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affeldt-aist committed Mar 28, 2024
1 parent ee75502 commit 913a929
Showing 1 changed file with 4 additions and 4 deletions.
8 changes: 4 additions & 4 deletions probability/proba.v
Original file line number Diff line number Diff line change
Expand Up @@ -1272,7 +1272,7 @@ Notation R := real_realType.

Variables (A I : finType) (k : nat) (d : R.-fdist A) (E : I -> {set A}).

Definition kwide_inde := forall (J : {set I}), (#|J| <= k)%nat ->
Definition kwise_inde := forall (J : {set I}), (#|J| <= k)%nat ->
Pr d (\bigcap_(i in J) E i) = \prod_(i in J) Pr d (E i).

End k_wise_independence.
Expand All @@ -1282,7 +1282,7 @@ Notation R := real_realType.

Variables (A I : finType) (d : R.-fdist A) (E : I -> {set A}).

Definition pairwise_inde := @kwide_inde A I 2%nat d E.
Definition pairwise_inde := @kwise_inde A I 2%nat d E.

Lemma pairwise_indeE :
pairwise_inde <-> (forall i j, i != j -> inde_events d (E i) (E j)).
Expand Down Expand Up @@ -1311,7 +1311,7 @@ Notation R := real_realType.

Variables (A I : finType) (d : R.-fdist A) (E : I -> {set A}).

Definition mutual_inde := (forall k, @kwide_inde A I k.+1 d E).
Definition mutual_inde := (forall k, @kwise_inde A I k.+1 d E).

Lemma mutual_indeE :
mutual_inde <-> (forall J : {set I}, J \subset I ->
Expand All @@ -1324,7 +1324,7 @@ rewrite /mutual_inde; split => [H J JI|H k J JI].
by rewrite H //; apply/subsetP => i ij; rewrite inE.
Qed.

Lemma mutual_indeE' : #|I| != O -> mutual_inde <-> kwide_inde #|I| d E.
Lemma mutual_indeE' : #|I| != O -> mutual_inde <-> kwise_inde #|I| d E.
Proof.
move=> I0.
rewrite /mutual_inde; split => [H J JI|].
Expand Down

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