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18841bdb
Commit
18841bdb
authored
Jan 20, 2016
by
Robbert Krebbers
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Plain Diff
Some more Forall3 lemmas.
And use more uniform variable names.
parent
aea3b304
Changes
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prelude/list.v
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18841bdb
...
...
@@ -1372,6 +1372,11 @@ Proof.
induction
l
as
[|
x
l
IH
];
[
done
|].
by
rewrite
reverse_cons
,
(
commutative
(++)),
IH
.
Qed
.
Lemma
delete_Permutation
l
i
x
:
l
!!
i
=
Some
x
→
l
≡ₚ
x
::
delete
i
l
.
Proof
.
revert
i
;
induction
l
as
[|
y
l
IH
];
intros
[|
i
]
?;
simplify_equality'
;
auto
.
by
rewrite
Permutation_swap
,
<-(
IH
i
).
Qed
.
(** ** Properties of the [prefix_of] and [suffix_of] predicates *)
Global
Instance
:
PreOrder
(@
prefix_of
A
).
...
...
@@ -2522,59 +2527,85 @@ End Forall2_order.
Section
Forall3
.
Context
{
A
B
C
}
(
P
:
A
→
B
→
C
→
Prop
).
Hint
Extern
0
(
Forall3
_
_
_
_)
=>
constructor
.
Lemma
Forall3_app
l1
k1
k1'
l2
k2
k2'
:
Lemma
Forall3_app
l1
l2
k1
k2
k1'
k2'
:
Forall3
P
l1
k1
k1'
→
Forall3
P
l2
k2
k2'
→
Forall3
P
(
l1
++
l2
)
(
k1
++
k2
)
(
k1'
++
k2'
).
Proof
.
induction
1
;
simpl
;
auto
.
Qed
.
Lemma
Forall3_cons_inv_m
l
y
l2'
k
:
Forall3
P
l
(
y
::
l2'
)
k
→
∃
x
l2
z
k2
,
l
=
x
::
l2
∧
k
=
z
::
k2
∧
P
x
y
z
∧
Forall3
P
l2
l2'
k2
.
Lemma
Forall3_cons_inv_l
x
l
k
k'
:
Forall3
P
(
x
::
l
)
k
k'
→
∃
y
k2
z
k2'
,
k
=
y
::
k2
∧
k'
=
z
::
k2'
∧
P
x
y
z
∧
Forall3
P
l
k2
k2'
.
Proof
.
inversion_clear
1
;
naive_solver
.
Qed
.
Lemma
Forall3_app_inv_l
l1
l2
k
k'
:
Forall3
P
(
l1
++
l2
)
k
k'
→
∃
k1
k2
k1'
k2'
,
k
=
k1
++
k2
∧
k'
=
k1'
++
k2'
∧
Forall3
P
l1
k1
k1'
∧
Forall3
P
l2
k2
k2'
.
Proof
.
revert
k
k'
.
induction
l1
as
[|
x
l1
IH
];
simpl
;
inversion_clear
1
.
*
by
repeat
eexists
;
eauto
.
*
by
repeat
eexists
;
eauto
.
*
edestruct
IH
as
(?&?&?&?&?&?&?&?);
eauto
;
naive_solver
.
Qed
.
Lemma
Forall3_cons_inv_m
l
y
k
k'
:
Forall3
P
l
(
y
::
k
)
k'
→
∃
x
l2
z
k2'
,
l
=
x
::
l2
∧
k'
=
z
::
k2'
∧
P
x
y
z
∧
Forall3
P
l2
k
k2'
.
Proof
.
inversion_clear
1
;
naive_solver
.
Qed
.
Lemma
Forall3_app_inv_m
l
k1
k2
k'
:
Forall3
P
l
(
k1
++
k2
)
k'
→
∃
l1
l2
k1'
k2'
,
l
=
l1
++
l2
∧
k'
=
k1'
++
k2'
∧
Forall3
P
l1
k1
k1'
∧
Forall3
P
l2
k2
k2'
.
Proof
.
revert
l
k'
.
induction
k1
as
[|
x
k1
IH
];
simpl
;
inversion_clear
1
.
*
by
repeat
eexists
;
eauto
.
*
by
repeat
eexists
;
eauto
.
*
edestruct
IH
as
(?&?&?&?&?&?&?&?);
eauto
;
naive_solver
.
Qed
.
Lemma
Forall3_cons_inv_r
l
k
z
k'
:
Forall3
P
l
k
(
z
::
k'
)
→
∃
x
l2
y
k2
,
l
=
x
::
l2
∧
k
=
y
::
k2
∧
P
x
y
z
∧
Forall3
P
l2
k2
k'
.
Proof
.
inversion_clear
1
;
naive_solver
.
Qed
.
Lemma
Forall3_app_inv_
m
l
l1'
l2'
k
:
Forall3
P
l
(
l1'
++
l2'
)
k
→
∃
l1
l2
k1
k2
,
l
=
l1
++
l2
∧
k
=
k1
++
k2
∧
Forall3
P
l1
l1'
k1
∧
Forall3
P
l2
l2'
k2
.
Lemma
Forall3_app_inv_
r
l
k
k1'
k2'
:
Forall3
P
l
k
(
k1'
++
k2'
)
→
∃
l1
l2
k1
k2
,
l
=
l1
++
l2
∧
k
=
k1
++
k2
∧
Forall3
P
l1
k1
k1'
∧
Forall3
P
l2
k2
k2'
.
Proof
.
revert
l
k
.
induction
l1'
as
[|
x
l
1'
IH
];
simpl
;
inversion_clear
1
.
revert
l
k
.
induction
k1'
as
[|
x
k
1'
IH
];
simpl
;
inversion_clear
1
.
*
by
repeat
eexists
;
eauto
.
*
by
repeat
eexists
;
eauto
.
*
edestruct
IH
as
(?&?&?&?&?&?&?&?);
eauto
;
naive_solver
.
Qed
.
Lemma
Forall3_impl
(
Q
:
A
→
B
→
C
→
Prop
)
l
l'
k
:
Forall3
P
l
l'
k
→
(
∀
x
y
z
,
P
x
y
z
→
Q
x
y
z
)
→
Forall3
Q
l
l'
k
.
Proof
.
intros
Hl
?
.
induction
Hl
;
auto
.
Defined
.
Lemma
Forall3_length_lm
l
l'
k
:
Forall3
P
l
l'
k
→
length
l
=
length
l'
.
Lemma
Forall3_impl
(
Q
:
A
→
B
→
C
→
Prop
)
l
k
k'
:
Forall3
P
l
k
k'
→
(
∀
x
y
z
,
P
x
y
z
→
Q
x
y
z
)
→
Forall3
Q
l
k
k'
.
Proof
.
intros
Hl
?
;
induction
Hl
;
auto
.
Defined
.
Lemma
Forall3_length_lm
l
k
k'
:
Forall3
P
l
k
k'
→
length
l
=
length
k
.
Proof
.
by
induction
1
;
f_equal'
.
Qed
.
Lemma
Forall3_length_lr
l
l'
k
:
Forall3
P
l
l'
k
→
length
l
=
length
k
.
Lemma
Forall3_length_lr
l
k
k'
:
Forall3
P
l
k
k'
→
length
l
=
length
k'
.
Proof
.
by
induction
1
;
f_equal'
.
Qed
.
Lemma
Forall3_lookup_lmr
l
l'
k
i
x
y
z
:
Forall3
P
l
l'
k
→
l
!!
i
=
Some
x
→
l'
!!
i
=
Some
y
→
k
!!
i
=
Some
z
→
P
x
y
z
.
Lemma
Forall3_lookup_lmr
l
k
k'
i
x
y
z
:
Forall3
P
l
k
k'
→
l
!!
i
=
Some
x
→
k
!!
i
=
Some
y
→
k'
!!
i
=
Some
z
→
P
x
y
z
.
Proof
.
intros
H
.
revert
i
.
induction
H
;
intros
[|?]
???;
simplify_equality'
;
eauto
.
Qed
.
Lemma
Forall3_lookup_l
l
l'
k
i
x
:
Forall3
P
l
l'
k
→
l
!!
i
=
Some
x
→
∃
y
z
,
l'
!!
i
=
Some
y
∧
k
!!
i
=
Some
z
∧
P
x
y
z
.
Lemma
Forall3_lookup_l
l
k
k'
i
x
:
Forall3
P
l
k
k'
→
l
!!
i
=
Some
x
→
∃
y
z
,
k
!!
i
=
Some
y
∧
k'
!!
i
=
Some
z
∧
P
x
y
z
.
Proof
.
intros
H
.
revert
i
.
induction
H
;
intros
[|?]
?;
simplify_equality'
;
eauto
.
Qed
.
Lemma
Forall3_lookup_m
l
l'
k
i
y
:
Forall3
P
l
l'
k
→
l'
!!
i
=
Some
y
→
∃
x
z
,
l
!!
i
=
Some
x
∧
k
!!
i
=
Some
z
∧
P
x
y
z
.
Lemma
Forall3_lookup_m
l
k
k'
i
y
:
Forall3
P
l
k
k'
→
k
!!
i
=
Some
y
→
∃
x
z
,
l
!!
i
=
Some
x
∧
k
'
!!
i
=
Some
z
∧
P
x
y
z
.
Proof
.
intros
H
.
revert
i
.
induction
H
;
intros
[|?]
?;
simplify_equality'
;
eauto
.
Qed
.
Lemma
Forall3_lookup_r
l
l'
k
i
z
:
Forall3
P
l
l'
k
→
k
!!
i
=
Some
z
→
∃
x
y
,
l
!!
i
=
Some
x
∧
l'
!!
i
=
Some
y
∧
P
x
y
z
.
Lemma
Forall3_lookup_r
l
k
k'
i
z
:
Forall3
P
l
k
k'
→
k'
!!
i
=
Some
z
→
∃
x
y
,
l
!!
i
=
Some
x
∧
k
!!
i
=
Some
y
∧
P
x
y
z
.
Proof
.
intros
H
.
revert
i
.
induction
H
;
intros
[|?]
?;
simplify_equality'
;
eauto
.
Qed
.
Lemma
Forall3_alter_lm
f
g
l
l'
k
i
:
Forall3
P
l
l'
k
→
(
∀
x
y
z
,
l
!!
i
=
Some
x
→
l'
!!
i
=
Some
y
→
k
!!
i
=
Some
z
→
Lemma
Forall3_alter_lm
f
g
l
k
k'
i
:
Forall3
P
l
k
k'
→
(
∀
x
y
z
,
l
!!
i
=
Some
x
→
k
!!
i
=
Some
y
→
k'
!!
i
=
Some
z
→
P
x
y
z
→
P
(
f
x
)
(
g
y
)
z
)
→
Forall3
P
(
alter
f
i
l
)
(
alter
g
i
l'
)
k
.
Forall3
P
(
alter
f
i
l
)
(
alter
g
i
k
)
k'
.
Proof
.
intros
Hl
.
revert
i
.
induction
Hl
;
intros
[|];
auto
.
Qed
.
End
Forall3
.
...
...
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