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add lemma lt0_adde #1715

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Sep 18, 2025
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add lemma lt0_adde #1715

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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -13,6 +13,9 @@
- in `num_normedtype.v`:
+ lemma `nbhs_infty_gtr`

- in `constructive_ereal.v`:
+ lemma `lt0_adde`

### Changed

- in `lebesgue_stieltjes_measure.v` specialized from `numFieldType` to `realFieldType`:
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11 changes: 11 additions & 0 deletions reals/constructive_ereal.v
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Expand Up @@ -1743,6 +1743,17 @@ Qed.
Lemma lte_add_pinfty x y : x < +oo -> y < +oo -> x + y < +oo.
Proof. by move: x y => -[r [r'| |]| |] // ? ?; rewrite -EFinD ltry. Qed.

Lemma lt0_adde x y : x + y < 0 -> (x < 0) || (y < 0).
Proof.
move: x y => [x| |] [y| |]//; rewrite ?lee_fin ?lte_fin.
- rewrite !ltNge -negb_and; apply: contra.
by move=> /andP[? ?]; rewrite addr_ge0.
- by move=> _; rewrite ltNyr orbT.
- by move=> _; rewrite ltNyr.
- by move=> _; rewrite ltNy0.
- by rewrite ltNy0.
Qed.

Lemma lte_sum_pinfty I (s : seq I) (P : pred I) (f : I -> \bar R) :
(forall i, P i -> f i < +oo) -> \sum_(i <- s | P i) f i < +oo.
Proof.
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