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Numerical Analysis in Coq
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Micaela Mayero
Numerical Analysis in Coq
Commits
e7aebdb7
Commit
e7aebdb7
authored
5 months ago
by
Houda Mouhcine
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Change in indices.
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c78ee4be
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Pipeline
#8616
waiting for manual action
Stage: test
Changes
2
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1
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2 changed files
FEM/geom_simplex.v
+14
-14
14 additions, 14 deletions
FEM/geom_simplex.v
FEM/poly_LagPd1_ref.v
+7
-4
7 additions, 4 deletions
FEM/poly_LagPd1_ref.v
with
21 additions
and
18 deletions
FEM/geom_simplex.v
+
14
−
14
View file @
e7aebdb7
...
...
@@ -247,18 +247,18 @@ Qed.
Lemma
vtx_cur_invalF
:
forall
vtx_cur
,
(
0
<
k
)
%
coq_nat
->
invalF
vtx_cur
(
node_cur
vtx_cur
).
Proof
.
intros
vtx_cur
Hk
i
.
destruct
(
ord_eq_dec
i
ord0
)
as
[
Hi
|
Hi
].
(
*
i
=
ord0
*
)
rewrite
Hi
.
intros
vtx_cur
Hk
i
pk
.
unfold
inF
.
destruct
(
ord_eq_dec
i
pk
ord0
)
as
[
Hi
pk
|
Hi
pk
].
(
*
i
pk
=
ord0
*
)
rewrite
Hi
pk
.
exists
ord0
.
unfold
node_cur
.
rewrite
(
Adk_0
d
k
)
!
scal_zero_r
!
lc_zero_l
minus_zero_r
scal_one
plus_zero_r
;
easy
.
(
*
i
<>
ord0
*
)
exists
(
Adk_inv
d
k
(
itemF
d
k
(
lower_S
Hi
))).
(
*
i
pk
<>
ord0
*
)
exists
(
Adk_inv
d
k
(
itemF
d
k
(
lower_S
Hi
pk
))).
unfold
node_cur
.
rewrite
Adk_kron
!
(
lc_eq_l
_
(
kronecker
(
lower_S
Hi
))).
rewrite
Adk_kron
!
(
lc_eq_l
_
(
kronecker
(
lower_S
Hi
pk
))).
rewrite
lc_ones_r
sum_kronecker_r
minus_eq_zero
scal_zero_l
plus_zero_l
lc_kronecker_l_in_r
liftF_lower_S
;
easy
.
...
...
@@ -319,27 +319,27 @@ Variable d : nat.
Lemma
vtx_node_Pd1_cur
:
forall
(
vtx_cur
:
'
R
^{
d
.
+
1
,
d
}
)
(
ipk
:
'
I_d
.
+
1
),
vtx_cur
ipk
=
node_cur
d
1
vtx_cur
(
cast_ord
(
eq_sym
(
pbinomS_1_r
d
))
ipk
).
Proof
.
intros
vtx_cur
i
;
unfold
node_cur
.
intros
vtx_cur
i
pk
;
unfold
node_cur
.
rewrite
Rinv_1
scal_one
.
destruct
(
ord_eq_dec
i
ord0
)
as
[
Hi
|
Hi
].
(
*
i
=
ord0
*
)
destruct
(
ord_eq_dec
i
pk
ord0
)
as
[
Hi
|
Hi
].
(
*
i
pk
=
ord0
*
)
rewrite
Hi
.
replace
(
mapF
INR
(
Adk
d
1
(
cast_ord
(
eq_sym
(
pbinomS_1_r
d
))
ord0
)))
with
(
mapF
INR
(
Adk
d
1
ord0
)).
rewrite
(
Adk_0
d
1
)
!
lc_zero_l
minus_zero_r
scal_one
plus_zero_r
;
auto
.
f_equal
;
f_equal
;
apply
ord_inj
;
easy
.
(
*
i
<>
ord0
*
)
(
*
i
pk
<>
ord0
*
)
rewrite
Ad1_eq
.
unfold
castF
.
rewrite
concatF_correct_r
;
simpl
.
apply
ord_n0_nlt_equiv
;
easy
.
intros
H
.
rewrite
mapF_itemF_0
// itemF_kronecker_eq; auto; simpl.
replace
(
fun
j
:
'
I_d
=>
1
*
kronecker
(
i
-
1
)
j
)
with
(
fun
j
:
'
I_d
=>
kronecker
(
i
-
1
)
j
).
replace
(
fun
j
:
'
I_d
=>
1
*
kronecker
(
i
pk
-
1
)
j
)
with
(
fun
j
:
'
I_d
=>
kronecker
(
i
pk
-
1
)
j
).
rewrite
lc_ones_r
.
replace
(
sum
(
fun
j
:
'
I_d
=>
kronecker
(
i
-
1
)
j
))
with
1.
replace
(
sum
(
fun
j
:
'
I_d
=>
kronecker
(
i
pk
-
1
)
j
))
with
1.
rewrite
minus_eq_zero
scal_zero_l
plus_zero_l
.
apply
eq_sym
.
rewrite
(
lc_kronecker_l_in_r
(
lower_S
Hi
))
...
...
This diff is collapsed.
Click to expand it.
FEM/poly_LagPd1_ref.v
+
7
−
4
View file @
e7aebdb7
...
...
@@ -319,8 +319,10 @@ Variable d : nat.
Lemma
Pd1_lin_span_LagPd1_ref
:
Pdk
d
1
=
lin_span
(
LagPd1_ref
d
).
Proof
.
apply
lin_span_ext
;
intro
ipk
.
unfold
Pdk
.
apply
lin_span_ext
.
(
*
<=
*
)
intro
ipk
.
destruct
(
ord_eq_dec
ipk
ord0
)
as
[
Hipk
|
Hipk
].
(
*
ipk
=
ord0
*
)
apply
lin_span_ex
.
...
...
@@ -359,7 +361,8 @@ apply cast_ordS_n0; easy.
intros
H1
;
apply
fun_ext
;
intros
x
.
f_equal
;
f_equal
;
easy
.
(
*
=>
*
)
destruct
(
ord_eq_dec
ipk
ord0
)
as
[
Hipk
|
Hipk
].
intros
j
.
destruct
(
ord_eq_dec
j
ord0
)
as
[
Hj
|
Hj
].
apply
lin_span_ex
.
(
*
ipk
=
ord0
*
)
pose
(
L
:=
concatF
(
singleF
1
)
(
opp
(
@
ones
_
d
))).
...
...
@@ -398,8 +401,8 @@ apply ord_inj; simpl.
rewrite
bump_r
;
auto
with
arith
.
f_equal
;
apply
ord_inj
;
easy
.
(
*
ipk
<>
ord0
*
)
replace
(
LagPd1_ref
d
ipk
)
with
(
BasisPdk
d
1
(
cast_ord
(
eq_sym
(
pbinomS_1_r
d
))
ipk
)).
replace
(
LagPd1_ref
d
j
)
with
(
BasisPdk
d
1
(
cast_ord
(
eq_sym
(
pbinomS_1_r
d
))
j
)).
apply
lin_span_inclF_diag
.
rewrite
BasisPd1_neq0
;
simpl
.
apply
cast_ordS_n0
;
easy
.
...
...
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