ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Continuous.matrix_updateColumn
@[continuity]
theorem Continuous.matrix_updateColumn [DecidableEq n] (i : n) {A : X → Matrix m n R}
    {B : X → m → R} (hA : Continuous A) (hB : Continuous B) :
    Continuous fun x => (A x).updateColumn i (B x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [DecidableEq n] (i : n) {A : X → Matrix m n R} {B : X → m → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => (A x).updateColumn i (B x)
-- Continuous.matrix_updateRow
@[continuity]
theorem Continuous.matrix_updateRow [DecidableEq m] (i : m) {A : X → Matrix m n R} {B : X → n → R}
    (hA : Continuous A) (hB : Continuous B) : Continuous fun x => (A x).updateRow i (B x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [DecidableEq m] (i : m) {A : X → Matrix m n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => (A x).updateRow i (B x)
-- Continuous.matrix_cramer
@[continuity]
theorem Continuous.matrix_cramer [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
    {A : X → Matrix n n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
    Continuous fun x => cramer (A x) (B x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R] {A : X → Matrix n n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => (A x).cramer (B x)
-- Continuous.matrix_adjugate
@[continuity]
theorem Continuous.matrix_adjugate [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
    {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).adjugate 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R] {A : X → Matrix n n R} (hA : Continuous A)
Goal: Continuous fun x => (A x).adjugate
-- continuousAt_matrix_inv
theorem continuousAt_matrix_inv [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
    (A : Matrix n n R) (h : ContinuousAt Ring.inverse A.det) : ContinuousAt Inv.inv A 
Arguments: {n : Type u_5} {R : Type u_8} [TopologicalSpace R] [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R] (A : Matrix n n R) (h : ContinuousAt Ring.inverse A.det)
Goal: ContinuousAt Inv.inv A
-- Continuous.matrix_fromBlocks
@[continuity]
theorem Continuous.matrix_fromBlocks {A : X → Matrix n l R} {B : X → Matrix n m R}
    {C : X → Matrix p l R} {D : X → Matrix p m R} (hA : Continuous A) (hB : Continuous B)
    (hC : Continuous C) (hD : Continuous D) :
    Continuous fun x => Matrix.fromBlocks (A x) (B x) (C x) (D x) 
Arguments: {X : Type u_1} {l : Type u_3} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix n l R} {B : X → Matrix n m R} {C : X → Matrix p l R} {D : X → Matrix p m R} (hA : Continuous A) (hB : Continuous B) (hC : Continuous C) (hD : Continuous D)
Goal: Continuous fun x => Matrix.fromBlocks (A x) (B x) (C x) (D x)
-- Continuous.matrix_blockDiagonal
@[continuity]
theorem Continuous.matrix_blockDiagonal [Zero R] [DecidableEq p] {A : X → p → Matrix m n R}
    (hA : Continuous A) : Continuous fun x => blockDiagonal (A x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Zero R] [DecidableEq p] {A : X → p → Matrix m n R} (hA : Continuous A)
Goal: Continuous fun x => Matrix.blockDiagonal (A x)
-- Continuous.matrix_blockDiag
@[continuity]
theorem Continuous.matrix_blockDiag {A : X → Matrix (m × p) (n × p) R} (hA : Continuous A) :
    Continuous fun x => blockDiag (A x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix (m × p) (n × p) R} (hA : Continuous A)
Goal: Continuous fun x => (A x).blockDiag
-- Continuous.matrix_blockDiagonal'
@[continuity]
theorem Continuous.matrix_blockDiagonal' [Zero R] [DecidableEq l]
    {A : X → ∀ i, Matrix (m' i) (n' i) R} (hA : Continuous A) :
    Continuous fun x => blockDiagonal' (A x) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [TopologicalSpace X] [TopologicalSpace R] [Zero R] [DecidableEq l] {A : X → (i : l) → Matrix (m' i) (n' i) R} (hA : Continuous A)
Goal: Continuous fun x => Matrix.blockDiagonal' (A x)
-- Continuous.matrix_blockDiag'
@[continuity]
theorem Continuous.matrix_blockDiag' {A : X → Matrix (Σi, m' i) (Σi, n' i) R} (hA : Continuous A) :
    Continuous fun x => blockDiag' (A x) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix ((i : l) × m' i) ((i : l) × n' i) R} (hA : Continuous A)
Goal: Continuous fun x => (A x).blockDiag'
-- HasSum.matrix_transpose
theorem HasSum.matrix_transpose {f : X → Matrix m n R} {a : Matrix m n R} (hf : HasSum f a) :
    HasSum (fun x => (f x)ᵀ) aᵀ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix m n R} {a : Matrix m n R} (hf : HasSum f a)
Goal: HasSum (fun x => (f x)ᵀ) aᵀ
-- Summable.matrix_transpose
theorem Summable.matrix_transpose {f : X → Matrix m n R} (hf : Summable f) :
    Summable fun x => (f x)ᵀ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix m n R} (hf : Summable f)
Goal: Summable fun x => (f x)ᵀ
-- summable_matrix_transpose
@[simp]
theorem summable_matrix_transpose {f : X → Matrix m n R} :
    (Summable fun x => (f x)ᵀ) ↔ Summable f 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix m n R}
Goal: (Summable fun x => (f x)ᵀ) ↔ Summable f
-- Matrix.transpose_tsum
theorem Matrix.transpose_tsum [T2Space R] {f : X → Matrix m n R} : (∑' x, f x)ᵀ = ∑' x, (f x)ᵀ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [T2Space R] {f : X → Matrix m n R}
Goal: (∑' (x : X), f x)ᵀ = ∑' (x : X), (f x)ᵀ
-- HasSum.matrix_conjTranspose
theorem HasSum.matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R}
    {a : Matrix m n R} (hf : HasSum f a) : HasSum (fun x => (f x)ᴴ) aᴴ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R} {a : Matrix m n R} (hf : HasSum f a)
Goal: HasSum (fun x => (f x)ᴴ) aᴴ
-- Summable.matrix_conjTranspose
theorem Summable.matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R}
    (hf : Summable f) : Summable fun x => (f x)ᴴ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R} (hf : Summable f)
Goal: Summable fun x => (f x)ᴴ
-- summable_matrix_conjTranspose
@[simp]
theorem summable_matrix_conjTranspose [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R} :
    (Summable fun x => (f x)ᴴ) ↔ Summable f 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [StarAddMonoid R] [ContinuousStar R] {f : X → Matrix m n R}
Goal: (Summable fun x => (f x)ᴴ) ↔ Summable f
-- Matrix.conjTranspose_tsum
theorem Matrix.conjTranspose_tsum [StarAddMonoid R] [ContinuousStar R] [T2Space R]
    {f : X → Matrix m n R} : (∑' x, f x)ᴴ = ∑' x, (f x)ᴴ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [StarAddMonoid R] [ContinuousStar R] [T2Space R] {f : X → Matrix m n R}
Goal: (∑' (x : X), f x)ᴴ = ∑' (x : X), (f x)ᴴ
-- HasSum.matrix_diagonal
theorem HasSum.matrix_diagonal [DecidableEq n] {f : X → n → R} {a : n → R} (hf : HasSum f a) :
    HasSum (fun x => diagonal (f x)) (diagonal a) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq n] {f : X → n → R} {a : n → R} (hf : HasSum f a)
Goal: HasSum (fun x => Matrix.diagonal (f x)) (Matrix.diagonal a)
-- Summable.matrix_diagonal
theorem Summable.matrix_diagonal [DecidableEq n] {f : X → n → R} (hf : Summable f) :
    Summable fun x => diagonal (f x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq n] {f : X → n → R} (hf : Summable f)
Goal: Summable fun x => Matrix.diagonal (f x)
-- summable_matrix_diagonal
@[simp]
theorem summable_matrix_diagonal [DecidableEq n] {f : X → n → R} :
    (Summable fun x => diagonal (f x)) ↔ Summable f 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq n] {f : X → n → R}
Goal: (Summable fun x => Matrix.diagonal (f x)) ↔ Summable f
-- Matrix.diagonal_tsum
theorem Matrix.diagonal_tsum [DecidableEq n] [T2Space R] {f : X → n → R} :
    diagonal (∑' x, f x) = ∑' x, diagonal (f x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq n] [T2Space R] {f : X → n → R}
Goal: Matrix.diagonal (∑' (x : X), f x) = ∑' (x : X), Matrix.diagonal (f x)
-- HasSum.matrix_diag
theorem HasSum.matrix_diag {f : X → Matrix n n R} {a : Matrix n n R} (hf : HasSum f a) :
    HasSum (fun x => diag (f x)) (diag a) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix n n R} {a : Matrix n n R} (hf : HasSum f a)
Goal: HasSum (fun x => (f x).diag) a.diag
-- Summable.matrix_diag
theorem Summable.matrix_diag {f : X → Matrix n n R} (hf : Summable f) :
    Summable fun x => diag (f x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix n n R} (hf : Summable f)
Goal: Summable fun x => (f x).diag
-- HasSum.matrix_blockDiagonal
theorem HasSum.matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R}
    {a : p → Matrix m n R} (hf : HasSum f a) :
    HasSum (fun x => blockDiagonal (f x)) (blockDiagonal a) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq p] {f : X → p → Matrix m n R} {a : p → Matrix m n R} (hf : HasSum f a)
Goal: HasSum (fun x => Matrix.blockDiagonal (f x)) (Matrix.blockDiagonal a)
-- Summable.matrix_blockDiagonal
theorem Summable.matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R} (hf : Summable f) :
    Summable fun x => blockDiagonal (f x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq p] {f : X → p → Matrix m n R} (hf : Summable f)
Goal: Summable fun x => Matrix.blockDiagonal (f x)
-- summable_matrix_blockDiagonal
theorem summable_matrix_blockDiagonal [DecidableEq p] {f : X → p → Matrix m n R} :
    (Summable fun x => blockDiagonal (f x)) ↔ Summable f 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq p] {f : X → p → Matrix m n R}
Goal: (Summable fun x => Matrix.blockDiagonal (f x)) ↔ Summable f
-- Matrix.blockDiagonal_tsum
theorem Matrix.blockDiagonal_tsum [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R} :
    blockDiagonal (∑' x, f x) = ∑' x, blockDiagonal (f x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R}
Goal: Matrix.blockDiagonal (∑' (x : X), f x) = ∑' (x : X), Matrix.blockDiagonal (f x)
-- HasSum.matrix_blockDiag
theorem HasSum.matrix_blockDiag {f : X → Matrix (m × p) (n × p) R} {a : Matrix (m × p) (n × p) R}
    (hf : HasSum f a) : HasSum (fun x => blockDiag (f x)) (blockDiag a) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix (m × p) (n × p) R} {a : Matrix (m × p) (n × p) R} (hf : HasSum f a)
Goal: HasSum (fun x => (f x).blockDiag) a.blockDiag
-- Summable.matrix_blockDiag
theorem Summable.matrix_blockDiag {f : X → Matrix (m × p) (n × p) R} (hf : Summable f) :
    Summable fun x => blockDiag (f x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {p : Type u_6} {R : Type u_8} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix (m × p) (n × p) R} (hf : Summable f)
Goal: Summable fun x => (f x).blockDiag
-- HasSum.matrix_blockDiagonal'
theorem HasSum.matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R}
    {a : ∀ i, Matrix (m' i) (n' i) R} (hf : HasSum f a) :
    HasSum (fun x => blockDiagonal' (f x)) (blockDiagonal' a) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq l] {f : X → (i : l) → Matrix (m' i) (n' i) R} {a : (i : l) → Matrix (m' i) (n' i) R} (hf : HasSum f a)
Goal: HasSum (fun x => Matrix.blockDiagonal' (f x)) (Matrix.blockDiagonal' a)
-- Summable.matrix_blockDiagonal'
theorem Summable.matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R}
    (hf : Summable f) : Summable fun x => blockDiagonal' (f x) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq l] {f : X → (i : l) → Matrix (m' i) (n' i) R} (hf : Summable f)
Goal: Summable fun x => Matrix.blockDiagonal' (f x)
-- summable_matrix_blockDiagonal'
theorem summable_matrix_blockDiagonal' [DecidableEq l] {f : X → ∀ i, Matrix (m' i) (n' i) R} :
    (Summable fun x => blockDiagonal' (f x)) ↔ Summable f 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq l] {f : X → (i : l) → Matrix (m' i) (n' i) R}
Goal: (Summable fun x => Matrix.blockDiagonal' (f x)) ↔ Summable f
-- Matrix.blockDiagonal'_tsum
theorem Matrix.blockDiagonal'_tsum [DecidableEq l] [T2Space R]
    {f : X → ∀ i, Matrix (m' i) (n' i) R} :
    blockDiagonal' (∑' x, f x) = ∑' x, blockDiagonal' (f x) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] [DecidableEq l] [T2Space R] {f : X → (i : l) → Matrix (m' i) (n' i) R}
Goal: Matrix.blockDiagonal' (∑' (x : X), f x) = ∑' (x : X), Matrix.blockDiagonal' (f x)
-- HasSum.matrix_blockDiag'
theorem HasSum.matrix_blockDiag' {f : X → Matrix (Σi, m' i) (Σi, n' i) R}
    {a : Matrix (Σi, m' i) (Σi, n' i) R} (hf : HasSum f a) :
    HasSum (fun x => blockDiag' (f x)) (blockDiag' a) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix ((i : l) × m' i) ((i : l) × n' i) R} {a : Matrix ((i : l) × m' i) ((i : l) × n' i) R} (hf : HasSum f a)
Goal: HasSum (fun x => (f x).blockDiag') a.blockDiag'
-- Summable.matrix_blockDiag'
theorem Summable.matrix_blockDiag' {f : X → Matrix (Σi, m' i) (Σi, n' i) R} (hf : Summable f) :
    Summable fun x => blockDiag' (f x) 
Arguments: {X : Type u_1} {l : Type u_3} {R : Type u_8} {m' : l → Type u_9} {n' : l → Type u_10} [AddCommMonoid R] [TopologicalSpace R] {f : X → Matrix ((i : l) × m' i) ((i : l) × n' i) R} (hf : Summable f)
Goal: Summable fun x => (f x).blockDiag'
-- Real.uniformContinuous_add
theorem Real.uniformContinuous_add : UniformContinuous fun p : ℝ × ℝ => p.1 + p.2 
Goal: UniformContinuous fun p => p.1 + p.2
-- Real.uniformContinuous_neg
theorem Real.uniformContinuous_neg : UniformContinuous (@Neg.neg ℝ _) 
Goal: UniformContinuous Neg.neg
-- Real.isTopologicalBasis_Ioo_rat
theorem Real.isTopologicalBasis_Ioo_rat :
    @IsTopologicalBasis ℝ _ (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) b}) 
Goal: TopologicalSpace.IsTopologicalBasis (⋃ a, ⋃ b, ⋃ (_ : a < b), {Set.Ioo ↑a ↑b})
-- Real.cobounded_eq
@[simp]
theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop 
Goal: Bornology.cobounded ℝ = Filter.atBot ⊔ Filter.atTop
-- Real.mem_closure_iff
theorem Real.mem_closure_iff {s : Set ℝ} {x : ℝ} :
    x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε 
Arguments: {s : Set ℝ} {x : ℝ}
Goal: x ∈ closure s ↔ ∀ ε > 0, ∃ y ∈ s, |y - x| < ε
-- Real.uniformContinuous_inv
theorem Real.uniformContinuous_inv (s : Set ℝ) {r : ℝ} (r0 : 0 < r) (H : ∀ x ∈ s, r ≤ |x|) :
    UniformContinuous fun p : s => p.1⁻¹ 
Arguments: (s : Set ℝ) {r : ℝ} (r0 : 0 < r) (H : ∀ x ∈ s, r ≤ |x|)
Goal: UniformContinuous fun p => (↑p)⁻¹
-- Real.uniformContinuous_abs
theorem Real.uniformContinuous_abs : UniformContinuous (abs : ℝ → ℝ) 
Goal: UniformContinuous abs
-- Real.continuous_inv
theorem Real.continuous_inv : Continuous fun a : { r : ℝ // r ≠ 0 } => a.val⁻¹ 
Goal: Continuous fun a => (↑a)⁻¹
-- Real.uniformContinuous_const_mul
theorem Real.uniformContinuous_const_mul {x : ℝ} : UniformContinuous (x * ·) 
Arguments: {x : ℝ}
Goal: UniformContinuous fun x_1 => x * x_1
-- Real.uniformContinuous_mul
theorem Real.uniformContinuous_mul (s : Set (ℝ × ℝ)) {r₁ r₂ : ℝ}
    (H : ∀ x ∈ s, |(x : ℝ × ℝ).1| < r₁ ∧ |x.2| < r₂) :
    UniformContinuous fun p : s => p.1.1 * p.1.2 
Arguments: (s : Set (ℝ × ℝ)) {r₁ r₂ : ℝ} (H : ∀ x ∈ s, |x.1| < r₁ ∧ |x.2| < r₂)
Goal: UniformContinuous fun p => (↑p).1 * (↑p).2
-- Real.totallyBounded_ball
theorem Real.totallyBounded_ball (x ε : ℝ) : TotallyBounded (ball x ε) 
Arguments: (x ε : ℝ)
Goal: TotallyBounded (Metric.ball x ε)
-- closure_of_rat_image_lt
theorem closure_of_rat_image_lt {q : ℚ} :
    closure (((↑) : ℚ → ℝ) '' { x | q < x }) = { r | ↑q ≤ r } 
Arguments: {q : ℚ}
Goal: closure (Rat.cast '' {x | q < x}) = {r | ↑q ≤ r}
-- Function.Periodic.compact_of_continuous
theorem Periodic.compact_of_continuous [TopologicalSpace α] {f : ℝ → α} {c : ℝ} (hp : Periodic f c)
    (hc : c ≠ 0) (hf : Continuous f) : IsCompact (range f) 
Arguments: {α : Type u} [TopologicalSpace α] {f : ℝ → α} {c : ℝ} (hp : Function.Periodic f c) (hc : c ≠ 0) (hf : Continuous f)
Goal: IsCompact (Set.range f)
-- Function.Periodic.isBounded_of_continuous
theorem Periodic.isBounded_of_continuous [PseudoMetricSpace α] {f : ℝ → α} {c : ℝ}
    (hp : Periodic f c) (hc : c ≠ 0) (hf : Continuous f) : IsBounded (range f) 
Arguments: {α : Type u} [PseudoMetricSpace α] {f : ℝ → α} {c : ℝ} (hp : Function.Periodic f c) (hc : c ≠ 0) (hf : Continuous f)
Goal: Bornology.IsBounded (Set.range f)
-- Int.tendsto_coe_cofinite
theorem tendsto_coe_cofinite : Tendsto ((↑) : ℤ → ℝ) cofinite (cocompact ℝ) 
Goal: Filter.Tendsto Int.cast Filter.cofinite (Filter.cocompact ℝ)
-- Int.tendsto_zmultiplesHom_cofinite
theorem tendsto_zmultiplesHom_cofinite {a : ℝ} (ha : a ≠ 0) :
    Tendsto (zmultiplesHom ℝ a) cofinite (cocompact ℝ) 
Arguments: {a : ℝ} (ha : a ≠ 0)
Goal: Filter.Tendsto (⇑((zmultiplesHom ℝ) a)) Filter.cofinite (Filter.cocompact ℝ)
-- AddSubgroup.tendsto_zmultiples_subtype_cofinite
theorem tendsto_zmultiples_subtype_cofinite (a : ℝ) :
    Tendsto (zmultiples a).subtype cofinite (cocompact ℝ) 
Arguments: (a : ℝ)
Goal: Filter.Tendsto (⇑(AddSubgroup.zmultiples a).subtype) Filter.cofinite (Filter.cocompact ℝ)
-- Rat.interior_compact_eq_empty
theorem interior_compact_eq_empty (hs : IsCompact s) : interior s = ∅ 
Arguments: {s : Set ℚ} (hs : IsCompact s)
Goal: interior s = ∅
-- Rat.dense_compl_compact
theorem dense_compl_compact (hs : IsCompact s) : Dense sᶜ 
Arguments: {s : Set ℚ} (hs : IsCompact s)
Goal: Dense sᶜ
-- Rat.not_countably_generated_cocompact
theorem not_countably_generated_cocompact : ¬IsCountablyGenerated (cocompact ℚ) 
Goal: ¬(Filter.cocompact ℚ).IsCountablyGenerated
-- Rat.not_countably_generated_nhds_infty_opc
theorem not_countably_generated_nhds_infty_opc : ¬IsCountablyGenerated (𝓝 (∞ : ℚ∞)) 
Goal: ¬(𝓝 ∞).IsCountablyGenerated
-- Rat.not_firstCountableTopology_opc
theorem not_firstCountableTopology_opc : ¬FirstCountableTopology ℚ∞ 
Goal: ¬FirstCountableTopology ℚ∞
-- Rat.not_secondCountableTopology_opc
theorem not_secondCountableTopology_opc : ¬SecondCountableTopology ℚ∞ 
Goal: ¬SecondCountableTopology ℚ∞
-- ExtremallyDisconnected.open_closure
class ExtremallyDisconnected : Prop where
    open_closure : ∀ U : Set X, IsOpen U → IsOpen (closure U)
Arguments: {X : Type u} [TopologicalSpace X] [self : ExtremallyDisconnected X] (U : Set X)
Goal: IsOpen U → IsOpen (closure U)
-- extremallyDisconnected_of_homeo
theorem extremallyDisconnected_of_homeo {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
    [ExtremallyDisconnected X] (e : X ≃ₜ Y) : ExtremallyDisconnected Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [ExtremallyDisconnected X] (e : X ≃ₜ Y)
Goal: ExtremallyDisconnected Y
-- StoneCech.projective
theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneCech X) 
Arguments: {X : Type u} [TopologicalSpace X] [DiscreteTopology X]
Goal: CompactT2.Projective (StoneCech X)
-- CompactT2.Projective.extremallyDisconnected
protected theorem CompactT2.Projective.extremallyDisconnected [CompactSpace X] [T2Space X]
    (h : CompactT2.Projective X) : ExtremallyDisconnected X 
Arguments: {X : Type u} [TopologicalSpace X] [CompactSpace X] [T2Space X] (h : CompactT2.Projective X)
Goal: ExtremallyDisconnected X
-- exists_compact_surjective_zorn_subset
lemma exists_compact_surjective_zorn_subset [T1Space A] [CompactSpace D] {π : D → A}
    (π_cont : Continuous π) (π_surj : π.Surjective) : ∃ E : Set D, CompactSpace E ∧ π '' E = univ ∧
    ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → E.restrict π '' E₀ ≠ univ 
Arguments: {A D : Type u} [TopologicalSpace A] [TopologicalSpace D] [T1Space A] [CompactSpace D] {π : D → A} (π_cont : Continuous π) (π_surj : Function.Surjective π)
Goal: ∃ E, CompactSpace ↑E ∧ π '' E = Set.univ ∧ ∀ (E₀ : Set ↑E), E₀ ≠ Set.univ → IsClosed E₀ → E.restrict π '' E₀ ≠ Set.univ
-- image_subset_closure_compl_image_compl_of_isOpen
lemma image_subset_closure_compl_image_compl_of_isOpen {ρ : E → A} (ρ_cont : Continuous ρ)
    (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ)
    {G : Set E} (hG : IsOpen G) : ρ '' G ⊆ closure ((ρ '' Gᶜ)ᶜ) 
Arguments: {A E : Type u} [TopologicalSpace A] [TopologicalSpace E] {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : Function.Surjective ρ) (zorn_subset : ∀ (E₀ : Set E), E₀ ≠ Set.univ → IsClosed E₀ → ρ '' E₀ ≠ Set.univ) {G : Set E} (hG : IsOpen G)
Goal: ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ
-- ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen
lemma ExtremallyDisconnected.disjoint_closure_of_disjoint_isOpen [ExtremallyDisconnected A]
    {U₁ U₂ : Set A} (h : Disjoint U₁ U₂) (hU₁ : IsOpen U₁) (hU₂ : IsOpen U₂) :
    Disjoint (closure U₁) (closure U₂) 
Arguments: {A : Type u} [TopologicalSpace A] [ExtremallyDisconnected A] {U₁ U₂ : Set A} (h : Disjoint U₁ U₂) (hU₁ : IsOpen U₁) (hU₂ : IsOpen U₂)
Goal: Disjoint (closure U₁) (closure U₂)
-- _private.Mathlib.Topology.ExtremallyDisconnected.0.ExtremallyDisconnected.homeoCompactToT2_injective
private lemma ExtremallyDisconnected.homeoCompactToT2_injective [ExtremallyDisconnected A]
    [T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ)
    (ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) :
    ρ.Injective 
Arguments: {A E : Type u} [TopologicalSpace A] [TopologicalSpace E] [ExtremallyDisconnected A] [T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ) (ρ_surj : Function.Surjective ρ) (zorn_subset : ∀ (E₀ : Set E), E₀ ≠ Set.univ → IsClosed E₀ → ρ '' E₀ ≠ Set.univ)
Goal: Function.Injective ρ
-- CompactT2.ExtremallyDisconnected.projective
protected theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconnected A]
    [CompactSpace A] [T2Space A] : CompactT2.Projective A 
Arguments: {A : Type u} [TopologicalSpace A] [ExtremallyDisconnected A] [CompactSpace A] [T2Space A]
Goal: CompactT2.Projective A
-- CompactT2.projective_iff_extremallyDisconnected
protected theorem CompactT2.projective_iff_extremallyDisconnected [CompactSpace A] [T2Space A] :
    Projective A ↔ ExtremallyDisconnected A 
Arguments: {A : Type u} [TopologicalSpace A] [CompactSpace A] [T2Space A]
Goal: CompactT2.Projective A ↔ ExtremallyDisconnected A