ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousMap.restrict_apply_mk
@[simp]
theorem restrict_apply_mk (f : C(α, β)) (s : Set α) (x : α) (hx : x ∈ s) :
    f.restrict s ⟨x, hx⟩ = f x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set α) (x : α) (hx : x ∈ s)
Goal: (ContinuousMap.restrict s f) ⟨x, hx⟩ = f x
-- ContinuousMap.injective_restrict
theorem injective_restrict [T2Space β] {s : Set α} (hs : Dense s) :
    Injective (restrict s : C(α, β) → C(s, β)) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [T2Space β] {s : Set α} (hs : Dense s)
Goal: Function.Injective (ContinuousMap.restrict s)
-- ContinuousMap.restrictPreimage_apply
@[simps]
def restrictPreimage (f : C(α, β)) (s : Set β) : C(f ⁻¹' s, s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (s : Set β)
Goal: ∀ (a : ↑(⇑f ⁻¹' s)), (f.restrictPreimage s) a = s.restrictPreimage (⇑f) a
-- ContinuousMap.liftCover_coe
@[simp]
theorem liftCover_coe {i : ι} (x : S i) : liftCover S φ hφ hS x = φ i x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Type u_5} {S : ι → Set α} {φ : (i : ι) → C(↑(S i), β)} {hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩} {hS : ∀ (x : α), ∃ i, S i ∈ 𝓝 x} {i : ι} (x : ↑(S i))
Goal: (ContinuousMap.liftCover S φ hφ hS) ↑x = (φ i) x
-- ContinuousMap.liftCover_restrict
theorem liftCover_restrict {i : ι} : (liftCover S φ hφ hS).restrict (S i) = φ i 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {ι : Type u_5} {S : ι → Set α} {φ : (i : ι) → C(↑(S i), β)} {hφ : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), (φ i) ⟨x, hxi⟩ = (φ j) ⟨x, hxj⟩} {hS : ∀ (x : α), ∃ i, S i ∈ 𝓝 x} {i : ι}
Goal: ContinuousMap.restrict (S i) (ContinuousMap.liftCover S φ hφ hS) = φ i
-- ContinuousMap.liftCover_coe'
@[simp]
theorem liftCover_coe' {s : Set α} {hs : s ∈ A} (x : s) : liftCover' A F hF hA x = F s hs x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Set (Set α)} {F : (s : Set α) → s ∈ A → C(↑s, β)} {hF :
  ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t),
    (F s hs) ⟨x, hxi⟩ = (F t ht) ⟨x, hxj⟩} {hA : ∀ (x : α), ∃ i ∈ A, i ∈ 𝓝 x} {s : Set α} {hs : s ∈ A} (x : ↑s)
Goal: (ContinuousMap.liftCover' A F hF hA) ↑x = (F s hs) x
-- ContinuousMap.liftCover_restrict'
@[simp]
theorem liftCover_restrict' {s : Set α} {hs : s ∈ A} :
    (liftCover' A F hF hA).restrict s = F s hs 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Set (Set α)} {F : (s : Set α) → s ∈ A → C(↑s, β)} {hF :
  ∀ (s : Set α) (hs : s ∈ A) (t : Set α) (ht : t ∈ A) (x : α) (hxi : x ∈ s) (hxj : x ∈ t),
    (F s hs) ⟨x, hxi⟩ = (F t ht) ⟨x, hxj⟩} {hA : ∀ (x : α), ∃ i ∈ A, i ∈ 𝓝 x} {s : Set α} {hs : s ∈ A}
Goal: ContinuousMap.restrict s (ContinuousMap.liftCover' A F hF hA) = F s hs
-- Function.RightInverse.homeomorph_symm_apply
@[simps!]
def Function.RightInverse.homeomorph {f' : C(Y, X)} (hf : Function.RightInverse f' f) :
    Quotient (Setoid.ker f) ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) (b : Y)
Goal: hf.homeomorph.symm b = Quotient.mk'' (f' b)
-- Function.RightInverse.homeomorph_apply
@[simps!]
def Function.RightInverse.homeomorph {f' : C(Y, X)} (hf : Function.RightInverse f' f) :
    Quotient (Setoid.ker f) ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} {f' : C(Y, X)} (hf : Function.RightInverse ⇑f' ⇑f) (a : Quotient (Setoid.ker ⇑f))
Goal: hf.homeomorph a = a.liftOn' ⇑f ⋯
-- QuotientMap.homeomorph_symm_apply
@[simps!]
noncomputable def homeomorph (hf : QuotientMap f) : Quotient (Setoid.ker f) ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : QuotientMap ⇑f) (b : Y)
Goal: hf.homeomorph.symm b = Quotient.mk'' (Function.surjInv ⋯ b)
-- QuotientMap.homeomorph_apply
@[simps!]
noncomputable def homeomorph (hf : QuotientMap f) : Quotient (Setoid.ker f) ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : C(X, Y)} (hf : QuotientMap ⇑f) (a : Quotient (Setoid.ker ⇑f))
Goal: hf.homeomorph a = a.liftOn' ⇑f ⋯
-- QuotientMap.lift_apply
@[simps]
noncomputable def lift : C(Y, Z) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f)
Goal: ∀ (a : Y), (hf.lift g h) a = ((fun i => i.liftOn' ⇑g ⋯) ∘ ⇑hf.homeomorph.symm) a
-- QuotientMap.lift_comp
@[simp]
theorem lift_comp : (hf.lift g h).comp f = g 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(X, Z)) (h : Function.FactorsThrough ⇑g ⇑f)
Goal: (hf.lift g h).comp f = g
-- QuotientMap.liftEquiv_apply
@[simps]
noncomputable def liftEquiv : { g : C(X, Z) // Function.FactorsThrough g f} ≃ C(Y, Z) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : { g // Function.FactorsThrough ⇑g ⇑f })
Goal: hf.liftEquiv g = hf.lift ↑g ⋯
-- QuotientMap.liftEquiv_symm_apply_coe
@[simps]
noncomputable def liftEquiv : { g : C(X, Z) // Function.FactorsThrough g f} ≃ C(Y, Z) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : C(X, Y)} (hf : QuotientMap ⇑f) (g : C(Y, Z))
Goal: ↑(hf.liftEquiv.symm g) = g.comp f
-- Homeomorph.toContinuousMap_apply
@[simps]
def toContinuousMap (e : α ≃ₜ β) : C(α, β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (e : α ≃ₜ β) (a : α)
Goal: e.toContinuousMap a = e a
-- Homeomorph.coe_refl
@[simp]
theorem coe_refl : (Homeomorph.refl α : C(α, α)) = ContinuousMap.id α 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: (Homeomorph.refl α).toContinuousMap = ContinuousMap.id α
-- Homeomorph.coe_trans
@[simp]
theorem coe_trans : (f.trans g : C(α, γ)) = (g : C(β, γ)).comp f 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : α ≃ₜ β) (g : β ≃ₜ γ)
Goal: (f.trans g).toContinuousMap = g.toContinuousMap.comp f.toContinuousMap
-- Homeomorph.symm_comp_toContinuousMap
@[simp]
theorem symm_comp_toContinuousMap :
    (f.symm : C(β, α)).comp (f : C(α, β)) = ContinuousMap.id α 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: f.symm.toContinuousMap.comp f.toContinuousMap = ContinuousMap.id α
-- Homeomorph.toContinuousMap_comp_symm
@[simp]
theorem toContinuousMap_comp_symm :
    (f : C(α, β)).comp (f.symm : C(β, α)) = ContinuousMap.id β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β)
Goal: f.toContinuousMap.comp f.symm.toContinuousMap = ContinuousMap.id β
-- ContinuousMap.embedding_sigmaMk_comp
theorem embedding_sigmaMk_comp [Nonempty X] :
    Embedding (fun g : Σ i, C(X, Y i) ↦ (sigmaMk g.1).comp g.2) 
Arguments: {X : Type u_1} {ι : Type u_2} {Y : ι → Type u_3} [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [Nonempty X]
Goal: Embedding fun g => (ContinuousMap.sigmaMk g.fst).comp g.snd
-- ContinuousMap.exists_lift_sigma
theorem exists_lift_sigma (f : C(X, Σ i, Y i)) : ∃ i g, f = (sigmaMk i).comp g 
Arguments: {X : Type u_1} {ι : Type u_2} {Y : ι → Type u_3} [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [ConnectedSpace X] (f : C(X, (i : ι) × Y i))
Goal: ∃ i g, f = (ContinuousMap.sigmaMk i).comp g
-- ContinuousMap.sigmaCodHomeomorph_symm_apply
@[simps! symm_apply]
def sigmaCodHomeomorph : C(X, Σ i, Y i) ≃ₜ Σ i, C(X, Y i) 
Arguments: (X : Type u_1) {ι : Type u_2} (Y : ι → Type u_3) [TopologicalSpace X] [(i : ι) → TopologicalSpace (Y i)] [ConnectedSpace X]
Goal: ∀ (a : (i : ι) × C(X, Y i)), (ContinuousMap.sigmaCodHomeomorph X Y).symm a = (ContinuousMap.sigmaMk a.fst).comp a.snd
-- LocallyConstant.toContinuousMapMonoidHom_apply
@[to_additive (attr := simps)
"The inclusion of locally-constant functions into continuous functions as an additive monoid hom."]
def toContinuousMapMonoidHom [Monoid Y] [ContinuousMul Y] : LocallyConstant X Y →* C(X, Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Monoid Y] [ContinuousMul Y] (f : LocallyConstant X Y)
Goal: LocallyConstant.toContinuousMapMonoidHom f = ↑f
-- LocallyConstant.toContinuousMapAddMonoidHom_apply
@[to_additive (attr := simps)
"The inclusion of locally-constant functions into continuous functions as an additive monoid hom."]
def toContinuousMapMonoidHom [Monoid Y] [ContinuousMul Y] : LocallyConstant X Y →* C(X, Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [AddMonoid Y] [ContinuousAdd Y] (f : LocallyConstant X Y)
Goal: LocallyConstant.toContinuousMapAddMonoidHom f = ↑f
-- LocallyConstant.toContinuousMapLinearMap_apply
@[simps]
def toContinuousMapLinearMap (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
    [ContinuousAdd Y] [ContinuousConstSMul R Y] : LocallyConstant X Y →ₗ[R] C(X, Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (R : Type u_3) [Semiring R] [AddCommMonoid Y] [Module R Y] [ContinuousAdd Y] [ContinuousConstSMul R Y] (f : LocallyConstant X Y)
Goal: (LocallyConstant.toContinuousMapLinearMap R) f = ↑f
-- LocallyConstant.toContinuousMapAlgHom_apply
@[simps]
def toContinuousMapAlgHom (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y]
    [TopologicalSemiring Y] : LocallyConstant X Y →ₐ[R] C(X, Y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (R : Type u_3) [CommSemiring R] [Semiring Y] [Algebra R Y] [TopologicalSemiring Y] (f : LocallyConstant X Y)
Goal: (LocallyConstant.toContinuousMapAlgHom R) f = ↑f
-- LowerSemicontinuousWithinAt.mono
theorem LowerSemicontinuousWithinAt.mono (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s) :
    LowerSemicontinuousWithinAt f t x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} {s t : Set α} (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s)
Goal: LowerSemicontinuousWithinAt f t x
-- lowerSemicontinuousWithinAt_univ_iff
theorem lowerSemicontinuousWithinAt_univ_iff :
    LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α}
Goal: LowerSemicontinuousWithinAt f Set.univ x ↔ LowerSemicontinuousAt f x
-- LowerSemicontinuousAt.lowerSemicontinuousWithinAt
theorem LowerSemicontinuousAt.lowerSemicontinuousWithinAt (s : Set α)
    (h : LowerSemicontinuousAt f x) : LowerSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} (s : Set α) (h : LowerSemicontinuousAt f x)
Goal: LowerSemicontinuousWithinAt f s x
-- LowerSemicontinuousOn.lowerSemicontinuousWithinAt
theorem LowerSemicontinuousOn.lowerSemicontinuousWithinAt (h : LowerSemicontinuousOn f s)
    (hx : x ∈ s) : LowerSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {x : α} {s : Set α} (h : LowerSemicontinuousOn f s) (hx : x ∈ s)
Goal: LowerSemicontinuousWithinAt f s x
-- LowerSemicontinuousOn.mono
theorem LowerSemicontinuousOn.mono (h : LowerSemicontinuousOn f s) (hst : t ⊆ s) :
    LowerSemicontinuousOn f t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} {s t : Set α} (h : LowerSemicontinuousOn f s) (hst : t ⊆ s)
Goal: LowerSemicontinuousOn f t
-- lowerSemicontinuousOn_univ_iff
theorem lowerSemicontinuousOn_univ_iff : LowerSemicontinuousOn f univ ↔ LowerSemicontinuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β}
Goal: LowerSemicontinuousOn f Set.univ ↔ LowerSemicontinuous f
-- LowerSemicontinuous.lowerSemicontinuousAt
theorem LowerSemicontinuous.lowerSemicontinuousAt (h : LowerSemicontinuous f) (x : α) :
    LowerSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : LowerSemicontinuous f) (x : α)
Goal: LowerSemicontinuousAt f x
-- LowerSemicontinuous.lowerSemicontinuousWithinAt
theorem LowerSemicontinuous.lowerSemicontinuousWithinAt (h : LowerSemicontinuous f) (s : Set α)
    (x : α) : LowerSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : LowerSemicontinuous f) (s : Set α) (x : α)
Goal: LowerSemicontinuousWithinAt f s x
-- LowerSemicontinuous.lowerSemicontinuousOn
theorem LowerSemicontinuous.lowerSemicontinuousOn (h : LowerSemicontinuous f) (s : Set α) :
    LowerSemicontinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (h : LowerSemicontinuous f) (s : Set α)
Goal: LowerSemicontinuousOn f s
-- lowerSemicontinuousWithinAt_const
theorem lowerSemicontinuousWithinAt_const : LowerSemicontinuousWithinAt (fun _x => z) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {z : β}
Goal: LowerSemicontinuousWithinAt (fun _x => z) s x
-- lowerSemicontinuousAt_const
theorem lowerSemicontinuousAt_const : LowerSemicontinuousAt (fun _x => z) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {z : β}
Goal: LowerSemicontinuousAt (fun _x => z) x
-- lowerSemicontinuousOn_const
theorem lowerSemicontinuousOn_const : LowerSemicontinuousOn (fun _x => z) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {z : β}
Goal: LowerSemicontinuousOn (fun _x => z) s
-- lowerSemicontinuous_const
theorem lowerSemicontinuous_const : LowerSemicontinuous fun _x : α => z 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {z : β}
Goal: LowerSemicontinuous fun _x => z
-- IsOpen.lowerSemicontinuous_indicator
theorem IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
    LowerSemicontinuous (indicator s fun _x => y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : 0 ≤ y)
Goal: LowerSemicontinuous (s.indicator fun _x => y)
-- IsOpen.lowerSemicontinuousOn_indicator
theorem IsOpen.lowerSemicontinuousOn_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
    LowerSemicontinuousOn (indicator s fun _x => y) t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s t : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : 0 ≤ y)
Goal: LowerSemicontinuousOn (s.indicator fun _x => y) t
-- IsOpen.lowerSemicontinuousAt_indicator
theorem IsOpen.lowerSemicontinuousAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
    LowerSemicontinuousAt (indicator s fun _x => y) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : 0 ≤ y)
Goal: LowerSemicontinuousAt (s.indicator fun _x => y) x
-- IsOpen.lowerSemicontinuousWithinAt_indicator
theorem IsOpen.lowerSemicontinuousWithinAt_indicator (hs : IsOpen s) (hy : 0 ≤ y) :
    LowerSemicontinuousWithinAt (indicator s fun _x => y) t x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s t : Set α} {y : β} [Zero β] (hs : IsOpen s) (hy : 0 ≤ y)
Goal: LowerSemicontinuousWithinAt (s.indicator fun _x => y) t x
-- IsClosed.lowerSemicontinuous_indicator
theorem IsClosed.lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) :
    LowerSemicontinuous (indicator s fun _x => y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : y ≤ 0)
Goal: LowerSemicontinuous (s.indicator fun _x => y)
-- IsClosed.lowerSemicontinuousOn_indicator
theorem IsClosed.lowerSemicontinuousOn_indicator (hs : IsClosed s) (hy : y ≤ 0) :
    LowerSemicontinuousOn (indicator s fun _x => y) t 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {s t : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : y ≤ 0)
Goal: LowerSemicontinuousOn (s.indicator fun _x => y) t
-- IsClosed.lowerSemicontinuousAt_indicator
theorem IsClosed.lowerSemicontinuousAt_indicator (hs : IsClosed s) (hy : y ≤ 0) :
    LowerSemicontinuousAt (indicator s fun _x => y) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : y ≤ 0)
Goal: LowerSemicontinuousAt (s.indicator fun _x => y) x
-- IsClosed.lowerSemicontinuousWithinAt_indicator
theorem IsClosed.lowerSemicontinuousWithinAt_indicator (hs : IsClosed s) (hy : y ≤ 0) :
    LowerSemicontinuousWithinAt (indicator s fun _x => y) t x 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {x : α} {s t : Set α} {y : β} [Zero β] (hs : IsClosed s) (hy : y ≤ 0)
Goal: LowerSemicontinuousWithinAt (s.indicator fun _x => y) t x
-- lowerSemicontinuous_iff_isOpen_preimage
theorem lowerSemicontinuous_iff_isOpen_preimage :
    LowerSemicontinuous f ↔ ∀ y, IsOpen (f ⁻¹' Ioi y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β}
Goal: LowerSemicontinuous f ↔ ∀ (y : β), IsOpen (f ⁻¹' Set.Ioi y)
-- LowerSemicontinuous.isOpen_preimage
theorem LowerSemicontinuous.isOpen_preimage (hf : LowerSemicontinuous f) (y : β) :
    IsOpen (f ⁻¹' Ioi y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {β : Type u_2} [Preorder β] {f : α → β} (hf : LowerSemicontinuous f) (y : β)
Goal: IsOpen (f ⁻¹' Set.Ioi y)
-- lowerSemicontinuous_iff_isClosed_preimage
theorem lowerSemicontinuous_iff_isClosed_preimage {f : α → γ} :
    LowerSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Iic y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] {f : α → γ}
Goal: LowerSemicontinuous f ↔ ∀ (y : γ), IsClosed (f ⁻¹' Set.Iic y)
-- LowerSemicontinuous.isClosed_preimage
theorem LowerSemicontinuous.isClosed_preimage {f : α → γ} (hf : LowerSemicontinuous f) (y : γ) :
    IsClosed (f ⁻¹' Iic y) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] {f : α → γ} (hf : LowerSemicontinuous f) (y : γ)
Goal: IsClosed (f ⁻¹' Set.Iic y)
-- ContinuousWithinAt.lowerSemicontinuousWithinAt
theorem ContinuousWithinAt.lowerSemicontinuousWithinAt {f : α → γ} (h : ContinuousWithinAt f s x) :
    LowerSemicontinuousWithinAt f s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousWithinAt f s x)
Goal: LowerSemicontinuousWithinAt f s x
-- ContinuousAt.lowerSemicontinuousAt
theorem ContinuousAt.lowerSemicontinuousAt {f : α → γ} (h : ContinuousAt f x) :
    LowerSemicontinuousAt f x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousAt f x)
Goal: LowerSemicontinuousAt f x
-- ContinuousOn.lowerSemicontinuousOn
theorem ContinuousOn.lowerSemicontinuousOn {f : α → γ} (h : ContinuousOn f s) :
    LowerSemicontinuousOn f s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : ContinuousOn f s)
Goal: LowerSemicontinuousOn f s
-- Continuous.lowerSemicontinuous
theorem Continuous.lowerSemicontinuous {f : α → γ} (h : Continuous f) : LowerSemicontinuous f 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ} (h : Continuous f)
Goal: LowerSemicontinuous f
-- lowerSemicontinuousWithinAt_iff_le_liminf
theorem lowerSemicontinuousWithinAt_iff_le_liminf {f : α → γ} :
    LowerSemicontinuousWithinAt f s x ↔ f x ≤ liminf f (𝓝[s] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: LowerSemicontinuousWithinAt f s x ↔ f x ≤ Filter.liminf f (𝓝[s] x)
-- lowerSemicontinuousAt_iff_le_liminf
theorem lowerSemicontinuousAt_iff_le_liminf {f : α → γ} :
    LowerSemicontinuousAt f x ↔ f x ≤ liminf f (𝓝 x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: LowerSemicontinuousAt f x ↔ f x ≤ Filter.liminf f (𝓝 x)
-- lowerSemicontinuous_iff_le_liminf
theorem lowerSemicontinuous_iff_le_liminf {f : α → γ} :
    LowerSemicontinuous f ↔ ∀ x, f x ≤ liminf f (𝓝 x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: LowerSemicontinuous f ↔ ∀ (x : α), f x ≤ Filter.liminf f (𝓝 x)
-- lowerSemicontinuousOn_iff_le_liminf
theorem lowerSemicontinuousOn_iff_le_liminf {f : α → γ} :
    LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] {f : α → γ}
Goal: LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ Filter.liminf f (𝓝[s] x)
-- lowerSemicontinuous_iff_isClosed_epigraph
theorem lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} :
    LowerSemicontinuous f ↔ IsClosed {p : α × γ | f p.1 ≤ p.2} 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [CompleteLinearOrder γ] [DenselyOrdered γ] [TopologicalSpace γ] [OrderTopology γ] {f : α → γ}
Goal: LowerSemicontinuous f ↔ IsClosed {p | f p.1 ≤ p.2}
-- ContinuousAt.comp_lowerSemicontinuousWithinAt
theorem ContinuousAt.comp_lowerSemicontinuousWithinAt {g : γ → δ} {f : α → γ}
    (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g) :
    LowerSemicontinuousWithinAt (g ∘ f) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousWithinAt f s x) (gmon : Monotone g)
Goal: LowerSemicontinuousWithinAt (g ∘ f) s x
-- ContinuousAt.comp_lowerSemicontinuousAt
theorem ContinuousAt.comp_lowerSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x))
    (hf : LowerSemicontinuousAt f x) (gmon : Monotone g) : LowerSemicontinuousAt (g ∘ f) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {x : α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x)) (hf : LowerSemicontinuousAt f x) (gmon : Monotone g)
Goal: LowerSemicontinuousAt (g ∘ f) x
-- Continuous.comp_lowerSemicontinuousOn
theorem Continuous.comp_lowerSemicontinuousOn {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : LowerSemicontinuousOn f s) (gmon : Monotone g) : LowerSemicontinuousOn (g ∘ f) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {s : Set α} {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuousOn f s) (gmon : Monotone g)
Goal: LowerSemicontinuousOn (g ∘ f) s
-- Continuous.comp_lowerSemicontinuous
theorem Continuous.comp_lowerSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : LowerSemicontinuous f) (gmon : Monotone g) : LowerSemicontinuous (g ∘ f) 
Arguments: {α : Type u_1} [TopologicalSpace α] {γ : Type u_3} [LinearOrder γ] [TopologicalSpace γ] [OrderTopology γ] {δ : Type u_4} [LinearOrder δ] [TopologicalSpace δ] [OrderTopology δ] {g : γ → δ} {f : α → γ} (hg : Continuous g) (hf : LowerSemicontinuous f) (gmon : Monotone g)
Goal: LowerSemicontinuous (g ∘ f)