Skip to content

added a few lemmas about sup #1964

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Open
holgerthies wants to merge 1 commit into
base: master
Choose a base branch
from
+74 −0
Open

added a few lemmas about sup #1964

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -70,6 +70,9 @@
- in `derive.v`:
+ lemmas `compact_EVT_max`, `compact_EVT_min`, `EVT_max_rV`, `EVT_min_rV`

- in `reals.v`:
+ lemmas `sup_ge0`, `has_sup_wpZl`, `gt0_has_supZl`, `has_sup_Mn`, `sup_Mn`

### Changed

- in `constructive_ereal.v`: fixed the infamous `%E` scope bug.
Expand Down
71 changes: 71 additions & 0 deletions reals/reals.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -632,7 +632,78 @@ move=> /inf_adherent/(_ hs)[_ [x ->]]; rewrite addrC subrK => ltFxl.
by exists x => //; rewrite (ge_inf hs.2)//; exists x.
Qed.


Lemma sup_ge0 (A : set R) : (forall x, A x -> 0 <= x) -> 0 <= sup A.
Proof.
move=> Ax.
have [->|/set0P[a Aa]] := eqVneq A set0; first by rewrite sup0.
have [[A0 Aub]|supA] := pselect (has_sup A).
by rewrite (le_trans (Ax _ Aa))// ub_le_sup.
by rewrite sup_out.
Qed.

Lemma has_sup_wpZl (A : set R) (a : R) : 0 <= a -> has_sup A -> has_sup [set a * x | x in A ].
Proof.
move => a0 [[x Ax] [b ub]].
split;first by exists (a*x); exists x.
exists (a * b) => _ [y Ay <-].
by rewrite ler_wpM2l //; apply ub.
Qed.

Lemma gt0_has_supZl (A : set R) (a : R) : 0 < a -> has_sup [set a * x | x in A ] -> has_sup A.
Proof.
move => a0 [[_ [x Ax _]] [b ub]].
split;first by exists x.
exists (b/a).
move => y Ay.
rewrite ler_pdivlMr // mulrC.
by apply ub;exists y.
Qed.
Lemma ge0_supZl (A : set R) (a : R) :
0 <= a -> sup [set a * x | x in A ] = a * sup A .
Proof.
move =>a0.
have [->|an0] := eqVneq a 0.
have [->| /negPf Anonempty] := eqVneq A set0; first by rewrite image_set0 sup0 mulr0.
suff -> : [set 0*x | x in A] = [set 0] by rewrite sup1 mul0r.
under eq_fun do rewrite mul0r.
by rewrite set_cst Anonempty.
have [->|/set0P Anonempty] := eqVneq A set0; first by rewrite image_set0 sup0 mulr0.
have [ex_sup | not_ex_sup] := pselect (has_sup A); last by rewrite !sup_out ?mulr0 // => -h;apply not_ex_sup; apply: gt0_has_supZl h;rewrite lt0r an0.
have [[x Ax] ub] := ex_sup.
apply /eqP;rewrite eq_le;apply /andP;split.
apply ge_sup; first by exists (a * x), x.
move => _ [x0 Axo <-].
by rewrite ler_wpM2l// ub_le_sup.
rewrite -ler_pdivlMl; last by rewrite lt0r an0.
apply ge_sup; first by apply ex_sup.
move => x0 Ax0.
rewrite ler_pdivlMl; last by rewrite lt0r an0.
rewrite ub_le_sup //; last by exists x0.
have [x1 ubx1] := ub.
exists (a * x1) => _ [x2 Ax2 <-].
by rewrite ler_wpM2l// ubx1.
Qed.

Lemma has_sup_Mn (A : set R) n :
has_sup A -> has_sup [set x *+n | x in A ].
Proof.
move => [-[] x Ax [y uby]].
split; first by exists (x *+ n), x.
exists (y *+ n).
move => _ [y0 Ay0 <-] .
by rewrite lerMn2r uby// orbT.
Qed.

Lemma sup_Mn (A : set R) n :
sup [set x *+n | x in A ] = sup A *+ n.
Proof.
rewrite -mulr_natl [X in sup X = _](_ : _ = [set n%:R * x | x in A]); first exact: ge0_supZl.
by under eq_fun do rewrite -mulr_natl.
Qed.

End Sup.

#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `inf_le`.")]
Notation le_inf := inf_le (only parsing).
#[deprecated(since="mathcomp-analysis 1.14.0", note="Renamed `sup_le`.")]
Expand Down
Loading