Supremum independence

In this file, we define supremum independence of indexed sets. An indexed family f : ι → α is sup-independent if, for all a, f a and the supremum of the rest are disjoint.

Main definitions

• Finset.SupIndep s f: a family of elements f are supremum independent on the finite set s.
• sSupIndep s: a set of elements are supremum independent.
• iSupIndep f: a family of elements are supremum independent.

Main statements

• In a distributive lattice, supremum independence is equivalent to pairwise disjointness:
  • Finset.supIndep_iff_pairwiseDisjoint
  • CompleteLattice.sSupIndep_iff_pairwiseDisjoint
  • CompleteLattice.iSupIndep_iff_pairwiseDisjoint
• Otherwise, supremum independence is stronger than pairwise disjointness:
  • Finset.SupIndep.pairwiseDisjoint
  • sSupIndep.pairwiseDisjoint
  • iSupIndep.pairwiseDisjoint

Implementation notes

For the finite version, we avoid the "obvious" definition ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) because erase would require decidable equality on ι.

On lattices with a bottom element, via Finset.sup

def Finset.SupIndep {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] (s : Finset ι) (f : ι → α) : Prop

Supremum independence of finite sets. We avoid the "obvious" definition using s.erase i because erase would require decidable equality on ι.

Equations

• s.SupIndep f = ∀ ⦃t : Finset ι⦄, t ⊆ s → ∀ ⦃i : ι⦄, i ∈ s → i ∉ t → Disjoint (f i) (t.sup f)

theorem Finset.supIndep_iff_disjoint_erase {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} [DecidableEq ι] : s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f)

The RHS looks like the definition of iSupIndep.

instance Finset.instDecidableSupIndepOfDecidableEq {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} [DecidableEq ι] [DecidableEq α] : Decidable (s.SupIndep f)

If both the index type and the lattice have decidable equality, then the SupIndep predicate is decidable.

TODO: speedup the definition and drop the [DecidableEq ι] assumption by iterating over the pairs (a, t) such that s = Finset.cons a t _ using something like List.eraseIdx or by generating both f i and (s.erase i).sup f in one loop over s. Yet another possible optimization is to precompute partial suprema of f over the inits and tails of the list representing s, store them in 2 Arrays, then compute each sup in 1 operation.

Equations

• Finset.instDecidableSupIndepOfDecidableEq = decidable_of_iff (∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f)) ⋯

theorem Finset.SupIndep.subset {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s t : Finset ι} {f : ι → α} (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f

@[simp]

theorem Finset.supIndep_empty {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] (f : ι → α) : ∅.SupIndep f

@[simp]

theorem Finset.supIndep_singleton {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] (i : ι) (f : ι → α) : {i}.SupIndep f

theorem Finset.SupIndep.pairwiseDisjoint {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} (hs : s.SupIndep f) : (↑s).PairwiseDisjoint f

@[deprecated Finset.SupIndep.pairwiseDisjoint (since := "2025-01-17")]

theorem Finset.sup_indep.pairwise_disjoint {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} (hs : s.SupIndep f) : (↑s).PairwiseDisjoint f

Alias of Finset.SupIndep.pairwiseDisjoint.

theorem Finset.SupIndep.le_sup_iff {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s t : Finset ι} {f : ι → α} {i : ι} (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ (i : ι), f i ≠ ⊥) : f i ≤ t.sup f ↔ i ∈ t

theorem Finset.SupIndep.antitone_fun {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f g : ι → α} (hle : ∀ x ∈ s, f x ≤ g x) (h : s.SupIndep g) : s.SupIndep f

theorem Finset.SupIndep.image {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [Lattice α] [OrderBot α] {f : ι → α} [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) : (image g s).SupIndep f

theorem Finset.supIndep_map {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [Lattice α] [OrderBot α] {f : ι → α} {s : Finset ι'} {g : ι' ↪ ι} : (map g s).SupIndep f ↔ s.SupIndep (f ∘ ⇑g)

@[simp]

theorem Finset.supIndep_pair {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {f : ι → α} [DecidableEq ι] {i j : ι} (hij : i ≠ j) : {i, j}.SupIndep f ↔ Disjoint (f i) (f j)

theorem Finset.supIndep_univ_bool {α : Type u_1} [Lattice α] [OrderBot α] (f : Bool → α) : univ.SupIndep f ↔ Disjoint (f false) (f true)

@[simp]

theorem Finset.supIndep_univ_fin_two {α : Type u_1} [Lattice α] [OrderBot α] (f : Fin 2 → α) : univ.SupIndep f ↔ Disjoint (f 0) (f 1)

@[simp]

theorem Finset.supIndep_attach {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : (s.attach.SupIndep fun (a : { x : ι // x ∈ s }) => f ↑a) ↔ s.SupIndep f

theorem Finset.SupIndep.attach {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : s.SupIndep f → s.attach.SupIndep fun (a : { x : ι // x ∈ s }) => f ↑a

Alias of the reverse direction of Finset.supIndep_attach.

theorem Finset.supIndep_iff_pairwiseDisjoint {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : s.SupIndep f ↔ (↑s).PairwiseDisjoint f

theorem Set.PairwiseDisjoint.supIndep {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} : (↑s).PairwiseDisjoint f → s.SupIndep f

Alias of the reverse direction of Finset.supIndep_iff_pairwiseDisjoint.

theorem Finset.SupIndep.sup {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun (i : ι') => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) : (s.sup g).SupIndep f

Bind operation for SupIndep.

theorem Finset.SupIndep.biUnion {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun (i : ι') => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) : (s.biUnion g).SupIndep f

Bind operation for SupIndep.

theorem Finset.SupIndep.sigma {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {β : ι → Type u_5} {s : Finset ι} {g : (i : ι) → Finset (β i)} {f : Sigma β → α} (hs : s.SupIndep fun (i : ι) => (g i).sup fun (b : β i) => f ⟨i, b⟩) (hg : ∀ i ∈ s, (g i).SupIndep fun (b : β i) => f ⟨i, b⟩) : (s.sigma g).SupIndep f

Bind operation for SupIndep.

theorem Finset.SupIndep.product {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} (hs : s.SupIndep fun (i : ι) => t.sup fun (i' : ι') => f (i, i')) (ht : t.SupIndep fun (i' : ι') => s.sup fun (i : ι) => f (i, i')) : (s ×ˢ t).SupIndep f

theorem Finset.supIndep_product_iff {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} : (s.product t).SupIndep f ↔ (s.SupIndep fun (i : ι) => t.sup fun (i' : ι') => f (i, i')) ∧ t.SupIndep fun (i' : ι') => s.sup fun (i : ι) => f (i, i')

On complete lattices via sSup

def sSupIndep {α : Type u_1} [CompleteLattice α] (s : Set α) : Prop

An independent set of elements in a complete lattice is one in which every element is disjoint from the Sup of the rest.

Equations

• sSupIndep s = ∀ ⦃a : α⦄, a ∈ s → Disjoint a (sSup (s \ {a}))

@[simp]

theorem sSupIndep_empty {α : Type u_1} [CompleteLattice α] : sSupIndep ∅

theorem sSupIndep.mono {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : sSupIndep s) {t : Set α} (hst : t ⊆ s) : sSupIndep t

theorem sSupIndep.pairwiseDisjoint {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : sSupIndep s) : s.PairwiseDisjoint id

If the elements of a set are independent, then any pair within that set is disjoint.

theorem sSupIndep_singleton {α : Type u_1} [CompleteLattice α] (a : α) : sSupIndep {a}

theorem sSupIndep_pair {α : Type u_1} [CompleteLattice α] {a b : α} (hab : a ≠ b) : sSupIndep {a, b} ↔ Disjoint a b

theorem sSupIndep.disjoint_sSup {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : sSupIndep s) {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y) : Disjoint x (sSup y)

If the elements of a set are independent, then any element is disjoint from the sSup of some subset of the rest.

def iSupIndep {ι : Sort u_5} {α : Type u_6} [CompleteLattice α] (t : ι → α) : Prop

An independent indexed family of elements in a complete lattice is one in which every element is disjoint from the iSup of the rest.

Example: an indexed family of non-zero elements in a vector space is linearly independent iff the indexed family of subspaces they generate is independent in this sense.

Example: an indexed family of submodules of a module is independent in this sense if and only the natural map from the direct sum of the submodules to the module is injective.

Equations

• iSupIndep t = ∀ (i : ι), Disjoint (t i) (⨆ (j : ι), ⨆ (_ : j ≠ i), t j)

theorem sSupIndep_iff {α : Type u_5} [CompleteLattice α] (s : Set α) : sSupIndep s ↔ iSupIndep Subtype.val

theorem iSupIndep_def {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} : iSupIndep t ↔ ∀ (i : ι), Disjoint (t i) (⨆ (j : ι), ⨆ (_ : j ≠ i), t j)

theorem iSupIndep_def' {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} : iSupIndep t ↔ ∀ (i : ι), Disjoint (t i) (sSup (t '' {j : ι | j ≠ i}))

theorem iSupIndep_def'' {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} : iSupIndep t ↔ ∀ (i : ι), Disjoint (t i) (sSup {a : α | ∃ (j : ι), j ≠ i ∧ t j = a})

@[simp]

theorem iSupIndep_empty {α : Type u_1} [CompleteLattice α] (t : Empty → α) : iSupIndep t

@[simp]

theorem iSupIndep_pempty {α : Type u_1} [CompleteLattice α] (t : PEmpty.{u_5} → α) : iSupIndep t

theorem iSupIndep.pairwiseDisjoint {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) : Pairwise (Function.onFun Disjoint t)

If the elements of a set are independent, then any pair within that set is disjoint.

theorem iSupIndep.mono {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s t : ι → α} (hs : iSupIndep s) (hst : t ≤ s) : iSupIndep t

theorem iSupIndep.comp {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {ι' : Sort u_6} {t : ι → α} {f : ι' → ι} (ht : iSupIndep t) (hf : Function.Injective f) : iSupIndep (t ∘ f)

Composing an independent indexed family with an injective function on the index results in another indepedendent indexed family.

theorem iSupIndep.comp' {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {ι' : Sort u_6} {t : ι → α} {f : ι' → ι} (ht : iSupIndep (t ∘ f)) (hf : Function.Surjective f) : iSupIndep t

theorem iSupIndep.sSupIndep_range {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) : sSupIndep (Set.range t)

@[simp]

theorem iSupIndep_ne_bot {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} : (iSupIndep fun (i : { i : ι // t i ≠ ⊥ }) => t ↑i) ↔ iSupIndep t

theorem iSupIndep.injOn {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) : Set.InjOn t {i : ι | t i ≠ ⊥}

theorem iSupIndep.injective {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) (h_ne_bot : ∀ (i : ι), t i ≠ ⊥) : Function.Injective t

theorem iSupIndep_pair {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} {i j : ι} (hij : i ≠ j) (huniv : ∀ (k : ι), k = i ∨ k = j) : iSupIndep t ↔ Disjoint (t i) (t j)

theorem iSupIndep.map_orderIso {ι : Sort u_5} {α : Type u_6} {β : Type u_7} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) {a : ι → α} (ha : iSupIndep a) : iSupIndep (⇑f ∘ a)

Composing an independent indexed family with an order isomorphism on the elements results in another independent indexed family.

@[simp]

theorem iSupIndep_map_orderIso_iff {ι : Sort u_5} {α : Type u_6} {β : Type u_7} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) {a : ι → α} : iSupIndep (⇑f ∘ a) ↔ iSupIndep a

theorem iSupIndep.disjoint_biSup {ι : Type u_5} {α : Type u_6} [CompleteLattice α] {t : ι → α} (ht : iSupIndep t) {x : ι} {y : Set ι} (hx : x ∉ y) : Disjoint (t x) (⨆ i ∈ y, t i)

If the elements of a set are independent, then any element is disjoint from the iSup of some subset of the rest.

theorem iSupIndep.of_coe_Iic_comp {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {a : α} {t : ι → ↑(Set.Iic a)} (ht : iSupIndep (Subtype.val ∘ t)) : iSupIndep t

theorem iSupIndep_iff_supIndep {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s : Finset ι} {f : ι → α} : iSupIndep (f ∘ Subtype.val) ↔ s.SupIndep f

theorem iSupIndep.supIndep {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s : Finset ι} {f : ι → α} : iSupIndep (f ∘ Subtype.val) → s.SupIndep f

Alias of the forward direction of iSupIndep_iff_supIndep.

theorem Finset.SupIndep.independent {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s : Finset ι} {f : ι → α} : s.SupIndep f → iSupIndep (f ∘ Subtype.val)

Alias of the reverse direction of iSupIndep_iff_supIndep.

theorem iSupIndep.supIndep' {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {f : ι → α} (s : Finset ι) (h : iSupIndep f) : s.SupIndep f

theorem iSupIndep_iff_supIndep_univ {α : Type u_1} {ι : Type u_3} [CompleteLattice α] [Fintype ι] {f : ι → α} : iSupIndep f ↔ Finset.univ.SupIndep f

A variant of CompleteLattice.iSupIndep_iff_supIndep for Fintypes.

theorem Finset.SupIndep.iSupIndep_of_univ {α : Type u_1} {ι : Type u_3} [CompleteLattice α] [Fintype ι] {f : ι → α} : univ.SupIndep f → iSupIndep f

Alias of the reverse direction of iSupIndep_iff_supIndep_univ.

---

A variant of CompleteLattice.iSupIndep_iff_supIndep for Fintypes.

theorem iSupIndep.sup_indep_univ {α : Type u_1} {ι : Type u_3} [CompleteLattice α] [Fintype ι] {f : ι → α} : iSupIndep f → Finset.univ.SupIndep f

Alias of the forward direction of iSupIndep_iff_supIndep_univ.

---

A variant of CompleteLattice.iSupIndep_iff_supIndep for Fintypes.

theorem sSupIndep_iff_pairwiseDisjoint {α : Type u_1} [Order.Frame α] {s : Set α} : sSupIndep s ↔ s.PairwiseDisjoint id

theorem Set.PairwiseDisjoint.sSupIndep {α : Type u_1} [Order.Frame α] {s : Set α} : s.PairwiseDisjoint id → _root_.sSupIndep s

Alias of the reverse direction of sSupIndep_iff_pairwiseDisjoint.

theorem iSupIndep_iff_pairwiseDisjoint {α : Type u_1} {ι : Type u_3} [Order.Frame α] {f : ι → α} : iSupIndep f ↔ Pairwise (Function.onFun Disjoint f)