Constructions involving sets of sets.

Finite Intersections

We define a structure FiniteInter which asserts that a set S of subsets of α is closed under finite intersections.

We define finiteInterClosure which, given a set S of subsets of α, is the smallest set of subsets of α which is closed under finite intersections.

finiteInterClosure S is endowed with a term of type FiniteInter using finiteInterClosure_finiteInter.

structure FiniteInter {α : Type u_1} (S : Set (Set α)) : Prop

A structure encapsulating the fact that a set of sets is closed under finite intersection.

• univ_mem : Set.univ ∈ S

  univ_mem states that Set.univ is in S.

• inter_mem : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S

  inter_mem states that any two intersections of sets in S is also in S.

theorem FiniteInter.univ_mem {α : Type u_1} {S : Set (Set α)} (self : FiniteInter S) : Set.univ ∈ S

univ_mem states that Set.univ is in S.

theorem FiniteInter.inter_mem {α : Type u_1} {S : Set (Set α)} (self : FiniteInter S) ⦃s : Set α⦄ : s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S

inter_mem states that any two intersections of sets in S is also in S.

inductive FiniteInter.finiteInterClosure {α : Type u_1} (S : Set (Set α)) : Set (Set α)

The smallest set of sets containing S which is closed under finite intersections.

• basic: ∀ {α : Type u_1} {S : Set (Set α)} {s : Set α}, s ∈ S → FiniteInter.finiteInterClosure S s
• univ: ∀ {α : Type u_1} {S : Set (Set α)}, FiniteInter.finiteInterClosure S Set.univ
• inter: ∀ {α : Type u_1} {S : Set (Set α)} {s t : Set α}, FiniteInter.finiteInterClosure S s → FiniteInter.finiteInterClosure S t → FiniteInter.finiteInterClosure S (s ∩ t)

theorem FiniteInter.finiteInterClosure_finiteInter {α : Type u_1} (S : Set (Set α)) : FiniteInter (FiniteInter.finiteInterClosure S)

theorem FiniteInter.finiteInter_mem {α : Type u_1} {S : Set (Set α)} (cond : FiniteInter S) (F : Finset (Set α)) : ↑F ⊆ S → ⋂₀ ↑F ∈ S

theorem FiniteInter.finiteInterClosure_insert {α : Type u_1} {S : Set (Set α)} {A : Set α} (cond : FiniteInter S) (P : Set α) (H : P ∈ FiniteInter.finiteInterClosure (insert A S)) : P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q

theorem FiniteInter.mk₂ {α : Type u_1} {S : Set (Set α)} (h : ∀ ⦃s : Set α⦄, s ∈ S → ∀ ⦃t : Set α⦄, t ∈ S → s ∩ t ∈ S) : FiniteInter (insert Set.univ S)