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Iris
stdpp
Commits
98e61928
Commit
98e61928
authored
Sep 04, 2014
by
Robbert Krebbers
Browse files
Strengthen induction lemma for reflexive transitive closure.
parent
6907a08a
Changes
1
Hide whitespace changes
Inline
Side-by-side
theories/ars.v
View file @
98e61928
...
...
@@ -61,19 +61,22 @@ Section rtc.
Lemma
rtc_inv
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
R
x
y
∧
rtc
R
y
z
.
Proof
.
inversion_clear
1
;
eauto
.
Qed
.
Lemma
rtc_ind_r
(
P
:
A
→
A
→
Prop
)
(
Prefl
:
∀
x
,
P
x
x
)
(
Pstep
:
∀
x
y
z
,
rtc
R
x
y
→
R
y
z
→
P
x
y
→
P
x
z
)
:
Lemma
rtc_ind_r_weak
(
P
:
A
→
A
→
Prop
)
(
Prefl
:
∀
x
,
P
x
x
)
(
Pstep
:
∀
x
y
z
,
rtc
R
x
y
→
R
y
z
→
P
x
y
→
P
x
z
)
:
∀
x
z
,
rtc
R
x
z
→
P
x
z
.
Proof
.
cut
(
∀
y
z
,
rtc
R
y
z
→
∀
x
,
rtc
R
x
y
→
P
x
y
→
P
x
z
).
{
eauto
using
rtc_refl
.
}
induction
1
;
eauto
using
rtc_r
.
Qed
.
Lemma
rtc_ind_r
(
P
:
A
→
Prop
)
(
x
:
A
)
(
Prefl
:
P
x
)
(
Pstep
:
∀
y
z
,
rtc
R
x
y
→
R
y
z
→
P
y
→
P
z
)
:
∀
z
,
rtc
R
x
z
→
P
z
.
Proof
.
intros
z
p
.
revert
x
z
p
Prefl
Pstep
.
refine
(
rtc_ind_r_weak
_
_
_
)
;
eauto
.
Qed
.
Lemma
rtc_inv_r
x
z
:
rtc
R
x
z
→
x
=
z
∨
∃
y
,
rtc
R
x
y
∧
R
y
z
.
Proof
.
revert
x
z
.
apply
rtc_ind_r
;
eauto
.
Qed
.
Proof
.
revert
z
.
apply
rtc_ind_r
;
eauto
.
Qed
.
Lemma
nsteps_once
x
y
:
R
x
y
→
nsteps
R
1
x
y
.
Proof
.
eauto
with
ars
.
Qed
.
...
...
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