Documentation

Mathlib.Logic.Denumerable

Denumerable types #

This file defines denumerable (countably infinite) types as a typeclass extending Encodable. This is used to provide explicit encode/decode functions from and to , with the information that those functions are inverses of each other.

Implementation notes #

This property already has a name, namely α ≃ ℕ, but here we are interested in using it as a typeclass.

class Denumerable (α : Type u_3) extends Encodable :
Type u_3

A denumerable type is (constructively) bijective with . Typeclass equivalent of α ≃ ℕ.

Instances
    def Denumerable.ofNat (α : Type u_3) [Denumerable α] (n : ) :
    α

    Returns the n-th element of α indexed by the decoding.

    Equations
    Instances For
      @[simp]
      theorem Denumerable.ofNat_of_decode {α : Type u_1} [Denumerable α] {n : } {b : α} (h : Encodable.decode n = some b) :
      @[simp]
      def Denumerable.eqv (α : Type u_3) [Denumerable α] :
      α

      A denumerable type is equivalent to .

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For
        instance Denumerable.instInfinite {α : Type u_1} [Denumerable α] :
        Equations
        def Denumerable.mk' {α : Type u_3} (e : α ) :

        A type equivalent to is denumerable.

        Equations
        Instances For
          def Denumerable.ofEquiv (α : Type u_3) {β : Type u_4} [Denumerable α] (e : β α) :

          Denumerability is conserved by equivalences. This is transitivity of equivalence the denumerable way.

          Equations
          Instances For
            @[simp]
            theorem Denumerable.ofEquiv_ofNat (α : Type u_3) {β : Type u_4} [Denumerable α] (e : β α) (n : ) :
            def Denumerable.equiv₂ (α : Type u_3) (β : Type u_4) [Denumerable α] [Denumerable β] :
            α β

            All denumerable types are equivalent.

            Equations
            Instances For
              instance Denumerable.option {α : Type u_1} [Denumerable α] :

              If α is denumerable, then so is Option α.

              Equations
              instance Denumerable.sum {α : Type u_1} {β : Type u_2} [Denumerable α] [Denumerable β] :

              If α and β are denumerable, then so is their sum.

              Equations
              instance Denumerable.sigma {α : Type u_1} [Denumerable α] {γ : αType u_3} [(a : α) → Denumerable (γ a)] :

              A denumerable collection of denumerable types is denumerable.

              Equations
              • One or more equations did not get rendered due to their size.
              @[simp]
              theorem Denumerable.sigma_ofNat_val {α : Type u_1} [Denumerable α] {γ : αType u_3} [(a : α) → Denumerable (γ a)] (n : ) :
              instance Denumerable.prod {α : Type u_1} {β : Type u_2} [Denumerable α] [Denumerable β] :
              Denumerable (α × β)

              If α and β are denumerable, then so is their product.

              Equations

              The lift of a denumerable type is denumerable.

              Equations
              instance Denumerable.plift {α : Type u_1} [Denumerable α] :

              The lift of a denumerable type is denumerable.

              Equations
              def Denumerable.pair {α : Type u_1} [Denumerable α] :
              α × α α

              If α is denumerable, then α × α and α are equivalent.

              Equations
              Instances For

                Subsets of #

                theorem Nat.Subtype.exists_succ {s : Set } [Infinite s] (x : s) :
                ∃ (n : ), x + n + 1 s
                def Nat.Subtype.succ {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] (x : s) :
                s

                Returns the next natural in a set, according to the usual ordering of .

                Equations
                Instances For
                  theorem Nat.Subtype.succ_le_of_lt {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] {x : s} {y : s} (h : y < x) :
                  theorem Nat.Subtype.le_succ_of_forall_lt_le {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] {x : s} {y : s} (h : z < x, z y) :
                  theorem Nat.Subtype.lt_succ_self {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] (x : s) :
                  theorem Nat.Subtype.lt_succ_iff_le {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] {x : s} {y : s} :
                  def Nat.Subtype.ofNat (s : Set ) [DecidablePred fun (x : ) => x s] [Infinite s] :
                  s

                  Returns the n-th element of a set, according to the usual ordering of .

                  Equations
                  Instances For
                    theorem Nat.Subtype.ofNat_surjective_aux {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] {x : } (hx : x s) :
                    ∃ (n : ), Nat.Subtype.ofNat s n = { val := x, property := hx }
                    @[simp]
                    theorem Nat.Subtype.ofNat_range {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] :
                    @[simp]
                    theorem Nat.Subtype.coe_comp_ofNat_range {s : Set } [Infinite s] [DecidablePred fun (x : ) => x s] :
                    Set.range (Subtype.val Nat.Subtype.ofNat s) = s
                    def Nat.Subtype.denumerable (s : Set ) [DecidablePred fun (x : ) => x s] [Infinite s] :

                    Any infinite set of naturals is denumerable.

                    Equations
                    • One or more equations did not get rendered due to their size.
                    Instances For

                      An infinite encodable type is denumerable.

                      Equations
                      Instances For
                        instance nonempty_equiv_of_countable {α : Type u_1} {β : Type u_2} [Countable α] [Infinite α] [Countable β] [Infinite β] :
                        Nonempty (α β)
                        Equations