File: Mathlib/Dynamics/Minimal.lean

full_name: AddAction.IsMinimal.dense_orbit
code: /-- An action of an additive monoid `M` on a topological space is called *minimal* if the `M`-orbit
of every point `x : α` is dense. -/
class AddAction.IsMinimal (M α : Type*) [AddMonoid M] [TopologicalSpace α] [AddAction M α] :
    Prop where
  dense_orbit : ∀ x : α, Dense (AddAction.orbit M x)
goal: Dense (AddAction.orbit M x)

full_name: AddAction.IsMinimal
code: /-- An action of an additive monoid `M` on a topological space is called *minimal* if the `M`-orbit
of every point `x : α` is dense. -/
class AddAction.IsMinimal (M α : Type*) [AddMonoid M] [TopologicalSpace α] [AddAction M α] :
    Prop where
  dense_orbit : ∀ x : α, Dense (AddAction.orbit M x)
goal: Prop

full_name: MulAction.IsMinimal.dense_orbit
code: /-- An action of a monoid `M` on a topological space is called *minimal* if the `M`-orbit of every
point `x : α` is dense. -/
@[to_additive]
class MulAction.IsMinimal (M α : Type*) [Monoid M] [TopologicalSpace α] [MulAction M α] :
    Prop where
  dense_orbit : ∀ x : α, Dense (MulAction.orbit M x)
goal: Dense (MulAction.orbit M x)

full_name: MulAction.IsMinimal
code: /-- An action of a monoid `M` on a topological space is called *minimal* if the `M`-orbit of every
point `x : α` is dense. -/
@[to_additive]
class MulAction.IsMinimal (M α : Type*) [Monoid M] [TopologicalSpace α] [MulAction M α] :
    Prop where
  dense_orbit : ∀ x : α, Dense (MulAction.orbit M x)
goal: Prop

full_name: MulAction.dense_orbit
code: @[to_additive]
theorem MulAction.dense_orbit [IsMinimal M α] (x : α) : Dense (orbit M x) 
goal: Dense (MulAction.orbit M x)

full_name: AddAction.dense_orbit
code: @[to_additive]
theorem MulAction.dense_orbit [IsMinimal M α] (x : α) : Dense (orbit M x) 
goal: Dense (AddAction.orbit M x)

full_name: denseRange_vadd
code: @[to_additive]
theorem denseRange_smul [IsMinimal M α] (x : α) : DenseRange fun c : M ↦ c • x 
goal: DenseRange fun c => c +ᵥ x

full_name: denseRange_smul
code: @[to_additive]
theorem denseRange_smul [IsMinimal M α] (x : α) : DenseRange fun c : M ↦ c • x 
goal: DenseRange fun c => c • x

full_name: AddAction.isMinimal_of_pretransitive
code: @[to_additive]
instance (priority := 100) MulAction.isMinimal_of_pretransitive [IsPretransitive M α] :
    IsMinimal M α 
goal: AddAction.IsMinimal M α

full_name: MulAction.isMinimal_of_pretransitive
code: @[to_additive]
instance (priority := 100) MulAction.isMinimal_of_pretransitive [IsPretransitive M α] :
    IsMinimal M α 
goal: MulAction.IsMinimal M α

full_name: IsOpen.exists_smul_mem
code: @[to_additive]
theorem IsOpen.exists_smul_mem [IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ∃ c : M, c • x ∈ U 
goal: ∃ c, c • x ∈ U

full_name: IsOpen.exists_vadd_mem
code: @[to_additive]
theorem IsOpen.exists_smul_mem [IsMinimal M α] (x : α) {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ∃ c : M, c • x ∈ U 
goal: ∃ c, c +ᵥ x ∈ U

full_name: IsOpen.iUnion_preimage_smul
code: @[to_additive]
theorem IsOpen.iUnion_preimage_smul [IsMinimal M α] {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ⋃ c : M, (c • ·) ⁻¹' U = univ 
goal: ⋃ c, (fun x => c • x) ⁻¹' U = Set.univ

full_name: IsOpen.iUnion_preimage_vadd
code: @[to_additive]
theorem IsOpen.iUnion_preimage_smul [IsMinimal M α] {U : Set α} (hUo : IsOpen U)
    (hne : U.Nonempty) : ⋃ c : M, (c • ·) ⁻¹' U = univ 
goal: ⋃ c, (fun x => c +ᵥ x) ⁻¹' U = Set.univ

full_name: IsOpen.iUnion_vadd
code: @[to_additive]
theorem IsOpen.iUnion_smul [IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) :
    ⋃ g : G, g • U = univ 
goal: ⋃ g, g +ᵥ U = Set.univ

full_name: IsOpen.iUnion_smul
code: @[to_additive]
theorem IsOpen.iUnion_smul [IsMinimal G α] {U : Set α} (hUo : IsOpen U) (hne : U.Nonempty) :
    ⋃ g : G, g • U = univ 
goal: ⋃ g, g • U = Set.univ

full_name: IsCompact.exists_finite_cover_smul
code: @[to_additive]
theorem IsCompact.exists_finite_cover_smul [IsMinimal G α] [ContinuousConstSMul G α]
    {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
    ∃ I : Finset G, K ⊆ ⋃ g ∈ I, g • U 
goal: ∃ I, K ⊆ ⋃ g ∈ I, g • U

full_name: IsCompact.exists_finite_cover_vadd
code: @[to_additive]
theorem IsCompact.exists_finite_cover_smul [IsMinimal G α] [ContinuousConstSMul G α]
    {K U : Set α} (hK : IsCompact K) (hUo : IsOpen U) (hne : U.Nonempty) :
    ∃ I : Finset G, K ⊆ ⋃ g ∈ I, g • U 
goal: ∃ I, K ⊆ ⋃ g ∈ I, g +ᵥ U

full_name: dense_of_nonempty_vadd_invariant
code: @[to_additive]
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s 
goal: Dense s

full_name: dense_of_nonempty_smul_invariant
code: @[to_additive]
theorem dense_of_nonempty_smul_invariant [IsMinimal M α] {s : Set α} (hne : s.Nonempty)
    (hsmul : ∀ c : M, c • s ⊆ s) : Dense s 
goal: Dense s

full_name: eq_empty_or_univ_of_vadd_invariant_closed
code: @[to_additive]
theorem eq_empty_or_univ_of_smul_invariant_closed [IsMinimal M α] {s : Set α} (hs : IsClosed s)
    (hsmul : ∀ c : M, c • s ⊆ s) : s = ∅ ∨ s = univ 
goal: s = ∅ ∨ s = Set.univ

full_name: eq_empty_or_univ_of_smul_invariant_closed
code: @[to_additive]
theorem eq_empty_or_univ_of_smul_invariant_closed [IsMinimal M α] {s : Set α} (hs : IsClosed s)
    (hsmul : ∀ c : M, c • s ⊆ s) : s = ∅ ∨ s = univ 
goal: s = ∅ ∨ s = Set.univ

full_name: isMinimal_iff_closed_smul_invariant
code: @[to_additive]
theorem isMinimal_iff_closed_smul_invariant [ContinuousConstSMul M α] :
    IsMinimal M α ↔ ∀ s : Set α, IsClosed s → (∀ c : M, c • s ⊆ s) → s = ∅ ∨ s = univ 
goal: MulAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = Set.univ

full_name: isMinimal_iff_closed_vadd_invariant
code: @[to_additive]
theorem isMinimal_iff_closed_smul_invariant [ContinuousConstSMul M α] :
    IsMinimal M α ↔ ∀ s : Set α, IsClosed s → (∀ c : M, c • s ⊆ s) → s = ∅ ∨ s = univ 
goal: AddAction.IsMinimal M α ↔ ∀ (s : Set α), IsClosed s → (∀ (c : M), c +ᵥ s ⊆ s) → s = ∅ ∨ s = Set.univ