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Rodolphe Lepigre
stdpp
Commits
87b2fe62
Commit
87b2fe62
authored
Feb 10, 2019
by
Robbert Krebbers
Browse files
Merge branch 'robbert/relations' into 'master'
Confluent relations See merge request
iris/stdpp!53
parents
824e9723
89454051
Changes
1
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theories/relations.v
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87b2fe62
...
...
@@ 2,8 +2,7 @@
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on abstract rewriting systems.
These are particularly useful as we define the operational semantics as a
small step semantics. This file defines a hint database [ars] containing
some theorems on abstract rewriting systems. *)
small step semantics. *)
From
Coq
Require
Import
Wf_nat
.
From
stdpp
Require
Export
tactics
base
.
Set
Default
Proof
Using
"Type"
.
...
...
@@ 18,6 +17,9 @@ Section definitions.
(** An element is in normal form if no further steps are possible. *)
Definition
nf
(
x
:
A
)
:
=
¬
red
x
.
(** The symmetric closure. *)
Definition
sc
:
relation
A
:
=
λ
x
y
,
R
x
y
∨
R
y
x
.
(** The reflexive transitive closure. *)
Inductive
rtc
:
relation
A
:
=

rtc_refl
x
:
rtc
x
x
...
...
@@ 54,13 +56,29 @@ Section definitions.

ex_loop_do_step
x
y
:
R
x
y
→
ex_loop
y
→
ex_loop
x
.
End
definitions
.
(* Strongly normalizing elements *)
(** The reflexive transitive symmetric closure. *)
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 *)
Section
rtc
.
Section
closure
.
Context
`
{
R
:
relation
A
}.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
core
.
...
...
@@ 79,6 +97,14 @@ Section rtc.
Global
Instance
rtc_po
:
PreOrder
(
rtc
R
)

10
.
Proof
.
split
.
exact
(@
rtc_refl
A
R
).
exact
rtc_transitive
.
Qed
.
(* Not an instance, related to the issue described above, this sometimes makes
[setoid_rewrite] queries loop. *)
Lemma
rtc_equivalence
:
Symmetric
R
→
Equivalence
(
rtc
R
).
Proof
.
split
;
try
apply
_
.
intros
x
y
.
induction
1
as
[
x1
x2
x3
]
;
[
done

trans
x2
;
eauto
].
Qed
.
Lemma
rtc_once
x
y
:
R
x
y
→
rtc
R
x
y
.
Proof
.
eauto
.
Qed
.
Lemma
rtc_r
x
y
z
:
rtc
R
x
y
→
R
y
z
→
rtc
R
x
z
.
...
...
@@ 106,6 +132,9 @@ Section rtc.
Lemma
rtc_inv_r
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
rtc
R
x
y
∧
R
y
z
.
Proof
.
revert
z
.
apply
rtc_ind_r
;
eauto
.
Qed
.
Lemma
rtc_nf
x
y
:
rtc
R
x
y
→
nf
R
x
→
x
=
y
.
Proof
.
destruct
1
as
[
x

x
y1
y2
].
done
.
intros
[]
;
eauto
.
Qed
.
Lemma
nsteps_once
x
y
:
R
x
y
→
nsteps
R
1
x
y
.
Proof
.
eauto
.
Qed
.
Lemma
nsteps_trans
n
m
x
y
z
:
...
...
@@ 172,6 +201,36 @@ Section rtc.
Lemma
tc_rtc
x
y
:
tc
R
x
y
→
rtc
R
x
y
.
Proof
.
induction
1
;
eauto
.
Qed
.
Global
Instance
sc_symmetric
:
Symmetric
(
sc
R
).
Proof
.
unfold
Symmetric
,
sc
.
naive_solver
.
Qed
.
Lemma
sc_lr
x
y
:
R
x
y
→
sc
R
x
y
.
Proof
.
by
left
.
Qed
.
Lemma
sc_rl
x
y
:
R
y
x
→
sc
R
x
y
.
Proof
.
by
right
.
Qed
.
End
closure
.
Section
more_closure
.
Context
`
{
R
:
relation
A
}.
Global
Instance
rtsc_equivalence
:
Equivalence
(
rtsc
R
)

10
.
Proof
.
apply
rtc_equivalence
,
_
.
Qed
.
Lemma
rtsc_lr
x
y
:
R
x
y
→
rtsc
R
x
y
.
Proof
.
unfold
rtsc
.
eauto
using
sc_lr
,
rtc_once
.
Qed
.
Lemma
rtsc_rl
x
y
:
R
y
x
→
rtsc
R
x
y
.
Proof
.
unfold
rtsc
.
eauto
using
sc_rl
,
rtc_once
.
Qed
.
Lemma
rtc_rtsc_rl
x
y
:
rtc
R
x
y
→
rtsc
R
x
y
.
Proof
.
induction
1
;
econstructor
;
eauto
using
sc_lr
.
Qed
.
Lemma
rtc_rtsc_lr
x
y
:
rtc
R
y
x
→
rtsc
R
x
y
.
Proof
.
intros
.
symmetry
.
eauto
using
rtc_rtsc_rl
.
Qed
.
End
more_closure
.
Section
properties
.
Context
`
{
R
:
relation
A
}.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
core
.
Lemma
acc_not_ex_loop
x
:
Acc
(
flip
R
)
x
→
¬
ex_loop
R
x
.
Proof
.
unfold
not
.
induction
1
;
destruct
1
;
eauto
.
Qed
.
...
...
@@ 188,11 +247,60 @@ Section rtc.
intros
H
.
cut
(
∀
z
,
rtc
R
x
z
→
all_loop
R
z
)
;
[
eauto
].
cofix
FIX
.
constructor
;
eauto
using
rtc_r
.
Qed
.
End
rtc
.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
ars
.
Hint
Resolve
rtc_once
rtc_r
tc_r
rtc_transitive
tc_rtc_l
tc_rtc_r
tc_rtc
bsteps_once
bsteps_r
bsteps_refl
bsteps_trans
:
ars
.
(** 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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