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Merged
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Apr 30, 2025
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Add Lemma cvge_ninftyP #1591

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33abd4c
add Lemma cvge_ninftyP
Yosuke-Ito-345 Apr 27, 2025
7126b1d
add to CHANGELOG
Yosuke-Ito-345 Apr 27, 2025
e6d8d3d
fix
affeldt-aist Apr 30, 2025
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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -36,6 +36,8 @@
- in `exp.v`:
+ lemmas `lnNy`, `powR_cvg0`, `derivable_powR`, `powR_derive1`
+ Instance `is_derive1_powR`
- in `realfun.v`:
+ lemma `cvge_ninftyP`

### Changed

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13 changes: 13 additions & 0 deletions theories/realfun.v
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Expand Up @@ -374,6 +374,19 @@ move /not_near_inftyP => eps_sep; exists (PosNum eps_gt0) => A /=.
by case: (eps_sep _ (num_real A)) => x ? ?; exists x.
Unshelve. all: by end_near. Qed.

(* NB: almost the same proof as cvg_ninftyP *)
Lemma cvge_ninftyP (f : R -> \bar R) (l : \bar R) :
f x @[x --> -oo] --> l <->
forall u : R^nat, (u n @[n --> \oo] --> -oo) -> f (u n) @[n --> \oo] --> l.
Proof.
rewrite cvgNy_compNP cvge_pinftyP/= (@bij_forall R^nat _ -%R)//.
have u_opp (u : R^nat) :
((- u) n @[n --> \oo] --> +oo) = (u n @[n --> \oo] --> -oo).
by rewrite propeqE cvgNry.
by under eq_forall do
(rewrite u_opp; under (eq_cvg _ (nbhs l)) do rewrite opprK).
Qed.

End cvge_fun_dvg_seq.

Section fun_cvg_realType.
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