Documentation

Mathlib.Order.Closure

Closure operators between preorders #

We define (bundled) closure operators on a preorder as monotone (increasing), extensive (inflationary) and idempotent functions. We define closed elements for the operator as elements which are fixed by it.

Lower adjoints to a function between preorders u : β → α allow to generalise closure operators to situations where the closure operator we are dealing with naturally decomposes as u ∘ l where l is a worthy function to have on its own. Typical examples include l : Set G → Subgroup G := Subgroup.closure, u : Subgroup G → Set G := (↑), where G is a group. This shows there is a close connection between closure operators, lower adjoints and Galois connections/insertions: every Galois connection induces a lower adjoint which itself induces a closure operator by composition (see GaloisConnection.lowerAdjoint and LowerAdjoint.closureOperator), and every closure operator on a partial order induces a Galois insertion from the set of closed elements to the underlying type (see ClosureOperator.gi).

Main definitions #

Implementation details #

Although LowerAdjoint is technically a generalisation of ClosureOperator (by defining toFun := id), it is desirable to have both as otherwise ids would be carried all over the place when using concrete closure operators such as ConvexHull.

LowerAdjoint really is a semibundled structure version of GaloisConnection.

References #

Closure operator #

structure ClosureOperator (α : Type u_1) [Preorder α] extends OrderHom :
Type u_1

A closure operator on the preorder α is a monotone function which is extensive (every x is less than its closure) and idempotent.

  • toFun : αα
  • monotone' : Monotone self.toFun
  • le_closure' : ∀ (x : α), x self.toFun x

    An element is less than or equal its closure

  • idempotent' : ∀ (x : α), self.toFun (self.toFun x) = self.toFun x

    Closures are idempotent

  • IsClosed : αProp

    Predicate for an element to be closed.

    By default, this is defined as c.IsClosed x := (c x = x) (see isClosed_iff). We allow an override to fix definitional equalities.

  • isClosed_iff : ∀ {x : α}, self.IsClosed x self.toFun x = x
Instances For
    theorem ClosureOperator.le_closure' {α : Type u_1} [Preorder α] (self : ClosureOperator α) (x : α) :
    x self.toFun x

    An element is less than or equal its closure

    theorem ClosureOperator.idempotent' {α : Type u_1} [Preorder α] (self : ClosureOperator α) (x : α) :
    self.toFun (self.toFun x) = self.toFun x

    Closures are idempotent

    theorem ClosureOperator.isClosed_iff {α : Type u_1} [Preorder α] (self : ClosureOperator α) {x : α} :
    self.IsClosed x self.toFun x = x
    instance ClosureOperator.instFunLike (α : Type u_1) [Preorder α] :
    Equations
    Equations
    • =
    @[simp]
    theorem ClosureOperator.conjBy_apply {α : Type u_4} {β : Type u_5} [Preorder α] [Preorder β] (c : ClosureOperator α) (e : α ≃o β) (a : β) :
    (c.conjBy e) a = (e.conj c) a
    def ClosureOperator.conjBy {α : Type u_4} {β : Type u_5} [Preorder α] [Preorder β] (c : ClosureOperator α) (e : α ≃o β) :

    If c is a closure operator on α and e an order-isomorphism between α and β then e ∘ c ∘ e⁻¹ is a closure operator on β.

    Equations
    • c.conjBy e = { toFun := (e.conj c), monotone' := , le_closure' := , idempotent' := , IsClosed := fun (b : β) => c.IsClosed (e.symm b), isClosed_iff := }
    Instances For
      theorem ClosureOperator.conjBy_refl {α : Type u_4} [Preorder α] (c : ClosureOperator α) :
      c.conjBy (OrderIso.refl α) = c
      theorem ClosureOperator.conjBy_trans {α : Type u_4} {β : Type u_5} {γ : Type u_6} [Preorder α] [Preorder β] [Preorder γ] (e₁ : α ≃o β) (e₂ : β ≃o γ) (c : ClosureOperator α) :
      c.conjBy (e₁.trans e₂) = (c.conjBy e₁).conjBy e₂
      @[simp]
      theorem ClosureOperator.id_isClosed (α : Type u_1) [PartialOrder α] :
      ∀ (x : α), (ClosureOperator.id α).IsClosed x = True
      @[simp]
      theorem ClosureOperator.id_apply (α : Type u_1) [PartialOrder α] (a : α) :

      The identity function as a closure operator.

      Equations
      • ClosureOperator.id α = { toOrderHom := OrderHom.id, le_closure' := , idempotent' := , IsClosed := fun (x : α) => True, isClosed_iff := }
      Instances For
        theorem ClosureOperator.ext_iff {α : Type u_1} [PartialOrder α] {c₁ : ClosureOperator α} {c₂ : ClosureOperator α} :
        c₁ = c₂ ∀ (x : α), c₁ x = c₂ x
        theorem ClosureOperator.ext {α : Type u_1} [PartialOrder α] (c₁ : ClosureOperator α) (c₂ : ClosureOperator α) :
        (∀ (x : α), c₁ x = c₂ x)c₁ = c₂
        @[simp]
        theorem ClosureOperator.mk'_apply {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) :
        ∀ (a : α), (ClosureOperator.mk' f hf₁ hf₂ hf₃) a = f a
        @[simp]
        theorem ClosureOperator.mk'_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) (x : α) :
        (ClosureOperator.mk' f hf₁ hf₂ hf₃).IsClosed x = (f x = x)
        def ClosureOperator.mk' {α : Type u_1} [PartialOrder α] (f : αα) (hf₁ : Monotone f) (hf₂ : ∀ (x : α), x f x) (hf₃ : ∀ (x : α), f (f x) f x) :

        Constructor for a closure operator using the weaker idempotency axiom: f (f x) ≤ f x.

        Equations
        • ClosureOperator.mk' f hf₁ hf₂ hf₃ = { toFun := f, monotone' := hf₁, le_closure' := hf₂, idempotent' := , IsClosed := fun (x : α) => f x = x, isClosed_iff := }
        Instances For
          @[simp]
          theorem ClosureOperator.mk₂_apply {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) :
          ∀ (a : α), (ClosureOperator.mk₂ f hf hmin) a = f a
          @[simp]
          theorem ClosureOperator.mk₂_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) (x : α) :
          (ClosureOperator.mk₂ f hf hmin).IsClosed x = (f x = x)
          def ClosureOperator.mk₂ {α : Type u_1} [PartialOrder α] (f : αα) (hf : ∀ (x : α), x f x) (hmin : ∀ ⦃x y : α⦄, x f yf x f y) :

          Convenience constructor for a closure operator using the weaker minimality axiom: x ≤ f y → f x ≤ f y, which is sometimes easier to prove in practice.

          Equations
          • ClosureOperator.mk₂ f hf hmin = { toFun := f, monotone' := , le_closure' := hf, idempotent' := , IsClosed := fun (x : α) => f x = x, isClosed_iff := }
          Instances For
            @[simp]
            theorem ClosureOperator.ofPred_apply {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :
            ∀ (a : α), (ClosureOperator.ofPred f p hf hfp hmin) a = f a
            @[simp]
            theorem ClosureOperator.ofPred_isClosed {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :
            ∀ (a : α), (ClosureOperator.ofPred f p hf hfp hmin).IsClosed a = p a
            def ClosureOperator.ofPred {α : Type u_1} [PartialOrder α] (f : αα) (p : αProp) (hf : ∀ (x : α), x f x) (hfp : ∀ (x : α), p (f x)) (hmin : ∀ ⦃x y : α⦄, x yp yf x y) :

            Construct a closure operator from an inflationary function f and a "closedness" predicate p witnessing minimality of f x among closed elements greater than x.

            Equations
            Instances For
              theorem ClosureOperator.le_closure {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
              x c x

              Every element is less than its closure. This property is sometimes referred to as extensivity or inflationarity.

              @[simp]
              theorem ClosureOperator.idempotent {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
              c (c x) = c x
              @[simp]
              theorem ClosureOperator.isClosed_closure {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
              c.IsClosed (c x)
              @[reducible, inline]
              abbrev ClosureOperator.Closeds {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
              Type (max 0 u_1)

              The type of elements closed under a closure operator.

              Equations
              • c.Closeds = { x : α // c.IsClosed x }
              Instances For
                def ClosureOperator.toCloseds {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) (x : α) :
                c.Closeds

                Send an element to a closed element (by taking the closure).

                Equations
                • c.toCloseds x = c x,
                Instances For
                  theorem ClosureOperator.IsClosed.closure_eq {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} :
                  c.IsClosed xc x = x
                  theorem ClosureOperator.isClosed_iff_closure_le {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} :
                  c.IsClosed x c x x
                  theorem ClosureOperator.setOf_isClosed_eq_range_closure {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} :
                  {x : α | c.IsClosed x} = Set.range c

                  The set of closed elements for c is exactly its range.

                  theorem ClosureOperator.le_closure_iff {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} :
                  x c y c x c y
                  @[simp]
                  theorem ClosureOperator.IsClosed.closure_le_iff {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} (hy : c.IsClosed y) :
                  c x y x y
                  theorem ClosureOperator.closure_min {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} {x : α} {y : α} (hxy : x y) (hy : c.IsClosed y) :
                  c x y
                  theorem ClosureOperator.closure_isGLB {α : Type u_1} [PartialOrder α] {c : ClosureOperator α} (x : α) :
                  IsGLB {y : α | x y c.IsClosed y} (c x)
                  theorem ClosureOperator.ext_isClosed {α : Type u_1} [PartialOrder α] (c₁ : ClosureOperator α) (c₂ : ClosureOperator α) (h : ∀ (x : α), c₁.IsClosed x c₂.IsClosed x) :
                  c₁ = c₂
                  theorem ClosureOperator.eq_ofPred_closed {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
                  c = ClosureOperator.ofPred (⇑c) c.IsClosed

                  A closure operator is equal to the closure operator obtained by feeding c.closed into the ofPred constructor.

                  @[simp]
                  @[simp]
                  theorem ClosureOperator.isClosed_top {α : Type u_1} [PartialOrder α] [OrderTop α] (c : ClosureOperator α) :
                  c.IsClosed
                  theorem ClosureOperator.closure_inf_le {α : Type u_1} [SemilatticeInf α] (c : ClosureOperator α) (x : α) (y : α) :
                  c (x y) c x c y
                  theorem ClosureOperator.closure_sup_closure_le {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                  c x c y c (x y)
                  theorem ClosureOperator.closure_sup_closure_left {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                  c (c x y) = c (x y)
                  theorem ClosureOperator.closure_sup_closure_right {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                  c (x c y) = c (x y)
                  theorem ClosureOperator.closure_sup_closure {α : Type u_1} [SemilatticeSup α] (c : ClosureOperator α) (x : α) (y : α) :
                  c (c x c y) = c (x y)
                  @[simp]
                  theorem ClosureOperator.ofCompletePred_apply {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (∀ as, p a)p (sInf s)) (a : α) :
                  (ClosureOperator.ofCompletePred p hsinf) a = ⨅ (b : { b : α // a b p b }), b
                  @[simp]
                  theorem ClosureOperator.ofCompletePred_isClosed {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (∀ as, p a)p (sInf s)) :
                  ∀ (a : α), (ClosureOperator.ofCompletePred p hsinf).IsClosed a = p a
                  def ClosureOperator.ofCompletePred {α : Type u_1} [CompleteLattice α] (p : αProp) (hsinf : ∀ (s : Set α), (∀ as, p a)p (sInf s)) :

                  Define a closure operator from a predicate that's preserved under infima.

                  Equations
                  Instances For
                    theorem ClosureOperator.sInf_isClosed {α : Type u_1} [CompleteLattice α] {c : ClosureOperator α} {S : Set α} (H : xS, c.IsClosed x) :
                    c.IsClosed (sInf S)
                    @[simp]
                    theorem ClosureOperator.closure_iSup_closure {α : Type u_1} {ι : Sort u_2} [CompleteLattice α] (c : ClosureOperator α) (f : ια) :
                    c (⨆ (i : ι), c (f i)) = c (⨆ (i : ι), f i)
                    @[simp]
                    theorem ClosureOperator.closure_iSup₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} [CompleteLattice α] (c : ClosureOperator α) (f : (i : ι) → κ iα) :
                    c (⨆ (i : ι), ⨆ (j : κ i), c (f i j)) = c (⨆ (i : ι), ⨆ (j : κ i), f i j)
                    @[simp]
                    theorem OrderIso.equivClosureOperator_symm_apply {α : Type u_4} {β : Type u_5} [Preorder α] [Preorder β] (e : α ≃o β) (c : ClosureOperator β) :
                    e.equivClosureOperator.symm c = c.conjBy e.symm
                    @[simp]
                    theorem OrderIso.equivClosureOperator_apply {α : Type u_4} {β : Type u_5} [Preorder α] [Preorder β] (e : α ≃o β) (c : ClosureOperator α) :
                    e.equivClosureOperator c = c.conjBy e
                    def OrderIso.equivClosureOperator {α : Type u_4} {β : Type u_5} [Preorder α] [Preorder β] (e : α ≃o β) :

                    Conjugating ClosureOperators on α and on β by a fixed isomorphism e : α ≃o β gives an equivalence ClosureOperator α ≃ ClosureOperator β.

                    Equations
                    • e.equivClosureOperator = { toFun := fun (c : ClosureOperator α) => c.conjBy e, invFun := fun (c : ClosureOperator β) => c.conjBy e.symm, left_inv := , right_inv := }
                    Instances For

                      Lower adjoint #

                      structure LowerAdjoint {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] (u : βα) :
                      Type (max u_1 u_4)

                      A lower adjoint of u on the preorder α is a function l such that l and u form a Galois connection. It allows us to define closure operators whose output does not match the input. In practice, u is often (↑) : β → α.

                      • toFun : αβ

                        The underlying function

                      • gc' : GaloisConnection self.toFun u

                        The underlying function is a lower adjoint.

                      Instances For
                        theorem LowerAdjoint.gc' {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (self : LowerAdjoint u) :
                        GaloisConnection self.toFun u

                        The underlying function is a lower adjoint.

                        @[simp]
                        theorem LowerAdjoint.id_toFun (α : Type u_1) [Preorder α] (x : α) :
                        (LowerAdjoint.id α).toFun x = x
                        def LowerAdjoint.id (α : Type u_1) [Preorder α] :

                        The identity function as a lower adjoint to itself.

                        Equations
                        Instances For
                          Equations
                          instance LowerAdjoint.instCoeFunForall {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} :
                          CoeFun (LowerAdjoint u) fun (x : LowerAdjoint u) => αβ
                          Equations
                          • LowerAdjoint.instCoeFunForall = { coe := LowerAdjoint.toFun }
                          theorem LowerAdjoint.gc {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                          theorem LowerAdjoint.ext_iff {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} {l₁ : LowerAdjoint u} {l₂ : LowerAdjoint u} :
                          l₁ = l₂ l₁.toFun = l₂.toFun
                          theorem LowerAdjoint.ext {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l₁ : LowerAdjoint u) (l₂ : LowerAdjoint u) :
                          l₁.toFun = l₂.toFunl₁ = l₂
                          theorem LowerAdjoint.monotone {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                          Monotone (u l.toFun)
                          theorem LowerAdjoint.le_closure {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          x u (l.toFun x)

                          Every element is less than its closure. This property is sometimes referred to as extensivity or inflationarity.

                          @[simp]
                          theorem LowerAdjoint.closureOperator_isClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          l.closureOperator.IsClosed x = (u (l.toFun x) = x)
                          @[simp]
                          theorem LowerAdjoint.closureOperator_apply {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                          l.closureOperator x = u (l.toFun x)
                          def LowerAdjoint.closureOperator {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :

                          Every lower adjoint induces a closure operator given by the composition. This is the partial order version of the statement that every adjunction induces a monad.

                          Equations
                          • l.closureOperator = { toFun := fun (x : α) => u (l.toFun x), monotone' := , le_closure' := , idempotent' := , IsClosed := fun (x : α) => u (l.toFun x) = x, isClosed_iff := }
                          Instances For
                            theorem LowerAdjoint.idempotent {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                            u (l.toFun (u (l.toFun x))) = u (l.toFun x)
                            theorem LowerAdjoint.le_closure_iff {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                            x u (l.toFun y) u (l.toFun x) u (l.toFun y)
                            def LowerAdjoint.closed {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                            Set α

                            An element x is closed for l : LowerAdjoint u if it is a fixed point: u (l x) = x

                            Equations
                            • l.closed = {x : α | u (l.toFun x) = x}
                            Instances For
                              theorem LowerAdjoint.mem_closed_iff {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                              x l.closed u (l.toFun x) = x
                              theorem LowerAdjoint.closure_eq_self_of_mem_closed {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {u : βα} (l : LowerAdjoint u) {x : α} (h : x l.closed) :
                              u (l.toFun x) = x
                              theorem LowerAdjoint.mem_closed_iff_closure_le {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                              x l.closed u (l.toFun x) x
                              @[simp]
                              theorem LowerAdjoint.closure_is_closed {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                              u (l.toFun x) l.closed
                              theorem LowerAdjoint.closed_eq_range_close {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) :
                              l.closed = Set.range (u l.toFun)

                              The set of closed elements for l is the range of u ∘ l.

                              def LowerAdjoint.toClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) :
                              l.closed

                              Send an x to an element of the set of closed elements (by taking the closure).

                              Equations
                              • l.toClosed x = u (l.toFun x),
                              Instances For
                                @[simp]
                                theorem LowerAdjoint.closure_le_closed_iff_le {α : Type u_1} {β : Type u_4} [PartialOrder α] [PartialOrder β] {u : βα} (l : LowerAdjoint u) (x : α) {y : α} (hy : l.closed y) :
                                u (l.toFun x) y x y
                                theorem LowerAdjoint.closure_top {α : Type u_1} {β : Type u_4} [PartialOrder α] [OrderTop α] [Preorder β] {u : βα} (l : LowerAdjoint u) :
                                u (l.toFun ) =
                                theorem LowerAdjoint.closure_inf_le {α : Type u_1} {β : Type u_4} [SemilatticeInf α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                                u (l.toFun (x y)) u (l.toFun x) u (l.toFun y)
                                theorem LowerAdjoint.closure_sup_closure_le {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                                u (l.toFun x) u (l.toFun y) u (l.toFun (x y))
                                theorem LowerAdjoint.closure_sup_closure_left {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                                u (l.toFun (u (l.toFun x) y)) = u (l.toFun (x y))
                                theorem LowerAdjoint.closure_sup_closure_right {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                                u (l.toFun (x u (l.toFun y))) = u (l.toFun (x y))
                                theorem LowerAdjoint.closure_sup_closure {α : Type u_1} {β : Type u_4} [SemilatticeSup α] [Preorder β] {u : βα} (l : LowerAdjoint u) (x : α) (y : α) :
                                u (l.toFun (u (l.toFun x) u (l.toFun y))) = u (l.toFun (x y))
                                theorem LowerAdjoint.closure_iSup_closure {α : Type u_1} {ι : Sort u_2} {β : Type u_4} [CompleteLattice α] [Preorder β] {u : βα} (l : LowerAdjoint u) (f : ια) :
                                u (l.toFun (⨆ (i : ι), u (l.toFun (f i)))) = u (l.toFun (⨆ (i : ι), f i))
                                theorem LowerAdjoint.closure_iSup₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} {β : Type u_4} [CompleteLattice α] [Preorder β] {u : βα} (l : LowerAdjoint u) (f : (i : ι) → κ iα) :
                                u (l.toFun (⨆ (i : ι), ⨆ (j : κ i), u (l.toFun (f i j)))) = u (l.toFun (⨆ (i : ι), ⨆ (j : κ i), f i j))
                                theorem LowerAdjoint.subset_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) :
                                s (l.toFun s)
                                theorem LowerAdjoint.not_mem_of_not_mem_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) {s : Set β} {P : β} (hP : Pl.toFun s) :
                                Ps
                                theorem LowerAdjoint.le_iff_subset {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) (S : α) :
                                l.toFun s S s S
                                theorem LowerAdjoint.mem_iff {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (s : Set β) (x : β) :
                                x l.toFun s ∀ (S : α), s Sx S
                                theorem LowerAdjoint.eq_of_le {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) {s : Set β} {S : α} (h₁ : s S) (h₂ : S l.toFun s) :
                                l.toFun s = S
                                theorem LowerAdjoint.closure_union_closure_subset {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                                (l.toFun x) (l.toFun y) (l.toFun (x y))
                                @[simp]
                                theorem LowerAdjoint.closure_union_closure_left {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                                l.toFun ((l.toFun x) y) = l.toFun (x y)
                                @[simp]
                                theorem LowerAdjoint.closure_union_closure_right {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                                l.toFun (x (l.toFun y)) = l.toFun (x y)
                                theorem LowerAdjoint.closure_union_closure {α : Type u_1} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (x : α) (y : α) :
                                l.toFun ((l.toFun x) (l.toFun y)) = l.toFun (x y)
                                @[simp]
                                theorem LowerAdjoint.closure_iUnion_closure {α : Type u_1} {ι : Sort u_2} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (f : ια) :
                                l.toFun (⋃ (i : ι), (l.toFun (f i))) = l.toFun (⋃ (i : ι), (f i))
                                @[simp]
                                theorem LowerAdjoint.closure_iUnion₂_closure {α : Type u_1} {ι : Sort u_2} {κ : ιSort u_3} {β : Type u_4} [SetLike α β] (l : LowerAdjoint SetLike.coe) (f : (i : ι) → κ iα) :
                                l.toFun (⋃ (i : ι), ⋃ (j : κ i), (l.toFun (f i j))) = l.toFun (⋃ (i : ι), ⋃ (j : κ i), (f i j))

                                Translations between GaloisConnection, LowerAdjoint, ClosureOperator #

                                @[simp]
                                theorem GaloisConnection.lowerAdjoint_toFun {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :
                                ∀ (a : α), gc.lowerAdjoint.toFun a = l a
                                def GaloisConnection.lowerAdjoint {α : Type u_1} {β : Type u_4} [Preorder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :

                                Every Galois connection induces a lower adjoint.

                                Equations
                                • gc.lowerAdjoint = { toFun := l, gc' := gc }
                                Instances For
                                  @[simp]
                                  theorem GaloisConnection.closureOperator_isClosed {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) (x : α) :
                                  gc.closureOperator.IsClosed x = (u (l x) = x)
                                  @[simp]
                                  theorem GaloisConnection.closureOperator_apply {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) (x : α) :
                                  gc.closureOperator x = u (l x)
                                  def GaloisConnection.closureOperator {α : Type u_1} {β : Type u_4} [PartialOrder α] [Preorder β] {l : αβ} {u : βα} (gc : GaloisConnection l u) :

                                  Every Galois connection induces a closure operator given by the composition. This is the partial order version of the statement that every adjunction induces a monad.

                                  Equations
                                  • gc.closureOperator = gc.lowerAdjoint.closureOperator
                                  Instances For
                                    def ClosureOperator.gi {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
                                    GaloisInsertion c.toCloseds Subtype.val

                                    The set of closed elements has a Galois insertion to the underlying type.

                                    Equations
                                    • c.gi = { choice := fun (x : α) (hx : (c.toCloseds x) x) => x, , gc := , le_l_u := , choice_eq := }
                                    Instances For
                                      @[simp]
                                      theorem closureOperator_gi_self {α : Type u_1} [PartialOrder α] (c : ClosureOperator α) :
                                      .closureOperator = c

                                      The Galois insertion associated to a closure operator can be used to reconstruct the closure operator. Note that the inverse in the opposite direction does not hold in general.