iris
.
prelude
Require
Export
countable
list
.
From
iris
.
prelude
Require
Export
countable
vector
.
Class
Finite
A
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
:
=
{
Class
Finite
A
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
:
=
{
enum
:
list
A
;
enum
:
list
A
;
...
@@ 61,6 +61,39 @@ Proof.
...
@@ 61,6 +61,39 @@ Proof.
exists
y
.
by
rewrite
!
Nat2Pos
.
id
by
done
.
exists
y
.
by
rewrite
!
Nat2Pos
.
id
by
done
.
Qed
.
Qed
.
Definition
encode_fin
`
{
Finite
A
}
(
x
:
A
)
:
fin
(
card
A
)
:
=
Fin
.
of_nat_lt
(
encode_lt_card
x
).
Program
Definition
decode_fin
`
{
Finite
A
}
(
i
:
fin
(
card
A
))
:
A
:
=
match
Some_dec
(
decode_nat
i
)
return
_
with

inleft
(
exist
x
_
)
=>
x

inright
_
=>
_
end
.
Next
Obligation
.
intros
A
??
i
?
;
exfalso
.
destruct
(
encode_decode
A
i
)
;
naive_solver
auto
using
fin_to_nat_lt
.
Qed
.
Lemma
decode_encode_fin
`
{
Finite
A
}
(
x
:
A
)
:
decode_fin
(
encode_fin
x
)
=
x
.
Proof
.
unfold
decode_fin
,
encode_fin
.
destruct
(
Some_dec
_
)
as
[[
x'
Hx
]
Hx
].
{
by
rewrite
fin_to_of_nat
,
decode_encode_nat
in
Hx
;
simplify_eq
.
}
exfalso
;
by
rewrite
>
fin_to_of_nat
,
decode_encode_nat
in
Hx
.
Qed
.
Lemma
fin_choice
{
n
}
{
B
:
fin
n
→
Type
}
(
P
:
∀
i
,
B
i
→
Prop
)
:
(
∀
i
,
∃
y
,
P
i
y
)
→
∃
f
,
∀
i
,
P
i
(
f
i
).
Proof
.
induction
n
as
[
n
IH
]
;
intros
Hex
.
{
exists
(
fin_0_inv
_
)
;
intros
i
;
inv_fin
i
.
}
destruct
(
IH
_
_
(
λ
i
,
Hex
(
FS
i
)))
as
[
f
Hf
],
(
Hex
0
%
fin
)
as
[
y
Hy
].
exists
(
fin_S_inv
_
y
f
)
;
intros
i
;
by
inv_fin
i
.
Qed
.
Lemma
finite_choice
`
{
Finite
A
}
{
B
:
A
→
Type
}
(
P
:
∀
x
,
B
x
→
Prop
)
:
(
∀
x
,
∃
y
,
P
x
y
)
→
∃
f
,
∀
x
,
P
x
(
f
x
).
Proof
.
intros
Hex
.
destruct
(
fin_choice
_
(
λ
i
,
Hex
(
decode_fin
i
)))
as
[
f
?].
exists
(
λ
x
,
eq_rect
_
_
(
f
(
encode_fin
x
))
_
(
decode_encode_fin
x
))
;
intros
x
.
destruct
(
decode_encode_fin
x
)
;
simpl
;
auto
.
Qed
.
Lemma
card_0_inv
P
`
{
finA
:
Finite
A
}
:
card
A
=
0
→
A
→
P
.
Lemma
card_0_inv
P
`
{
finA
:
Finite
A
}
:
card
A
=
0
→
A
→
P
.
Proof
.
Proof
.
intros
?
x
.
destruct
finA
as
[[??]
??]
;
simplify_eq
.
intros
?
x
.
destruct
finA
as
[[??]
??]
;
simplify_eq
.
...
@@ 297,3 +330,18 @@ Proof.
...
@@ 297,3 +330,18 @@ Proof.
induction
(
enum
A
)
as
[
x
xs
IH
]
;
intros
l
;
simpl
;
auto
.
induction
(
enum
A
)
as
[
x
xs
IH
]
;
intros
l
;
simpl
;
auto
.
by
rewrite
app_length
,
fmap_length
,
IH
.
by
rewrite
app_length
,
fmap_length
,
IH
.
Qed
.
Qed
.
Fixpoint
fin_enum
(
n
:
nat
)
:
list
(
fin
n
)
:
=
match
n
with
0
=>
[]

S
n
=>
0
%
fin
::
FS
<$>
fin_enum
n
end
.
Program
Instance
fin_finite
n
:
Finite
(
fin
n
)
:
=
{
enum
:
=
fin_enum
n
}.
Next
Obligation
.
intros
n
.
induction
n
;
simpl
;
constructor
.

rewrite
elem_of_list_fmap
.
by
intros
(?&?&?).

by
apply
(
NoDup_fmap
_
).
Qed
.
Next
Obligation
.
intros
n
i
.
induction
i
as
[
n
i
IH
]
;
simpl
;
rewrite
elem_of_cons
,
?elem_of_list_fmap
;
eauto
.
Qed
.
Lemma
fin_card
n
:
card
(
fin
n
)
=
n
.
Proof
.
unfold
card
;
simpl
.
induction
n
;
simpl
;
rewrite
?fmap_length
;
auto
.
Qed
.
prelude/vector.v
View file @
e15c090e
...
@@ 5,7 +5,7 @@
...
@@ 5,7 +5,7 @@
definitions from the standard library, but renames or changes their notations,
definitions from the standard library, but renames or changes their notations,
so that it becomes more consistent with the naming conventions in this
so that it becomes more consistent with the naming conventions in this
development. *)
development. *)
From
iris
.
prelude
Require
Import
list
finite
.
From
iris
.
prelude
Require
Export
list
.
Open
Scope
vector_scope
.
Open
Scope
vector_scope
.
(** * The fin type *)
(** * The fin type *)
...
@@ 82,21 +82,6 @@ Proof.
...
@@ 82,21 +82,6 @@ Proof.
revert
m
H
.
induction
n
;
intros
[?]
;
simpl
;
auto
;
intros
;
exfalso
;
lia
.
revert
m
H
.
induction
n
;
intros
[?]
;
simpl
;
auto
;
intros
;
exfalso
;
lia
.
Qed
.
Qed
.
Fixpoint
fin_enum
(
n
:
nat
)
:
list
(
fin
n
)
:
=
match
n
with
0
=>
[]

S
n
=>
0
%
fin
::
FS
<$>
fin_enum
n
end
.
Program
Instance
fin_finite
n
:
Finite
(
fin
n
)
:
=
{
enum
:
=
fin_enum
n
}.
Next
Obligation
.
intros
n
.
induction
n
;
simpl
;
constructor
.

rewrite
elem_of_list_fmap
.
by
intros
(?&?&?).

by
apply
(
NoDup_fmap
_
).
Qed
.
Next
Obligation
.
intros
n
i
.
induction
i
as
[
n
i
IH
]
;
simpl
;
rewrite
elem_of_cons
,
?elem_of_list_fmap
;
eauto
.
Qed
.
Lemma
fin_card
n
:
card
(
fin
n
)
=
n
.
Proof
.
unfold
card
;
simpl
.
induction
n
;
simpl
;
rewrite
?fmap_length
;
auto
.
Qed
.
(** * Vectors *)
(** * Vectors *)
(** The type [vec n] represents lists of consisting of exactly [n] elements.
(** The type [vec n] represents lists of consisting of exactly [n] elements.
Whereas the standard library declares exactly the same notations for vectors as
Whereas the standard library declares exactly the same notations for vectors as
...
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