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stdpp
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c73f285d
Commit
c73f285d
authored
Feb 05, 2019
by
Robbert Krebbers
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The symmetric and reflexive/transitive/symmetric closure.
parent
51d69170
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theories/relations.v
theories/relations.v
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theories/relations.v
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c73f285d
...
...
@@ -17,6 +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
...
...
@@ -53,13 +56,16 @@ 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
)).
Hint
Unfold
nf
red
:
core
.
(** * General theorems *)
Section
rtc
.
Section
closure
.
Context
`
{
R
:
relation
A
}.
Hint
Constructors
rtc
nsteps
bsteps
tc
:
core
.
...
...
@@ -78,6 +84,14 @@ 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
.
...
...
@@ -105,6 +119,9 @@ 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
:
...
...
@@ -171,6 +188,36 @@ 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
.
...
...
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