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Add generic facts about einfs and limn_einf #2107

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Add generic facts about einfs and limn_einf #2107

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33 changes: 33 additions & 0 deletions theories/sequences.v
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Original file line number Diff line number Diff line change
Expand Up @@ -2419,6 +2419,20 @@ by rewrite [in RHS](_ : u = -%E \o -%E \o u);
rewrite ?esupsN funeqE => n /=; rewrite oppeK.
Qed.

Lemma einfs_le (* {R : realType}*) u n m :
(n <= m)%N -> (einfs u n <= u m)%E.
Proof. by move=> nm; apply: ereal_inf_lbound; exists m; [exact: nm | by []]. Qed.

Lemma einfs_lift u p n :
einfs (fun k => u (k + p)%N) n = einfs u (n + p)%N.
Proof.
congr (ereal_inf _); apply/seteqP; split => _ /= [k /= nk] <-.
- by exists (k + p)%N => //=; rewrite leq_add2r.
- have pk : (p <= k)%N by apply: leq_trans nk; exact: leq_addl.
exists (k - p)%N => /=; last by rewrite subnK.
by rewrite -(leq_add2r p) subnK.
Qed.

Lemma nonincreasing_esups u : nonincreasing_seq (esups u).
Proof.
move=> m n mn; apply: ereal_sup_le => _ /= [k nk <-]; exists k => //=.
Expand Down Expand Up @@ -2642,6 +2656,25 @@ move=> /cvg_ex[l ul]; have [_ ->] := cvg_limn_einf_sup ul.
by move/cvg_lim : ul => ->.
Qed.

Lemma limn_einf_lift u p :
limn_einf (fun n => u (n + p)%N) = limn_einf u.
Proof.
rewrite !limn_einf_lim.
have -> : einfs (fun k => u (k + p)%N) = (fun n => einfs u (n + p)%N).
by apply/funext => n; exact: einfs_lift.
by apply/cvg_lim => //; rewrite (cvg_shiftn p (einfs u)); exact: is_cvg_einfs.
Qed.

Lemma limn_einf_bump u :
limn_einf (fun n => u n.+1) = limn_einf u.
Proof.
rewrite -(limn_einf_lift u 1); congr limn_einf.
by apply/funext => n; rewrite addn1.
Qed.

Lemma limn_einf_cst (c : \bar R) : limn_einf (fun=> c) = c.
Proof. by rewrite is_cvg_limn_einfE ?lim_cst//; exact: is_cvg_cst. Qed.

End lim_esup_inf.

Lemma geometric_le_lim {R : realType} (n : nat) (a x : R) :
Expand Down
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