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Numerical Analysis in Coq
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Micaela Mayero
Numerical Analysis in Coq
Commits
8303aca0
Commit
8303aca0
authored
2 years ago
by
Mouhcine
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work on LagPQ_is_non_neg lemma
parent
d5e504e9
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FEM/finite_element.v
+16
-7
16 additions, 7 deletions
FEM/finite_element.v
FEM/poly_Lagrange.v
+4
-4
4 additions, 4 deletions
FEM/poly_Lagrange.v
with
20 additions
and
11 deletions
FEM/finite_element.v
+
16
−
7
View file @
8303aca0
...
...
@@ -462,23 +462,30 @@ apply LagQ1_kron.
Qed
.
(
*
TODO
:
Not
really
!
we
should
add
some
hypothesis
on
x_i
*
)
(
*
ajouter
une
hyp
qu
'
on
est
dans
l
'
element
de
ref
*
)
Lemma
LagPQ_is_non_neg
:
forall
(
i
:
'
I_nnvtx
)
(
x
:
'
R
^
dd
),
0
<=
LagPQ
i
x
.
K_geom_ref
x
->
0
<=
LagPQ
i
x
.
Proof
.
intros
i
x
;
unfold
LagPQ
.
intros
i
x
Hx
;
unfold
LagPQ
.
case
(
nvtx_dec
FE_ref
);
intros
H
.
apply
LagP1_is_non_neg
.
apply
LagQ1_is_non_neg
;
easy
.
Qed
.
unfold
K_geom_ref
,
K_geom
in
Hx
.
unfold
dd
.
admit
.
admit
.
(
*
apply
LagQ1_is_non_neg
;
easy
.
Qed
.
*
)
Admitted
.
(
*
TODO
verify
if
this
lemma
is
true
,
we
apply
it
to
T_geom_is_bij_id
*
)
Lemma
LagPQ_sum_1
:
forall
x
,
comb_lin
(
fun
i
:
'
I_nnvtx
=>
LagPQ
i
x
)(
fun
_
=>
1
)
=
1.
Proof
.
(
*
intros
x
.
unfold
LagPQ
.
case
(
nvtx_dec
FE_ref
);
intros
H
.
admit
.
admit
.
*
)
(
*
apply
LagP1_sum_1
.
apply
LagQ1_sum_1
.
*
)
...
...
@@ -767,6 +774,7 @@ unfold range; intros x_cur H.
apply
K_geom_ref_correct
;
easy
.
(
*
*
)
intros
i
;
apply
LagPQ_is_non_neg
.
(
*
apply
LagPQ_sum_1
.
(
*
*
)
unfold
identity
;
intros
x_cur
H
;
symmetry
.
...
...
@@ -774,7 +782,8 @@ apply T_geom_comp; easy.
(
*
*
)
unfold
identity
;
intros
x_ref
H
;
symmetry
.
apply
T_geom_inv_comp
;
easy
.
Qed
.
Qed
.
*
)
Admitted
.
Lemma
theta_cur_correct
:
forall
i
:
'
I_
(
ndof
FE_cur
),
theta_ref
i
=
cur_to_ref
(
theta_cur
i
).
...
...
This diff is collapsed.
Click to expand it.
FEM/poly_Lagrange.v
+
4
−
4
View file @
8303aca0
...
...
@@ -107,9 +107,9 @@ 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
),
0
<=
LagP1
i
x
.
convex_envelop
d
.
+
1
vtxP1
x
->
0
<=
LagP1
i
x
.
Proof
.
intros
i
x
;
unfold
LagP1
.
intros
i
x
Hx
;
unfold
LagP1
.
Admitted
.
Lemma
LagP1_sum_1
:
forall
x
,
...
...
@@ -160,9 +160,9 @@ Proof.
Admitted
.
Lemma
LagQ1_is_non_neg
:
forall
(
i
:
'
I_
(
2
^
d
))
(
x
:
'
R
^
d
),
0
<=
LagQ1
i
x
.
convex_envelop
(
2
^
d
)
vtxQ1
x
->
0
<=
LagQ1
i
x
.
Proof
.
intros
i
x
;
unfold
LagQ1
.
intros
i
x
Hx
;
unfold
LagQ1
.
Admitted
.
...
...
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