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Numerical Analysis in Coq
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Micaela Mayero
Numerical Analysis in Coq
Commits
cbdd03bc
Commit
cbdd03bc
authored
2 years ago
by
Mouhcine
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Add LagP1_is_affine and LagQ1_is_LagP1 lemmas.
parent
c1658422
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FEM/poly_Lagrange.v
+43
-1
43 additions, 1 deletion
FEM/poly_Lagrange.v
with
43 additions
and
1 deletion
FEM/poly_Lagrange.v
+
43
−
1
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cbdd03bc
...
...
@@ -105,7 +105,6 @@ contradict H.
admit
.
Admitted
.
(
*
TODO
:
Faut
des
hypoth
è
ses
de
plus
sur
x_i
>
0
*
)
Lemma
LagP1_is_non_neg
:
forall
(
i
:
'
I_d
.
+
1
)
(
x
:
'
R
^
d
),
convex_envelop
d
.
+
1
vtxP1
x
->
0
<=
LagP1
i
x
.
Proof
.
...
...
@@ -119,6 +118,36 @@ intros.
unfold
comb_lin
.
Admitted
.
Lemma
LagP1_is_affine
:
forall
i
,
is_affine
(
LagP1
i
).
Proof
.
intros
i
.
pose
(
L
:=
fun
x
=>
minus
(
LagP1
i
x
)
(
LagP1
i
zero
)).
apply
is_affine_ext
with
(
fun
x
=>
plus
(
L
x
)
(
LagP1
i
zero
)).
intros
x
.
unfold
L
.
admit
.
(
*
easy
*
)
apply
(
Is_affine
L
).
unfold
L
,
LagP1
.
case
(
eq_nat_dec
i
0
);
intros
Hi
.
rewrite
bigop_op_idx
;
try
easy
.
apply
is_linear_mapping_ext
with
(
fun
x
:
'
R
^
d
=>
\
big
[
+%
R
/
0
]
_
(
i
<
d
)
x
i
).
intros
x
.
admit
.
(
*
easy
*
)
apply
is_linear_mapping_ext
with
(
fun
x
:
'
R
^
d
=>
comb_lin
(
fun
=>
1
)
x
).
admit
.
(
*
easy
*
)
apply
component_sum_is_linear_mapping
.
(
*
*
)
apply
is_linear_mapping_ext
with
(
fun
x
:
'
R
^
d
=>
(
x
(
Ordinal
(
n
:=
d
)
(
m
:=
i
-
1
)
(
lt_minus_1
i
Hi
)))).
admit
.
(
*
easy
*
)
apply
component_is_linear_mapping
.
Admitted
.
End
Poly_Lagrange_Simplex
.
...
...
@@ -174,3 +203,16 @@ unfold comb_lin.
Admitted
.
End
Poly_Lagrange_Quad
.
Section
LagPQ
.
(
*
TODO
Faut
le
phi
*
)
Lemma
LagQ1_is_LagP1
:
forall
(
i
:
'
I_2
)
x
,
LagQ1
1
i
x
=
LagP1
1
i
x
.
Proof
.
intros
i
x
.
unfold
LagQ1
.
Admitted
.
End
LagPQ
.
\ No newline at end of file
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