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Thibaut Pérami
stdpp
Commits
a62a60ae
Commit
a62a60ae
authored
7 years ago
by
Johannes Kloos
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Compile fixes and addition of wf_guard.
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theories/infinite.v
+28
-26
28 additions, 26 deletions
theories/infinite.v
with
28 additions
and
26 deletions
theories/infinite.v
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26
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a62a60ae
(* Copyright (c) 2012-2017, Coq-std++ developers. *)
(* This file is distributed under the terms of the BSD license. *)
From
stdpp
Require
Import
pretty
fin_collections
.
From
stdpp
Require
Import
pretty
fin_collections
relations
prelude
.
(** The class [Infinite] axiomatizes types with infinitely many elements
by giving an injection from the natural numbers into the type. It is mostly
...
...
@@ -25,7 +25,7 @@ Qed.
(** * Fresh elements *)
Section
Fresh
.
Context
`{
Infinite
A
,
∀
(
x
:
A
)
(
s
:
C
),
Decision
(
x
∈
s
)}
.
Context
`{
FinCollection
A
C
}
`{
Infinite
A
,
!
RelDecision
(
@
elem_of
A
C
_
)}
.
Lemma
subset_difference_in
{
x
:
A
}
{
s
:
C
}
(
inx
:
x
∈
s
):
s
∖
{[
x
]}
⊂
s
.
Proof
.
set_solver
.
Qed
.
...
...
@@ -47,36 +47,35 @@ Section Fresh.
apply
relfg
.
Qed
.
Definition
fresh_generic_fix
:=
Fix
collection_wf
(
const
(
nat
→
A
))
fresh_generic_body
.
Definition
fresh_generic_fix
u
:=
Fix
(
wf_guard
u
collection_wf
)
(
const
(
nat
→
A
))
fresh_generic_body
.
Lemma
fresh_generic_fixpoint_unfold
s
n
:
fresh_generic_fix
s
n
=
fresh_generic_body
s
(
λ
s'
_
n
,
fresh_generic_fix
s'
n
)
n
.
Lemma
fresh_generic_fixpoint_unfold
u
s
n
:
fresh_generic_fix
u
s
n
=
fresh_generic_body
s
(
λ
s'
_
n
,
fresh_generic_fix
u
s'
n
)
n
.
Proof
.
apply
(
Fix_unfold_rel
collection_wf
(
const
(
nat
→
A
))
(
const
(
pointwise_relation
nat
(
=
)))
apply
(
Fix_unfold_rel
(
wf_guard
u
collection_wf
)
(
const
(
nat
→
A
))
(
const
(
pointwise_relation
nat
(
=
)))
fresh_generic_body
fresh_generic_body_proper
s
n
)
.
Qed
.
Lemma
fresh_generic_fixpoint_spec
s
n
:
∃
m
,
n
≤
m
∧
fresh_generic_fix
s
n
=
inject
m
∧
inject
m
∉
s
∧
∀
i
,
n
≤
i
<
m
→
inject
i
∈
s
.
Lemma
fresh_generic_fixpoint_spec
u
s
n
:
∃
m
,
n
≤
m
∧
fresh_generic_fix
u
s
n
=
inject
m
∧
inject
m
∉
s
∧
∀
i
,
n
≤
i
<
m
→
inject
i
∈
s
.
Proof
.
revert
n
.
induction
s
as
[
s
IH
]
using
(
well_founded_ind
collection_wf
);
intro
.
setoid_rewrite
fresh_generic_fixpoint_unfold
;
unfold
fresh_generic_body
.
destruct
decide
as
[
case
|
case
]
.
-
destruct
(
IH
_
(
subset_difference_in
case
)
(
S
n
))
as
[
m
[
mbound
[
eqfix
[
notin
inbelow
]]]]
.
exists
m
;
repeat
split
;
auto
.
+
omega
.
+
rewrite
not_elem_of_difference
,
elem_of_singleton
in
notin
.
destruct
notin
as
[?|?
%
inject_injective
];
auto
;
omega
.
+
intros
i
ibound
.
destruct
(
decide
(
i
=
n
))
as
[
<-|
neq
];
auto
.
enough
(
inject
i
∈
s
∖
{[
inject
n
]})
by
set_solver
.
apply
inbelow
;
omega
.
-
exists
n
;
repeat
split
;
auto
.
intros
;
omega
.
destruct
decide
as
[
case
|
case
];
eauto
with
omega
.
destruct
(
IH
_
(
subset_difference_in
case
)
(
S
n
))
as
[
m
[
mbound
[
eqfix
[
notin
inbelow
]]]]
.
exists
m
;
repeat
split
;
auto
with
omega
.
-
rewrite
not_elem_of_difference
,
elem_of_singleton
in
notin
.
destruct
notin
as
[?|?
%
inject_injective
];
auto
with
omega
.
-
intros
i
ibound
.
destruct
(
decide
(
i
=
n
))
as
[
<-|
neq
];
auto
.
enough
(
inject
i
∈
s
∖
{[
inject
n
]})
by
set_solver
.
apply
inbelow
;
omega
.
Qed
.
Instance
fresh_generic
:
Fresh
A
C
|
20
:=
λ
s
,
fresh_generic_fix
s
0
.
Instance
fresh_generic
:
Fresh
A
C
|
20
:=
λ
s
,
fresh_generic_fix
(
1
+
Nat
.
log2
(
size
s
))
s
0
.
Instance
fresh_generic_spec
:
FreshSpec
A
C
.
Proof
.
...
...
@@ -84,20 +83,23 @@ Section Fresh.
-
apply
_
.
-
intros
*
eqXY
.
unfold
fresh
,
fresh_generic
.
destruct
(
fresh_generic_fixpoint_spec
X
0
)
as
[
mX
[_
[
->
[
notinX
belowinX
]]]]
.
destruct
(
fresh_generic_fixpoint_spec
Y
0
)
as
[
mY
[_
[
->
[
notinY
belowinY
]]]]
.
destruct
(
fresh_generic_fixpoint_spec
(
1
+
Nat
.
log2
(
size
X
))
X
0
)
as
[
mX
[_
[
->
[
notinX
belowinX
]]]]
.
destruct
(
fresh_generic_fixpoint_spec
(
1
+
Nat
.
log2
(
size
Y
))
Y
0
)
as
[
mY
[_
[
->
[
notinY
belowinY
]]]]
.
destruct
(
Nat
.
lt_trichotomy
mX
mY
)
as
[
case
|[
->
|
case
]];
auto
.
+
contradict
notinX
;
rewrite
eqXY
;
apply
belowinY
;
omega
.
+
contradict
notinY
;
rewrite
<-
eqXY
;
apply
belowinX
;
omega
.
-
intro
.
unfold
fresh
,
fresh_generic
.
destruct
(
fresh_generic_fixpoint_spec
X
0
)
as
[
m
[_
[
->
[
notinX
belowinX
]]]];
auto
.
destruct
(
fresh_generic_fixpoint_spec
(
1
+
Nat
.
log2
(
size
X
))
X
0
)
as
[
m
[_
[
->
[
notinX
belowinX
]]]];
auto
.
Qed
.
End
Fresh
.
(** Derive fresh instances. *)
Section
StringFresh
.
Context
`{
FinCollection
string
C
,
∀
(
x
:
string
)
(
s
:
C
),
Decision
(
x
∈
s
)
}
.
Context
`{
FinCollection
string
C
,
!
Rel
Decision
elem_of
}
.
Global
Instance
string_fresh
:
Fresh
string
C
:=
fresh_generic
.
Global
Instance
string_fresh_spec
:
FreshSpec
string
C
:=
_
.
Global
Instance
string_fresh_spec
:
FreshSpec
string
C
:=
fresh_generic_spec
.
End
StringFresh
.
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