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Micaela Mayero
Numerical Analysis in Coq
Commits
b28ab820
Commit
b28ab820
authored
1 year ago
by
Sylvie Boldo
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Preuve shape_fun_L_cur_eq
parent
7c300eb9
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Pipeline
#7289
waiting for manual action
Stage: test
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FEM/FE_LagP.v
+17
-29
17 additions, 29 deletions
FEM/FE_LagP.v
with
17 additions
and
29 deletions
FEM/FE_LagP.v
+
17
−
29
View file @
b28ab820
...
...
@@ -682,40 +682,28 @@ rewrite -T_geom_transports_nodes; try easy; f_equal.
unfold
node_ref_aux
;
f_equal
;
apply
ord_inj
;
easy
.
Qed
.
Lemma
shape_fun_ext
:
forall
(
fe1
fe2
:
FE
)
(
H
:
fe1
=
fe2
),
shape_fun
fe1
=
(
castF
(
eq_sym
(
f_equal
ndof
H
))
(
mapF
(
castF_fun
(
eq_sym
(
f_equal
d
H
)))
(
shape_fun
fe2
))).
Proof
.
intros
fe1
fe2
H
;
subst
.
apply
extF
;
intros
i
;
apply
fun_ext
;
intros
x
.
rewrite
castF_refl
.
rewrite
mapF_correct
;
unfold
castF_fun
;
f_equal
.
now
rewrite
castF_refl
.
Qed
.
Lemma
shape_fun_L_cur_eq
:
forall
(
vtx_cur
:
'
R
^{
dL
.
+
1
,
dL
}
)
(
Hvtx
:
affine_independent
vtx_cur
),
shape_fun
(
FE_LagPk_d_cur
dL
kL
dL_pos
kL_pos
vtx_cur
Hvtx
)
=
shape_fun_cur
FE_LagPk_d_ref
shape_Lag_ref
vtx_cur
Hvtx
.
Proof
.
intros
vtx_cur
Hvtx
;
unfold
shape_fun_cur
.
apply
extF
;
intros
i
.
generalize
(
FE_cur_eq
vtx_cur
Hvtx
);
intros
T
.
(
*
pose
(
t
:=
FE_LagPk_d_cur
dL
kL
dL_pos
kL_pos
vtx_cur
Hvtx
);
fold
t
in
i
,
T
|-
*
.
assert
(
V
:
ndof
t
=
ndof
(
FE_cur
FE_LagPk_d_ref
shape_Lag_ref
vtx_cur
Hvtx
))
by
easy
.
apply
trans_eq
with
(
shape_fun
(
FE_cur
FE_LagPk_d_ref
shape_Lag_ref
vtx_cur
Hvtx
)
(
cast_ord
V
i
)).
subst
t
.
generalize
(
eq_sym
T
);
intros
T
'
.
*
)
(
*
apply
is_local_shape_fun_inj
.
apply
shape_fun_correct
.
generalize
(
shape_fun_correct
(
FE_cur
FE_LagPk_d_ref
shape_Lag_ref
vtx_cur
Hvtx
)).
intros
T
.
rewrite
-
FE_cur_eq
in
T
.
unfold
is_local_shape_fun
in
*
.
rewrite
-
FE_cur_eq
.
apply
shape_fun_correct
.
unfold
FE_LagPk_d_ref
.
*
)
(
*
rewrite
->
(
FE_cur_eq
vtx_cur
Hvtx
).
apply
shape_fun_correct
.
*
)
(
*
16
/
10
/
23
:
Ce
n
'
est
pas
urgent
*
)
Admitted
.
intros
vtx_cur
Hvtx
;
unfold
shape_fun_cur
;
eapply
trans_eq
.
apply
shape_fun_ext
with
(
H
:=
FE_cur_eq
vtx_cur
Hvtx
).
rewrite
castF_id
.
apply
extF
;
intros
i
;
rewrite
mapF_correct
;
apply
fun_ext
;
intros
x
.
unfold
castF_fun
;
f_equal
;
now
rewrite
castF_id
.
Qed
.
(
*
From
Aide
-
memoire
EF
Alexandre
Ern
:
Eq
3.37
p
.
63
*
)
(
*
"Ip (v o T) = (Ip v) o T"
*
)
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