File: Mathlib/Topology/Bornology/Absorbs.lean

full_name: Absorbs
code: /-- A set `s` absorbs another set `t` if `t` is contained in all scalings of `s`
by all but a bounded set of elements. -/
def Absorbs (s t : Set α) : Prop 
goal: Prop

full_name: Absorbent
code: /-- A set is *absorbent* if it absorbs every singleton. -/
def Absorbent (s : Set α) : Prop 
goal: Prop

full_name: Absorbs.empty
code: protected lemma empty : Absorbs M s ∅ 
goal: Absorbs M s ∅

full_name: Absorbs.eventually
code: protected lemma eventually (h : Absorbs M s t) : ∀ᶠ a in cobounded M, t ⊆ a • s 
goal: ∀ᶠ (a : M) in Bornology.cobounded M, t ⊆ a • s

full_name: Absorbs.of_boundedSpace
code: @[simp] lemma of_boundedSpace [BoundedSpace M] : Absorbs M s t 
goal: Absorbs M s t

full_name: Absorbs.mono_left
code: lemma mono_left (h : Absorbs M s₁ t) (hs : s₁ ⊆ s₂) : Absorbs M s₂ t 
goal: Absorbs M s₂ t

full_name: Absorbs.mono_right
code: lemma mono_right (h : Absorbs M s t₁) (ht : t₂ ⊆ t₁) : Absorbs M s t₂ 
goal: Absorbs M s t₂

full_name: Absorbs.mono
code: lemma mono (h : Absorbs M s₁ t₁) (hs : s₁ ⊆ s₂) (ht : t₂ ⊆ t₁) : Absorbs M s₂ t₂ 
goal: Absorbs M s₂ t₂

full_name: absorbs_union
code: @[simp]
lemma _root_.absorbs_union : Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂ 
goal: Absorbs M s (t₁ ∪ t₂) ↔ Absorbs M s t₁ ∧ Absorbs M s t₂

full_name: Absorbs.union
code: protected lemma union (h₁ : Absorbs M s t₁) (h₂ : Absorbs M s t₂) : Absorbs M s (t₁ ∪ t₂) 
goal: Absorbs M s (t₁ ∪ t₂)

full_name: Set.Finite.absorbs_sUnion
code: lemma _root_.Set.Finite.absorbs_sUnion {T : Set (Set α)} (hT : T.Finite) :
    Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t 
goal: Absorbs M s (⋃₀ T) ↔ ∀ t ∈ T, Absorbs M s t

full_name: Absorbs.sUnion
code: protected lemma sUnion (hT : T.Finite) (hs : ∀ t ∈ T, Absorbs M s t) :
    Absorbs M s (⋃₀ T) 
goal: Absorbs M s (⋃₀ T)

full_name: absorbs_iUnion
code: @[simp]
lemma _root_.absorbs_iUnion {ι : Sort*} [Finite ι] {t : ι → Set α} :
    Absorbs M s (⋃ i, t i) ↔ ∀ i, Absorbs M s (t i) 
goal: Absorbs M s (⋃ i, t i) ↔ ∀ (i : ι), Absorbs M s (t i)

full_name: Set.Finite.absorbs_biUnion
code: lemma _root_.Set.Finite.absorbs_biUnion {ι : Type*} {t : ι → Set α} {I : Set ι} (hI : I.Finite) :
    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) 
goal: Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i)

full_name: absorbs_biUnion_finset
code: @[simp]
lemma _root_.absorbs_biUnion_finset {ι : Type*} {t : ι → Set α} {I : Finset ι} :
    Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i) 
goal: Absorbs M s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, Absorbs M s (t i)

full_name: Absorbs.add
code: protected lemma add [AddZeroClass E] [DistribSMul M E]
    (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) : Absorbs M (s₁ + s₂) (t₁ + t₂) 
goal: Absorbs M (s₁ + s₂) (t₁ + t₂)

full_name: Absorbs.zero
code: protected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : Absorbs M s 0 
goal: Absorbs M s 0

full_name: Absorbs.univ
code: @[simp]
protected lemma univ : Absorbs G₀ univ s 
goal: Absorbs G₀ Set.univ s

full_name: absorbs_iff_eventually_cobounded_mapsTo
code: lemma _root_.absorbs_iff_eventually_cobounded_mapsTo :
    Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s 
goal: Absorbs G₀ s t ↔ ∀ᶠ (c : G₀) in Bornology.cobounded G₀, Set.MapsTo (fun x => c⁻¹ • x) t s

full_name: Set.Finite.absorbs_sInter
code: lemma _root_.Set.Finite.absorbs_sInter (hS : S.Finite) :
    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t 
goal: Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t

full_name: absorbs_inter
code: @[simp]
lemma _root_.absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u 
goal: Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u

full_name: Absorbs.inter
code: protected lemma inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u 
goal: Absorbs G₀ (s ∩ t) u

full_name: absorbs_iInter
code: @[simp]
lemma _root_.absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :
    Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t 
goal: Absorbs G₀ (⋂ i, s i) t ↔ ∀ (i : ι), Absorbs G₀ (s i) t

full_name: Set.Finite.absorbs_biInter
code: lemma _root_.Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t 
goal: Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t

full_name: absorbs_zero_iff
code: @[simp]
lemma _root_.absorbs_zero_iff [NeBot (cobounded G₀)] {E : Type*} [AddMonoid E]
    [DistribMulAction G₀ E] {s : Set E} : Absorbs G₀ s 0 ↔ 0 ∈ s 
goal: Absorbs G₀ s 0 ↔ 0 ∈ s

full_name: absorbs_neg_neg
code: @[simp]
lemma absorbs_neg_neg {s t : Set E} : Absorbs M (-s) (-t) ↔ Absorbs M s t 
goal: Absorbs M (-s) (-t) ↔ Absorbs M s t

full_name: Absorbs.sub
code: lemma Absorbs.sub {s₁ s₂ t₁ t₂ : Set E} (h₁ : Absorbs M s₁ t₁) (h₂ : Absorbs M s₂ t₂) :
    Absorbs M (s₁ - s₂) (t₁ - t₂) 
goal: Absorbs M (s₁ - s₂) (t₁ - t₂)

full_name: Absorbent.mono
code: protected theorem mono (ht : Absorbent M s) (hsub : s ⊆ t) : Absorbent M t 
goal: Absorbent M t

full_name: absorbent_iff_forall_absorbs_singleton
code: theorem _root_.absorbent_iff_forall_absorbs_singleton : Absorbent M s ↔ ∀ x, Absorbs M s {x} 
goal: Absorbent M s ↔ ∀ (x : α), Absorbs M s {x}

full_name: Absorbent.absorbs
code: protected theorem absorbs (hs : Absorbent M s) {x : α} : Absorbs M s {x} 
goal: Absorbs M s {x}

full_name: Absorbent.absorbs_finite
code: theorem absorbs_finite (hs : Absorbent M s) (ht : t.Finite) : Absorbs M s t 
goal: Absorbs M s t

full_name: Absorbent.vadd_absorbs
code: theorem vadd_absorbs {M E : Type*} [Bornology M] [AddZeroClass E] [DistribSMul M E]
    {s₁ s₂ t : Set E} {x : E} (h₁ : Absorbent M s₁) (h₂ : Absorbs M s₂ t) :
    Absorbs M (s₁ + s₂) (x +ᵥ t) 
goal: Absorbs M (s₁ + s₂) (x +ᵥ t)

full_name: absorbent_univ
code: lemma absorbent_univ : Absorbent G₀ (univ : Set α) 
goal: Absorbent G₀ Set.univ

full_name: absorbent_iff_inv_smul
code: lemma absorbent_iff_inv_smul {s : Set α} :
    Absorbent G₀ s ↔ ∀ x, ∀ᶠ c in cobounded G₀, c⁻¹ • x ∈ s 
goal: Absorbent G₀ s ↔ ∀ (x : α), ∀ᶠ (c : G₀) in Bornology.cobounded G₀, c⁻¹ • x ∈ s

full_name: Absorbent.zero_mem
code: lemma Absorbent.zero_mem [NeBot (cobounded G₀)] [AddMonoid E] [DistribMulAction G₀ E]
    {s : Set E} (hs : Absorbent G₀ s) : (0 : E) ∈ s 
goal: 0 ∈ s