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stdpp
Commits
db6bf449
Commit
db6bf449
authored
Mar 11, 2016
by
Robbert Krebbers
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Choice principle for finite types.
parent
5fa4d3e1
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17
theories/finite.v
theories/finite.v
+49
1
theories/vector.v
theories/vector.v
+1
16
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theories/finite.v
View file @
db6bf449
(* Copyright (c) 20122015, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
From
stdpp
Require
Export
countable
list
.
From
stdpp
Require
Export
countable
vector
.
Class
Finite
A
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}
:
=
{
enum
:
list
A
;
...
...
@@ 61,6 +61,39 @@ Proof.
exists
y
.
by
rewrite
!
Nat2Pos
.
id
by
done
.
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
.
Proof
.
intros
?
x
.
destruct
finA
as
[[??]
??]
;
simplify_eq
.
...
...
@@ 297,3 +330,18 @@ Proof.
induction
(
enum
A
)
as
[
x
xs
IH
]
;
intros
l
;
simpl
;
auto
.
by
rewrite
app_length
,
fmap_length
,
IH
.
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
.
theories/vector.v
View file @
db6bf449
...
...
@@ 5,7 +5,7 @@
definitions from the standard library, but renames or changes their notations,
so that it becomes more consistent with the naming conventions in this
development. *)
From
stdpp
Require
Import
list
finite
.
From
stdpp
Require
Export
list
.
Open
Scope
vector_scope
.
(** * The fin type *)
...
...
@@ 82,21 +82,6 @@ Proof.
revert
m
H
.
induction
n
;
intros
[?]
;
simpl
;
auto
;
intros
;
exfalso
;
lia
.
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 *)
(** The type [vec n] represents lists of consisting of exactly [n] elements.
Whereas the standard library declares exactly the same notations for vectors as
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