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Rice Wine
Iris
Commits
6092efe9
Commit
6092efe9
authored
Apr 10, 2016
by
Robbert Krebbers
Browse files
Prove iff_equiv in the logic (instead of the model).
parent
7a822edf
Changes
1
Hide whitespace changes
Inline
Side-by-side
algebra/upred.v
View file @
6092efe9
...
...
@@ -509,11 +509,6 @@ Proof.
unseal
=>
Hab
;
apply
equiv_dist
;
intros
n
;
apply
Hab
with
∅
;
last
done
.
apply
cmra_valid_validN
,
cmra_unit_valid
.
Qed
.
Lemma
iff_equiv
P
Q
:
True
⊢
(
P
↔
Q
)
→
P
⊣
⊢
Q
.
Proof
.
rewrite
/
uPred_iff
;
unseal
=>
HPQ
.
split
=>
n
x
?
;
split
;
intros
;
by
apply
HPQ
with
n
x
.
Qed
.
(* Derived logical stuff *)
Lemma
True_intro
P
:
P
⊢
True
.
...
...
@@ -550,6 +545,16 @@ Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
Lemma
entails_impl
P
Q
:
(
P
⊢
Q
)
→
True
⊢
(
P
→
Q
).
Proof
.
auto
using
impl_intro_l
.
Qed
.
Lemma
iff_refl
Q
P
:
Q
⊢
(
P
↔
P
).
Proof
.
rewrite
/
uPred_iff
;
apply
and_intro
;
apply
impl_intro_l
;
auto
.
Qed
.
Lemma
iff_equiv
P
Q
:
True
⊢
(
P
↔
Q
)
→
P
⊣
⊢
Q
.
Proof
.
intros
HPQ
;
apply
(
anti_symm
(
⊢
))
;
apply
impl_entails
;
rewrite
HPQ
/
uPred_iff
;
auto
.
Qed
.
Lemma
equiv_iff
P
Q
:
P
⊣
⊢
Q
→
True
⊢
(
P
↔
Q
).
Proof
.
intros
->
;
apply
iff_refl
.
Qed
.
Lemma
const_mono
φ
1
φ
2
:
(
φ
1
→
φ
2
)
→
■
φ
1
⊢
■
φ
2
.
Proof
.
intros
;
apply
const_elim
with
φ
1
;
eauto
using
const_intro
.
Qed
.
Lemma
and_mono
P
P'
Q
Q'
:
P
⊢
Q
→
P'
⊢
Q'
→
(
P
∧
P'
)
⊢
(
Q
∧
Q'
).
...
...
@@ -633,8 +638,6 @@ Proof.
-
by
rewrite
-(
left_id
True
%
I
uPred_and
(
_
→
_
)%
I
)
impl_elim_r
.
-
by
apply
impl_intro_l
;
rewrite
left_id
.
Qed
.
Lemma
iff_refl
Q
P
:
Q
⊢
(
P
↔
P
).
Proof
.
rewrite
/
uPred_iff
;
apply
and_intro
;
apply
impl_intro_l
;
auto
.
Qed
.
Lemma
or_and_l
P
Q
R
:
(
P
∨
Q
∧
R
)
⊣
⊢
((
P
∨
Q
)
∧
(
P
∨
R
)).
Proof
.
...
...
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