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stdpp
Commits
f6e466d1
Commit
f6e466d1
authored
Jul 21, 2020
by
Robbert Krebbers
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Add lemma `tc_finite_sn`.
parent
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#31760
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theories/relations.v
theories/relations.v
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theories/relations.v
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f6e466d1
...
...
@@ -2,7 +2,7 @@
These are particularly useful as we define the operational semantics as a
small step semantics. *)
From
Coq
Require
Import
Wf_nat
.
From
stdpp
Require
Export
tactics
base
.
From
stdpp
Require
Export
sets
.
Set
Default
Proof
Using
"Type"
.
(** * Definitions *)
...
...
@@ -280,6 +280,22 @@ Section properties.
Lemma
wn_step_rtc
x
y
:
wn
R
y
→
rtc
R
x
y
→
wn
R
x
.
Proof
.
induction
2
;
eauto
using
wn_step
.
Qed
.
(** An acyclic relation that can only take finitely many steps is strongly
normalizing *)
Lemma
tc_finite_sn
x
:
Irreflexive
(
tc
R
)
→
pred_finite
(
tc
R
x
)
→
sn
R
x
.
Proof
.
intros
Hirr
[
xs
Hfin
].
remember
(
length
xs
)
as
n
eqn
:
Hn
.
revert
x
xs
Hn
Hfin
.
induction
(
lt_wf
n
)
as
[
n
_
IH
]
;
intros
x
xs
->
Hfin
.
constructor
;
simpl
;
intros
x'
Hxx'
.
assert
(
x'
∈
xs
)
as
(
xs1
&
xs2
&->)%
elem_of_list_split
by
eauto
using
tc_once
.
refine
(
IH
(
length
xs1
+
length
xs2
)
_
_
(
xs1
++
xs2
)
_
_
)
;
[
rewrite
app_length
;
simpl
;
lia
..|].
intros
x''
Hx'x''
.
feed
pose
proof
(
Hfin
x''
)
as
Hx''
;
[
by
econstructor
|].
cut
(
x'
≠
x''
)
;
[
set_solver
|].
intros
->.
by
apply
(
Hirr
x''
).
Qed
.
(** The following theorem requires that [red R] is decidable. The intuition
for this requirement is that [wn R] is a very "positive" statement as it
points out a particular trace. In contrast, [sn R] just says "this also holds
...
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