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stdpp
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89454051
Commit
89454051
authored
Feb 05, 2019
by
Robbert Krebbers
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The different notions of confluence and properties about them.
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theories/relations.v
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theories/relations.v
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89454051
...
...
@@ 62,6 +62,19 @@ Definition rtsc {A} (R : relation A) := rtc (sc R).
(** Strongly normalizing elements. *)
Notation
sn
R
:
=
(
Acc
(
flip
R
)).
(** The various kinds of "confluence" properties. Any relation that has the
diamond property is confluent, and any confluent relation is locally confluent.
The naming convention are taken from "Term Rewriting and All That" by Baader and
Nipkow. *)
Definition
diamond
{
A
}
(
R
:
relation
A
)
:
=
∀
x
y1
y2
,
R
x
y1
→
R
x
y2
→
∃
z
,
R
y1
z
∧
R
y2
z
.
Definition
confluent
{
A
}
(
R
:
relation
A
)
:
=
diamond
(
rtc
R
).
Definition
locally_confluent
{
A
}
(
R
:
relation
A
)
:
=
∀
x
y1
y2
,
R
x
y1
→
R
x
y2
→
∃
z
,
rtc
R
y1
z
∧
rtc
R
y2
z
.
Hint
Unfold
nf
red
:
core
.
(** * General theorems *)
...
...
@@ 234,7 +247,60 @@ Section properties.
intros
H
.
cut
(
∀
z
,
rtc
R
x
z
→
all_loop
R
z
)
;
[
eauto
].
cofix
FIX
.
constructor
;
eauto
using
rtc_r
.
Qed
.
End
rtc
.
(** An alternative definition of confluence; also known as the ChurchRosser
property. *)
Lemma
confluent_alt
:
confluent
R
↔
(
∀
x
y
,
rtsc
R
x
y
→
∃
z
,
rtc
R
x
z
∧
rtc
R
y
z
).
Proof
.
split
.

intros
Hcr
.
induction
1
as
[
x

x
y1
y1'
[
Hy1

Hy1
]
Hy1'
(
z
&
IH1
&
IH2
)]
;
eauto
.
destruct
(
Hcr
y1
x
z
)
as
(
z'
&?&?)
;
eauto
using
rtc_transitive
.

intros
Hcr
x
y1
y2
Hy1
Hy2
.
apply
Hcr
;
trans
x
;
eauto
using
rtc_rtsc_rl
,
rtc_rtsc_lr
.
Qed
.
Lemma
confluent_nf_r
x
y
:
confluent
R
→
rtsc
R
x
y
→
nf
R
y
→
rtc
R
x
y
.
Proof
.
rewrite
confluent_alt
.
intros
Hcr
??.
destruct
(
Hcr
x
y
)
as
(
z
&
Hx
&
Hy
)
;
auto
.
by
apply
rtc_nf
in
Hy
as
>.
Qed
.
Lemma
confluent_nf_l
x
y
:
confluent
R
→
rtsc
R
x
y
→
nf
R
x
→
rtc
R
y
x
.
Proof
.
intros
.
by
apply
(
confluent_nf_r
y
x
).
Qed
.
Lemma
diamond_confluent
:
diamond
R
→
confluent
R
.
Proof
.
intros
Hdiam
.
assert
(
∀
x
y1
y2
,
rtc
R
x
y1
→
R
x
y2
→
∃
z
,
rtc
R
y1
z
∧
rtc
R
y2
z
)
as
Hstrip
.
{
intros
x
y1
y2
Hy1
;
revert
y2
.
induction
Hy1
as
[
x

x
y1
y1'
Hy1
Hy1'
IH
]
;
[
by
eauto
]
;
intros
y2
Hy2
.
destruct
(
Hdiam
x
y1
y2
)
as
(
z
&
Hy1z
&
Hy2z
)
;
auto
.
destruct
(
IH
z
)
as
(
z'
&?&?)
;
eauto
.
}
intros
x
y1
y2
Hy1
;
revert
y2
.
induction
Hy1
as
[
x

x
y1
y1'
Hy1
Hy1'
IH
]
;
[
by
eauto
]
;
intros
y2
Hy2
.
destruct
(
Hstrip
x
y2
y1
)
as
(
z
&?&?)
;
eauto
.
destruct
(
IH
z
)
as
(
z'
&?&?)
;
eauto
using
rtc_transitive
.
Qed
.
Lemma
confluent_locally_confluent
:
confluent
R
→
locally_confluent
R
.
Proof
.
unfold
confluent
,
locally_confluent
;
eauto
.
Qed
.
(** The following is also known as Newman's lemma *)
Lemma
locally_confluent_confluent
:
(
∀
x
,
sn
R
x
)
→
locally_confluent
R
→
confluent
R
.
Proof
.
intros
Hsn
Hcr
x
.
induction
(
Hsn
x
)
as
[
x
_
IH
].
intros
y1
y2
Hy1
Hy2
.
destruct
Hy1
as
[
x

x
y1
y1'
Hy1
Hy1'
]
;
[
by
eauto
].
destruct
Hy2
as
[
x

x
y2
y2'
Hy2
Hy2'
]
;
[
by
eauto
].
destruct
(
Hcr
x
y1
y2
)
as
(
z
&
Hy1z
&
Hy2z
)
;
auto
.
destruct
(
IH
_
Hy1
y1'
z
)
as
(
z1
&?&?)
;
auto
.
destruct
(
IH
_
Hy2
y2'
z1
)
as
(
z2
&?&?)
;
eauto
using
rtc_transitive
.
Qed
.
End
properties
.
(** * Theorems on sub relations *)
Section
subrel
.
...
...
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