Skip to content
GitLab
Explore
Sign in
Primary navigation
Search or go to…
Project
N
Numerical Analysis in Coq
Manage
Activity
Members
Labels
Plan
Issues
Issue boards
Milestones
Wiki
Code
Merge requests
Repository
Branches
Commits
Tags
Repository graph
Compare revisions
Snippets
Build
Pipelines
Jobs
Pipeline schedules
Artifacts
Deploy
Releases
Package Registry
Model registry
Operate
Environments
Terraform modules
Monitor
Incidents
Analyze
Value stream analytics
Contributor analytics
CI/CD analytics
Repository analytics
Model experiments
Help
Help
Support
GitLab documentation
Compare GitLab plans
Community forum
Contribute to GitLab
Provide feedback
Keyboard shortcuts
?
Snippets
Groups
Projects
Show more breadcrumbs
Micaela Mayero
Numerical Analysis in Coq
Commits
425462d4
Commit
425462d4
authored
1 week ago
by
François Clément
Browse files
Options
Downloads
Patches
Plain Diff
Rename LagPd1_ref_kron_vtx -> LagPd1_ref_kron_vtxF.
Add LagPd1_ref_kron_vtx, LagPd1_ref_vtx_skipF. Proof of Hface_ref_is_Ker.
parent
0e7c8f4d
No related branches found
Branches containing commit
No related tags found
Tags containing commit
No related merge requests found
Pipeline
#10657
waiting for manual action
Stage: test
Stage: deploy
Changes
3
Pipelines
1
Hide whitespace changes
Inline
Side-by-side
Showing
3 changed files
FEM/geom_transf_affine.v
+1
-1
1 addition, 1 deletion
FEM/geom_transf_affine.v
FEM/poly_LagPd1.v
+1
-1
1 addition, 1 deletion
FEM/poly_LagPd1.v
FEM/poly_LagPd1_ref.v
+27
-5
27 additions, 5 deletions
FEM/poly_LagPd1_ref.v
with
29 additions
and
7 deletions
FEM/geom_transf_affine.v
+
1
−
1
View file @
425462d4
...
...
@@ -81,7 +81,7 @@ Qed.
Lemma
T_geom_vtx
:
forall
i
,
T_geom
(
vtx_ref
i
)
=
vtx
i
.
Proof
.
intro
;
unfold
T_geom
;
rewrite
LagPd1_ref_kron_vtx
baryc_ms_correct
baryc_l_kron_r
;
easy
.
rewrite
LagPd1_ref_kron_vtx
F
baryc_ms_correct
baryc_l_kron_r
;
easy
.
Qed
.
Lemma
T_geom_vtxF
:
mapF
T_geom
vtx_ref
=
vtx
.
...
...
This diff is collapsed.
Click to expand it.
FEM/poly_LagPd1.v
+
1
−
1
View file @
425462d4
...
...
@@ -100,7 +100,7 @@ Proof. intros; apply LagPd1_ref_sum. Qed.
#
<
A
HREF
=
"##RR9557v1"
>
#[[
RR9557v1
]]#
</
A
>
#
Lem
1554
,
Eq
(
9.45
),
p
.
85.
*
)
Lemma
LagPd1_kron_vtx
:
forall
i
,
LagPd1
^~
(
vtx
i
)
=
kronR
i
.
Proof
.
intro
;
rewrite
-
LagPd1_ref_kron_vtx
-
(
T_geom_inv_vtx
Hvtx
);
easy
.
Qed
.
Proof
.
intro
;
rewrite
-
LagPd1_ref_kron_vtx
F
-
(
T_geom_inv_vtx
Hvtx
);
easy
.
Qed
.
Lemma
LagPd1_am
:
forall
i
,
aff_map_ms
(
LagPd1
i
).
Proof
.
...
...
This diff is collapsed.
Click to expand it.
FEM/poly_LagPd1_ref.v
+
27
−
5
View file @
425462d4
...
...
@@ -207,7 +207,7 @@ Lemma LagPd1_ref_bc_gather : baryc_coord_ms vtx_ref = gather LagPd1_ref.
Proof
.
rewrite
LagPd1_ref_bc_scatter
gather_scatter
;
easy
.
Qed
.
Lemma
LagPd1_ref_barycF_rev
:
forall
(
L
:
'
R
^
d
.
+
1
)
,
sum
L
=
1
->
forall
{
L
:
'
R
^
d
.
+
1
}
,
sum
L
=
1
->
LagPd1_ref
^~
(
barycenter_ms
L
vtx_ref
)
=
L
.
Proof
.
intros
;
rewrite
LagPd1_ref_bcF
bc_ms_correct
;
...
...
@@ -222,9 +222,18 @@ Qed.
(
**
#
<
A
HREF
=
"##RR9557v1"
>
#[[
RR9557v1
]]#
</
A
>
#
Lem
1543
,
Eq
(
9.32
),
p
.
81.
*
)
Lemma
LagPd1_ref_kron_vtx
:
forall
i
,
LagPd1_ref
^~
(
vtx_ref
i
)
=
kronR
i
.
Lemma
LagPd1_ref_kron_vtx
F
:
forall
i
,
LagPd1_ref
^~
(
vtx_ref
i
)
=
kronR
i
.
Proof
.
intro
;
rewrite
LagPd1_ref_bcF
;
apply
bc_kronF_r
,
vtx_ref_aff_indep
.
Qed
.
Lemma
LagPd1_ref_kron_vtx
:
forall
i
j
,
LagPd1_ref
j
(
vtx_ref
i
)
=
kronR
i
j
.
Proof
.
intros
i
j
;
move
:
j
;
apply
extF_rev
,
LagPd1_ref_kron_vtxF
.
Qed
.
Lemma
LagPd1_ref_vtx_skipF
:
forall
i
j
,
LagPd1_ref
i
(
skipF
vtx_ref
i
j
)
=
0.
Proof
.
intros
;
unfold
skipF
;
rewrite
LagPd1_ref_kron_vtx
.
apply
kron_is_0
,
ord_neq_compat
,
skip_ord_correct_m
.
Qed
.
Lemma
LagPd1_ref_S_Monom
:
liftF_S
LagPd1_ref
=
castF
(
pbinom_1_r
d
)
(
liftF_S
(
Monom_dk
d
1
)).
Proof
.
...
...
@@ -257,7 +266,7 @@ Qed.
Lemma
LagPd1_ref_lin_indep
:
lin_indep
LagPd1_ref
.
Proof
.
intros
L
HL
;
extF
;
rewrite
-
lc_r_kron_r
.
rewrite
-
LagPd1_ref_kron_vtx
-
fct_lc_r_eq
HL
;
easy
.
rewrite
-
LagPd1_ref_kron_vtx
F
-
fct_lc_r_eq
HL
;
easy
.
Qed
.
(
**
...
...
@@ -274,7 +283,9 @@ Proof. apply (Dim _ _ _ LagPd1_ref_basis). Qed.
Lemma
LagPd1_ref_baryc_lc
:
forall
L
i
,
sum
L
=
1
->
LagPd1_ref
i
(
lin_comb
L
vtx_ref
)
=
L
i
.
Proof
.
intros
;
rewrite
-
baryc_ms_eq
// -{2}(LagPd1_ref_barycF_rev L); easy. Qed.
Proof
.
move
=>>
HL
;
rewrite
-
baryc_ms_eq
// -{2}(LagPd1_ref_barycF_rev HL); easy.
Qed
.
Let
K_geom_ref
:
'
R
^
d
->
Prop
:=
convex_envelop
vtx_ref
.
...
...
@@ -328,7 +339,18 @@ Definition Hface_ref (i : 'I_d.+1) : 'R^d -> Prop :=
Lem
1564
,
p
.
88.
*
)
Lemma
Hface_ref_is_Ker
:
forall
i
,
Hface_ref
i
=
Ker
(
LagPd1_ref
i
).
Proof
.
Admitted
.
intro
;
unfold
Hface_ref
;
apply
subset_ext_equiv
;
split
.
(
*
*
)
intros
_
[
L
HL
];
apply
(
Ker_cas
(
LagPd1_ref_am
_
));
[
easy
|
].
intros
j
;
rewrite
Ker_equiv
;
apply
LagPd1_ref_vtx_skipF
.
(
*
*
)
intros
x_ref
;
move
:
(
vtx_ref_aff_span
x_ref
)
=>
/
aff_span_EX
[
L
[
HL
->
]].
rewrite
Ker_equiv
.
move:
(
LagPd1_ref_barycF_rev
HL
)
=>
/
extF_rev
HL1
;
rewrite
(
HL1
i
)
=>
HLi
.
rewrite
(
baryc_skip_zero
_
_
i
)
//; [apply Aff_span |].
rewrite
-
(
plus_zero_l
(
sum
_
))
-
HLi
-
sum_skipF
.
1
,
2
:
apply
invertible_eq_one
;
easy
.
Qed
.
Lemma
Hface_ref_equiv
:
forall
i
x_ref
,
Hface_ref
i
x_ref
<->
LagPd1_ref
i
x_ref
=
0.
...
...
This diff is collapsed.
Click to expand it.
Preview
0%
Loading
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Save comment
Cancel
Please
register
or
sign in
to comment