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e7d2b424
Commit
e7d2b424
authored
Dec 08, 2015
by
Robbert Krebbers
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Clean-up in prelude.numbers.
parent
d011f232
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prelude/numbers.v
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e7d2b424
...
...
@@ -144,24 +144,18 @@ Proof. intros ?? p. by induction p; intros; f_equal'. Qed.
Global
Instance
:
∀
p
:
positive
,
Injective
(=)
(=)
(++
p
).
Proof
.
intros
p
???.
induction
p
;
simplify_equality
;
auto
.
Qed
.
Lemma
Preverse_go_app_cont
p1
p2
p3
:
Preverse_go
(
p2
++
p1
)
p3
=
p2
++
Preverse_go
p1
p3
.
Proof
.
revert
p1
.
induction
p3
;
simpl
;
intros
.
*
apply
(
IHp3
(
_
~
1
)).
*
apply
(
IHp3
(
_
~
0
)).
*
done
.
Qed
.
Lemma
Preverse_go_app
p1
p2
p3
:
Preverse_go
p1
(
p2
++
p3
)
=
Preverse_go
p1
p3
++
Preverse_go
1
p2
.
Proof
.
revert
p1
.
induction
p3
;
intros
p1
;
simpl
;
auto
.
by
rewrite
<-
Preverse_go_app_cont
.
revert
p3
p1
p2
.
cut
(
∀
p1
p2
p3
,
Preverse_go
(
p2
++
p3
)
p1
=
p2
++
Preverse_go
p3
p1
).
{
by
intros
go
p3
;
induction
p3
;
intros
p1
p2
;
simpl
;
auto
;
rewrite
<-
?go
.
}
intros
p1
;
induction
p1
as
[
p1
IH
|
p1
IH
|]
;
intros
p2
p3
;
simpl
;
auto
.
*
apply
(
IH
_
(
_
~
1
)).
*
apply
(
IH
_
(
_
~
0
)).
Qed
.
Lemma
Preverse_app
p1
p2
:
Preverse
(
p1
++
p2
)
=
Preverse
p2
++
Preverse
p1
.
Lemma
Preverse_app
p1
p2
:
Preverse
(
p1
++
p2
)
=
Preverse
p2
++
Preverse
p1
.
Proof
.
unfold
Preverse
.
by
rewrite
Preverse_go_app
.
Qed
.
Lemma
Preverse_xO
p
:
Preverse
(
p
~
0
)
=
(
1
~
0
)
++
Preverse
p
.
Proof
Preverse_app
p
(
1
~
0
).
Lemma
Preverse_xI
p
:
Preverse
(
p
~
1
)
=
(
1
~
1
)
++
Preverse
p
.
...
...
@@ -169,8 +163,7 @@ Proof Preverse_app p (1~1).
Fixpoint
Plength
(
p
:
positive
)
:
nat
:
=
match
p
with
1
=>
0
%
nat
|
p
~
0
|
p
~
1
=>
S
(
Plength
p
)
end
.
Lemma
Papp_length
p1
p2
:
Plength
(
p1
++
p2
)
=
(
Plength
p2
+
Plength
p1
)%
nat
.
Lemma
Papp_length
p1
p2
:
Plength
(
p1
++
p2
)
=
(
Plength
p2
+
Plength
p1
)%
nat
.
Proof
.
by
induction
p2
;
f_equal'
.
Qed
.
Close
Scope
positive_scope
.
...
...
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