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Lemma nondecreasing_right_continuousP now has e: {posnum R} #764

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Lemma nondecreasing_right_continuousP now has e: {posnum R} #764

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8 changes: 4 additions & 4 deletions theories/lebesgue_stieltjes_measure.v
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Expand Up @@ -39,12 +39,12 @@ HB.structure Definition CF (R : realType) := { f of isCumulative R f }.
Arguments cumulative_is_nondecreasing {R} _.
Arguments cumulative_is_right_continuous {R} _.

Lemma nondecreasing_right_continuousP (R : realType) (a : R) (e : R)
Lemma nondecreasing_right_continuousP (R : realType) (a : R) (e : {posnum R})
(f : cumulative R) :
e > 0 -> exists d : {posnum R}, f (a + d%:num) <= f a + e.
exists d : {posnum R}, f (a + d%:num) <= f a + (e%:num).
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@proux01 proux01 Oct 7, 2022

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Suggested change
exists d : {posnum R}, f (a + d%:num) <= f a + (e%:num).
exists d : {posnum R}, f (a + d%:num) <= f a + e%:num.

Proof.
move=> e0; move: (cumulative_is_right_continuous f) => /(_ a)/(@cvg_dist _ [normedModType R of R^o]).
move=> /(_ _ e0)[] _ /posnumP[d] => h.
move: (cumulative_is_right_continuous f) => /(_ a) /(@cvg_dist _ [normedModType R of R^o]).
move=> /(_ _ (gt0 e))[] _ /posnumP[d] => h.
exists (PosNum [gt0 of (d%:num / 2)]) => //=.
move: h => /(_ (a + d%:num / 2)) /=.
rewrite opprD addrA subrr distrC subr0 ger0_norm //.
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