Theorems · Theorem · order theory

IncidenceAlgebra.moebius_inversion_bot

∀ {𝕜 : Type u_2} {α : Type u_5} [inst : Ring 𝕜] [inst_1 : PartialOrder α] [inst_2 : OrderBot α]
  [inst_3 : LocallyFiniteOrder α] [inst_4 : DecidableEq α] (f g : α → 𝕜),
  (∀ (x : α), g x = ∑ y ≤ x, f y) → ∀ (x : α), f x = ∑ y ≤ x, (IncidenceAlgebra.mu 𝕜) y x * g y

A general form of Möbius inversion. Based on lemma 2.1.3 of Incidence Algebras by Spiegel and O'Donnell.

Defined in

Mathlib.Combinatorics.Enumerative.IncidenceAlgebra

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingPartialOrderOrderBotLocallyFiniteOrderDecidableEq