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Numerical Analysis in Coq
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Micaela Mayero
Numerical Analysis in Coq
Commits
81687d55
Commit
81687d55
authored
2 years ago
by
Mouhcine
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Prove vtx_in_convex_envelop lemma.
move the comment to FE.v
parent
ef72d37e
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FEM/geometry.v
+5
-60
5 additions, 60 deletions
FEM/geometry.v
with
5 additions
and
60 deletions
FEM/geometry.v
+
5
−
60
View file @
81687d55
...
...
@@ -52,72 +52,17 @@ intros i.
replace
(
vtx
i
)
with
(
comb_lin
(
kronecker
i
)
vtx
).
apply
Cvx
.
(
*
;
unfold
elt_geom
.
exists
(
kronecker
i
).
repeat
split
*
)
replace
(
vtx
i
)
with
(
\
big
[
plus
/
zero
]
_
(
j
<
nvtx
)
(
scal
(
kronecker
i
j
)
(
vtx
j
))).
(
*
apply
Cvx
.
intros
j
;
apply
kronecker_bound
.
*
)
(
*
apply
kronecker_sum_r
;
easy
.
(
*
*
)
induction
nvtx
;
try
easy
.
rewrite
sum_pn_Sn
.
case
(
lt_dec
i
n
);
intros
Hn
.
(
*
case1
*
)
rewrite
<-
IHn
;
try
easy
.
replace
(
kronecker
i
n
)
with
(
@
zero
R_Ring
).
rewrite
(
scal_zero_l
).
rewrite
plus_zero_r
;
easy
.
unfold
kronecker
.
case
(
eq_nat_dec
i
n
);
try
easy
.
intros
;
lia
.
(
*
case2
*
)
unfold
kronecker
at
2.
case
(
eq_nat_dec
i
n
);
try
lia
.
intros
H1
.
replace
(
sum_pn
_
_
)
with
(
@
zero
E
).
replace
1
with
(
@
one
R_Ring
)
by
easy
.
rewrite
scal_one
.
rewrite
H1
.
rewrite
plus_zero_l
;
easy
.
rewrite
sum_pn_zero
;
try
easy
.
intros
m
Hm
.
unfold
kronecker
.
case
(
eq_nat_dec
i
m
);
try
lia
.
intros
;
replace
0
with
(
@
zero
R_Ring
)
by
easy
.
apply
(
scal_zero_l
(
vtx
m
)).
intros
j
;
apply
kronecker_bound
.
apply
comb_lin_kronecker_r
.
apply
comb_lin_kronecker_in_r
.
Qed
.
*
)
Admitted
.
(
*
TODO
:
hypothesis
on
vertices
:
geometrical
form
not
degenerate
*
)
(
*
TODO
:
define
basic
geometric
shapes
:
(
simplices
)
Seg
Tria
,
Tetra
,
(
others
)
Quad
,
Hexa
,
Prism
?
...
Define
the
geometric
transformation
T_geom
(
from
the
reference
mesh
to
the
current
mesh
)
for
all
previous
geometric
shapes
,
and
its
Jacobian
matrix
and
Jacobian
determinant
.
For
the
simplex
in
dim
E
=
d
,
T_geom
is
affine
:
x
^
->
x
=
a0
+
J_geom
x
^
where
J_geom
=
(
a1
-
a0
a2
-
a0
...
ad
-
a0
)
is
the
Jacobian
matrix
of
T_geom
,
made
of
the
column
vectors
ai
-
a0
,
where
(
a0
,
a1
...
ad
)
are
the
vertices
of
the
current
mesh
.
The
vertices
of
the
reference
mesh
are
a0
^
=
(
0
,
0.
.
.0
),
a1
^
=
(
1
,
0.
.
.0
),
a2
^
=
(
0
,
1
,
0.
.
.0
),...,
ad
^
=
(
0
,..
.0
,
1
).
In
each
case
,
prove
that
J_geom
is
indeed
the
Jacobian
matrix
of
T_geom
.
Prove
that
T_geom
is
invertible
,
ie
its
Jacobian
determinant
is
nonzero
.
Then
,
in
the
simple
cases
,
T_geom
{-
1
}
:
x
->
x
^
=
J_geom
{-
1
}
(
x
-
a0
).
(
others
)
Quad
,
Hexa
,
Prism
?
...
*
)
Maybe
prove
correction
lemmas
of
the
form
:
forall
x
^
,
K_geom
(
T_geom
x
^
)
<->
K_geom
^
x
^
,
ie
image
T_geom
K_geom
^
=
K_geom
.
forall
x
,
K_geom
x
<->
K_geom
^
(
T_geom
{-
1
}
x
),
ie
image
T_geom
{-
1
}
K_geom
=
K_geom
^
.
*
)
End
Geometry
.
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