ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearEquiv.compContinuousAlternatingMap_coe
@[simp]
theorem _root_.ContinuousLinearEquiv.compContinuousAlternatingMap_coe
    (e : N ≃L[R] N') (f : M [⋀^ι]→L[R] N) : ⇑(e.compContinuousAlternatingMap f) = e ∘ f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {N' : Type u_5} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [AddCommMonoid N'] [Module R N'] [TopologicalSpace N'] (e : N ≃L[R] N') (f : M [⋀^ι]→L[R] N)
Goal: ⇑(e.compContinuousAlternatingMap f) = ⇑e ∘ ⇑f
-- ContinuousAlternatingMap.piEquiv_apply
@[simps]
def piEquiv {ι' : Type*} {N : ι' → Type*} [∀ i, AddCommMonoid (N i)] [∀ i, TopologicalSpace (N i)]
    [∀ i, Module R (N i)] : (∀ i, M [⋀^ι]→L[R] N i) ≃ M [⋀^ι]→L[R] ∀ i, N i 
Arguments: {R : Type u_1} {M : Type u_2} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] {ι' : Type u_7} {N : ι' → Type u_8} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → TopologicalSpace (N i)] [(i : ι') → Module R (N i)] (f : (i : ι') → M [⋀^ι]→L[R] N i)
Goal: ContinuousAlternatingMap.piEquiv f = ContinuousAlternatingMap.pi f
-- ContinuousAlternatingMap.piEquiv_symm_apply
@[simps]
def piEquiv {ι' : Type*} {N : ι' → Type*} [∀ i, AddCommMonoid (N i)] [∀ i, TopologicalSpace (N i)]
    [∀ i, Module R (N i)] : (∀ i, M [⋀^ι]→L[R] N i) ≃ M [⋀^ι]→L[R] ∀ i, N i 
Arguments: {R : Type u_1} {M : Type u_2} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] {ι' : Type u_7} {N : ι' → Type u_8} [(i : ι') → AddCommMonoid (N i)] [(i : ι') → TopologicalSpace (N i)] [(i : ι') → Module R (N i)] (f : M [⋀^ι]→L[R] ((i : ι') → N i)) (i : ι')
Goal: ContinuousAlternatingMap.piEquiv.symm f i = (ContinuousLinearMap.proj i).compContinuousAlternatingMap f
-- ContinuousAlternatingMap.cons_add
theorem cons_add (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (x y : M) :
    f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {n : ℕ} (f : M [⋀^Fin (n + 1)]→L[R] N) (m : Fin n → M) (x y : M)
Goal: f (Fin.cons (x + y) m) = f (Fin.cons x m) + f (Fin.cons y m)
-- ContinuousAlternatingMap.vecCons_add
theorem vecCons_add (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (x y : M) :
    f (vecCons (x + y) m) = f (vecCons x m) + f (vecCons y m) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {n : ℕ} (f : M [⋀^Fin (n + 1)]→L[R] N) (m : Fin n → M) (x y : M)
Goal: f (Matrix.vecCons (x + y) m) = f (Matrix.vecCons x m) + f (Matrix.vecCons y m)
-- ContinuousAlternatingMap.cons_smul
theorem cons_smul (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (c : R)
    (x : M) : f (Fin.cons (c • x) m) = c • f (Fin.cons x m) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {n : ℕ} (f : M [⋀^Fin (n + 1)]→L[R] N) (m : Fin n → M) (c : R) (x : M)
Goal: f (Fin.cons (c • x) m) = c • f (Fin.cons x m)
-- ContinuousAlternatingMap.vecCons_smul
theorem vecCons_smul (f : ContinuousAlternatingMap R M N (Fin (n + 1))) (m : Fin n → M) (c : R)
    (x : M) : f (vecCons (c • x) m) = c • f (vecCons x m) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {n : ℕ} (f : M [⋀^Fin (n + 1)]→L[R] N) (m : Fin n → M) (c : R) (x : M)
Goal: f (Matrix.vecCons (c • x) m) = c • f (Matrix.vecCons x m)
-- ContinuousAlternatingMap.map_piecewise_add
theorem map_piecewise_add [DecidableEq ι] (m m' : ι → M) (t : Finset ι) :
    f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m') 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m m' : ι → M) (t : Finset ι)
Goal: f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')
-- ContinuousAlternatingMap.map_add_univ
theorem map_add_univ [DecidableEq ι] [Fintype ι] (m m' : ι → M) :
    f (m + m') = ∑ s : Finset ι, f (s.piecewise m m') 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] [Fintype ι] (m m' : ι → M)
Goal: f (m + m') = ∑ s : Finset ι, f (s.piecewise m m')
-- ContinuousAlternatingMap.map_sum_finset
theorem map_sum_finset :
    (f fun i => ∑ j ∈ A i, g' i j) = ∑ r ∈ piFinset A, f fun i => g' i (r i) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) {α : ι → Type u_7} [Fintype ι] [DecidableEq ι] (g' : (i : ι) → α i → M) (A : (i : ι) → Finset (α i))
Goal: (f fun i => ∑ j ∈ A i, g' i j) = ∑ r ∈ Fintype.piFinset A, f fun i => g' i (r i)
-- ContinuousAlternatingMap.map_sum
theorem map_sum [∀ i, Fintype (α i)] :
    (f fun i => ∑ j, g' i j) = ∑ r : ∀ i, α i, f fun i => g' i (r i) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) {α : ι → Type u_7} [Fintype ι] [DecidableEq ι] (g' : (i : ι) → α i → M) [(i : ι) → Fintype (α i)]
Goal: (f fun i => ∑ j : α i, g' i j) = ∑ r : (i : ι) → α i, f fun i => g' i (r i)
-- ContinuousAlternatingMap.coe_restrictScalars
@[simp]
theorem coe_restrictScalars (f : M [⋀^ι]→L[A] N) : ⇑(f.restrictScalars R) = f 
Arguments: (R : Type u_1) {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] {A : Type u_7} [Semiring A] [SMul R A] [Module A M] [Module A N] [IsScalarTower R A M] [IsScalarTower R A N] (f : M [⋀^ι]→L[A] N)
Goal: ⇑(ContinuousAlternatingMap.restrictScalars R f) = ⇑f
-- ContinuousAlternatingMap.map_sub
@[simp]
theorem map_sub [DecidableEq ι] (m : ι → M) (i : ι) (x y : M) :
    f (update m i (x - y)) = f (update m i x) - f (update m i y) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (m : ι → M) (i : ι) (x y : M)
Goal: f (Function.update m i (x - y)) = f (Function.update m i x) - f (Function.update m i y)
-- ContinuousAlternatingMap.coe_neg
@[simp]
theorem coe_neg : ⇑(-f) = -f 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [TopologicalAddGroup N]
Goal: ⇑(-f) = -⇑f
-- ContinuousAlternatingMap.neg_apply
theorem neg_apply (m : ι → M) : (-f) m = -f m 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [TopologicalAddGroup N] (m : ι → M)
Goal: (-f) m = -f m
-- ContinuousAlternatingMap.coe_sub
@[simp] theorem coe_sub : ⇑(f - g) = ⇑f - ⇑g 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] (f g : M [⋀^ι]→L[R] N) [TopologicalAddGroup N]
Goal: ⇑(f - g) = ⇑f - ⇑g
-- ContinuousAlternatingMap.sub_apply
theorem sub_apply (m : ι → M) : (f - g) m = f m - g m 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] (f g : M [⋀^ι]→L[R] N) [TopologicalAddGroup N] (m : ι → M)
Goal: (f - g) m = f m - g m
-- ContinuousAlternatingMap.map_piecewise_smul
theorem map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ι → M) (s : Finset ι) :
    f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [DecidableEq ι] (c : ι → R) (m : ι → M) (s : Finset ι)
Goal: f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m
-- ContinuousAlternatingMap.map_smul_univ
theorem map_smul_univ [Fintype ι] (c : ι → R) (m : ι → M) :
    (f fun i => c i • m i) = (∏ i, c i) • f m 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_4} {ι : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N] (f : M [⋀^ι]→L[R] N) [Fintype ι] (c : ι → R) (m : ι → M)
Goal: (f fun i => c i • m i) = (∏ i : ι, c i) • f m
-- ContinuousAlternatingMap.toContinuousMultilinearMapLinear_apply
@[simps]
def toContinuousMultilinearMapLinear :
    M [⋀^ι]→L[A] N →ₗ[R] ContinuousMultilinearMap A (fun _ : ι => M) N 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [Semiring R] [Semiring A] [AddCommMonoid M] [AddCommMonoid N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousAdd N] [Module A M] [Module A N] [Module R N] [ContinuousConstSMul R N] [SMulCommClass A R N] (self : M [⋀^ι]→L[A] N)
Goal: ContinuousAlternatingMap.toContinuousMultilinearMapLinear self = self.toContinuousMultilinearMap
-- ContinuousAlternatingMap.toAlternatingMapLinear_apply
@[simps (config := .asFn) apply]
def toAlternatingMapLinear : (M [⋀^ι]→L[A] N) →ₗ[R] (M [⋀^ι]→ₗ[A] N) 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [Semiring R] [Semiring A] [AddCommMonoid M] [AddCommMonoid N] [TopologicalSpace M] [TopologicalSpace N] [ContinuousAdd N] [Module A M] [Module A N] [Module R N] [ContinuousConstSMul R N] [SMulCommClass A R N]
Goal: ⇑ContinuousAlternatingMap.toAlternatingMapLinear = ContinuousAlternatingMap.toAlternatingMap
-- ContinuousAlternatingMap.piLinearEquiv_apply
@[simps (config := { simpRhs := true })]
def piLinearEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, ContinuousAdd (M' i)] [∀ i, Module R (M' i)]
    [∀ i, Module A (M' i)] [∀ i, SMulCommClass A R (M' i)] [∀ i, ContinuousConstSMul R (M' i)] :
    (∀ i, M [⋀^ι]→L[A] M' i) ≃ₗ[R] M [⋀^ι]→L[A] ∀ i, M' i 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} {ι : Type u_5} [Semiring R] [Semiring A] [AddCommMonoid M] [TopologicalSpace M] [Module A M] {ι' : Type u_6} {M' : ι' → Type u_7} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [∀ (i : ι'), ContinuousAdd (M' i)] [(i : ι') → Module R (M' i)] [(i : ι') → Module A (M' i)] [∀ (i : ι'), SMulCommClass A R (M' i)] [∀ (i : ι'), ContinuousConstSMul R (M' i)]
Goal: ∀ (a : (i : ι') → M [⋀^ι]→L[A] M' i), ContinuousAlternatingMap.piLinearEquiv a = ContinuousAlternatingMap.pi a
-- ContinuousAlternatingMap.piLinearEquiv_symm_apply
@[simps (config := { simpRhs := true })]
def piLinearEquiv {ι' : Type*} {M' : ι' → Type*} [∀ i, AddCommMonoid (M' i)]
    [∀ i, TopologicalSpace (M' i)] [∀ i, ContinuousAdd (M' i)] [∀ i, Module R (M' i)]
    [∀ i, Module A (M' i)] [∀ i, SMulCommClass A R (M' i)] [∀ i, ContinuousConstSMul R (M' i)] :
    (∀ i, M [⋀^ι]→L[A] M' i) ≃ₗ[R] M [⋀^ι]→L[A] ∀ i, M' i 
Arguments: {R : Type u_1} {A : Type u_2} {M : Type u_3} {ι : Type u_5} [Semiring R] [Semiring A] [AddCommMonoid M] [TopologicalSpace M] [Module A M] {ι' : Type u_6} {M' : ι' → Type u_7} [(i : ι') → AddCommMonoid (M' i)] [(i : ι') → TopologicalSpace (M' i)] [∀ (i : ι'), ContinuousAdd (M' i)] [(i : ι') → Module R (M' i)] [(i : ι') → Module A (M' i)] [∀ (i : ι'), SMulCommClass A R (M' i)] [∀ (i : ι'), ContinuousConstSMul R (M' i)]
Goal: ∀ (a : M [⋀^ι]→L[A] ((i : ι') → M' i)) (i : ι'),
  ContinuousAlternatingMap.piLinearEquiv.symm a i = (ContinuousLinearMap.proj i).compContinuousAlternatingMap a
-- ContinuousAlternatingMap.smulRight_apply
@[simps! toContinuousMultilinearMap apply]
def smulRight : M [⋀^ι]→L[R] N 
Arguments: {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [TopologicalSpace R] [TopologicalSpace M] [TopologicalSpace N] [ContinuousSMul R N] (f : M [⋀^ι]→L[R] R) (z : N)
Goal: ∀ (a : (i : ι) → (fun x => M) i), (f.smulRight z) a = f a • z
-- ContinuousAlternatingMap.smulRight_toContinuousMultilinearMap
@[simps! toContinuousMultilinearMap apply]
def smulRight : M [⋀^ι]→L[R] N 
Arguments: {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [TopologicalSpace R] [TopologicalSpace M] [TopologicalSpace N] [ContinuousSMul R N] (f : M [⋀^ι]→L[R] R) (z : N)
Goal: (f.smulRight z).toContinuousMultilinearMap = f.smulRight z
-- ContinuousAlternatingMap.compContinuousLinearMapₗ_apply
@[simps]
def compContinuousLinearMapₗ (f : M →L[R] M') : (M' [⋀^ι]→L[R] N) →ₗ[R] (M [⋀^ι]→L[R] N) 
Arguments: {R : Type u_1} {M : Type u_2} {M' : Type u_3} {N : Type u_4} {ι : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommMonoid M'] [Module R M'] [TopologicalSpace M'] [AddCommMonoid N] [Module R N] [TopologicalSpace N] [ContinuousAdd N] [ContinuousConstSMul R N] (f : M →L[R] M') (g : M' [⋀^ι]→L[R] N)
Goal: (ContinuousAlternatingMap.compContinuousLinearMapₗ f) g = g.compContinuousLinearMap f
-- ContinuousMultilinearMap.alternatization_apply_toContinuousMultilinearMap
@[simps (config := .lemmasOnly) apply_toContinuousMultilinearMap]
def alternatization : ContinuousMultilinearMap R (fun _ : ι => M) N →+ M [⋀^ι]→L[R] N 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} {ι : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] [TopologicalAddGroup N] [Fintype ι] [DecidableEq ι] (f : ContinuousMultilinearMap R (fun x => M) N)
Goal: (ContinuousMultilinearMap.alternatization f).toContinuousMultilinearMap =
  ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • ContinuousMultilinearMap.domDomCongr σ f
-- ContinuousMultilinearMap.alternatization_apply_apply
theorem alternatization_apply_apply (v : ι → M) :
    alternatization f v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f (v ∘ σ) 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} {ι : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] [TopologicalAddGroup N] [Fintype ι] [DecidableEq ι] (f : ContinuousMultilinearMap R (fun x => M) N) (v : ι → M)
Goal: (ContinuousMultilinearMap.alternatization f) v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f (v ∘ ⇑σ)
-- ContinuousMultilinearMap.alternatization_apply_toAlternatingMap
@[simp]
theorem alternatization_apply_toAlternatingMap :
    (alternatization f).toAlternatingMap = MultilinearMap.alternatization f.1 
Arguments: {R : Type u_1} {M : Type u_2} {N : Type u_3} {ι : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [AddCommGroup N] [Module R N] [TopologicalSpace N] [TopologicalAddGroup N] [Fintype ι] [DecidableEq ι] (f : ContinuousMultilinearMap R (fun x => M) N)
Goal: (ContinuousMultilinearMap.alternatization f).toAlternatingMap = MultilinearMap.alternatization f.toMultilinearMap
-- isTotallyDisconnected_empty
theorem isTotallyDisconnected_empty : IsTotallyDisconnected (∅ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: IsTotallyDisconnected ∅
-- isTotallyDisconnected_singleton
theorem isTotallyDisconnected_singleton {x} : IsTotallyDisconnected ({x} : Set α) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsTotallyDisconnected {x}
-- TotallyDisconnectedSpace.isTotallyDisconnected_univ
@[mk_iff]
class TotallyDisconnectedSpace (α : Type u) [TopologicalSpace α] : Prop where
    isTotallyDisconnected_univ : IsTotallyDisconnected (univ : Set α)
Arguments: {α : Type u} [TopologicalSpace α] [self : TotallyDisconnectedSpace α]
Goal: IsTotallyDisconnected Set.univ
-- totallyDisconnectedSpace_iff
@[mk_iff]
class TotallyDisconnectedSpace (α : Type u) [TopologicalSpace α] : Prop where
    isTotallyDisconnected_univ : IsTotallyDisconnected (univ : Set α)
Arguments: (α : Type u) [TopologicalSpace α]
Goal: TotallyDisconnectedSpace α ↔ IsTotallyDisconnected Set.univ
-- IsPreconnected.subsingleton
theorem IsPreconnected.subsingleton [TotallyDisconnectedSpace α] {s : Set α}
    (h : IsPreconnected s) : s.Subsingleton 
Arguments: {α : Type u} [TopologicalSpace α] [TotallyDisconnectedSpace α] {s : Set α} (h : IsPreconnected s)
Goal: s.Subsingleton
-- isTotallyDisconnected_of_isClopen_set
theorem isTotallyDisconnected_of_isClopen_set {X : Type*} [TopologicalSpace X]
    (hX : Pairwise fun x y => ∃ (U : Set X), IsClopen U ∧ x ∈ U ∧ y ∉ U) :
    IsTotallyDisconnected (Set.univ : Set X) 
Arguments: {X : Type u_3} [TopologicalSpace X] (hX : Pairwise fun x y => ∃ U, IsClopen U ∧ x ∈ U ∧ y ∉ U)
Goal: IsTotallyDisconnected Set.univ
-- totallyDisconnectedSpace_iff_connectedComponent_subsingleton
theorem totallyDisconnectedSpace_iff_connectedComponent_subsingleton :
    TotallyDisconnectedSpace α ↔ ∀ x : α, (connectedComponent x).Subsingleton 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: TotallyDisconnectedSpace α ↔ ∀ (x : α), (connectedComponent x).Subsingleton
-- totallyDisconnectedSpace_iff_connectedComponent_singleton
theorem totallyDisconnectedSpace_iff_connectedComponent_singleton :
    TotallyDisconnectedSpace α ↔ ∀ x : α, connectedComponent x = {x} 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: TotallyDisconnectedSpace α ↔ ∀ (x : α), connectedComponent x = {x}
-- connectedComponent_eq_singleton
@[simp] theorem connectedComponent_eq_singleton [TotallyDisconnectedSpace α] (x : α) :
    connectedComponent x = {x} 
Arguments: {α : Type u} [TopologicalSpace α] [TotallyDisconnectedSpace α] (x : α)
Goal: connectedComponent x = {x}
-- Continuous.image_connectedComponent_eq_singleton
@[simp]
theorem Continuous.image_connectedComponent_eq_singleton {β : Type*} [TopologicalSpace β]
    [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f) (a : α) :
    f '' connectedComponent a = {f a} 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f) (a : α)
Goal: f '' connectedComponent a = {f a}
-- isTotallyDisconnected_of_totallyDisconnectedSpace
theorem isTotallyDisconnected_of_totallyDisconnectedSpace [TotallyDisconnectedSpace α] (s : Set α) :
    IsTotallyDisconnected s 
Arguments: {α : Type u} [TopologicalSpace α] [TotallyDisconnectedSpace α] (s : Set α)
Goal: IsTotallyDisconnected s
-- isTotallyDisconnected_of_image
theorem isTotallyDisconnected_of_image [TopologicalSpace β] {f : α → β} (hf : ContinuousOn f s)
    (hf' : Injective f) (h : IsTotallyDisconnected (f '' s)) : IsTotallyDisconnected s 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] {s : Set α} [TopologicalSpace β] {f : α → β} (hf : ContinuousOn f s) (hf' : Function.Injective f) (h : IsTotallyDisconnected (f '' s))
Goal: IsTotallyDisconnected s
-- Embedding.isTotallyDisconnected
theorem Embedding.isTotallyDisconnected [TopologicalSpace β] {f : α → β} (hf : Embedding f)
    {s : Set α} (h : IsTotallyDisconnected (f '' s)) : IsTotallyDisconnected s 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Embedding f) {s : Set α} (h : IsTotallyDisconnected (f '' s))
Goal: IsTotallyDisconnected s
-- Embedding.isTotallyDisconnected_image
lemma Embedding.isTotallyDisconnected_image [TopologicalSpace β] {f : α → β} (hf : Embedding f)
    {s : Set α} : IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Embedding f) {s : Set α}
Goal: IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s
-- Embedding.isTotallyDisconnected_range
lemma Embedding.isTotallyDisconnected_range [TopologicalSpace β] {f : α → β} (hf : Embedding f) :
    IsTotallyDisconnected (range f) ↔ TotallyDisconnectedSpace α 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Embedding f)
Goal: IsTotallyDisconnected (Set.range f) ↔ TotallyDisconnectedSpace α
-- totallyDisconnectedSpace_subtype_iff
lemma totallyDisconnectedSpace_subtype_iff {s : Set α} :
    TotallyDisconnectedSpace s ↔ IsTotallyDisconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α}
Goal: TotallyDisconnectedSpace ↑s ↔ IsTotallyDisconnected s
-- isTotallySeparated_empty
theorem isTotallySeparated_empty : IsTotallySeparated (∅ : Set α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: IsTotallySeparated ∅
-- isTotallySeparated_singleton
theorem isTotallySeparated_singleton {x} : IsTotallySeparated ({x} : Set α) 
Arguments: {α : Type u} [TopologicalSpace α] {x : α}
Goal: IsTotallySeparated {x}
-- isTotallyDisconnected_of_isTotallySeparated
theorem isTotallyDisconnected_of_isTotallySeparated {s : Set α} (H : IsTotallySeparated s) :
    IsTotallyDisconnected s 
Arguments: {α : Type u} [TopologicalSpace α] {s : Set α} (H : IsTotallySeparated s)
Goal: IsTotallyDisconnected s
-- TotallySeparatedSpace.isTotallySeparated_univ
class TotallySeparatedSpace (α : Type u) [TopologicalSpace α] : Prop where
    isTotallySeparated_univ : IsTotallySeparated (univ : Set α)
Arguments: {α : Type u} [TopologicalSpace α] [self : TotallySeparatedSpace α]
Goal: IsTotallySeparated Set.univ
-- exists_isClopen_of_totally_separated
theorem exists_isClopen_of_totally_separated {α : Type*} [TopologicalSpace α]
    [TotallySeparatedSpace α] {x y : α} (hxy : x ≠ y) :
    ∃ U : Set α, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ 
Arguments: {α : Type u_3} [TopologicalSpace α] [TotallySeparatedSpace α] {x y : α} (hxy : x ≠ y)
Goal: ∃ U, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ
-- Continuous.image_eq_of_connectedComponent_eq
theorem Continuous.image_eq_of_connectedComponent_eq (h : Continuous f) (a b : α)
    (hab : connectedComponent a = connectedComponent b) : f a = f b 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f) (a b : α) (hab : connectedComponent a = connectedComponent b)
Goal: f a = f b
-- Continuous.connectedComponentsLift_continuous
@[continuity]
theorem Continuous.connectedComponentsLift_continuous (h : Continuous f) :
    Continuous h.connectedComponentsLift 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f)
Goal: Continuous h.connectedComponentsLift
-- Continuous.connectedComponentsLift_apply_coe
@[simp]
theorem Continuous.connectedComponentsLift_apply_coe (h : Continuous f) (x : α) :
    h.connectedComponentsLift x = f x 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f) (x : α)
Goal: h.connectedComponentsLift (ConnectedComponents.mk x) = f x
-- Continuous.connectedComponentsLift_comp_coe
@[simp]
theorem Continuous.connectedComponentsLift_comp_coe (h : Continuous f) :
    h.connectedComponentsLift ∘ (↑) = f 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f)
Goal: h.connectedComponentsLift ∘ ConnectedComponents.mk = f
-- connectedComponents_lift_unique'
theorem connectedComponents_lift_unique' {β : Sort*} {g₁ g₂ : ConnectedComponents α → β}
    (hg : g₁ ∘ ((↑) : α → ConnectedComponents α) = g₂ ∘ (↑)) : g₁ = g₂ 
Arguments: {α : Type u} [TopologicalSpace α] {β : Sort u_3} {g₁ g₂ : ConnectedComponents α → β} (hg : g₁ ∘ ConnectedComponents.mk = g₂ ∘ ConnectedComponents.mk)
Goal: g₁ = g₂