ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- InnerProductGeometry.angle_self
@[simp]
theorem angle_self {x : V} (hx : x ≠ 0) : angle x x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x : V} (hx : x ≠ 0)
Goal: InnerProductGeometry.angle x x = 0
-- InnerProductGeometry.angle_self_neg_of_nonzero
@[simp]
theorem angle_self_neg_of_nonzero {x : V} (hx : x ≠ 0) : angle x (-x) = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x : V} (hx : x ≠ 0)
Goal: InnerProductGeometry.angle x (-x) = π
-- InnerProductGeometry.angle_neg_self_of_nonzero
@[simp]
theorem angle_neg_self_of_nonzero {x : V} (hx : x ≠ 0) : angle (-x) x = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x : V} (hx : x ≠ 0)
Goal: InnerProductGeometry.angle (-x) x = π
-- InnerProductGeometry.angle_smul_right_of_pos
@[simp]
theorem angle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle x (r • y) = angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V) {r : ℝ} (hr : 0 < r)
Goal: InnerProductGeometry.angle x (r • y) = InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_smul_left_of_pos
@[simp]
theorem angle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) : angle (r • x) y = angle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V) {r : ℝ} (hr : 0 < r)
Goal: InnerProductGeometry.angle (r • x) y = InnerProductGeometry.angle x y
-- InnerProductGeometry.angle_smul_right_of_neg
@[simp]
theorem angle_smul_right_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
    angle x (r • y) = angle x (-y) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V) {r : ℝ} (hr : r < 0)
Goal: InnerProductGeometry.angle x (r • y) = InnerProductGeometry.angle x (-y)
-- InnerProductGeometry.angle_smul_left_of_neg
@[simp]
theorem angle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) : angle (r • x) y = angle (-x) y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V) {r : ℝ} (hr : r < 0)
Goal: InnerProductGeometry.angle (r • x) y = InnerProductGeometry.angle (-x) y
-- InnerProductGeometry.cos_angle_mul_norm_mul_norm
theorem cos_angle_mul_norm_mul_norm (x y : V) : Real.cos (angle x y) * (‖x‖ * ‖y‖) = ⟪x, y⟫ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: Real.cos (InnerProductGeometry.angle x y) * (‖x‖ * ‖y‖) = ⟪x, y⟫_ℝ
-- InnerProductGeometry.sin_angle_mul_norm_mul_norm
theorem sin_angle_mul_norm_mul_norm (x y : V) :
    Real.sin (angle x y) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: Real.sin (InnerProductGeometry.angle x y) * (‖x‖ * ‖y‖) = √(⟪x, x⟫_ℝ * ⟪y, y⟫_ℝ - ⟪x, y⟫_ℝ * ⟪x, y⟫_ℝ)
-- InnerProductGeometry.angle_eq_zero_iff
theorem angle_eq_zero_iff {x y : V} : angle x y = 0 ↔ x ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: InnerProductGeometry.angle x y = 0 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x
-- InnerProductGeometry.angle_eq_pi_iff
theorem angle_eq_pi_iff {x y : V} : angle x y = π ↔ x ≠ 0 ∧ ∃ r : ℝ, r < 0 ∧ y = r • x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: InnerProductGeometry.angle x y = π ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x
-- InnerProductGeometry.angle_add_angle_eq_pi_of_angle_eq_pi
theorem angle_add_angle_eq_pi_of_angle_eq_pi {x y : V} (z : V) (h : angle x y = π) :
    angle x z + angle y z = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (z : V) (h : InnerProductGeometry.angle x y = π)
Goal: InnerProductGeometry.angle x z + InnerProductGeometry.angle y z = π
-- InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two
theorem inner_eq_zero_iff_angle_eq_pi_div_two (x y : V) : ⟪x, y⟫ = 0 ↔ angle x y = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: ⟪x, y⟫_ℝ = 0 ↔ InnerProductGeometry.angle x y = π / 2
-- InnerProductGeometry.inner_eq_neg_mul_norm_of_angle_eq_pi
theorem inner_eq_neg_mul_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) :
    ⟪x, y⟫ = -(‖x‖ * ‖y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : InnerProductGeometry.angle x y = π)
Goal: ⟪x, y⟫_ℝ = -(‖x‖ * ‖y‖)
-- InnerProductGeometry.inner_eq_mul_norm_of_angle_eq_zero
theorem inner_eq_mul_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) : ⟪x, y⟫ = ‖x‖ * ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : InnerProductGeometry.angle x y = 0)
Goal: ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖
-- InnerProductGeometry.inner_eq_neg_mul_norm_iff_angle_eq_pi
theorem inner_eq_neg_mul_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ⟪x, y⟫ = -(‖x‖ * ‖y‖) ↔ angle x y = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ⟪x, y⟫_ℝ = -(‖x‖ * ‖y‖) ↔ InnerProductGeometry.angle x y = π
-- InnerProductGeometry.inner_eq_mul_norm_iff_angle_eq_zero
theorem inner_eq_mul_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ⟪x, y⟫ = ‖x‖ * ‖y‖ ↔ angle x y = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ ↔ InnerProductGeometry.angle x y = 0
-- InnerProductGeometry.norm_sub_eq_add_norm_of_angle_eq_pi
theorem norm_sub_eq_add_norm_of_angle_eq_pi {x y : V} (h : angle x y = π) :
    ‖x - y‖ = ‖x‖ + ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : InnerProductGeometry.angle x y = π)
Goal: ‖x - y‖ = ‖x‖ + ‖y‖
-- InnerProductGeometry.norm_add_eq_add_norm_of_angle_eq_zero
theorem norm_add_eq_add_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) :
    ‖x + y‖ = ‖x‖ + ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : InnerProductGeometry.angle x y = 0)
Goal: ‖x + y‖ = ‖x‖ + ‖y‖
-- InnerProductGeometry.norm_sub_eq_abs_sub_norm_of_angle_eq_zero
theorem norm_sub_eq_abs_sub_norm_of_angle_eq_zero {x y : V} (h : angle x y = 0) :
    ‖x - y‖ = |‖x‖ - ‖y‖| 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : InnerProductGeometry.angle x y = 0)
Goal: ‖x - y‖ = |‖x‖ - ‖y‖|
-- InnerProductGeometry.norm_sub_eq_add_norm_iff_angle_eq_pi
theorem norm_sub_eq_add_norm_iff_angle_eq_pi {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ‖x - y‖ = ‖x‖ + ‖y‖ ↔ angle x y = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ‖x - y‖ = ‖x‖ + ‖y‖ ↔ InnerProductGeometry.angle x y = π
-- InnerProductGeometry.norm_add_eq_add_norm_iff_angle_eq_zero
theorem norm_add_eq_add_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ‖x + y‖ = ‖x‖ + ‖y‖ ↔ angle x y = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ‖x + y‖ = ‖x‖ + ‖y‖ ↔ InnerProductGeometry.angle x y = 0
-- InnerProductGeometry.norm_sub_eq_abs_sub_norm_iff_angle_eq_zero
theorem norm_sub_eq_abs_sub_norm_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    ‖x - y‖ = |‖x‖ - ‖y‖| ↔ angle x y = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: ‖x - y‖ = |‖x‖ - ‖y‖| ↔ InnerProductGeometry.angle x y = 0
-- InnerProductGeometry.norm_add_eq_norm_sub_iff_angle_eq_pi_div_two
theorem norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (x y : V) :
    ‖x + y‖ = ‖x - y‖ ↔ angle x y = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] (x y : V)
Goal: ‖x + y‖ = ‖x - y‖ ↔ InnerProductGeometry.angle x y = π / 2
-- InnerProductGeometry.cos_eq_one_iff_angle_eq_zero
theorem cos_eq_one_iff_angle_eq_zero : cos (angle x y) = 1 ↔ angle x y = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: Real.cos (InnerProductGeometry.angle x y) = 1 ↔ InnerProductGeometry.angle x y = 0
-- InnerProductGeometry.cos_eq_zero_iff_angle_eq_pi_div_two
theorem cos_eq_zero_iff_angle_eq_pi_div_two : cos (angle x y) = 0 ↔ angle x y = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: Real.cos (InnerProductGeometry.angle x y) = 0 ↔ InnerProductGeometry.angle x y = π / 2
-- InnerProductGeometry.cos_eq_neg_one_iff_angle_eq_pi
theorem cos_eq_neg_one_iff_angle_eq_pi : cos (angle x y) = -1 ↔ angle x y = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: Real.cos (InnerProductGeometry.angle x y) = -1 ↔ InnerProductGeometry.angle x y = π
-- InnerProductGeometry.sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi
theorem sin_eq_zero_iff_angle_eq_zero_or_angle_eq_pi :
    sin (angle x y) = 0 ↔ angle x y = 0 ∨ angle x y = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: Real.sin (InnerProductGeometry.angle x y) = 0 ↔ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π
-- InnerProductGeometry.sin_eq_one_iff_angle_eq_pi_div_two
theorem sin_eq_one_iff_angle_eq_pi_div_two : sin (angle x y) = 1 ↔ angle x y = π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V}
Goal: Real.sin (InnerProductGeometry.angle x y) = 1 ↔ InnerProductGeometry.angle x y = π / 2
-- ContinuousSMul.of_nhds_zero
theorem ContinuousSMul.of_nhds_zero [TopologicalRing R] [TopologicalAddGroup M]
    (hmul : Tendsto (fun p : R × M => p.1 • p.2) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0))
    (hmulleft : ∀ m : M, Tendsto (fun a : R => a • m) (𝓝 0) (𝓝 0))
    (hmulright : ∀ a : R, Tendsto (fun m : M => a • m) (𝓝 0) (𝓝 0)) : ContinuousSMul R M 
Arguments: {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [Module R M] [TopologicalRing R] [TopologicalAddGroup M] (hmul : Filter.Tendsto (fun p => p.1 • p.2) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)) (hmulleft : ∀ (m : M), Filter.Tendsto (fun a => a • m) (𝓝 0) (𝓝 0)) (hmulright : ∀ (a : R), Filter.Tendsto (fun m => a • m) (𝓝 0) (𝓝 0))
Goal: ContinuousSMul R M
-- Submodule.eq_top_of_nonempty_interior'
theorem Submodule.eq_top_of_nonempty_interior' [NeBot (𝓝[{ x : R | IsUnit x }] 0)]
    (s : Submodule R M) (hs : (interior (s : Set M)).Nonempty) : s = ⊤ 
Arguments: {R : Type u_1} {M : Type u_2} [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [(𝓝[{x | IsUnit x}] 0).NeBot] (s : Submodule R M) (hs : (interior ↑s).Nonempty)
Goal: s = ⊤
-- Module.punctured_nhds_neBot
theorem Module.punctured_nhds_neBot [Nontrivial M] [NeBot (𝓝[≠] (0 : R))] [NoZeroSMulDivisors R M]
    (x : M) : NeBot (𝓝[≠] x) 
Arguments: (R : Type u_1) (M : Type u_2) [Ring R] [TopologicalSpace R] [TopologicalSpace M] [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M] [Nontrivial M] [(𝓝[≠] 0).NeBot] [NoZeroSMulDivisors R M] (x : M)
Goal: (𝓝[≠] x).NeBot
-- continuousSMul_induced
theorem continuousSMul_induced : @ContinuousSMul R M₁ _ u (t.induced f) 
Arguments: {R : Type u_2} {M₁ : Type u_3} {M₂ : Type u_4} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module R M₂] [u : TopologicalSpace R] {t : TopologicalSpace M₂} [ContinuousSMul R M₂] (f : M₁ →ₗ[R] M₂)
Goal: ContinuousSMul R M₁
-- TopologicalSpace.IsSeparable.span
lemma TopologicalSpace.IsSeparable.span {R M : Type*} [AddCommMonoid M] [Semiring R] [Module R M]
    [TopologicalSpace M] [TopologicalSpace R] [SeparableSpace R]
    [ContinuousAdd M] [ContinuousSMul R M] {s : Set M} (hs : IsSeparable s) :
    IsSeparable (Submodule.span R s : Set M) 
Arguments: {R : Type u_1} {M : Type u_2} [AddCommMonoid M] [Semiring R] [Module R M] [TopologicalSpace M] [TopologicalSpace R] [TopologicalSpace.SeparableSpace R] [ContinuousAdd M] [ContinuousSMul R M] {s : Set M} (hs : TopologicalSpace.IsSeparable s)
Goal: TopologicalSpace.IsSeparable ↑(Submodule.span R s)
-- Submodule.mapsTo_smul_closure
theorem Submodule.mapsTo_smul_closure (s : Submodule R M) (c : R) :
    Set.MapsTo (c • ·) (closure s : Set M) (closure s) 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R)
Goal: Set.MapsTo (fun x => c • x) (closure ↑s) (closure ↑s)
-- Submodule.smul_closure_subset
theorem Submodule.smul_closure_subset (s : Submodule R M) (c : R) :
    c • closure (s : Set M) ⊆ closure (s : Set M) 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] (s : Submodule R M) (c : R)
Goal: c • closure ↑s ⊆ closure ↑s
-- Submodule.topologicalClosure_coe
@[simp]
theorem Submodule.topologicalClosure_coe (s : Submodule R M) :
    (s.topologicalClosure : Set M) = closure (s : Set M) 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M)
Goal: ↑s.topologicalClosure = closure ↑s
-- Submodule.le_topologicalClosure
theorem Submodule.le_topologicalClosure (s : Submodule R M) : s ≤ s.topologicalClosure 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M)
Goal: s ≤ s.topologicalClosure
-- Submodule.closure_subset_topologicalClosure_span
theorem Submodule.closure_subset_topologicalClosure_span (s : Set M) :
    closure s ⊆ (span R s).topologicalClosure 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Set M)
Goal: closure s ⊆ ↑(Submodule.span R s).topologicalClosure
-- Submodule.isClosed_topologicalClosure
theorem Submodule.isClosed_topologicalClosure (s : Submodule R M) :
    IsClosed (s.topologicalClosure : Set M) 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M)
Goal: IsClosed ↑s.topologicalClosure
-- Submodule.topologicalClosure_minimal
theorem Submodule.topologicalClosure_minimal (s : Submodule R M) {t : Submodule R M} (h : s ≤ t)
    (ht : IsClosed (t : Set M)) : s.topologicalClosure ≤ t 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) {t : Submodule R M} (h : s ≤ t) (ht : IsClosed ↑t)
Goal: s.topologicalClosure ≤ t
-- Submodule.topologicalClosure_mono
theorem Submodule.topologicalClosure_mono {s : Submodule R M} {t : Submodule R M} (h : s ≤ t) :
    s.topologicalClosure ≤ t.topologicalClosure 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s t : Submodule R M} (h : s ≤ t)
Goal: s.topologicalClosure ≤ t.topologicalClosure
-- IsClosed.submodule_topologicalClosure_eq
theorem IsClosed.submodule_topologicalClosure_eq {s : Submodule R M} (hs : IsClosed (s : Set M)) :
    s.topologicalClosure = s 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M} (hs : IsClosed ↑s)
Goal: s.topologicalClosure = s
-- Submodule.dense_iff_topologicalClosure_eq_top
theorem Submodule.dense_iff_topologicalClosure_eq_top {s : Submodule R M} :
    Dense (s : Set M) ↔ s.topologicalClosure = ⊤ 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] {s : Submodule R M}
Goal: Dense ↑s ↔ s.topologicalClosure = ⊤
-- Submodule.isClosed_or_dense_of_isCoatom
theorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :
    IsClosed (s : Set M) ∨ Dense (s : Set M) 
Arguments: {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousConstSMul R M] [ContinuousAdd M] (s : Submodule R M) (hs : IsCoatom s)
Goal: IsClosed ↑s ∨ Dense ↑s
-- LinearMap.continuous_on_pi
theorem LinearMap.continuous_on_pi {ι : Type*} {R : Type*} {M : Type*} [Finite ι] [Semiring R]
    [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]
    [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M) : Continuous f 
Arguments: {ι : Type u_1} {R : Type u_2} {M : Type u_3} [Finite ι] [Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M] [ContinuousSMul R M] (f : (ι → R) →ₗ[R] M)
Goal: Continuous ⇑f
-- ContinuousLinearMap.cont
structure ContinuousLinearMap {R : Type*} {S : Type*} [Semiring R] [Semiring S] (σ : R →+* S)
    (M : Type*) [TopologicalSpace M] [AddCommMonoid M] (M₂ : Type*) [TopologicalSpace M₂]
    [AddCommMonoid M₂] [Module R M] [Module S M₂] extends M →ₛₗ[σ] M₂ where
  cont : Continuous toFun := by continuity
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {σ : R →+* S} {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M →SL[σ] M₂)
Goal: Continuous (↑self).toFun
-- ContinuousLinearEquiv.continuous_invFun
-- Porting note (#5171): linter not ported yet; was @[nolint has_nonempty_instance]
structure ContinuousLinearEquiv {R : Type*} {S : Type*} [Semiring R] [Semiring S] (σ : R →+* S)
    {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type*) [TopologicalSpace M]
    [AddCommMonoid M] (M₂ : Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
    [Module S M₂] extends M ≃ₛₗ[σ] M₂ where
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃SL[σ] M₂)
Goal: Continuous self.invFun
-- ContinuousLinearEquiv.continuous_toFun
-- Porting note (#5171): linter not ported yet; was @[nolint has_nonempty_instance]
structure ContinuousLinearEquiv {R : Type*} {S : Type*} [Semiring R] [Semiring S] (σ : R →+* S)
    {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type*) [TopologicalSpace M]
    [AddCommMonoid M] (M₂ : Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
    [Module S M₂] extends M ≃ₛₗ[σ] M₂ where
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {R : Type u_1} {S : Type u_2} [Semiring R] [Semiring S] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : Type u_3} [TopologicalSpace M] [AddCommMonoid M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] (self : M ≃SL[σ] M₂)
Goal: Continuous (↑self.toLinearEquiv).toFun
-- ContinuousSemilinearEquivClass.map_continuous
class ContinuousSemilinearEquivClass (F : Type*) {R : outParam Type*} {S : outParam Type*}
    [Semiring R] [Semiring S] (σ : outParam <| R →+* S) {σ' : outParam <| S →+* R}
    [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam Type*) [TopologicalSpace M]
    [AddCommMonoid M] (M₂ : outParam Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
    [Module S M₂] [EquivLike F M M₂] extends SemilinearEquivClass F σ M M₂ : Prop where
  map_continuous : ∀ f : F, Continuous f := by continuity
  inv_continuous : ∀ f : F, Continuous (EquivLike.inv f) := by continuity
Arguments: {F : Type u_1} {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : outParam (Type u_4)} [TopologicalSpace M] [AddCommMonoid M] {M₂ : outParam (Type u_5)} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] [self : ContinuousSemilinearEquivClass F σ M M₂] (f : F)
Goal: Continuous ⇑f
-- ContinuousSemilinearEquivClass.inv_continuous
class ContinuousSemilinearEquivClass (F : Type*) {R : outParam Type*} {S : outParam Type*}
    [Semiring R] [Semiring S] (σ : outParam <| R →+* S) {σ' : outParam <| S →+* R}
    [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam Type*) [TopologicalSpace M]
    [AddCommMonoid M] (M₂ : outParam Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
    [Module S M₂] [EquivLike F M M₂] extends SemilinearEquivClass F σ M M₂ : Prop where
  map_continuous : ∀ f : F, Continuous f := by continuity
  inv_continuous : ∀ f : F, Continuous (EquivLike.inv f) := by continuity
Arguments: {F : Type u_1} {R : outParam (Type u_2)} {S : outParam (Type u_3)} [Semiring R] [Semiring S] {σ : outParam (R →+* S)} {σ' : outParam (S →+* R)} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] {M : outParam (Type u_4)} [TopologicalSpace M] [AddCommMonoid M] {M₂ : outParam (Type u_5)} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module S M₂] [EquivLike F M M₂] [self : ContinuousSemilinearEquivClass F σ M M₂] (f : F)
Goal: Continuous (EquivLike.inv f)
-- linearMapOfMemClosureRangeCoe_apply
@[simps (config := .asFn)]
def linearMapOfMemClosureRangeCoe (f : M₁ → M₂)
    (hf : f ∈ closure (Set.range ((↑) : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂))) : M₁ →ₛₗ[σ] M₂ 
Arguments: {M₁ : Type u_1} {M₂ : Type u_2} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} (f : M₁ → M₂) (hf : f ∈ closure (Set.range DFunLike.coe))
Goal: ⇑(linearMapOfMemClosureRangeCoe f hf) = (↑(addMonoidHomOfMemClosureRangeCoe f hf)).toFun
-- linearMapOfTendsto_apply
@[simps! (config := .asFn)]
def linearMapOfTendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.NeBot]
    (h : Tendsto (fun a x => g a x) l (𝓝 f)) : M₁ →ₛₗ[σ] M₂ 
Arguments: {M₁ : Type u_1} {M₂ : Type u_2} {α : Type u_3} {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] {σ : R →+* S} {l : Filter α} (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.NeBot] (h : Filter.Tendsto (fun a x => (g a) x) l (𝓝 f))
Goal: ⇑(linearMapOfTendsto f g h) = f
-- LinearMap.isClosed_range_coe
theorem LinearMap.isClosed_range_coe : IsClosed (Set.range ((↑) : (M₁ →ₛₗ[σ] M₂) → M₁ → M₂)) 
Arguments: (M₁ : Type u_1) (M₂ : Type u_2) {R : Type u_4} {S : Type u_5} [TopologicalSpace M₂] [T2Space M₂] [Semiring R] [Semiring S] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M₁] [Module S M₂] [ContinuousConstSMul S M₂] [ContinuousAdd M₂] (σ : R →+* S)
Goal: IsClosed (Set.range DFunLike.coe)
-- ContinuousLinearMap.coe_injective
theorem coe_injective : Function.Injective ((↑) : (M₁ →SL[σ₁₂] M₂) → M₁ →ₛₗ[σ₁₂] M₂) 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂]
Goal: Function.Injective ContinuousLinearMap.toLinearMap
-- ContinuousLinearMap.coe_mk
--instance toFun' : CoeFun (M₁ →SL[σ₁₂] M₂) fun _ => M₁ → M₂ := ⟨DFunLike.coe⟩

-- porting note (#10618): was `simp`, now `simp only` proves it
theorem coe_mk (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ →ₛₗ[σ₁₂] M₂) = f 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun)
Goal: ↑{ toLinearMap := f, cont := h } = f
-- ContinuousLinearMap.coe_mk'
@[simp]
theorem coe_mk' (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ → M₂) = f 
Arguments: {R₁ : Type u_1} {R₂ : Type u_2} [Semiring R₁] [Semiring R₂] {σ₁₂ : R₁ →+* R₂} {M₁ : Type u_4} [TopologicalSpace M₁] [AddCommMonoid M₁] {M₂ : Type u_6} [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R₁ M₁] [Module R₂ M₂] (f : M₁ →ₛₗ[σ₁₂] M₂) (h : Continuous f.toFun)
Goal: ⇑{ toLinearMap := f, cont := h } = ⇑f