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Simon Spies
stdpp
Commits
18e47df6
Commit
18e47df6
authored
Sep 06, 2014
by
Robbert Krebbers
Browse files
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Lemmas on Forall3 and tweak list tactics.
parent
93de71b2
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theories/list.v
theories/list.v
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theories/list.v
View file @
18e47df6
...
...
@@ 306,7 +306,7 @@ a list equality involving [(::)] and [(++)] of two lists that have a different
length as one of its hypotheses. *)
Tactic
Notation
"discriminate_list_equality"
hyp
(
H
)
:
=
apply
(
f_equal
length
)
in
H
;
repeat
(
simpl
in
H

rewrite
app_length
in
H
)
;
exfalso
;
lia
.
repeat
(
c
simpl
in
H

rewrite
app_length
in
H
)
;
exfalso
;
lia
.
Tactic
Notation
"discriminate_list_equality"
:
=
match
goal
with

H
:
@
eq
(
list
_
)
_
_

_
=>
discriminate_list_equality
H
...
...
@@ 324,19 +324,16 @@ Proof.
intros
?
Hl
.
apply
app_injective_1
;
auto
.
apply
(
f_equal
length
)
in
Hl
.
rewrite
!
app_length
in
Hl
.
lia
.
Qed
.
Ltac
simplify_list_equality
:
=
Ltac
simpl
e_simpl
ify_list_equality
:
=
repeat
match
goal
with

_
=>
progress
simplify_equality

_
=>
progress
simplify_equality
'

H
:
_
++
_
=
_
++
_

_
=>
first
[
apply
app_inv_head
in
H

apply
app_inv_tail
in
H

apply
app_injective_1
in
H
;
[
destruct
H

done
]

apply
app_injective_2
in
H
;
[
destruct
H

done
]
]

H
:
[
?x
]
!!
?i
=
Some
?y

_
=>
destruct
i
;
[
change
(
Some
x
=
Some
y
)
in
H

discriminate
]
end
;
try
discriminate_list_equality
.
Ltac
simplify_list_equality'
:
=
repeat
(
progress
simpl
in
*

simplify_list_equality
).
end
.
(** * General theorems *)
Section
general_properties
.
...
...
@@ 1358,7 +1355,7 @@ Proof. by rewrite prefix_suffix_reverse, !reverse_involutive. Qed.
Lemma
suffix_of_nil
l
:
[]
`
suffix_of
`
l
.
Proof
.
exists
l
.
by
rewrite
(
right_id_L
[]
(++)).
Qed
.
Lemma
suffix_of_nil_inv
l
:
l
`
suffix_of
`
[]
→
l
=
[].
Proof
.
by
intros
[[?]
?]
;
simplify_list_equality
.
Qed
.
Proof
.
by
intros
[[?]
?]
;
simpl
e_simpl
ify_list_equality
.
Qed
.
Lemma
suffix_of_cons_nil_inv
x
l
:
¬
x
::
l
`
suffix_of
`
[].
Proof
.
by
intros
[[]
?].
Qed
.
Lemma
suffix_of_snoc
l1
l2
x
:
...
...
@@ 1375,19 +1372,20 @@ Proof. intros >. apply suffix_of_app. Qed.
Lemma
suffix_of_snoc_inv_1
x
y
l1
l2
:
l1
++
[
x
]
`
suffix_of
`
l2
++
[
y
]
→
x
=
y
.
Proof
.
intros
[
k'
E
].
rewrite
(
associative_L
(++))
in
E
.
by
simplify_list_equality
.
intros
[
k'
E
].
rewrite
(
associative_L
(++))
in
E
.
by
simple_simplify_list_equality
.
Qed
.
Lemma
suffix_of_snoc_inv_2
x
y
l1
l2
:
l1
++
[
x
]
`
suffix_of
`
l2
++
[
y
]
→
l1
`
suffix_of
`
l2
.
Proof
.
intros
[
k'
E
].
exists
k'
.
rewrite
(
associative_L
(++))
in
E
.
by
simplify_list_equality
.
by
simpl
e_simpl
ify_list_equality
.
Qed
.
Lemma
suffix_of_app_inv
l1
l2
k
:
l1
++
k
`
suffix_of
`
l2
++
k
→
l1
`
suffix_of
`
l2
.
Proof
.
intros
[
k'
E
].
exists
k'
.
rewrite
(
associative_L
(++))
in
E
.
by
simplify_list_equality
.
by
simpl
e_simpl
ify_list_equality
.
Qed
.
Lemma
suffix_of_cons_l
l1
l2
x
:
x
::
l1
`
suffix_of
`
l2
→
l1
`
suffix_of
`
l2
.
Proof
.
intros
[
k
>].
exists
(
k
++
[
x
]).
by
rewrite
<(
associative_L
(++)).
Qed
.
...
...
@@ 1550,7 +1548,7 @@ Proof.
{
by
rewrite
<!(
associative_L
(++)).
}
rewrite
sublist_app_l
in
Hl12
.
destruct
Hl12
as
(
k1
&
k2
&
E
&?&
Hk2
).
destruct
k2
as
[
z
k2
]
using
rev_ind
;
[
inversion
Hk2
].
rewrite
(
associative_L
(++))
in
E
.
simplify_list_equality
.
rewrite
(
associative_L
(++))
in
E
;
simple_
simplify_list_equality
.
eauto
using
sublist_inserts_r
.
Qed
.
Global
Instance
:
PartialOrder
(@
sublist
A
).
...
...
@@ 2341,15 +2339,31 @@ 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'
:
Forall3
P
l1
k1
k1'
→
Forall3
P
l2
k2
k2'
→
Forall3
P
(
l1
++
l2
)
(
k1
++
k2
)
(
k1'
++
k2'
).
Proof
.
induction
1
;
simpl
;
[
done

constructor
]
;
auto
.
Qed
.
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
.
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
.
Proof
.
revert
l
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_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
;
constructor
;
auto
.
Defined
.
Proof
.
intros
Hl
?.
induction
Hl
;
auto
.
Defined
.
Lemma
Forall3_length_lm
l
l'
k
:
Forall3
P
l
l'
k
→
length
l
=
length
l'
.
Proof
.
by
induction
1
;
f_equal'
.
Qed
.
Lemma
Forall3_length_lr
l
l'
k
:
Forall3
P
l
l'
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
.
...
...
@@ 2379,7 +2393,7 @@ Section Forall3.
(
∀
x
y
z
,
l
!!
i
=
Some
x
→
l'
!!
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
.
Proof
.
intros
Hl
.
revert
i
.
induction
Hl
;
intros
[]
;
constructor
;
auto
.
Qed
.
Proof
.
intros
Hl
.
revert
i
.
induction
Hl
;
intros
[]
;
auto
.
Qed
.
End
Forall3
.
(** * Properties of the monadic operations *)
...
...
@@ 3080,9 +3094,9 @@ Ltac decompose_elem_of_list := repeat

H
:
_
∈
_
::
_

_
=>
apply
elem_of_cons
in
H
;
destruct
H

H
:
_
∈
_
++
_

_
=>
apply
elem_of_app
in
H
;
destruct
H
end
.
Ltac
simplify_list_
fmap_
equality
:
=
repeat
Ltac
simplify_list_equality
:
=
repeat
match
goal
with

_
=>
progress
simpl
ify
_equality

_
=>
progress
simpl
e_simplify_list
_equality

H
:
_
<$>
_
=
[]

_
=>
apply
fmap_nil_inv
in
H

H
:
[]
=
_
<$>
_

_
=>
symmetry
in
H
;
apply
fmap_nil_inv
in
H

H
:
_
<$>
_
=
_
::
_

_
=>
...
...
@@ 3091,10 +3105,6 @@ Ltac simplify_list_fmap_equality := repeat

H
:
_
<$>
_
=
_
++
_

_
=>
apply
fmap_app_inv
in
H
;
destruct
H
as
(?&?&?&?&?)

H
:
_
++
_
=
_
<$>
_

_
=>
symmetry
in
H
end
.
Ltac
simplify_zip_equality
:
=
repeat
match
goal
with

_
=>
progress
simplify_equality

H
:
zip_with
_
_
_
=
[]

_
=>
apply
zip_with_nil_inv
in
H
;
destruct
H

H
:
[]
=
zip_with
_
_
_

_
=>
symmetry
in
H

H
:
zip_with
_
_
_
=
_
::
_

_
=>
...
...
@@ 3116,15 +3126,17 @@ Ltac decompose_Forall_hyps := repeat

H
:
Forall2
_
?l
[]

_
=>
apply
Forall2_nil_inv_r
in
H
;
subst
l

H
:
Forall2
_
(
_
::
_
)
(
_
::
_
)

_
=>
apply
Forall2_cons_inv
in
H
;
destruct
H

_
=>
progress
simplify_equality'

H
:
Forall2
_
(
_
::
_
)
?k

_
=>
let
k_hd
:
=
fresh
k
"_hd"
in
let
k_tl
:
=
fresh
k
"_tl"
in
apply
Forall2_cons_inv_l
in
H
;
destruct
H
as
(
k_hd
&
k_tl
&?&?&>)
;
rename
k_tl
into
k

H
:
Forall2
_
?l
(
_
::
_
)

_
=>
let
l_hd
:
=
fresh
l
"_hd"
in
let
l_tl
:
=
fresh
l
"_tl"
in
apply
Forall2_cons_inv_r
in
H
;
destruct
H
as
(
l_hd
&
l_tl
&?&?&>)
;
rename
l_tl
into
l

_
=>
progress
simplify_list_equality

H
:
Forall2
_
(
_
::
_
)
?k

_
=>
first
[
let
k_hd
:
=
fresh
k
"_hd"
in
let
k_tl
:
=
fresh
k
"_tl"
in
apply
Forall2_cons_inv_l
in
H
;
destruct
H
as
(
k_hd
&
k_tl
&?&?&>)
;
rename
k_tl
into
k

apply
Forall2_cons_inv_l
in
H
;
destruct
H
as
(?&?&?&?&?)]

H
:
Forall2
_
?l
(
_
::
_
)

_
=>
first
[
let
l_hd
:
=
fresh
l
"_hd"
in
let
l_tl
:
=
fresh
l
"_tl"
in
apply
Forall2_cons_inv_r
in
H
;
destruct
H
as
(
l_hd
&
l_tl
&?&?&>)
;
rename
l_tl
into
l

apply
Forall2_cons_inv_r
in
H
;
destruct
H
as
(?&?&?&?&?)]

H
:
Forall2
_
(
_
++
_
)
(
_
++
_
)

_
=>
apply
Forall2_app_inv
in
H
;
[
destruct
H

first
[
by
eauto
using
Forall2_length

by
symmetry
;
eauto
using
Forall2_length
]]
...
...
@@ 3136,6 +3148,10 @@ Ltac decompose_Forall_hyps := repeat
[
let
l1
:
=
fresh
l
"_1"
in
let
l2
:
=
fresh
l
"_2"
in
apply
Forall2_app_inv_r
in
H
;
destruct
H
as
(
l1
&
l2
&?&?&>)

apply
Forall2_app_inv_r
in
H
;
destruct
H
as
(?&?&?&?&?)
]

H
:
Forall3
_
_
(
_
::
_
)
_

_
=>
apply
Forall3_cons_inv_m
in
H
;
destruct
H
as
(?&?&?&?&?&?&?&?)

H
:
Forall3
_
_
(
_
++
_
)
_

_
=>
apply
Forall3_app_inv_m
in
H
;
destruct
H
as
(?&?&?&?&?&?&?&?)

H
:
Forall
?P
?l
,
H1
:
?l
!!
_
=
Some
?x

_
=>
(* to avoid some stupid loops, not fool proof *)
unless
(
P
x
)
by
auto
using
Forall_app_2
,
Forall_nil_2
;
...
...
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