FormalConjecturesForMathlib.Combinatorics.SetFamily.UnionFree

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module


public import FormalConjecturesForMathlib.Data.Set.Triplewise
public import Mathlib.Data.Finset.Lattice.Basic

@[expose] public section

Union-free families

A family of sets is union-free if no member is the union of two other distinct members.

namespace Finset

A family F of finsets is union-free if there are no solutions to A ∪ B = C with A, B, C distinct members of F.

def UnionFree {α : Type*} [DecidableEq α] (F : Finset (Finset α)) : Prop :=
  (F : Set (Finset α)).Triplewise fun A B C ↦ A ∪ B ≠ C

A family F of finsets is subfamily union-free if no member is the union of a nonempty subfamily of the others. This is the arbitrary-subfamily strengthening of UnionFree, which only rules out unions of two members.

def SubfamilyUnionFree {α : Type*} [DecidableEq α] (F : Finset (Finset α)) : Prop :=
  ∀ A ∈ F, ∀ T ⊆ F.erase A, T.Nonempty → T.sup id ≠ A

end Finset