ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FormalMultilinearSeries.changeOriginSeries_summable_aux₁
theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
    Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
      ‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {r r' : ℝ≥0} (hr : ↑r + ↑r' < p.radius)
Goal: Summable fun s => ‖p (s.fst + s.snd.fst)‖₊ * r ^ s.snd.fst * r' ^ s.fst
-- FormalMultilinearSeries.changeOriginSeries_summable_aux₂
theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
    Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
      ‖p (k + s.1)‖₊ * r ^ s.1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ)
Goal: Summable fun s => ‖p (k + s.fst)‖₊ * r ^ s.fst
-- FormalMultilinearSeries.changeOriginSeries_summable_aux₃
theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
    Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ)
Goal: Summable fun l => ‖p.changeOriginSeries k l‖₊ * r ^ l
-- FormalMultilinearSeries.le_changeOriginSeries_radius
theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (k : ℕ)
Goal: p.radius ≤ (p.changeOriginSeries k).radius
-- FormalMultilinearSeries.nnnorm_changeOrigin_le
theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
    ‖p.changeOrigin x k‖₊ ≤
      ∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {x : E} (k : ℕ) (h : ↑‖x‖₊ < p.radius)
Goal: ‖p.changeOrigin x k‖₊ ≤ ∑' (s : (l : ℕ) × { s // s.card = l }), ‖p (k + s.fst)‖₊ * ‖x‖₊ ^ s.fst
-- FormalMultilinearSeries.changeOrigin_radius
theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {x : E}
Goal: p.radius - ↑‖x‖₊ ≤ (p.changeOrigin x).radius
-- FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin
theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
    HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) (k : ℕ) (hr : 0 < p.radius)
Goal: HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius
-- FormalMultilinearSeries.changeOrigin_eval
theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
    (p.changeOrigin x).sum y = p.sum (x + y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} (h : ↑‖x‖₊ + ↑‖y‖₊ < p.radius)
Goal: (p.changeOrigin x).sum y = p.sum (x + y)
-- FormalMultilinearSeries.analyticAt_changeOrigin
theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
    AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ)
Goal: AnalyticAt 𝕜 (fun x => p.changeOrigin x n) 0
-- HasFPowerSeriesOnBall.changeOrigin
theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
    (h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r) (h : ↑‖y‖₊ < r)
Goal: HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ↑‖y‖₊)
-- HasFPowerSeriesOnBall.analyticAt_of_mem
theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
    (h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r) (h : y ∈ EMetric.ball x r)
Goal: AnalyticAt 𝕜 f y
-- HasFPowerSeriesOnBall.analyticOn
theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
    AnalyticOn 𝕜 f (EMetric.ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFPowerSeriesOnBall f p x r)
Goal: AnalyticOn 𝕜 f (EMetric.ball x r)
-- isOpen_analyticAt
theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] (f : E → F)
Goal: IsOpen {x | AnalyticAt 𝕜 f x}
-- AnalyticAt.eventually_analyticAt
theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
    ∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x)
Goal: ∀ᶠ (y : E) in 𝓝 x, AnalyticAt 𝕜 f y
-- AnalyticAt.exists_mem_nhds_analyticOn
theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
    ∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x)
Goal: ∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s
-- AnalyticAt.exists_ball_analyticOn
theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
    ∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace F] {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x)
Goal: ∃ r, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r)
-- hasFPowerSeriesAt_iff
theorem hasFPowerSeriesAt_iff :
    HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
Goal: HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n => z ^ n • p.coeff n) (f (z₀ + z))
-- hasFPowerSeriesAt_iff'
theorem hasFPowerSeriesAt_iff' :
    HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ z in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E} {z₀ : 𝕜}
Goal: HasFPowerSeriesAt f p z₀ ↔ ∀ᶠ (z : 𝕜) in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • p.coeff n) (f z)
-- Geometry.SimplicialComplex.not_empty_mem
@[ext]
structure SimplicialComplex where
    faces : Set (Finset E)
    not_empty_mem : ∅ ∉ faces
    indep : ∀ {s}, s ∈ faces → AffineIndependent 𝕜 ((↑) : s → E)
    down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ≠ ∅ → t ∈ faces
  inter_subset_convexHull : ∀ {s t}, s ∈ faces → t ∈ faces →
    convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t ⊆ convexHull 𝕜 (s ∩ t : Set E)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (self : Geometry.SimplicialComplex 𝕜 E)
Goal: ∅ ∉ self.faces
-- Geometry.SimplicialComplex.indep
@[ext]
structure SimplicialComplex where
    faces : Set (Finset E)
    not_empty_mem : ∅ ∉ faces
    indep : ∀ {s}, s ∈ faces → AffineIndependent 𝕜 ((↑) : s → E)
    down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ≠ ∅ → t ∈ faces
  inter_subset_convexHull : ∀ {s t}, s ∈ faces → t ∈ faces →
    convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t ⊆ convexHull 𝕜 (s ∩ t : Set E)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (self : Geometry.SimplicialComplex 𝕜 E) {s : Finset E}
Goal: s ∈ self.faces → AffineIndependent 𝕜 Subtype.val
-- Geometry.SimplicialComplex.inter_subset_convexHull
@[ext]
structure SimplicialComplex where
    faces : Set (Finset E)
    not_empty_mem : ∅ ∉ faces
    indep : ∀ {s}, s ∈ faces → AffineIndependent 𝕜 ((↑) : s → E)
    down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ≠ ∅ → t ∈ faces
  inter_subset_convexHull : ∀ {s t}, s ∈ faces → t ∈ faces →
    convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t ⊆ convexHull 𝕜 (s ∩ t : Set E)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (self : Geometry.SimplicialComplex 𝕜 E) {s t : Finset E}
Goal: s ∈ self.faces → t ∈ self.faces → (convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t ⊆ (convexHull 𝕜) (↑s ∩ ↑t)
-- Geometry.SimplicialComplex.down_closed
@[ext]
structure SimplicialComplex where
    faces : Set (Finset E)
    not_empty_mem : ∅ ∉ faces
    indep : ∀ {s}, s ∈ faces → AffineIndependent 𝕜 ((↑) : s → E)
    down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ≠ ∅ → t ∈ faces
  inter_subset_convexHull : ∀ {s t}, s ∈ faces → t ∈ faces →
    convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t ⊆ convexHull 𝕜 (s ∩ t : Set E)
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (self : Geometry.SimplicialComplex 𝕜 E) {s t : Finset E}
Goal: s ∈ self.faces → t ⊆ s → t ≠ ∅ → t ∈ self.faces
-- Geometry.SimplicialComplex.mem_space_iff
theorem mem_space_iff : x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ convexHull 𝕜 (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {x : E}
Goal: x ∈ K.space ↔ ∃ s ∈ K.faces, x ∈ (convexHull 𝕜) ↑s
-- Geometry.SimplicialComplex.convexHull_subset_space
theorem convexHull_subset_space (hs : s ∈ K.faces) : convexHull 𝕜 ↑s ⊆ K.space 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s : Finset E} (hs : s ∈ K.faces)
Goal: (convexHull 𝕜) ↑s ⊆ K.space
-- Geometry.SimplicialComplex.subset_space
protected theorem subset_space (hs : s ∈ K.faces) : (s : Set E) ⊆ K.space 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s : Finset E} (hs : s ∈ K.faces)
Goal: ↑s ⊆ K.space
-- Geometry.SimplicialComplex.convexHull_inter_convexHull
theorem convexHull_inter_convexHull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
    convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t = convexHull 𝕜 (s ∩ t : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s t : Finset E} (hs : s ∈ K.faces) (ht : t ∈ K.faces)
Goal: (convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t = (convexHull 𝕜) (↑s ∩ ↑t)
-- Geometry.SimplicialComplex.disjoint_or_exists_inter_eq_convexHull
theorem disjoint_or_exists_inter_eq_convexHull (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
    Disjoint (convexHull 𝕜 (s : Set E)) (convexHull 𝕜 ↑t) ∨
      ∃ u ∈ K.faces, convexHull 𝕜 (s : Set E) ∩ convexHull 𝕜 ↑t = convexHull 𝕜 ↑u 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s t : Finset E} (hs : s ∈ K.faces) (ht : t ∈ K.faces)
Goal: Disjoint ((convexHull 𝕜) ↑s) ((convexHull 𝕜) ↑t) ∨
  ∃ u ∈ K.faces, (convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t = (convexHull 𝕜) ↑u
-- Geometry.SimplicialComplex.ofErase_faces
@[simps]
def ofErase (faces : Set (Finset E)) (indep : ∀ s ∈ faces, AffineIndependent 𝕜 ((↑) : s → E))
    (down_closed : ∀ s ∈ faces, ∀ t ⊆ s, t ∈ faces)
    (inter_subset_convexHull : ∀ᵉ (s ∈ faces) (t ∈ faces),
      convexHull 𝕜 ↑s ∩ convexHull 𝕜 ↑t ⊆ convexHull 𝕜 (s ∩ t : Set E)) :
    SimplicialComplex 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (faces : Set (Finset E)) (indep : ∀ s ∈ faces, AffineIndependent 𝕜 Subtype.val) (down_closed : ∀ s ∈ faces, ∀ t ⊆ s, t ∈ faces) (inter_subset_convexHull : ∀ s ∈ faces, ∀ t ∈ faces, (convexHull 𝕜) ↑s ∩ (convexHull 𝕜) ↑t ⊆ (convexHull 𝕜) (↑s ∩ ↑t))
Goal: (Geometry.SimplicialComplex.ofErase faces indep down_closed inter_subset_convexHull).faces = faces \ {∅}
-- Geometry.SimplicialComplex.ofSubcomplex_faces
@[simps]
def ofSubcomplex (K : SimplicialComplex 𝕜 E) (faces : Set (Finset E)) (subset : faces ⊆ K.faces)
    (down_closed : ∀ {s t}, s ∈ faces → t ⊆ s → t ∈ faces) : SimplicialComplex 𝕜 E 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (K : Geometry.SimplicialComplex 𝕜 E) (faces : Set (Finset E)) (subset : faces ⊆ K.faces) (down_closed : ∀ {s t : Finset E}, s ∈ faces → t ⊆ s → t ∈ faces)
Goal: (K.ofSubcomplex faces subset down_closed).faces = faces
-- Geometry.SimplicialComplex.mem_vertices
theorem mem_vertices : x ∈ K.vertices ↔ {x} ∈ K.faces 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {x : E}
Goal: x ∈ K.vertices ↔ {x} ∈ K.faces
-- Geometry.SimplicialComplex.vertices_eq
theorem vertices_eq : K.vertices = ⋃ k ∈ K.faces, (k : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E}
Goal: K.vertices = ⋃ k ∈ K.faces, ↑k
-- Geometry.SimplicialComplex.vertices_subset_space
theorem vertices_subset_space : K.vertices ⊆ K.space 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E}
Goal: K.vertices ⊆ K.space
-- Geometry.SimplicialComplex.vertex_mem_convexHull_iff
theorem vertex_mem_convexHull_iff (hx : x ∈ K.vertices) (hs : s ∈ K.faces) :
    x ∈ convexHull 𝕜 (s : Set E) ↔ x ∈ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s : Finset E} {x : E} (hx : x ∈ K.vertices) (hs : s ∈ K.faces)
Goal: x ∈ (convexHull 𝕜) ↑s ↔ x ∈ s
-- Geometry.SimplicialComplex.face_subset_face_iff
theorem face_subset_face_iff (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
    convexHull 𝕜 (s : Set E) ⊆ convexHull 𝕜 ↑t ↔ s ⊆ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s t : Finset E} (hs : s ∈ K.faces) (ht : t ∈ K.faces)
Goal: (convexHull 𝕜) ↑s ⊆ (convexHull 𝕜) ↑t ↔ s ⊆ t
-- Geometry.SimplicialComplex.mem_facets
theorem mem_facets : s ∈ K.facets ↔ s ∈ K.faces ∧ ∀ t ∈ K.faces, s ⊆ t → s = t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s : Finset E}
Goal: s ∈ K.facets ↔ s ∈ K.faces ∧ ∀ t ∈ K.faces, s ⊆ t → s = t
-- Geometry.SimplicialComplex.facets_subset
theorem facets_subset : K.facets ⊆ K.faces 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E}
Goal: K.facets ⊆ K.faces
-- Geometry.SimplicialComplex.not_facet_iff_subface
theorem not_facet_iff_subface (hs : s ∈ K.faces) : s ∉ K.facets ↔ ∃ t, t ∈ K.faces ∧ s ⊂ t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {K : Geometry.SimplicialComplex 𝕜 E} {s : Finset E} (hs : s ∈ K.faces)
Goal: s ∉ K.facets ↔ ∃ t ∈ K.faces, s ⊂ t
-- Geometry.SimplicialComplex.faces_bot
theorem faces_bot : (⊥ : SimplicialComplex 𝕜 E).faces = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
Goal: ⊥.faces = ∅
-- Geometry.SimplicialComplex.space_bot
theorem space_bot : (⊥ : SimplicialComplex 𝕜 E).space = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
Goal: ⊥.space = ∅
-- Geometry.SimplicialComplex.facets_bot
theorem facets_bot : (⊥ : SimplicialComplex 𝕜 E).facets = ∅ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [OrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
Goal: ⊥.facets = ∅
-- Real.convex_iff_isPreconnected
theorem Real.convex_iff_isPreconnected {s : Set ℝ} : Convex ℝ s ↔ IsPreconnected s 
Arguments: {s : Set ℝ}
Goal: Convex ℝ s ↔ IsPreconnected s
-- stdSimplex_subset_closedBall
theorem stdSimplex_subset_closedBall : stdSimplex ℝ ι ⊆ Metric.closedBall 0 1 
Arguments: {ι : Type u_1} [Fintype ι]
Goal: stdSimplex ℝ ι ⊆ Metric.closedBall 0 1
-- bounded_stdSimplex
theorem bounded_stdSimplex : IsBounded (stdSimplex ℝ ι) 
Arguments: (ι : Type u_1) [Fintype ι]
Goal: Bornology.IsBounded (stdSimplex ℝ ι)
-- isClosed_stdSimplex
theorem isClosed_stdSimplex : IsClosed (stdSimplex ℝ ι) 
Arguments: (ι : Type u_1) [Fintype ι]
Goal: IsClosed (stdSimplex ℝ ι)
-- isCompact_stdSimplex
theorem isCompact_stdSimplex : IsCompact (stdSimplex ℝ ι) 
Arguments: (ι : Type u_1) [Fintype ι]
Goal: IsCompact (stdSimplex ℝ ι)
-- stdSimplexHomeomorphUnitInterval_symm_apply_coe
@[simps! (config := .asFn)]
def stdSimplexHomeomorphUnitInterval : stdSimplex ℝ (Fin 2) ≃ₜ unitInterval 
Arguments: (x : ↑(Set.Icc 0 1))
Goal: ↑(stdSimplexHomeomorphUnitInterval.symm x) = ![↑x, 1 - ↑x]
-- stdSimplexHomeomorphUnitInterval_apply_coe
@[simps! (config := .asFn)]
def stdSimplexHomeomorphUnitInterval : stdSimplex ℝ (Fin 2) ≃ₜ unitInterval 
Arguments: (f : ↑(stdSimplex ℝ (Fin 2)))
Goal: ↑(stdSimplexHomeomorphUnitInterval f) = ↑f 0
-- segment_subset_closure_openSegment
theorem segment_subset_closure_openSegment : [x -[𝕜] y] ⊆ closure (openSegment 𝕜 x y) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedRing 𝕜] [DenselyOrdered 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜] [AddCommGroup E] [TopologicalSpace E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] {x y : E}
Goal: [x-[𝕜]y] ⊆ closure (openSegment 𝕜 x y)
-- closure_openSegment
@[simp]
theorem closure_openSegment (x y : E) : closure (openSegment 𝕜 x y) = [x -[𝕜] y] 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedRing 𝕜] [DenselyOrdered 𝕜] [PseudoMetricSpace 𝕜] [OrderTopology 𝕜] [ProperSpace 𝕜] [CompactIccSpace 𝕜] [AddCommGroup E] [TopologicalSpace E] [T2Space E] [ContinuousAdd E] [Module 𝕜 E] [ContinuousSMul 𝕜 E] (x y : E)
Goal: closure (openSegment 𝕜 x y) = [x-[𝕜]y]
-- Convex.combo_interior_closure_subset_interior
theorem Convex.combo_interior_closure_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
    (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • interior s + b • closure s ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a • interior s + b • closure s ⊆ interior s
-- Convex.combo_interior_self_subset_interior
theorem Convex.combo_interior_self_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
    (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) : a • interior s + b • s ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a • interior s + b • s ⊆ interior s
-- Convex.combo_closure_interior_subset_interior
theorem Convex.combo_closure_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
    (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • closure s + b • interior s ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
Goal: a • closure s + b • interior s ⊆ interior s
-- Convex.combo_self_interior_subset_interior
theorem Convex.combo_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
    (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) : a • s + b • interior s ⊆ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
Goal: a • s + b • interior s ⊆ interior s
-- Convex.combo_interior_closure_mem_interior
theorem Convex.combo_interior_closure_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ interior s) (hy : y ∈ closure s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b)
    (hab : a + b = 1) : a • x + b • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ closure s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a • x + b • y ∈ interior s
-- Convex.combo_interior_self_mem_interior
theorem Convex.combo_interior_self_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ interior s) (hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :
    a • x + b • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s) {a b : 𝕜} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1)
Goal: a • x + b • y ∈ interior s
-- Convex.combo_closure_interior_mem_interior
theorem Convex.combo_closure_interior_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E}
    (hx : x ∈ closure s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b)
    (hab : a + b = 1) : a • x + b • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
Goal: a • x + b • y ∈ interior s
-- Convex.combo_self_interior_mem_interior
theorem Convex.combo_self_interior_mem_interior {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s)
    (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :
    a • x + b • y ∈ interior s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {a b : 𝕜} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1)
Goal: a • x + b • y ∈ interior s