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Sophie Quinton
rtproofs
Commits
d3d75a3b
Commit
d3d75a3b
authored
Aug 09, 2016
by
Felipe Cerqueira
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Add more lemmas about \max
parent
6461aa51
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util/sum.v
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d3d75a3b
...
...
@@ 94,6 +94,74 @@ Section ExtraLemmasSumMax.
by
destruct
(
m2
<
n2
)
eqn
:
LT
;
[
by
apply
/
orP
;
right

by
apply
/
orP
;
left
].
Qed
.
Lemma
bigmax_ord_ltn_identity
n
:
n
>
0
>
\
max_
(
i
<
n
)
i
<
n
.
Proof
.
intros
LT
.
destruct
n
;
first
by
rewrite
ltn0
in
LT
.
clear
LT
.
induction
n
;
first
by
rewrite
big_ord_recr
/=
big_ord0
maxn0
.
rewrite
big_ord_recr
/=.
unfold
maxn
at
1
;
desf
.
by
apply
leq_trans
with
(
n
:
=
n
.+
1
).
Qed
.
Lemma
bigmax_ltn_ord
n
(
P
:
pred
nat
)
(
i0
:
'
I_n
)
:
P
i0
>
\
max_
(
i
<
n

P
i
)
i
<
n
.
Proof
.
intros
LT
.
destruct
n
;
first
by
destruct
i0
as
[
i0
P0
]
;
move
:
(
P0
)
=>
P0'
;
rewrite
ltn0
in
P0'
.
rewrite
big_mkcond
.
apply
leq_ltn_trans
with
(
n
:
=
\
max_
(
i
<
n
.+
1
)
i
).
{
apply
/
bigmax_leqP
;
ins
.
destruct
(
P
i
)
;
last
by
done
.
by
apply
leq_bigmax_cond
.
}
by
apply
bigmax_ord_ltn_identity
.
Qed
.
Lemma
bigmax_pred
n
(
P
:
pred
nat
)
(
i0
:
'
I_n
)
:
P
(
i0
)
>
P
(
\
max_
(
i
<
n

P
i
)
i
).
Proof
.
intros
PRED
.
induction
n
.
{
destruct
i0
as
[
i0
P0
].
by
move
:
(
P0
)
=>
P1
;
rewrite
ltn0
in
P1
.
}
rewrite
big_mkcond
big_ord_recr
/=
;
desf
.
{
destruct
n
;
first
by
rewrite
big_ord0
maxn0
.
unfold
maxn
at
1
;
desf
.
exfalso
.
apply
negbT
in
Heq0
;
move
:
Heq0
=>
/
negP
BUG
.
apply
BUG
.
apply
leq_ltn_trans
with
(
n
:
=
\
max_
(
i
<
n
.+
1
)
i
).
{
apply
/
bigmax_leqP
;
ins
.
destruct
(
P
i
)
;
last
by
done
.
by
apply
leq_bigmax_cond
.
}
by
apply
bigmax_ord_ltn_identity
.
}
{
rewrite
maxn0
.
rewrite

big_mkcond
/=.
have
LT
:
i0
<
n
.
{
rewrite
ltn_neqAle
;
apply
/
andP
;
split
;
last
by
rewrite

ltnS
;
apply
ltn_ord
.
apply
/
negP
;
move
=>
/
eqP
BUG
.
by
rewrite

BUG
PRED
in
Heq
.
}
by
rewrite
(
IHn
(
Ordinal
LT
)).
}
Qed
.
Lemma
sum_nat_eq0_nat
(
T
:
eqType
)
(
F
:
T
>
nat
)
(
r
:
seq
T
)
:
all
(
fun
x
=>
F
x
==
0
)
r
=
(
\
sum_
(
i
<
r
)
F
i
==
0
).
Proof
.
...
...
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