ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NNReal.hasSum_nat_add_iff
nonrec theorem hasSum_nat_add_iff {f : ℕ → ℝ≥0} (k : ℕ) {a : ℝ≥0} :
    HasSum (fun n => f (n + k)) a ↔ HasSum f (a + ∑ i ∈ range k, f i) 
Arguments: {f : ℕ → NNReal} (k : ℕ) {a : NNReal}
Goal: HasSum (fun n => f (n + k)) a ↔ HasSum f (a + ∑ i ∈ Finset.range k, f i)
-- NNReal.sum_add_tsum_nat_add
theorem sum_add_tsum_nat_add {f : ℕ → ℝ≥0} (k : ℕ) (hf : Summable f) :
    ∑' i, f i = (∑ i ∈ range k, f i) + ∑' i, f (i + k) 
Arguments: {f : ℕ → NNReal} (k : ℕ) (hf : Summable f)
Goal: ∑' (i : ℕ), f i = ∑ i ∈ Finset.range k, f i + ∑' (i : ℕ), f (i + k)
-- NNReal.iInf_real_pos_eq_iInf_nnreal_pos
theorem iInf_real_pos_eq_iInf_nnreal_pos [CompleteLattice α] {f : ℝ → α} :
    ⨅ (n : ℝ) (_ : 0 < n), f n = ⨅ (n : ℝ≥0) (_ : 0 < n), f n 
Arguments: {α : Type u_1} [CompleteLattice α] {f : ℝ → α}
Goal: ⨅ n, ⨅ (_ : 0 < n), f n = ⨅ n, ⨅ (_ : 0 < n), f ↑n
-- NNReal.tendsto_cofinite_zero_of_summable
theorem tendsto_cofinite_zero_of_summable {α} {f : α → ℝ≥0} (hf : Summable f) :
    Tendsto f cofinite (𝓝 0) 
Arguments: {α : Type u_1} {f : α → NNReal} (hf : Summable f)
Goal: Filter.Tendsto f Filter.cofinite (𝓝 0)
-- NNReal.tendsto_atTop_zero_of_summable
theorem tendsto_atTop_zero_of_summable {f : ℕ → ℝ≥0} (hf : Summable f) : Tendsto f atTop (𝓝 0) 
Arguments: {f : ℕ → NNReal} (hf : Summable f)
Goal: Filter.Tendsto f Filter.atTop (𝓝 0)
-- NNReal.tendsto_tsum_compl_atTop_zero
nonrec theorem tendsto_tsum_compl_atTop_zero {α : Type*} (f : α → ℝ≥0) :
    Tendsto (fun s : Finset α => ∑' b : { x // x ∉ s }, f b) atTop (𝓝 0) 
Arguments: {α : Type u_1} (f : α → NNReal)
Goal: Filter.Tendsto (fun s => ∑' (b : { x // x ∉ s }), f ↑b) Filter.atTop (𝓝 0)
-- Real.tendsto_of_bddAbove_monotone
theorem _root_.Real.tendsto_of_bddAbove_monotone {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f))
    (h_mon : Monotone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) 
Arguments: {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f)) (h_mon : Monotone f)
Goal: ∃ r, Filter.Tendsto f Filter.atTop (𝓝 r)
-- Real.tendsto_of_bddBelow_antitone
theorem _root_.Real.tendsto_of_bddBelow_antitone {f : ℕ → ℝ} (h_bdd : BddBelow (Set.range f))
    (h_ant : Antitone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) 
Arguments: {f : ℕ → ℝ} (h_bdd : BddBelow (Set.range f)) (h_ant : Antitone f)
Goal: ∃ r, Filter.Tendsto f Filter.atTop (𝓝 r)
-- NNReal.tendsto_of_antitone
theorem tendsto_of_antitone {f : ℕ → ℝ≥0} (h_ant : Antitone f) :
    ∃ r : ℝ≥0, Tendsto f atTop (𝓝 r) 
Arguments: {f : ℕ → NNReal} (h_ant : Antitone f)
Goal: ∃ r, Filter.Tendsto f Filter.atTop (𝓝 r)
-- DiscreteTopology.secondCountableTopology_of_encodable
@[deprecated DiscreteTopology.secondCountableTopology_of_countable (since := "2024-03-11")]
theorem DiscreteTopology.secondCountableTopology_of_encodable {α : Type*}
    [TopologicalSpace α] [DiscreteTopology α] [Countable α] : SecondCountableTopology α 
Arguments: {α : Type u_2} [TopologicalSpace α] [DiscreteTopology α] [Countable α]
Goal: SecondCountableTopology α
-- bot_topologicalSpace_eq_generateFrom_of_pred_succOrder
theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
    [SuccOrder α] [NoMinOrder α] [NoMaxOrder α] :
    (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [SuccOrder α] [NoMinOrder α] [NoMaxOrder α]
Goal: ⊥ = TopologicalSpace.generateFrom {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a}
-- discreteTopology_iff_orderTopology_of_pred_succ'
theorem discreteTopology_iff_orderTopology_of_pred_succ' [PartialOrder α] [PredOrder α]
    [SuccOrder α] [NoMinOrder α] [NoMaxOrder α] : DiscreteTopology α ↔ OrderTopology α 
Arguments: {α : Type u_1} [TopologicalSpace α] [PartialOrder α] [PredOrder α] [SuccOrder α] [NoMinOrder α] [NoMaxOrder α]
Goal: DiscreteTopology α ↔ OrderTopology α
-- LinearOrder.bot_topologicalSpace_eq_generateFrom
theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
    [SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] [SuccOrder α]
Goal: ⊥ = TopologicalSpace.generateFrom {s | ∃ a, s = Set.Ioi a ∨ s = Set.Iio a}
-- discreteTopology_iff_orderTopology_of_pred_succ
theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
    [SuccOrder α] : DiscreteTopology α ↔ OrderTopology α 
Arguments: {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [PredOrder α] [SuccOrder α]
Goal: DiscreteTopology α ↔ OrderTopology α
-- intermediate_value_univ₂
theorem intermediate_value_univ₂ [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f)
    (hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b) : ∃ x, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [PreconnectedSpace X] {a b : X} {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (hb : g b ≤ f b)
Goal: ∃ x, f x = g x
-- intermediate_value_univ₂_eventually₁
theorem intermediate_value_univ₂_eventually₁ [PreconnectedSpace X] {a : X} {l : Filter X} [NeBot l]
    {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f) :
    ∃ x, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [PreconnectedSpace X] {a : X} {l : Filter X} [l.NeBot] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (ha : f a ≤ g a) (he : g ≤ᶠ[l] f)
Goal: ∃ x, f x = g x
-- intermediate_value_univ₂_eventually₂
theorem intermediate_value_univ₂_eventually₂ [PreconnectedSpace X] {l₁ l₂ : Filter X} [NeBot l₁]
    [NeBot l₂] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g)
    (he₂ : g ≤ᶠ[l₂] f) : ∃ x, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [PreconnectedSpace X] {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] {f g : X → α} (hf : Continuous f) (hg : Continuous g) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f)
Goal: ∃ x, f x = g x
-- IsPreconnected.intermediate_value₂
theorem IsPreconnected.intermediate_value₂ {s : Set X} (hs : IsPreconnected s) {a b : X}
    (ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
    (ha' : f a ≤ g a) (hb' : g b ≤ f b) : ∃ x ∈ s, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (hb' : g b ≤ f b)
Goal: ∃ x ∈ s, f x = g x
-- IsPreconnected.intermediate_value₂_eventually₁
theorem IsPreconnected.intermediate_value₂_eventually₁ {s : Set X} (hs : IsPreconnected s) {a : X}
    {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f) : ∃ x ∈ s, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [l.NeBot] (hl : l ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (ha' : f a ≤ g a) (he : g ≤ᶠ[l] f)
Goal: ∃ x ∈ s, f x = g x
-- IsPreconnected.intermediate_value₂_eventually₂
theorem IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s)
    {l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
    (hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
    ∃ x ∈ s, f x = g x 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α} (hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f)
Goal: ∃ x ∈ s, f x = g x
-- IsPreconnected.intermediate_value
theorem IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s)
    (hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s) : Icc (f a) (f b) ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a b : X} (ha : a ∈ s) (hb : b ∈ s) {f : X → α} (hf : ContinuousOn f s)
Goal: Set.Icc (f a) (f b) ⊆ f '' s
-- IsPreconnected.intermediate_value_Ico
theorem IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X}
    {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
    (ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [l.NeBot] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Filter.Tendsto f l (𝓝 v))
Goal: Set.Ico (f a) v ⊆ f '' s
-- IsPreconnected.intermediate_value_Ioc
theorem IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X}
    {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
    (ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [l.NeBot] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht : Filter.Tendsto f l (𝓝 v))
Goal: Set.Ioc v (f a) ⊆ f '' s
-- IsPreconnected.intermediate_value_Ioo
theorem IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
    [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    {v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) :
    Ioo v₁ v₂ ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v₁ v₂ : α} (ht₁ : Filter.Tendsto f l₁ (𝓝 v₁)) (ht₂ : Filter.Tendsto f l₂ (𝓝 v₂))
Goal: Set.Ioo v₁ v₂ ⊆ f '' s
-- IsPreconnected.intermediate_value_Ici
theorem IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X}
    {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    (ht : Tendsto f l atTop) : Ici (f a) ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [l.NeBot] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Filter.Tendsto f l Filter.atTop)
Goal: Set.Ici (f a) ⊆ f '' s
-- IsPreconnected.intermediate_value_Iic
theorem IsPreconnected.intermediate_value_Iic {s : Set X} (hs : IsPreconnected s) {a : X}
    {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    (ht : Tendsto f l atBot) : Iic (f a) ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {a : X} {l : Filter X} (ha : a ∈ s) [l.NeBot] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht : Filter.Tendsto f l Filter.atBot)
Goal: Set.Iic (f a) ⊆ f '' s
-- IsPreconnected.intermediate_value_Ioi
theorem IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
    [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    {v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Filter.Tendsto f l₁ (𝓝 v)) (ht₂ : Filter.Tendsto f l₂ Filter.atTop)
Goal: Set.Ioi v ⊆ f '' s
-- IsPreconnected.intermediate_value_Iio
theorem IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
    [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    {v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α} (ht₁ : Filter.Tendsto f l₁ Filter.atBot) (ht₂ : Filter.Tendsto f l₂ (𝓝 v))
Goal: Set.Iio v ⊆ f '' s
-- IsPreconnected.intermediate_value_Iii
theorem IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
    [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
    (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X} [l₁.NeBot] [l₂.NeBot] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) (ht₁ : Filter.Tendsto f l₁ Filter.atBot) (ht₂ : Filter.Tendsto f l₂ Filter.atTop)
Goal: Set.univ ⊆ f '' s
-- intermediate_value_univ
theorem intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) :
    Icc (f a) (f b) ⊆ range f 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f)
Goal: Set.Icc (f a) (f b) ⊆ Set.range f
-- mem_range_of_exists_le_of_exists_ge
theorem mem_range_of_exists_le_of_exists_ge [PreconnectedSpace X] {c : α} {f : X → α}
    (hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b) : c ∈ range f 
Arguments: {X : Type u} {α : Type v} [TopologicalSpace X] [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] [PreconnectedSpace X] {c : α} {f : X → α} (hf : Continuous f) (h₁ : ∃ a, f a ≤ c) (h₂ : ∃ b, c ≤ f b)
Goal: c ∈ Set.range f
-- IsPreconnected.Icc_subset
theorem IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s)
    (hb : b ∈ s) : Icc a b ⊆ s 
Arguments: {α : Type v} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: Set.Icc a b ⊆ s
-- IsPreconnected.ordConnected
theorem IsPreconnected.ordConnected {s : Set α} (h : IsPreconnected s) : OrdConnected s 
Arguments: {α : Type v} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set α} (h : IsPreconnected s)
Goal: s.OrdConnected
-- IsConnected.Icc_subset
theorem IsConnected.Icc_subset {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s)
    (hb : b ∈ s) : Icc a b ⊆ s 
Arguments: {α : Type v} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set α} (hs : IsConnected s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: Set.Icc a b ⊆ s
-- IsPreconnected.eq_univ_of_unbounded
theorem IsPreconnected.eq_univ_of_unbounded {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
    (ha : ¬BddAbove s) : s = univ 
Arguments: {α : Type v} [LinearOrder α] [TopologicalSpace α] [OrderClosedTopology α] {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : ¬BddAbove s)
Goal: s = Set.univ
-- IsConnected.Ioo_csInf_csSup_subset
theorem IsConnected.Ioo_csInf_csSup_subset {s : Set α} (hs : IsConnected s) (hb : BddBelow s)
    (ha : BddAbove s) : Ioo (sInf s) (sSup s) ⊆ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : IsConnected s) (hb : BddBelow s) (ha : BddAbove s)
Goal: Set.Ioo (sInf s) (sSup s) ⊆ s
-- eq_Icc_csInf_csSup_of_connected_bdd_closed
theorem eq_Icc_csInf_csSup_of_connected_bdd_closed {s : Set α} (hc : IsConnected s)
    (hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s) : s = Icc (sInf s) (sSup s) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hc : IsConnected s) (hb : BddBelow s) (ha : BddAbove s) (hcl : IsClosed s)
Goal: s = Set.Icc (sInf s) (sSup s)
-- IsPreconnected.Ioi_csInf_subset
theorem IsPreconnected.Ioi_csInf_subset {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s)
    (ha : ¬BddAbove s) : Ioi (sInf s) ⊆ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : IsPreconnected s) (hb : BddBelow s) (ha : ¬BddAbove s)
Goal: Set.Ioi (sInf s) ⊆ s
-- IsPreconnected.Iio_csSup_subset
theorem IsPreconnected.Iio_csSup_subset {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s)
    (ha : BddAbove s) : Iio (sSup s) ⊆ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : IsPreconnected s) (hb : ¬BddBelow s) (ha : BddAbove s)
Goal: Set.Iio (sSup s) ⊆ s
-- IsPreconnected.mem_intervals
theorem IsPreconnected.mem_intervals {s : Set α} (hs : IsPreconnected s) :
    s ∈
      ({Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (sInf s) (sSup s),
          Ici (sInf s), Ioi (sInf s), Iic (sSup s), Iio (sSup s), univ, ∅} : Set (Set α)) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (hs : IsPreconnected s)
Goal: s ∈
  {Set.Icc (sInf s) (sSup s), Set.Ico (sInf s) (sSup s), Set.Ioc (sInf s) (sSup s), Set.Ioo (sInf s) (sSup s),
    Set.Ici (sInf s), Set.Ioi (sInf s), Set.Iic (sSup s), Set.Iio (sSup s), Set.univ, ∅}
-- setOf_isPreconnected_subset_of_ordered
theorem setOf_isPreconnected_subset_of_ordered :
    { s : Set α | IsPreconnected s } ⊆
      -- bounded intervals
      (range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo)) ∪
      -- unbounded intervals and `univ`
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
Goal: {s | IsPreconnected s} ⊆
  Set.range (Function.uncurry Set.Icc) ∪ Set.range (Function.uncurry Set.Ico) ∪ Set.range (Function.uncurry Set.Ioc) ∪
      Set.range (Function.uncurry Set.Ioo) ∪
    (Set.range Set.Ici ∪ Set.range Set.Ioi ∪ Set.range Set.Iic ∪ Set.range Set.Iio ∪ {Set.univ, ∅})
-- IsClosed.mem_of_ge_of_forall_exists_gt
theorem IsClosed.mem_of_ge_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
    (ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty) : b ∈ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b : α} {s : Set α} (hs : IsClosed (s ∩ Set.Icc a b)) (ha : a ∈ s) (hab : a ≤ b) (hgt : ∀ x ∈ s ∩ Set.Ico a b, (s ∩ Set.Ioc x b).Nonempty)
Goal: b ∈ s
-- IsClosed.Icc_subset_of_forall_exists_gt
theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
    (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {a b : α} {s : Set α} (hs : IsClosed (s ∩ Set.Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Set.Ico a b, ∀ y ∈ Set.Ioi x, (s ∩ Set.Ioc x y).Nonempty)
Goal: Set.Icc a b ⊆ s
-- IsClosed.Icc_subset_of_forall_mem_nhdsWithin
theorem IsClosed.Icc_subset_of_forall_mem_nhdsWithin {a b : α} {s : Set α}
    (hs : IsClosed (s ∩ Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, s ∈ 𝓝[>] x) :
    Icc a b ⊆ s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} {s : Set α} (hs : IsClosed (s ∩ Set.Icc a b)) (ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Set.Ico a b, s ∈ 𝓝[>] x)
Goal: Set.Icc a b ⊆ s
-- isPreconnected_Icc_aux
theorem isPreconnected_Icc_aux (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s)
    (ht : IsClosed t) (hab : Icc a b ⊆ s ∪ t) (hx : x ∈ Icc a b ∩ s) (hy : y ∈ Icc a b ∩ t) :
    (Icc a b ∩ (s ∩ t)).Nonempty 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} (x y : α) (s t : Set α) (hxy : x ≤ y) (hs : IsClosed s) (ht : IsClosed t) (hab : Set.Icc a b ⊆ s ∪ t) (hx : x ∈ Set.Icc a b ∩ s) (hy : y ∈ Set.Icc a b ∩ t)
Goal: (Set.Icc a b ∩ (s ∩ t)).Nonempty
-- isPreconnected_Icc
theorem isPreconnected_Icc : IsPreconnected (Icc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α}
Goal: IsPreconnected (Set.Icc a b)
-- isPreconnected_uIcc
theorem isPreconnected_uIcc : IsPreconnected (uIcc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α}
Goal: IsPreconnected (Set.uIcc a b)
-- Set.OrdConnected.isPreconnected
theorem Set.OrdConnected.isPreconnected {s : Set α} (h : s.OrdConnected) : IsPreconnected s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {s : Set α} (h : s.OrdConnected)
Goal: IsPreconnected s
-- isPreconnected_iff_ordConnected
theorem isPreconnected_iff_ordConnected {s : Set α} : IsPreconnected s ↔ OrdConnected s 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {s : Set α}
Goal: IsPreconnected s ↔ s.OrdConnected
-- isPreconnected_Ici
theorem isPreconnected_Ici : IsPreconnected (Ici a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsPreconnected (Set.Ici a)
-- isPreconnected_Iic
theorem isPreconnected_Iic : IsPreconnected (Iic a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsPreconnected (Set.Iic a)
-- isPreconnected_Iio
theorem isPreconnected_Iio : IsPreconnected (Iio a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsPreconnected (Set.Iio a)
-- isPreconnected_Ioi
theorem isPreconnected_Ioi : IsPreconnected (Ioi a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsPreconnected (Set.Ioi a)
-- isPreconnected_Ioo
theorem isPreconnected_Ioo : IsPreconnected (Ioo a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α}
Goal: IsPreconnected (Set.Ioo a b)
-- isPreconnected_Ioc
theorem isPreconnected_Ioc : IsPreconnected (Ioc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α}
Goal: IsPreconnected (Set.Ioc a b)
-- isPreconnected_Ico
theorem isPreconnected_Ico : IsPreconnected (Ico a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α}
Goal: IsPreconnected (Set.Ico a b)
-- isConnected_Ici
theorem isConnected_Ici : IsConnected (Ici a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsConnected (Set.Ici a)
-- isConnected_Iic
theorem isConnected_Iic : IsConnected (Iic a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α}
Goal: IsConnected (Set.Iic a)
-- isConnected_Ioi
theorem isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α} [NoMaxOrder α]
Goal: IsConnected (Set.Ioi a)