ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.VectorMeasure.measurable_of_not_restrict_le_zero
theorem measurable_of_not_restrict_le_zero (hi : ¬v ≤[i] 0) : MeasurableSet i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬v.restrict i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: MeasurableSet i
-- MeasureTheory.VectorMeasure.zero_le_restrict_subset
theorem zero_le_restrict_subset (hi₁ : MeasurableSet i) (hij : j ⊆ i) (hi₂ : 0 ≤[i] v) : 0 ≤[j] v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i j : Set α} (hi₁ : MeasurableSet i) (hij : j ⊆ i) (hi₂ : MeasureTheory.VectorMeasure.restrict 0 i ≤ v.restrict i)
Goal: MeasureTheory.VectorMeasure.restrict 0 j ≤ v.restrict j
-- MeasureTheory.VectorMeasure.restrict_le_zero_subset
theorem restrict_le_zero_subset (hi₁ : MeasurableSet i) (hij : j ⊆ i) (hi₂ : v ≤[i] 0) : v ≤[j] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [OrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i j : Set α} (hi₁ : MeasurableSet i) (hij : j ⊆ i) (hi₂ : v.restrict i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: v.restrict j ≤ MeasureTheory.VectorMeasure.restrict 0 j
-- MeasureTheory.VectorMeasure.exists_pos_measure_of_not_restrict_le_zero
theorem exists_pos_measure_of_not_restrict_le_zero (hi : ¬v ≤[i] 0) :
    ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ 0 < v j 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_3} [TopologicalSpace M] [LinearOrderedAddCommMonoid M] (v : MeasureTheory.VectorMeasure α M) {i : Set α} (hi : ¬v.restrict i ≤ MeasureTheory.VectorMeasure.restrict 0 i)
Goal: ∃ j, MeasurableSet j ∧ j ⊆ i ∧ 0 < ↑v j
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk
theorem mk (h : ∀ ⦃s : Set α⦄, MeasurableSet s → w s = 0 → v s = 0) : v ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : ∀ ⦃s : Set α⦄, MeasurableSet s → ↑w s = 0 → ↑v s = 0)
Goal: v ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.eq
theorem eq {w : VectorMeasure α M} (h : v = w) : v ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {v w : MeasureTheory.VectorMeasure α M} (h : v = w)
Goal: v ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.refl
@[refl]
theorem refl (v : VectorMeasure α M) : v ≪ᵥ v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] (v : MeasureTheory.VectorMeasure α M)
Goal: v ≪ᵥ v
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.trans
@[trans]
theorem trans {u : VectorMeasure α L} {v : VectorMeasure α M} {w : VectorMeasure α N} (huv : u ≪ᵥ v)
    (hvw : v ≪ᵥ w) : u ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {L : Type u_3} {M : Type u_4} {N : Type u_5} [AddCommMonoid L] [TopologicalSpace L] [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {u : MeasureTheory.VectorMeasure α L} {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (huv : u ≪ᵥ v) (hvw : v ≪ᵥ w)
Goal: u ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.zero
theorem zero (v : VectorMeasure α N) : (0 : VectorMeasure α M) ≪ᵥ v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] (v : MeasureTheory.VectorMeasure α N)
Goal: 0 ≪ᵥ v
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_left
theorem neg_left {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]
    {v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ≪ᵥ w) : -v ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {N : Type u_5} [AddCommMonoid N] [TopologicalSpace N] {M : Type u_6} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ≪ᵥ w)
Goal: -v ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.neg_right
theorem neg_right {N : Type*} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N]
    {v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ≪ᵥ w) : v ≪ᵥ -w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {N : Type u_6} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ≪ᵥ w)
Goal: v ≪ᵥ -w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.add
theorem add [ContinuousAdd M] {v₁ v₂ : VectorMeasure α M} {w : VectorMeasure α N} (hv₁ : v₁ ≪ᵥ w)
    (hv₂ : v₂ ≪ᵥ w) : v₁ + v₂ ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] [ContinuousAdd M] {v₁ v₂ : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (hv₁ : v₁ ≪ᵥ w) (hv₂ : v₂ ≪ᵥ w)
Goal: v₁ + v₂ ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.sub
theorem sub {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]
    {v₁ v₂ : VectorMeasure α M} {w : VectorMeasure α N} (hv₁ : v₁ ≪ᵥ w) (hv₂ : v₂ ≪ᵥ w) :
    v₁ - v₂ ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {N : Type u_5} [AddCommMonoid N] [TopologicalSpace N] {M : Type u_6} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] {v₁ v₂ : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (hv₁ : v₁ ≪ᵥ w) (hv₂ : v₂ ≪ᵥ w)
Goal: v₁ - v₂ ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.smul
theorem smul {R : Type*} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] {r : R}
    {v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ≪ᵥ w) : r • v ≪ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {R : Type u_6} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] {r : R} {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ≪ᵥ w)
Goal: r • v ≪ᵥ w
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.map
theorem map [MeasureSpace β] (h : v ≪ᵥ w) (f : α → β) : v.map f ≪ᵥ w.map f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} [MeasureTheory.MeasureSpace β] (h : v ≪ᵥ w) (f : α → β)
Goal: v.map f ≪ᵥ w.map f
-- MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure
theorem ennrealToMeasure {μ : VectorMeasure α ℝ≥0∞} :
    (∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → v s = 0) ↔ v ≪ᵥ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {v : MeasureTheory.VectorMeasure α M} {μ : MeasureTheory.VectorMeasure α ℝ≥0∞}
Goal: (∀ ⦃s : Set α⦄, μ.ennrealToMeasure s = 0 → ↑v s = 0) ↔ v ≪ᵥ μ
-- MeasureTheory.VectorMeasure.MutuallySingular.mk
theorem mk (s : Set α) (hs : MeasurableSet s) (h₁ : ∀ t ⊆ s, MeasurableSet t → v t = 0)
    (h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → w t = 0) : v ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (s : Set α) (hs : MeasurableSet s) (h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0) (h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0)
Goal: v ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.symm
theorem symm (h : v ⟂ᵥ w) : w ⟂ᵥ v 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ⟂ᵥ w)
Goal: w ⟂ᵥ v
-- MeasureTheory.VectorMeasure.MutuallySingular.zero_right
theorem zero_right : v ⟂ᵥ (0 : VectorMeasure α N) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M}
Goal: v ⟂ᵥ 0
-- MeasureTheory.VectorMeasure.MutuallySingular.zero_left
theorem zero_left : (0 : VectorMeasure α M) ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {w : MeasureTheory.VectorMeasure α N}
Goal: 0 ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.add_left
theorem add_left [T2Space N] [ContinuousAdd M] (h₁ : v₁ ⟂ᵥ w) (h₂ : v₂ ⟂ᵥ w) : v₁ + v₂ ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v₁ v₂ : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} [T2Space N] [ContinuousAdd M] (h₁ : v₁ ⟂ᵥ w) (h₂ : v₂ ⟂ᵥ w)
Goal: v₁ + v₂ ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.add_right
theorem add_right [T2Space M] [ContinuousAdd N] (h₁ : v ⟂ᵥ w₁) (h₂ : v ⟂ᵥ w₂) : v ⟂ᵥ w₁ + w₂ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w₁ w₂ : MeasureTheory.VectorMeasure α N} [T2Space M] [ContinuousAdd N] (h₁ : v ⟂ᵥ w₁) (h₂ : v ⟂ᵥ w₂)
Goal: v ⟂ᵥ w₁ + w₂
-- MeasureTheory.VectorMeasure.MutuallySingular.smul_right
theorem smul_right {R : Type*} [Semiring R] [DistribMulAction R N] [ContinuousConstSMul R N]
    (r : R) (h : v ⟂ᵥ w) : v ⟂ᵥ r • w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} {R : Type u_6} [Semiring R] [DistribMulAction R N] [ContinuousConstSMul R N] (r : R) (h : v ⟂ᵥ w)
Goal: v ⟂ᵥ r • w
-- MeasureTheory.VectorMeasure.MutuallySingular.smul_left
theorem smul_left {R : Type*} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] (r : R)
    (h : v ⟂ᵥ w) : r • v ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [TopologicalSpace M] [AddCommMonoid N] [TopologicalSpace N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} {R : Type u_6} [Semiring R] [DistribMulAction R M] [ContinuousConstSMul R M] (r : R) (h : v ⟂ᵥ w)
Goal: r • v ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.neg_left
theorem neg_left {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]
    {v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ⟂ᵥ w) : -v ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {N : Type u_5} [AddCommMonoid N] [TopologicalSpace N] {M : Type u_6} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ⟂ᵥ w)
Goal: -v ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.neg_right
theorem neg_right {N : Type*} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N]
    {v : VectorMeasure α M} {w : VectorMeasure α N} (h : v ⟂ᵥ w) : v ⟂ᵥ -w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {N : Type u_6} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N} (h : v ⟂ᵥ w)
Goal: v ⟂ᵥ -w
-- MeasureTheory.VectorMeasure.MutuallySingular.neg_left_iff
@[simp]
theorem neg_left_iff {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]
    {v : VectorMeasure α M} {w : VectorMeasure α N} : -v ⟂ᵥ w ↔ v ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {N : Type u_5} [AddCommMonoid N] [TopologicalSpace N] {M : Type u_6} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N}
Goal: -v ⟂ᵥ w ↔ v ⟂ᵥ w
-- MeasureTheory.VectorMeasure.MutuallySingular.neg_right_iff
@[simp]
theorem neg_right_iff {N : Type*} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N]
    {v : VectorMeasure α M} {w : VectorMeasure α N} : v ⟂ᵥ -w ↔ v ⟂ᵥ w 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {N : Type u_6} [AddCommGroup N] [TopologicalSpace N] [TopologicalAddGroup N] {v : MeasureTheory.VectorMeasure α M} {w : MeasureTheory.VectorMeasure α N}
Goal: v ⟂ᵥ -w ↔ v ⟂ᵥ w
-- MeasureTheory.VectorMeasure.trim_apply
@[simps]
def trim {m n : MeasurableSpace α} (v : VectorMeasure α M) (hle : m ≤ n) :
    @VectorMeasure α m M _ _ 
Arguments: {α : Type u_1} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {m n : MeasurableSpace α} (v : MeasureTheory.VectorMeasure α M) (hle : m ≤ n) (i : Set α)
Goal: ↑(v.trim hle) i = if MeasurableSet i then ↑v i else 0
-- MeasureTheory.VectorMeasure.trim_eq_self
theorem trim_eq_self : v.trim le_rfl = v 
Arguments: {α : Type u_1} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {n : MeasurableSpace α} {v : MeasureTheory.VectorMeasure α M}
Goal: v.trim ⋯ = v
-- MeasureTheory.VectorMeasure.zero_trim
@[simp]
theorem zero_trim (hle : m ≤ n) : (0 : VectorMeasure α M).trim hle = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {n : MeasurableSpace α} (hle : m ≤ n)
Goal: MeasureTheory.VectorMeasure.trim 0 hle = 0
-- MeasureTheory.VectorMeasure.trim_measurableSet_eq
theorem trim_measurableSet_eq (hle : m ≤ n) {i : Set α} (hi : MeasurableSet[m] i) :
    v.trim hle i = v i 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {n : MeasurableSpace α} {v : MeasureTheory.VectorMeasure α M} (hle : m ≤ n) {i : Set α} (hi : MeasurableSet i)
Goal: ↑(v.trim hle) i = ↑v i
-- MeasureTheory.VectorMeasure.restrict_trim
theorem restrict_trim (hle : m ≤ n) {i : Set α} (hi : MeasurableSet[m] i) :
    @VectorMeasure.restrict α m M _ _ (v.trim hle) i = (v.restrict i).trim hle 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {M : Type u_4} [AddCommMonoid M] [TopologicalSpace M] {n : MeasurableSpace α} {v : MeasureTheory.VectorMeasure α M} (hle : m ≤ n) {i : Set α} (hi : MeasurableSet i)
Goal: (v.trim hle).restrict i = (v.restrict i).trim hle
-- MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply
theorem toMeasureOfZeroLE_apply (hi : 0 ≤[i] s) (hi₁ : MeasurableSet i) (hj₁ : MeasurableSet j) :
    s.toMeasureOfZeroLE i hi₁ hi j = ((↑) : ℝ≥0 → ℝ≥0∞) ⟨s (i ∩ j), nonneg_of_zero_le_restrict
      s (zero_le_restrict_subset s hi₁ Set.inter_subset_left hi)⟩ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) {i j : Set α} (hi : MeasureTheory.VectorMeasure.restrict 0 i ≤ MeasureTheory.VectorMeasure.restrict s i) (hi₁ : MeasurableSet i) (hj₁ : MeasurableSet j)
Goal: (s.toMeasureOfZeroLE i hi₁ hi) j = ↑⟨↑s (i ∩ j), ⋯⟩
-- MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply
theorem toMeasureOfLEZero_apply (hi : s ≤[i] 0) (hi₁ : MeasurableSet i) (hj₁ : MeasurableSet j) :
    s.toMeasureOfLEZero i hi₁ hi j = ((↑) : ℝ≥0 → ℝ≥0∞) ⟨-s (i ∩ j), neg_apply s (i ∩ j) ▸
      nonneg_of_zero_le_restrict _ (zero_le_restrict_subset _ hi₁ Set.inter_subset_left
      (@neg_zero (VectorMeasure α ℝ) _ ▸ neg_le_neg _ _ hi₁ hi))⟩ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) {i j : Set α} (hi : MeasureTheory.VectorMeasure.restrict s i ≤ MeasureTheory.VectorMeasure.restrict 0 i) (hi₁ : MeasurableSet i) (hj₁ : MeasurableSet j)
Goal: (s.toMeasureOfLEZero i hi₁ hi) j = ↑⟨-↑s (i ∩ j), ⋯⟩
-- MeasureTheory.SignedMeasure.toMeasureOfZeroLE_toSignedMeasure
theorem toMeasureOfZeroLE_toSignedMeasure (hs : 0 ≤[Set.univ] s) :
    (s.toMeasureOfZeroLE Set.univ MeasurableSet.univ hs).toSignedMeasure = s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (hs : MeasureTheory.VectorMeasure.restrict 0 Set.univ ≤ MeasureTheory.VectorMeasure.restrict s Set.univ)
Goal: (s.toMeasureOfZeroLE Set.univ ⋯ hs).toSignedMeasure = s
-- MeasureTheory.SignedMeasure.toMeasureOfLEZero_toSignedMeasure
theorem toMeasureOfLEZero_toSignedMeasure (hs : s ≤[Set.univ] 0) :
    (s.toMeasureOfLEZero Set.univ MeasurableSet.univ hs).toSignedMeasure = -s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (hs : MeasureTheory.VectorMeasure.restrict s Set.univ ≤ MeasureTheory.VectorMeasure.restrict 0 Set.univ)
Goal: (s.toMeasureOfLEZero Set.univ ⋯ hs).toSignedMeasure = -s
-- MeasureTheory.Measure.zero_le_toSignedMeasure
theorem zero_le_toSignedMeasure : 0 ≤ μ.toSignedMeasure 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
Goal: 0 ≤ μ.toSignedMeasure
-- MeasureTheory.Measure.toSignedMeasure_toMeasureOfZeroLE
theorem toSignedMeasure_toMeasureOfZeroLE :
    μ.toSignedMeasure.toMeasureOfZeroLE Set.univ MeasurableSet.univ
      ((le_restrict_univ_iff_le _ _).2 (zero_le_toSignedMeasure μ)) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ]
Goal: μ.toSignedMeasure.toMeasureOfZeroLE Set.univ ⋯ ⋯ = μ
-- ContinuousLinearMap.fpowerSeries_radius
@[simp]
theorem fpowerSeries_radius (f : E →L[𝕜] F) (x : E) : (f.fpowerSeries x).radius = ∞ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: (f.fpowerSeries x).radius = ⊤
-- ContinuousLinearMap.hasFPowerSeriesOnBall
protected theorem hasFPowerSeriesOnBall (f : E →L[𝕜] F) (x : E) :
    HasFPowerSeriesOnBall f (f.fpowerSeries x) x ∞ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: HasFPowerSeriesOnBall (⇑f) (f.fpowerSeries x) x ⊤
-- ContinuousLinearMap.hasFPowerSeriesAt
protected theorem hasFPowerSeriesAt (f : E →L[𝕜] F) (x : E) :
    HasFPowerSeriesAt f (f.fpowerSeries x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: HasFPowerSeriesAt (⇑f) (f.fpowerSeries x) x
-- ContinuousLinearMap.analyticAt
protected theorem analyticAt (f : E →L[𝕜] F) (x : E) : AnalyticAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (x : E)
Goal: AnalyticAt 𝕜 (⇑f) x
-- ContinuousLinearMap.uncurryBilinear_apply
@[simp]
theorem uncurryBilinear_apply (f : E →L[𝕜] F →L[𝕜] G) (m : Fin 2 → E × F) :
    f.uncurryBilinear m = f (m 0).1 (m 1).2 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (m : Fin 2 → E × F)
Goal: f.uncurryBilinear m = (f (m 0).1) (m 1).2
-- ContinuousLinearMap.fpowerSeriesBilinear_apply_zero
theorem fpowerSeriesBilinear_apply_zero (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    fpowerSeriesBilinear f x 0 = ContinuousMultilinearMap.curry0 𝕜 _ (f x.1 x.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: f.fpowerSeriesBilinear x 0 = ContinuousMultilinearMap.curry0 𝕜 (E × F) ((f x.1) x.2)
-- ContinuousLinearMap.fpowerSeriesBilinear_apply_one
theorem fpowerSeriesBilinear_apply_one (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    fpowerSeriesBilinear f x 1 = (continuousMultilinearCurryFin1 𝕜 (E × F) G).symm (f.deriv₂ x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: f.fpowerSeriesBilinear x 1 = (continuousMultilinearCurryFin1 𝕜 (E × F) G).symm (f.deriv₂ x)
-- ContinuousLinearMap.fpowerSeriesBilinear_apply_two
theorem fpowerSeriesBilinear_apply_two (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    fpowerSeriesBilinear f x 2 = f.uncurryBilinear 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: f.fpowerSeriesBilinear x 2 = f.uncurryBilinear
-- ContinuousLinearMap.fpowerSeriesBilinear_apply_add_three
theorem fpowerSeriesBilinear_apply_add_three (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) (n) :
    fpowerSeriesBilinear f x (n + 3) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) (n : ℕ)
Goal: f.fpowerSeriesBilinear x (n + 3) = 0
-- ContinuousLinearMap.fpowerSeriesBilinear_radius
@[simp]
theorem fpowerSeriesBilinear_radius (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    (f.fpowerSeriesBilinear x).radius = ∞ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: (f.fpowerSeriesBilinear x).radius = ⊤
-- ContinuousLinearMap.hasFPowerSeriesOnBall_bilinear
protected theorem hasFPowerSeriesOnBall_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    HasFPowerSeriesOnBall (fun x : E × F => f x.1 x.2) (f.fpowerSeriesBilinear x) x ∞ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: HasFPowerSeriesOnBall (fun x => (f x.1) x.2) (f.fpowerSeriesBilinear x) x ⊤
-- ContinuousLinearMap.hasFPowerSeriesAt_bilinear
protected theorem hasFPowerSeriesAt_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    HasFPowerSeriesAt (fun x : E × F => f x.1 x.2) (f.fpowerSeriesBilinear x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: HasFPowerSeriesAt (fun x => (f x.1) x.2) (f.fpowerSeriesBilinear x) x
-- ContinuousLinearMap.analyticAt_bilinear
protected theorem analyticAt_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
    AnalyticAt 𝕜 (fun x : E × F => f x.1 x.2) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : E →L[𝕜] F →L[𝕜] G) (x : E × F)
Goal: AnalyticAt 𝕜 (fun x => (f x.1) x.2) x
-- analyticAt_id
lemma analyticAt_id (z : E) : AnalyticAt 𝕜 (id : E → E) z 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (z : E)
Goal: AnalyticAt 𝕜 id z
-- analyticOn_id
theorem analyticOn_id {s : Set E} : AnalyticOn 𝕜 (fun x : E ↦ x) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: AnalyticOn 𝕜 (fun x => x) s