ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isConnected_Iio
theorem isConnected_Iio [NoMinOrder α] : IsConnected (Iio a) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a : α} [NoMinOrder α]
Goal: IsConnected (Set.Iio a)
-- isConnected_Icc
theorem isConnected_Icc (h : a ≤ b) : IsConnected (Icc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a ≤ b)
Goal: IsConnected (Set.Icc a b)
-- isConnected_Ioo
theorem isConnected_Ioo (h : a < b) : IsConnected (Ioo a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: IsConnected (Set.Ioo a b)
-- isConnected_Ioc
theorem isConnected_Ioc (h : a < b) : IsConnected (Ioc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: IsConnected (Set.Ioc a b)
-- isConnected_Ico
theorem isConnected_Ico (h : a < b) : IsConnected (Ico a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: IsConnected (Set.Ico a b)
-- setOf_isPreconnected_eq_of_ordered
theorem setOf_isPreconnected_eq_of_ordered :
    { s : Set α | IsPreconnected s } =
      -- bounded intervals
      range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      -- unbounded intervals and `univ`
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α]
Goal: {s | IsPreconnected s} =
  Set.range (Function.uncurry Set.Icc) ∪ Set.range (Function.uncurry Set.Ico) ∪ Set.range (Function.uncurry Set.Ioc) ∪
      Set.range (Function.uncurry Set.Ioo) ∪
    (Set.range Set.Ici ∪ Set.range Set.Ioi ∪ Set.range Set.Iic ∪ Set.range Set.Iio ∪ {Set.univ, ∅})
-- isTotallyDisconnected_iff_lt
lemma isTotallyDisconnected_iff_lt {s : Set α} :
    IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Ioo x y 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {s : Set α}
Goal: IsTotallyDisconnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x < y → ∃ z ∉ s, z ∈ Set.Ioo x y
-- intermediate_value_Icc
theorem intermediate_value_Icc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
    Icc (f a) (f b) ⊆ f '' Icc a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Icc (f a) (f b) ⊆ f '' Set.Icc a b
-- intermediate_value_Icc'
theorem intermediate_value_Icc' {a b : α} (hab : a ≤ b) {f : α → δ}
    (hf : ContinuousOn f (Icc a b)) : Icc (f b) (f a) ⊆ f '' Icc a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Icc (f b) (f a) ⊆ f '' Set.Icc a b
-- intermediate_value_uIcc
theorem intermediate_value_uIcc {a b : α} {f : α → δ} (hf : ContinuousOn f (uIcc a b)) :
    uIcc (f a) (f b) ⊆ f '' uIcc a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} {f : α → δ} (hf : ContinuousOn f (Set.uIcc a b))
Goal: Set.uIcc (f a) (f b) ⊆ f '' Set.uIcc a b
-- intermediate_value_Ico
theorem intermediate_value_Ico {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
    Ico (f a) (f b) ⊆ f '' Ico a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ico (f a) (f b) ⊆ f '' Set.Ico a b
-- intermediate_value_Ico'
theorem intermediate_value_Ico' {a b : α} (hab : a ≤ b) {f : α → δ}
    (hf : ContinuousOn f (Icc a b)) : Ioc (f b) (f a) ⊆ f '' Ico a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ioc (f b) (f a) ⊆ f '' Set.Ico a b
-- intermediate_value_Ioc
theorem intermediate_value_Ioc {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
    Ioc (f a) (f b) ⊆ f '' Ioc a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ioc (f a) (f b) ⊆ f '' Set.Ioc a b
-- intermediate_value_Ioc'
theorem intermediate_value_Ioc' {a b : α} (hab : a ≤ b) {f : α → δ}
    (hf : ContinuousOn f (Icc a b)) : Ico (f b) (f a) ⊆ f '' Ioc a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ico (f b) (f a) ⊆ f '' Set.Ioc a b
-- intermediate_value_Ioo
theorem intermediate_value_Ioo {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Icc a b)) :
    Ioo (f a) (f b) ⊆ f '' Ioo a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ioo (f a) (f b) ⊆ f '' Set.Ioo a b
-- intermediate_value_Ioo'
theorem intermediate_value_Ioo' {a b : α} (hab : a ≤ b) {f : α → δ}
    (hf : ContinuousOn f (Icc a b)) : Ioo (f b) (f a) ⊆ f '' Ioo a b 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b))
Goal: Set.Ioo (f b) (f a) ⊆ f '' Set.Ioo a b
-- ContinuousOn.surjOn_Icc
theorem ContinuousOn.surjOn_Icc {s : Set α} [hs : OrdConnected s] {f : α → δ}
    (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) : SurjOn f s (Icc (f a) (f b)) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {s : Set α} [hs : s.OrdConnected] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: Set.SurjOn f s (Set.Icc (f a) (f b))
-- ContinuousOn.surjOn_uIcc
theorem ContinuousOn.surjOn_uIcc {s : Set α} [hs : OrdConnected s] {f : α → δ}
    (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) :
    SurjOn f s (uIcc (f a) (f b)) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {s : Set α} [hs : s.OrdConnected] {f : α → δ} (hf : ContinuousOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: Set.SurjOn f s (Set.uIcc (f a) (f b))
-- Continuous.surjective
theorem Continuous.surjective {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atTop atTop)
    (h_bot : Tendsto f atBot atBot) : Function.Surjective f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} (hf : Continuous f) (h_top : Filter.Tendsto f Filter.atTop Filter.atTop) (h_bot : Filter.Tendsto f Filter.atBot Filter.atBot)
Goal: Function.Surjective f
-- Continuous.surjective'
theorem Continuous.surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop)
    (h_bot : Tendsto f atTop atBot) : Function.Surjective f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} (hf : Continuous f) (h_top : Filter.Tendsto f Filter.atBot Filter.atTop) (h_bot : Filter.Tendsto f Filter.atTop Filter.atBot)
Goal: Function.Surjective f
-- ContinuousOn.surjOn_of_tendsto
theorem ContinuousOn.surjOn_of_tendsto {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
    (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atBot)
    (htop : Tendsto (fun x : s => f x) atTop atTop) : SurjOn f s univ 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} {s : Set α} [s.OrdConnected] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Filter.Tendsto (fun x => f ↑x) Filter.atBot Filter.atBot) (htop : Filter.Tendsto (fun x => f ↑x) Filter.atTop Filter.atTop)
Goal: Set.SurjOn f s Set.univ
-- ContinuousOn.surjOn_of_tendsto'
theorem ContinuousOn.surjOn_of_tendsto' {f : α → δ} {s : Set α} [OrdConnected s] (hs : s.Nonempty)
    (hf : ContinuousOn f s) (hbot : Tendsto (fun x : s => f x) atBot atTop)
    (htop : Tendsto (fun x : s => f x) atTop atBot) : SurjOn f s univ 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} {s : Set α} [s.OrdConnected] (hs : s.Nonempty) (hf : ContinuousOn f s) (hbot : Filter.Tendsto (fun x => f ↑x) Filter.atBot Filter.atTop) (htop : Filter.Tendsto (fun x => f ↑x) Filter.atTop Filter.atBot)
Goal: Set.SurjOn f s Set.univ
-- Continuous.strictMono_of_inj_boundedOrder
theorem Continuous.strictMono_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
    (hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Injective f) : StrictMono f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊥ ≤ f ⊤) (hf_i : Function.Injective f)
Goal: StrictMono f
-- Continuous.strictAnti_of_inj_boundedOrder
theorem Continuous.strictAnti_of_inj_boundedOrder [BoundedOrder α] {f : α → δ}
    (hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Injective f) : StrictAnti f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf : f ⊤ ≤ f ⊥) (hf_i : Function.Injective f)
Goal: StrictAnti f
-- Continuous.strictMono_of_inj_boundedOrder'
theorem Continuous.strictMono_of_inj_boundedOrder' [BoundedOrder α] {f : α → δ}
    (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] [BoundedOrder α] {f : α → δ} (hf_c : Continuous f) (hf_i : Function.Injective f)
Goal: StrictMono f ∨ StrictAnti f
-- Continuous.strictMonoOn_of_inj_rigidity
theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
    (hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
    (hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} (hf_c : Continuous f) (hf_i : Function.Injective f) {a b : α} (hab : a < b) (hf_mono : StrictMonoOn f (Set.Icc a b))
Goal: StrictMono f
-- ContinuousOn.strictMonoOn_of_injOn_Icc
theorem ContinuousOn.strictMonoOn_of_injOn_Icc {a b : α} {f : α → δ}
    (hab : a ≤ b) (hfab : f a ≤ f b)
    (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
    StrictMonoOn f (Icc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f a ≤ f b) (hf_c : ContinuousOn f (Set.Icc a b)) (hf_i : Set.InjOn f (Set.Icc a b))
Goal: StrictMonoOn f (Set.Icc a b)
-- ContinuousOn.strictAntiOn_of_injOn_Icc
theorem ContinuousOn.strictAntiOn_of_injOn_Icc {a b : α} {f : α → δ}
    (hab : a ≤ b) (hfab : f b ≤ f a)
    (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
    StrictAntiOn f (Icc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} {f : α → δ} (hab : a ≤ b) (hfab : f b ≤ f a) (hf_c : ContinuousOn f (Set.Icc a b)) (hf_i : Set.InjOn f (Set.Icc a b))
Goal: StrictAntiOn f (Set.Icc a b)
-- ContinuousOn.strictMonoOn_of_injOn_Icc'
theorem ContinuousOn.strictMonoOn_of_injOn_Icc' {a b : α} {f : α → δ} (hab : a ≤ b)
    (hf_c : ContinuousOn f (Icc a b)) (hf_i : InjOn f (Icc a b)) :
    StrictMonoOn f (Icc a b) ∨ StrictAntiOn f (Icc a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} {f : α → δ} (hab : a ≤ b) (hf_c : ContinuousOn f (Set.Icc a b)) (hf_i : Set.InjOn f (Set.Icc a b))
Goal: StrictMonoOn f (Set.Icc a b) ∨ StrictAntiOn f (Set.Icc a b)
-- Continuous.strictMono_of_inj
theorem Continuous.strictMono_of_inj {f : α → δ}
    (hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {f : α → δ} (hf_c : Continuous f) (hf_i : Function.Injective f)
Goal: StrictMono f ∨ StrictAnti f
-- ContinuousOn.strictMonoOn_of_injOn_Ioo
theorem ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b)
    (hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) :
    StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) 
Arguments: {α : Type u} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type u_1} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} {f : α → δ} (hab : a < b) (hf_c : ContinuousOn f (Set.Ioo a b)) (hf_i : Set.InjOn f (Set.Ioo a b))
Goal: StrictMonoOn f (Set.Ioo a b) ∨ StrictAntiOn f (Set.Ioo a b)
-- map_real_smul
theorem map_real_smul {G} [FunLike G E F] [AddMonoidHomClass G E F] (f : G) (hf : Continuous f)
    (c : ℝ) (x : E) :
    f (c • x) = c • f x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] {G : Type u_3} [FunLike G E F] [AddMonoidHomClass G E F] (f : G) (hf : Continuous ⇑f) (c : ℝ) (x : E)
Goal: f (c • x) = c • f x
-- AddMonoidHom.coe_toRealLinearMap
@[simp]
theorem coe_toRealLinearMap (f : E →+ F) (hf : Continuous f) : ⇑(f.toRealLinearMap hf) = f 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousSMul ℝ E] {F : Type u_2} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F] [ContinuousSMul ℝ F] [T2Space F] (f : E →+ F) (hf : Continuous ⇑f)
Goal: ⇑(f.toRealLinearMap hf) = ⇑f
-- continuous_matrix
@[continuity]
theorem continuous_matrix [TopologicalSpace α] {f : α → Matrix m n R}
    (h : ∀ i j, Continuous fun a => f a i j) : Continuous f 
Arguments: {α : Type u_2} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace R] [TopologicalSpace α] {f : α → Matrix m n R} (h : ∀ (i : m) (j : n), Continuous fun a => f a i j)
Goal: Continuous f
-- Continuous.matrix_elem
theorem Continuous.matrix_elem {A : X → Matrix m n R} (hA : Continuous A) (i : m) (j : n) :
    Continuous fun x => A x i j 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix m n R} (hA : Continuous A) (i : m) (j : n)
Goal: Continuous fun x => A x i j
-- Continuous.matrix_map
@[continuity]
theorem Continuous.matrix_map [TopologicalSpace S] {A : X → Matrix m n S} {f : S → R}
    (hA : Continuous A) (hf : Continuous f) : Continuous fun x => (A x).map f 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {S : Type u_7} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [TopologicalSpace S] {A : X → Matrix m n S} {f : S → R} (hA : Continuous A) (hf : Continuous f)
Goal: Continuous fun x => (A x).map f
-- Continuous.matrix_transpose
@[continuity]
theorem Continuous.matrix_transpose {A : X → Matrix m n R} (hA : Continuous A) :
    Continuous fun x => (A x)ᵀ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix m n R} (hA : Continuous A)
Goal: Continuous fun x => (A x)ᵀ
-- Continuous.matrix_conjTranspose
theorem Continuous.matrix_conjTranspose [Star R] [ContinuousStar R] {A : X → Matrix m n R}
    (hA : Continuous A) : Continuous fun x => (A x)ᴴ 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Star R] [ContinuousStar R] {A : X → Matrix m n R} (hA : Continuous A)
Goal: Continuous fun x => (A x)ᴴ
-- Continuous.matrix_col
@[continuity]
theorem Continuous.matrix_col {ι : Type*} {A : X → n → R} (hA : Continuous A) :
    Continuous fun x => col ι (A x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {ι : Type u_11} {A : X → n → R} (hA : Continuous A)
Goal: Continuous fun x => Matrix.col ι (A x)
-- Continuous.matrix_row
@[continuity]
theorem Continuous.matrix_row {ι : Type*} {A : X → n → R} (hA : Continuous A) :
    Continuous fun x => row ι (A x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {ι : Type u_11} {A : X → n → R} (hA : Continuous A)
Goal: Continuous fun x => Matrix.row ι (A x)
-- Continuous.matrix_diagonal
@[continuity]
theorem Continuous.matrix_diagonal [Zero R] [DecidableEq n] {A : X → n → R} (hA : Continuous A) :
    Continuous fun x => diagonal (A x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Zero R] [DecidableEq n] {A : X → n → R} (hA : Continuous A)
Goal: Continuous fun x => Matrix.diagonal (A x)
-- Continuous.matrix_dotProduct
@[continuity]
theorem Continuous.matrix_dotProduct [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
    [ContinuousMul R] {A : X → n → R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
    Continuous fun x => dotProduct (A x) (B x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R] [ContinuousMul R] {A B : X → n → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => A x ⬝ᵥ B x
-- Continuous.matrix_mul
@[continuity]
theorem Continuous.matrix_mul [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
    [ContinuousMul R] {A : X → Matrix m n R} {B : X → Matrix n p R} (hA : Continuous A)
    (hB : Continuous B) : Continuous fun x => A x * B 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] [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R] [ContinuousMul R] {A : X → Matrix m n R} {B : X → Matrix n p R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => A x * B x
-- Continuous.matrix_vecMulVec
@[continuity]
theorem Continuous.matrix_vecMulVec [Mul R] [ContinuousMul R] {A : X → m → R} {B : X → n → R}
    (hA : Continuous A) (hB : Continuous B) : Continuous fun x => vecMulVec (A x) (B x) 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Mul R] [ContinuousMul R] {A : X → m → R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => Matrix.vecMulVec (A x) (B x)
-- Continuous.matrix_mulVec
@[continuity]
theorem Continuous.matrix_mulVec [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
    [Fintype n] {A : X → Matrix m n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
    Continuous fun x => A x *ᵥ B x 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R] [Fintype n] {A : X → Matrix m n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => A x *ᵥ B x
-- Continuous.matrix_vecMul
@[continuity]
theorem Continuous.matrix_vecMul [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
    [Fintype m] {A : X → m → R} {B : X → Matrix m n R} (hA : Continuous A) (hB : Continuous B) :
    Continuous fun x => A x ᵥ* B x 
Arguments: {X : Type u_1} {m : Type u_4} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R] [Fintype m] {A : X → m → R} {B : X → Matrix m n R} (hA : Continuous A) (hB : Continuous B)
Goal: Continuous fun x => A x ᵥ* B x
-- Continuous.matrix_submatrix
@[continuity]
theorem Continuous.matrix_submatrix {A : X → Matrix l n R} (hA : Continuous A) (e₁ : m → l)
    (e₂ : p → n) : Continuous fun x => (A x).submatrix e₁ e₂ 
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 l n R} (hA : Continuous A) (e₁ : m → l) (e₂ : p → n)
Goal: Continuous fun x => (A x).submatrix e₁ e₂
-- Continuous.matrix_reindex
@[continuity]
theorem Continuous.matrix_reindex {A : X → Matrix l n R} (hA : Continuous A) (e₁ : l ≃ m)
    (e₂ : n ≃ p) : Continuous fun x => reindex e₁ e₂ (A 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 l n R} (hA : Continuous A) (e₁ : l ≃ m) (e₂ : n ≃ p)
Goal: Continuous fun x => (Matrix.reindex e₁ e₂) (A x)
-- Continuous.matrix_diag
@[continuity]
theorem Continuous.matrix_diag {A : X → Matrix n n R} (hA : Continuous A) :
    Continuous fun x => Matrix.diag (A x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] {A : X → Matrix n n R} (hA : Continuous A)
Goal: Continuous fun x => (A x).diag
-- continuous_matrix_diag
theorem continuous_matrix_diag : Continuous (Matrix.diag : Matrix n n R → n → R) 
Arguments: {n : Type u_5} {R : Type u_8} [TopologicalSpace R]
Goal: Continuous Matrix.diag
-- Continuous.matrix_trace
@[continuity]
theorem Continuous.matrix_trace [Fintype n] [AddCommMonoid R] [ContinuousAdd R]
    {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => trace (A x) 
Arguments: {X : Type u_1} {n : Type u_5} {R : Type u_8} [TopologicalSpace X] [TopologicalSpace R] [Fintype n] [AddCommMonoid R] [ContinuousAdd R] {A : X → Matrix n n R} (hA : Continuous A)
Goal: Continuous fun x => (A x).trace
-- Continuous.matrix_det
@[continuity]
theorem Continuous.matrix_det [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
    {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).det 
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).det