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EFinf is actually er_map
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affeldt-aist committed Apr 26, 2023
1 parent 6be8d39 commit a7fd96f
Showing 1 changed file with 5 additions and 8 deletions.
13 changes: 5 additions & 8 deletions theories/lebesgue_stieltjes_measure.v
Original file line number Diff line number Diff line change
Expand Up @@ -82,11 +82,8 @@ move=> S21 S12 S1i; rewrite ler_oppl opprK le_sup// ?has_inf_supN//.
exact/nonemptyN.
Qed.

Definition EFinf {R : numDomainType} (f : R -> R) : \bar R -> \bar R :=
fun x => if x is r%:E then (f r)%:E else x.

Lemma nondecreasing_EFinf (R : realDomainType) (f : R -> R) :
{homo f : x y / (x <= y)%R} -> {homo EFinf f : x y / (x <= y)%E}.
Lemma nondecreasing_er_map (R : realDomainType) (f : R -> R) :
{homo f : x y / (x <= y)%R} -> {homo er_map f : x y / (x <= y)%E}.
Proof.
move=> ndf.
by move=> [r| |] [l| |]//=; rewrite ?leey ?leNye// !lee_fin; exact: ndf.
Expand All @@ -95,7 +92,7 @@ Qed.
Section hlength.
Local Open Scope ereal_scope.
Variables (R : realType) (f : R -> R).
Let g : \bar R -> \bar R := EFinf f.
Let g : \bar R -> \bar R := er_map f.

Implicit Types i j : interval R.
Definition itvs : Type := R.
Expand Down Expand Up @@ -179,7 +176,7 @@ Qed.
Lemma hlength_ge0 (ndf : {homo f : x y / (x <= y)%R}) i : 0 <= hlength [set` i].
Proof.
rewrite hlength_itv; case: ifPn => //; case: (i.1 : \bar _) => [r| |].
- by rewrite suber_ge0// => /ltW /(nondecreasing_EFinf ndf).
- by rewrite suber_ge0// => /ltW /(nondecreasing_er_map ndf).
- by rewrite ltNge leey.
- by case: (i.2 : \bar _) => //= [r _]; rewrite leey.
Qed.
Expand Down Expand Up @@ -504,7 +501,7 @@ Variables (d : measure_display) (R : realType) (f : cumulative R).

Let m := [the measure _ _ of lebesgue_stieltjes_measure d f].

Let g : \bar R -> \bar R := EFinf f.
Let g : \bar R -> \bar R := er_map f.

Let lebesgue_stieltjes_measure_itvoc (a b : R) :
(m `]a, b] = hlength f `]a, b])%classic.
Expand Down

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