Skip to content

Proof of interchange_psum using esum #2061

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
lyonel2017 wants to merge 2 commits into
base: master
Choose a base branch
from
+321 −14
Open

Proof of interchange_psum using esum #2061

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
68b3cfc
Add set of lemmas for esum
lyonel2017 Jul 29, 2026
374a39c
Add proof for interchange_psum using esum
lyonel2017 Jun 30, 2026
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
43 changes: 41 additions & 2 deletions experimental_reals/realsum.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -721,6 +721,11 @@ case=> J ->; exists J.
by under [in RHS]eq_bigr do rewrite normr_id.
Qed.

Lemma esum_psum_abs S :
summable \`| S | ->
(PosSum.psum (fun y => S y))%:E = \esum_(y in [set: T]) `|S y|%:E.
Proof. by move => ?; rewrite -psum_abs -esum_psum. Qed.

Lemma eq_psum_abs S1 S2 : \`|S1| =1 \`|S2| -> PosSum.psum S1 = PosSum.psum S2.
Proof.
by move=> eqS; rewrite -[LHS]psum_abs -[RHS]psum_abs; apply/eq_psum.
Expand Down Expand Up @@ -1104,8 +1109,42 @@ Qed.

End PSumInterchange.

#[deprecated(since="1.17.0", note="use `interchange_psum` instead")]
Notation __admitted__interchange_psum := interchange_psum (only parsing).
(* -------------------------------------------------------------------- *)
Section PSumInterchangeEsum.
Context {R : realType} {T U : choiceType} (S: T -> U -> R).
Hypothesis H1 : (forall x, summable (S x)).
Hypothesis H2 : summable (PosSum.psum \o S).

Lemma esum_psum_abs2:
(PosSum.psum (fun x => PosSum.psum (fun y => S x y)))%:E =
\esum_(x in [set: T]) \esum_(y in [set: U]) `|S x y|%:E.
Proof.
rewrite -esum_psum//; first by move=> x; exact: ge0_psum.
by apply: eq_esum => x _; rewrite esum_psum_abs // summable_abs.
Qed.

Lemma interchange_psum_alt :
PosSum.psum (PosSum.psum \o S) =
PosSum.psum (fun y => PosSum.psum (S ^~ y)).
Proof.
apply: EFin_inj; rewrite esum_psum_abs2 exchange_esum.
+ by move => ? ?; rewrite lee_fin.
have summable_y y : summable (fun x : T => `|S x y|).
+ apply: (le_summable (F2 := fun x => PosSum.psum (fun y' => S x y'))) => // x.
by apply/andP; split; [exact: normr_ge0 | exact: ger1_psum y (H1 x)].
under eq_esum => y _ do rewrite -esum_psum_abs //.
rewrite -esum_psum//.
+ by move=> y; exact: ge0_psum.
apply /esum_summableP; rewrite /esum.summable.
apply: (@le_lt_trans _ _ (\esum_(y in [set: U]) (PosSum.psum (fun x => S x y))%:E)).
+ by apply le_esum=> y _; rewrite /comp/= ger0_norm// ge0_psum.
under eq_esum => y _ do rewrite esum_psum_abs//.
rewrite exchange_esum.
+ by move => ? ?; rewrite lee_fin.
+ by rewrite -esum_psum_abs2 ltey.
Qed.

End PSumInterchangeEsum.

(* -------------------------------------------------------------------- *)
Section SumTheory.
Expand Down
Loading
Loading