ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isComplete_image_iff
theorem isComplete_image_iff {m : α → β} {s : Set α} (hm : UniformInducing m) :
    IsComplete (m '' s) ↔ IsComplete s 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {m : α → β} {s : Set α} (hm : UniformInducing m)
Goal: IsComplete (m '' s) ↔ IsComplete s
-- completeSpace_iff_isComplete_range
theorem completeSpace_iff_isComplete_range {f : α → β} (hf : UniformInducing f) :
    CompleteSpace α ↔ IsComplete (range f) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} (hf : UniformInducing f)
Goal: CompleteSpace α ↔ IsComplete (Set.range f)
-- UniformInducing.isComplete_range
theorem UniformInducing.isComplete_range [CompleteSpace α] {f : α → β} (hf : UniformInducing f) :
    IsComplete (range f) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] [CompleteSpace α] {f : α → β} (hf : UniformInducing f)
Goal: IsComplete (Set.range f)
-- SeparationQuotient.completeSpace_iff
theorem SeparationQuotient.completeSpace_iff :
    CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α 
Arguments: {α : Type u} [UniformSpace α]
Goal: CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α
-- completeSpace_congr
theorem completeSpace_congr {e : α ≃ β} (he : UniformEmbedding e) :
    CompleteSpace α ↔ CompleteSpace β 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {e : α ≃ β} (he : UniformEmbedding ⇑e)
Goal: CompleteSpace α ↔ CompleteSpace β
-- completeSpace_coe_iff_isComplete
theorem completeSpace_coe_iff_isComplete {s : Set α} : CompleteSpace s ↔ IsComplete s 
Arguments: {α : Type u} [UniformSpace α] {s : Set α}
Goal: CompleteSpace ↑s ↔ IsComplete s
-- IsClosed.completeSpace_coe
theorem IsClosed.completeSpace_coe [CompleteSpace α] {s : Set α} (hs : IsClosed s) :
    CompleteSpace s 
Arguments: {α : Type u} [UniformSpace α] [CompleteSpace α] {s : Set α} (hs : IsClosed s)
Goal: CompleteSpace ↑s
-- completeSpace_extension
theorem completeSpace_extension {m : β → α} (hm : UniformInducing m) (dense : DenseRange m)
    (h : ∀ f : Filter β, Cauchy f → ∃ x : α, map m f ≤ 𝓝 x) : CompleteSpace α 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {m : β → α} (hm : UniformInducing m) (dense : DenseRange m) (h : ∀ (f : Filter β), Cauchy f → ∃ x, Filter.map m f ≤ 𝓝 x)
Goal: CompleteSpace α
-- totallyBounded_image_iff
lemma totallyBounded_image_iff {f : α → β} {s : Set α} (hf : UniformInducing f) :
    TotallyBounded (f '' s) ↔ TotallyBounded s 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} (hf : UniformInducing f)
Goal: TotallyBounded (f '' s) ↔ TotallyBounded s
-- totallyBounded_preimage
theorem totallyBounded_preimage {f : α → β} {s : Set β} (hf : UniformInducing f)
    (hs : TotallyBounded s) : TotallyBounded (f ⁻¹' s) 
Arguments: {α : Type u} {β : Type v} [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set β} (hf : UniformInducing f) (hs : TotallyBounded s)
Goal: TotallyBounded (f ⁻¹' s)
-- uniformEmbedding_comap
theorem uniformEmbedding_comap {α : Type*} {β : Type*} {f : α → β} [u : UniformSpace β]
    (hf : Function.Injective f) : @UniformEmbedding α β (UniformSpace.comap f u) u f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [u : UniformSpace β] (hf : Function.Injective f)
Goal: UniformEmbedding f
-- Embedding.to_uniformEmbedding
theorem Embedding.to_uniformEmbedding {α β} [TopologicalSpace α] [u : UniformSpace β] (f : α → β)
    (h : Embedding f) : @UniformEmbedding α β (h.comapUniformSpace f) u f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [u : UniformSpace β] (f : α → β) (h : Embedding f)
Goal: UniformEmbedding f
-- uniformly_extend_exists
theorem uniformly_extend_exists [CompleteSpace γ] (a : α) : ∃ c, Tendsto f (comap e (𝓝 a)) (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α)
Goal: ∃ c, Filter.Tendsto f (Filter.comap e (𝓝 a)) (𝓝 c)
-- uniform_extend_subtype
theorem uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
    {s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : UniformEmbedding e)
    (hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
    (hp : ∀ x ∈ s, p x) : ∃ c, Tendsto f (comap e (𝓝 b)) (𝓝 c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β} {s : Set α} (hf : UniformContinuous fun x => f ↑x) (he : UniformEmbedding e) (hd : ∀ (x : β), x ∈ closure (Set.range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s) (hp : ∀ x ∈ s, p x)
Goal: ∃ c, Filter.Tendsto f (Filter.comap e (𝓝 b)) (𝓝 c)
-- uniformly_extend_spec
theorem uniformly_extend_spec [CompleteSpace γ] (a : α) : Tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ] (a : α)
Goal: Filter.Tendsto f (Filter.comap e (𝓝 a)) (𝓝 (⋯.extend f a))
-- uniformContinuous_uniformly_extend
theorem uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuous ψ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [CompleteSpace γ]
Goal: UniformContinuous (⋯.extend f)
-- uniformly_extend_of_ind
theorem uniformly_extend_of_ind (b : β) : ψ (e b) = f b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ} (h_f : UniformContinuous f) [T0Space γ] (b : β)
Goal: ⋯.extend f (e b) = f b
-- uniformly_extend_unique
theorem uniformly_extend_unique {g : α → γ} (hg : ∀ b, g (e b) = f b) (hc : Continuous g) : ψ = g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ} [T0Space γ] {g : α → γ} (hg : ∀ (b : β), g (e b) = f b) (hc : Continuous g)
Goal: ⋯.extend f = g
-- IsUltrametricDist.dist_triangle_max
class IsUltrametricDist (X : Type*) [Dist X] : Prop where
  dist_triangle_max : ∀ x y z : X, dist x z ≤ max (dist x y) (dist y z)
Arguments: {X : Type u_2} [Dist X] [self : IsUltrametricDist X] (x y z : X)
Goal: dist x z ≤ max (dist x y) (dist y z)
-- dist_triangle_max
lemma dist_triangle_max : dist x z ≤ max (dist x y) (dist y z) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x y z : X)
Goal: dist x z ≤ max (dist x y) (dist y z)
-- IsUltrametricDist.ball_eq_of_mem
lemma ball_eq_of_mem {x y : X} {r : ℝ} (h : y ∈ ball x r) : ball x r = ball y r 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] {x y : X} {r : ℝ} (h : y ∈ Metric.ball x r)
Goal: Metric.ball x r = Metric.ball y r
-- IsUltrametricDist.mem_ball_iff
lemma mem_ball_iff {x y : X} {r : ℝ} : y ∈ ball x r ↔ x ∈ ball y r 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] {x y : X} {r : ℝ}
Goal: y ∈ Metric.ball x r ↔ x ∈ Metric.ball y r
-- IsUltrametricDist.ball_subset_trichotomy
lemma ball_subset_trichotomy :
    ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x y : X) (r s : ℝ)
Goal: Metric.ball x r ⊆ Metric.ball y s ∨ Metric.ball y s ⊆ Metric.ball x r ∨ Disjoint (Metric.ball x r) (Metric.ball y s)
-- IsUltrametricDist.ball_eq_or_disjoint
lemma ball_eq_or_disjoint :
    ball x r = ball y r ∨ Disjoint (ball x r) (ball y r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x y : X) (r : ℝ)
Goal: Metric.ball x r = Metric.ball y r ∨ Disjoint (Metric.ball x r) (Metric.ball y r)
-- IsUltrametricDist.closedBall_eq_of_mem
lemma closedBall_eq_of_mem {x y: X} {r : ℝ} (h : y ∈ closedBall x r) :
    closedBall x r = closedBall y r 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] {x y : X} {r : ℝ} (h : y ∈ Metric.closedBall x r)
Goal: Metric.closedBall x r = Metric.closedBall y r
-- IsUltrametricDist.mem_closedBall_iff
lemma mem_closedBall_iff {x y: X} {r : ℝ} :
    y ∈ closedBall x r ↔ x ∈ closedBall y r 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] {x y : X} {r : ℝ}
Goal: y ∈ Metric.closedBall x r ↔ x ∈ Metric.closedBall y r
-- IsUltrametricDist.closedBall_subset_trichotomy
lemma closedBall_subset_trichotomy :
    closedBall x r ⊆ closedBall y s ∨ closedBall y s ⊆ closedBall x r ∨
    Disjoint (closedBall x r) (closedBall y s) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x y : X) (r s : ℝ)
Goal: Metric.closedBall x r ⊆ Metric.closedBall y s ∨
  Metric.closedBall y s ⊆ Metric.closedBall x r ∨ Disjoint (Metric.closedBall x r) (Metric.closedBall y s)
-- IsUltrametricDist.isClosed_ball
lemma isClosed_ball (x : X) (r : ℝ) : IsClosed (ball x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) (r : ℝ)
Goal: IsClosed (Metric.ball x r)
-- IsUltrametricDist.isClopen_ball
lemma isClopen_ball : IsClopen (ball x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) (r : ℝ)
Goal: IsClopen (Metric.ball x r)
-- IsUltrametricDist.frontier_ball_eq_empty
lemma frontier_ball_eq_empty : frontier (ball x r) = ∅ 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) (r : ℝ)
Goal: frontier (Metric.ball x r) = ∅
-- IsUltrametricDist.closedBall_eq_or_disjoint
lemma closedBall_eq_or_disjoint :
    closedBall x r = closedBall y r ∨ Disjoint (closedBall x r) (closedBall y r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x y : X) (r : ℝ)
Goal: Metric.closedBall x r = Metric.closedBall y r ∨ Disjoint (Metric.closedBall x r) (Metric.closedBall y r)
-- IsUltrametricDist.isOpen_closedBall
lemma isOpen_closedBall {r : ℝ} (hr : r ≠ 0) : IsOpen (closedBall x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ} (hr : r ≠ 0)
Goal: IsOpen (Metric.closedBall x r)
-- IsUltrametricDist.isClopen_closedBall
lemma isClopen_closedBall {r : ℝ} (hr : r ≠ 0) : IsClopen (closedBall x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ} (hr : r ≠ 0)
Goal: IsClopen (Metric.closedBall x r)
-- IsUltrametricDist.frontier_closedBall_eq_empty
lemma frontier_closedBall_eq_empty {r : ℝ} (hr : r ≠ 0) : frontier (closedBall x r) = ∅ 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ} (hr : r ≠ 0)
Goal: frontier (Metric.closedBall x r) = ∅
-- IsUltrametricDist.isOpen_sphere
lemma isOpen_sphere {r : ℝ} (hr : r ≠ 0) : IsOpen (sphere x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ} (hr : r ≠ 0)
Goal: IsOpen (Metric.sphere x r)
-- IsUltrametricDist.isClopen_sphere
lemma isClopen_sphere {r : ℝ} (hr : r ≠ 0) : IsClopen (sphere x r) 
Arguments: {X : Type u_1} [PseudoMetricSpace X] [IsUltrametricDist X] (x : X) {r : ℝ} (hr : r ≠ 0)
Goal: IsClopen (Metric.sphere x r)
-- ContractingWith.toLipschitzWith
theorem toLipschitzWith (hf : ContractingWith K f) : LipschitzWith K f 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f)
Goal: LipschitzWith K f
-- ContractingWith.one_sub_K_pos'
theorem one_sub_K_pos' (hf : ContractingWith K f) : (0 : ℝ≥0∞) < 1 - K 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f)
Goal: 0 < 1 - ↑K
-- ContractingWith.one_sub_K_ne_zero
theorem one_sub_K_ne_zero (hf : ContractingWith K f) : (1 : ℝ≥0∞) - K ≠ 0 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f)
Goal: 1 - ↑K ≠ 0
-- ContractingWith.one_sub_K_ne_top
theorem one_sub_K_ne_top : (1 : ℝ≥0∞) - K ≠ ∞ 
Arguments: {K : ℝ≥0}
Goal: 1 - ↑K ≠ ⊤
-- ContractingWith.edist_inequality
theorem edist_inequality (hf : ContractingWith K f) {x y} (h : edist x y ≠ ∞) :
    edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x y : α} (h : edist x y ≠ ⊤)
Goal: edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - ↑K)
-- ContractingWith.edist_le_of_fixedPoint
theorem edist_le_of_fixedPoint (hf : ContractingWith K f) {x y} (h : edist x y ≠ ∞)
    (hy : IsFixedPt f y) : edist x y ≤ edist x (f x) / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x y : α} (h : edist x y ≠ ⊤) (hy : Function.IsFixedPt f y)
Goal: edist x y ≤ edist x (f x) / (1 - ↑K)
-- ContractingWith.eq_or_edist_eq_top_of_fixedPoints
theorem eq_or_edist_eq_top_of_fixedPoints (hf : ContractingWith K f) {x y} (hx : IsFixedPt f x)
    (hy : IsFixedPt f y) : x = y ∨ edist x y = ∞ 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x y : α} (hx : Function.IsFixedPt f x) (hy : Function.IsFixedPt f y)
Goal: x = y ∨ edist x y = ⊤
-- ContractingWith.restrict
theorem restrict (hf : ContractingWith K f) {s : Set α} (hs : MapsTo f s s) :
    ContractingWith K (hs.restrict f s s) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {s : Set α} (hs : Set.MapsTo f s s)
Goal: ContractingWith K (Set.MapsTo.restrict f s s hs)
-- ContractingWith.exists_fixedPoint
theorem exists_fixedPoint (hf : ContractingWith K f) (x : α) (hx : edist x (f x) ≠ ∞) :
    ∃ y, IsFixedPt f y ∧ Tendsto (fun n ↦ f^[n] x) atTop (𝓝 y) ∧
      ∀ n : ℕ, edist (f^[n] x) y ≤ edist x (f x) * (K : ℝ≥0∞) ^ n / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) (x : α) (hx : edist x (f x) ≠ ⊤)
Goal: ∃ y,
  Function.IsFixedPt f y ∧
    Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 y) ∧
      ∀ (n : ℕ), edist (f^[n] x) y ≤ edist x (f x) * ↑K ^ n / (1 - ↑K)
-- ContractingWith.efixedPoint_isFixedPt
theorem efixedPoint_isFixedPt (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞) :
    IsFixedPt f (efixedPoint f hf x hx) 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤)
Goal: Function.IsFixedPt f (ContractingWith.efixedPoint f hf x hx)
-- ContractingWith.tendsto_iterate_efixedPoint
theorem tendsto_iterate_efixedPoint (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞) :
    Tendsto (fun n ↦ f^[n] x) atTop (𝓝 <| efixedPoint f hf x hx) 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤)
Goal: Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 (ContractingWith.efixedPoint f hf x hx))
-- ContractingWith.apriori_edist_iterate_efixedPoint_le
theorem apriori_edist_iterate_efixedPoint_le (hf : ContractingWith K f) {x : α}
    (hx : edist x (f x) ≠ ∞) (n : ℕ) :
    edist (f^[n] x) (efixedPoint f hf x hx) ≤ edist x (f x) * (K : ℝ≥0∞) ^ n / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤) (n : ℕ)
Goal: edist (f^[n] x) (ContractingWith.efixedPoint f hf x hx) ≤ edist x (f x) * ↑K ^ n / (1 - ↑K)
-- ContractingWith.edist_efixedPoint_le
theorem edist_efixedPoint_le (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞) :
    edist x (efixedPoint f hf x hx) ≤ edist x (f x) / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤)
Goal: edist x (ContractingWith.efixedPoint f hf x hx) ≤ edist x (f x) / (1 - ↑K)
-- ContractingWith.edist_efixedPoint_lt_top
theorem edist_efixedPoint_lt_top (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞) :
    edist x (efixedPoint f hf x hx) < ∞ 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤)
Goal: edist x (ContractingWith.efixedPoint f hf x hx) < ⊤
-- ContractingWith.efixedPoint_eq_of_edist_lt_top
theorem efixedPoint_eq_of_edist_lt_top (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞)
    {y : α} (hy : edist y (f y) ≠ ∞) (h : edist x y ≠ ∞) :
    efixedPoint f hf x hx = efixedPoint f hf y hy 
Arguments: {α : Type u_1} [EMetricSpace α] [cs : CompleteSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ⊤) {y : α} (hy : edist y (f y) ≠ ⊤) (h : edist x y ≠ ⊤)
Goal: ContractingWith.efixedPoint f hf x hx = ContractingWith.efixedPoint f hf y hy
-- ContractingWith.exists_fixedPoint'
theorem exists_fixedPoint' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    ∃ y ∈ s, IsFixedPt f y ∧ Tendsto (fun n ↦ f^[n] x) atTop (𝓝 y) ∧
      ∀ n : ℕ, edist (f^[n] x) y ≤ edist x (f x) * (K : ℝ≥0∞) ^ n / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: ∃ y ∈ s,
  Function.IsFixedPt f y ∧
    Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 y) ∧
      ∀ (n : ℕ), edist (f^[n] x) y ≤ edist x (f x) * ↑K ^ n / (1 - ↑K)
-- ContractingWith.efixedPoint_mem'
theorem efixedPoint_mem' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    efixedPoint' f hsc hsf hf x hxs hx ∈ s 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: ContractingWith.efixedPoint' f hsc hsf hf x hxs hx ∈ s
-- ContractingWith.efixedPoint_isFixedPt'
theorem efixedPoint_isFixedPt' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    IsFixedPt f (efixedPoint' f hsc hsf hf x hxs hx) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: Function.IsFixedPt f (ContractingWith.efixedPoint' f hsc hsf hf x hxs hx)
-- ContractingWith.tendsto_iterate_efixedPoint'
theorem tendsto_iterate_efixedPoint' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    Tendsto (fun n ↦ f^[n] x) atTop (𝓝 <| efixedPoint' f hsc hsf hf x hxs hx) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: Filter.Tendsto (fun n => f^[n] x) Filter.atTop (𝓝 (ContractingWith.efixedPoint' f hsc hsf hf x hxs hx))
-- ContractingWith.apriori_edist_iterate_efixedPoint_le'
theorem apriori_edist_iterate_efixedPoint_le' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞)
    (n : ℕ) :
    edist (f^[n] x) (efixedPoint' f hsc hsf hf x hxs hx) ≤
      edist x (f x) * (K : ℝ≥0∞) ^ n / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤) (n : ℕ)
Goal: edist (f^[n] x) (ContractingWith.efixedPoint' f hsc hsf hf x hxs hx) ≤ edist x (f x) * ↑K ^ n / (1 - ↑K)
-- ContractingWith.edist_efixedPoint_le'
theorem edist_efixedPoint_le' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    edist x (efixedPoint' f hsc hsf hf x hxs hx) ≤ edist x (f x) / (1 - K) 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: edist x (ContractingWith.efixedPoint' f hsc hsf hf x hxs hx) ≤ edist x (f x) / (1 - ↑K)
-- ContractingWith.edist_efixedPoint_lt_top'
theorem edist_efixedPoint_lt_top' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    edist x (efixedPoint' f hsc hsf hf x hxs hx) < ∞ 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hf : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤)
Goal: edist x (ContractingWith.efixedPoint' f hsc hsf hf x hxs hx) < ⊤
-- ContractingWith.efixedPoint_eq_of_edist_lt_top'
theorem efixedPoint_eq_of_edist_lt_top' (hf : ContractingWith K f) {s : Set α} (hsc : IsComplete s)
    (hsf : MapsTo f s s) (hfs : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s)
    (hx : edist x (f x) ≠ ∞) {t : Set α} (htc : IsComplete t) (htf : MapsTo f t t)
    (hft : ContractingWith K <| htf.restrict f t t) {y : α} (hyt : y ∈ t) (hy : edist y (f y) ≠ ∞)
    (hxy : edist x y ≠ ∞) :
    efixedPoint' f hsc hsf hfs x hxs hx = efixedPoint' f htc htf hft y hyt hy 
Arguments: {α : Type u_1} [EMetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {s : Set α} (hsc : IsComplete s) (hsf : Set.MapsTo f s s) (hfs : ContractingWith K (Set.MapsTo.restrict f s s hsf)) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ⊤) {t : Set α} (htc : IsComplete t) (htf : Set.MapsTo f t t) (hft : ContractingWith K (Set.MapsTo.restrict f t t htf)) {y : α} (hyt : y ∈ t) (hy : edist y (f y) ≠ ⊤) (hxy : edist x y ≠ ⊤)
Goal: ContractingWith.efixedPoint' f hsc hsf hfs x hxs hx = ContractingWith.efixedPoint' f htc htf hft y hyt hy
-- ContractingWith.one_sub_K_pos
theorem one_sub_K_pos (hf : ContractingWith K f) : (0 : ℝ) < 1 - K 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f)
Goal: 0 < 1 - ↑K
-- ContractingWith.dist_le_mul
theorem dist_le_mul (x y : α) : dist (f x) (f y) ≤ K * dist x y 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) (x y : α)
Goal: dist (f x) (f y) ≤ ↑K * dist x y
-- ContractingWith.dist_inequality
theorem dist_inequality (x y) : dist x y ≤ (dist x (f x) + dist y (f y)) / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) (x y : α)
Goal: dist x y ≤ (dist x (f x) + dist y (f y)) / (1 - ↑K)
-- ContractingWith.dist_le_of_fixedPoint
theorem dist_le_of_fixedPoint (x) {y} (hy : IsFixedPt f y) : dist x y ≤ dist x (f x) / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) (x : α) {y : α} (hy : Function.IsFixedPt f y)
Goal: dist x y ≤ dist x (f x) / (1 - ↑K)
-- ContractingWith.fixedPoint_unique'
theorem fixedPoint_unique' {x y} (hx : IsFixedPt f x) (hy : IsFixedPt f y) : x = y 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) {x y : α} (hx : Function.IsFixedPt f x) (hy : Function.IsFixedPt f y)
Goal: x = y
-- ContractingWith.dist_fixedPoint_fixedPoint_of_dist_le'
theorem dist_fixedPoint_fixedPoint_of_dist_le' (g : α → α) {x y} (hx : IsFixedPt f x)
    (hy : IsFixedPt g y) {C} (hfg : ∀ z, dist (f z) (g z) ≤ C) : dist x y ≤ C / (1 - K) 
Arguments: {α : Type u_1} [MetricSpace α] {K : ℝ≥0} {f : α → α} (hf : ContractingWith K f) (g : α → α) {x y : α} (hx : Function.IsFixedPt f x) (hy : Function.IsFixedPt g y) {C : ℝ} (hfg : ∀ (z : α), dist (f z) (g z) ≤ C)
Goal: dist x y ≤ C / (1 - ↑K)