Documentation

Mathlib.Data.Set.Pointwise.Finite

Finiteness lemmas for pointwise operations on sets #

@[simp]
theorem Set.finite_zero {α : Type u_2} [Zero α] :
@[simp]
theorem Set.finite_one {α : Type u_2} [One α] :
theorem Set.Finite.neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} (hs : Set.Finite s) :
theorem Set.Finite.inv {α : Type u_2} [InvolutiveInv α] {s : Set α} (hs : Set.Finite s) :
theorem Set.Finite.add {α : Type u_2} [Add α] {s : Set α} {t : Set α} :
Set.Finite sSet.Finite tSet.Finite (s + t)
theorem Set.Finite.mul {α : Type u_2} [Mul α] {s : Set α} {t : Set α} :
Set.Finite sSet.Finite tSet.Finite (s * t)
def Set.fintypeAdd {α : Type u_2} [Add α] [DecidableEq α] (s : Set α) (t : Set α) [Fintype s] [Fintype t] :
Fintype (s + t)

Addition preserves finiteness.

Equations
Instances For
    def Set.fintypeMul {α : Type u_2} [Mul α] [DecidableEq α] (s : Set α) (t : Set α) [Fintype s] [Fintype t] :
    Fintype (s * t)

    Multiplication preserves finiteness.

    Equations
    Instances For
      instance Set.decidableMemAdd {α : Type u_2} [AddMonoid α] {s : Set α} {t : Set α} [Fintype α] [DecidableEq α] [DecidablePred fun (x : α) => x s] [DecidablePred fun (x : α) => x t] :
      DecidablePred fun (x : α) => x s + t
      Equations
      theorem Set.decidableMemAdd.proof_1 {α : Type u_1} [AddMonoid α] {s : Set α} {t : Set α} :
      ∀ (x : α), (∃ x_1 ∈ s, ∃ y ∈ t, x_1 + y = x) x s + t
      instance Set.decidableMemMul {α : Type u_2} [Monoid α] {s : Set α} {t : Set α} [Fintype α] [DecidableEq α] [DecidablePred fun (x : α) => x s] [DecidablePred fun (x : α) => x t] :
      DecidablePred fun (x : α) => x s * t
      Equations
      theorem Set.decidableMemNSMul.proof_1 {α : Type u_1} [AddMonoid α] {s : Set α} :
      (DecidablePred fun (x : α) => x Nat.zero s) = DecidablePred fun (x : α) => x = 0
      theorem Set.decidableMemNSMul.proof_2 {α : Type u_1} [AddMonoid α] {s : Set α} (n : ) :
      (DecidablePred fun (x : α) => x Nat.succ n s) = DecidablePred fun (x : α) => x n s + s
      instance Set.decidableMemNSMul {α : Type u_2} [AddMonoid α] {s : Set α} [Fintype α] [DecidableEq α] [DecidablePred fun (x : α) => x s] (n : ) :
      DecidablePred fun (x : α) => x n s
      Equations
      instance Set.decidableMemPow {α : Type u_2} [Monoid α] {s : Set α} [Fintype α] [DecidableEq α] [DecidablePred fun (x : α) => x s] (n : ) :
      DecidablePred fun (x : α) => x s ^ n
      Equations
      theorem Set.Finite.vadd {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set α} {t : Set β} :
      theorem Set.Finite.smul {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set α} {t : Set β} :
      theorem Set.Finite.vadd_set {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α} :
      theorem Set.Finite.smul_set {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α} :
      theorem Set.Infinite.of_vadd_set {α : Type u_2} {β : Type u_3} [VAdd α β] {s : Set β} {a : α} :
      theorem Set.Infinite.of_smul_set {α : Type u_2} {β : Type u_3} [SMul α β] {s : Set β} {a : α} :
      theorem Set.Finite.vsub {α : Type u_2} {β : Type u_3} [VSub α β] {s : Set β} {t : Set β} (hs : Set.Finite s) (ht : Set.Finite t) :
      theorem Set.finite_add {α : Type u_2} [Add α] [IsLeftCancelAdd α] [IsRightCancelAdd α] {s : Set α} {t : Set α} :
      theorem Set.finite_mul {α : Type u_2} [Mul α] [IsLeftCancelMul α] [IsRightCancelMul α] {s : Set α} {t : Set α} :
      @[simp]
      theorem Set.finite_vadd_set {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {a : α} {s : Set β} :
      @[simp]
      theorem Set.finite_smul_set {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {a : α} {s : Set β} :
      @[simp]
      theorem Set.infinite_vadd_set {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {a : α} {s : Set β} :
      @[simp]
      theorem Set.infinite_smul_set {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {a : α} {s : Set β} :
      theorem Set.Finite.of_smul_set {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {a : α} {s : Set β} :

      Alias of the forward direction of Set.finite_smul_set.

      theorem Set.Infinite.smul_set {α : Type u_2} {β : Type u_3} [Group α] [MulAction α β] {a : α} {s : Set β} :

      Alias of the reverse direction of Set.infinite_smul_set.

      theorem Set.Infinite.vadd_set {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {a : α} {s : Set β} :
      theorem Set.Finite.of_vadd_set {α : Type u_2} {β : Type u_3} [AddGroup α] [AddAction α β] {a : α} {s : Set β} :
      theorem AddGroup.card_nsmul_eq_card_nsmul_card_univ {G : Type u_5} [AddGroup G] [Fintype G] (S : Set G) [(k : ) → DecidablePred fun (x : G) => x k S] (k : ) :
      abbrev AddGroup.card_nsmul_eq_card_nsmul_card_univ.match_1 {G : Type u_1} (s : Set G) (motive : sSort u_2) :
      (x : s) → ((b : G) → (hb : b s) → motive { val := b, property := hb })motive x
      Equations
      Instances For
        theorem Group.card_pow_eq_card_pow_card_univ {G : Type u_5} [Group G] [Fintype G] (S : Set G) [(k : ) → DecidablePred fun (x : G) => x S ^ k] (k : ) :