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Merged
motikaku merged 1 commit into from
Dec 17, 2025
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continuous_at_sorgenfrey_to_Rtopo_at_rightP #6

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12 changes: 12 additions & 0 deletions theories/showcase/sorgenfreyline.v
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Expand Up @@ -303,6 +303,18 @@ Definition continuous_at_sorgenfrey_to_Rtopo x (f : sorgenfrey -> R) :=

Let Rmetric := GRing_regular__canonical__pseudometric_structure_PseudoMetric R.

Lemma continuous_at_sorgenfrey_to_Rtopo_at_rightP (f : sorgenfrey -> Rtopo) x :
continuous_at_sorgenfrey_to_Rtopo x f <-> f z @[z --> x^'+] --> f x.
Proof.
split => [H|/(@cvgrPdist_lt _ Rmetric) rcf eps eps0].
- apply/(@cvgrPdist_lt _ Rmetric) => e e0; have [d d0 {}H] := H e e0; near=> xd.
apply/H/andP; split=> //;rewrite -ltrBlDl; near: xd; exact: nbhs_right_ltDr.
- have [d d0 {}rcf] := rcf eps eps0; exists d => // z/andP[].
rewrite le_eqVlt => /predU1P[<- _|xz zxd]; first by rewrite subrr normr0.
apply: rcf => //=; rewrite lter_distl; apply/andP.
by split; [rewrite ltrBlDr|apply: (lt_trans xz); rewrite ltrDl].
Unshelve. end_near. Qed.

Lemma continuous_at_sorgenfrey_to_RtopoP f x :
continuous_at_sorgenfrey_to_Rtopo x f <-> @continuous_at sorgenfrey Rtopo x f.
Proof.
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