ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top
theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞)
    (hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set α → β} {μ' : MeasureTheory.Measure α} (h : ∀ (s : Set α), MeasurableSet s → μ s = ⊤ → μ' s = ⊤) (hT : MeasureTheory.FinMeasAdditive μ T)
Goal: MeasureTheory.FinMeasAdditive μ' T
-- MeasureTheory.FinMeasAdditive.of_smul_measure
theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
    FinMeasAdditive μ T 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set α → β} (c : ℝ≥0∞) (hc_ne_top : c ≠ ⊤) (hT : MeasureTheory.FinMeasAdditive (c • μ) T)
Goal: MeasureTheory.FinMeasAdditive μ T
-- MeasureTheory.FinMeasAdditive.smul_measure
theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
    FinMeasAdditive (c • μ) T 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set α → β} (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : MeasureTheory.FinMeasAdditive μ T)
Goal: MeasureTheory.FinMeasAdditive (c • μ) T
-- MeasureTheory.FinMeasAdditive.smul_measure_iff
theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) :
    FinMeasAdditive (c • μ) T ↔ FinMeasAdditive μ T 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] {T : Set α → β} (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ⊤)
Goal: MeasureTheory.FinMeasAdditive (c • μ) T ↔ MeasureTheory.FinMeasAdditive μ T
-- MeasureTheory.FinMeasAdditive.map_empty_eq_zero
theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) :
    T ∅ = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_8} [AddCancelMonoid β] {T : Set α → β} (hT : MeasureTheory.FinMeasAdditive μ T)
Goal: T ∅ = 0
-- MeasureTheory.FinMeasAdditive.map_iUnion_fin_meas_set_eq_sum
theorem map_iUnion_fin_meas_set_eq_sum (T : Set α → β) (T_empty : T ∅ = 0)
    (h_add : FinMeasAdditive μ T) {ι} (S : ι → Set α) (sι : Finset ι)
    (hS_meas : ∀ i, MeasurableSet (S i)) (hSp : ∀ i ∈ sι, μ (S i) ≠ ∞)
    (h_disj : ∀ᵉ (i ∈ sι) (j ∈ sι), i ≠ j → Disjoint (S i) (S j)) :
    T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [AddCommMonoid β] (T : Set α → β) (T_empty : T ∅ = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) {ι : Type u_8} (S : ι → Set α) (sι : Finset ι) (hS_meas : ∀ (i : ι), MeasurableSet (S i)) (hSp : ∀ i ∈ sι, μ (S i) ≠ ⊤) (h_disj : ∀ i ∈ sι, ∀ j ∈ sι, i ≠ j → Disjoint (S i) (S j))
Goal: T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i)
-- MeasureTheory.DominatedFinMeasAdditive.zero
theorem zero {m : MeasurableSpace α} (μ : Measure α) (hC : 0 ≤ C) :
    DominatedFinMeasAdditive μ (0 : Set α → β) C 
Arguments: {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {C : ℝ} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (hC : 0 ≤ C)
Goal: MeasureTheory.DominatedFinMeasAdditive μ 0 C
-- MeasureTheory.DominatedFinMeasAdditive.eq_zero_of_measure_zero
theorem eq_zero_of_measure_zero {β : Type*} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hs_zero : μ s = 0) :
    T s = 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_8} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) {s : Set α} (hs : MeasurableSet s) (hs_zero : μ s = 0)
Goal: T s = 0
-- MeasureTheory.DominatedFinMeasAdditive.eq_zero
theorem eq_zero {β : Type*} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ} {m : MeasurableSpace α}
    (hT : DominatedFinMeasAdditive (0 : Measure α) T C) {s : Set α} (hs : MeasurableSet s) :
    T s = 0 
Arguments: {α : Type u_1} {β : Type u_8} [NormedAddCommGroup β] {T : Set α → β} {C : ℝ} {m : MeasurableSpace α} (hT : MeasureTheory.DominatedFinMeasAdditive 0 T C) {s : Set α} (hs : MeasurableSet s)
Goal: T s = 0
-- MeasureTheory.DominatedFinMeasAdditive.add
theorem add (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C') :
    DominatedFinMeasAdditive μ (T + T') (C + C') 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T T' : Set α → β} {C C' : ℝ} (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C')
Goal: MeasureTheory.DominatedFinMeasAdditive μ (T + T') (C + C')
-- MeasureTheory.DominatedFinMeasAdditive.smul
theorem smul [NormedField 𝕜] [NormedSpace 𝕜 β] (hT : DominatedFinMeasAdditive μ T C) (c : 𝕜) :
    DominatedFinMeasAdditive μ (fun s => c • T s) (‖c‖ * C) 
Arguments: {α : Type u_1} {𝕜 : Type u_6} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ} [NormedField 𝕜] [NormedSpace 𝕜 β] (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (c : 𝕜)
Goal: MeasureTheory.DominatedFinMeasAdditive μ (fun s => c • T s) (‖c‖ * C)
-- MeasureTheory.DominatedFinMeasAdditive.of_measure_le
theorem of_measure_le {μ' : Measure α} (h : μ ≤ μ') (hT : DominatedFinMeasAdditive μ T C)
    (hC : 0 ≤ C) : DominatedFinMeasAdditive μ' T C 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ} {μ' : MeasureTheory.Measure α} (h : μ ≤ μ') (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C)
Goal: MeasureTheory.DominatedFinMeasAdditive μ' T C
-- MeasureTheory.DominatedFinMeasAdditive.add_measure_right
theorem add_measure_right {_ : MeasurableSpace α} (μ ν : Measure α)
    (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive (μ + ν) T C 
Arguments: {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ}
Goal: ∀ {x : MeasurableSpace α} (μ ν : MeasureTheory.Measure α),
  MeasureTheory.DominatedFinMeasAdditive μ T C → 0 ≤ C → MeasureTheory.DominatedFinMeasAdditive (μ + ν) T C
-- MeasureTheory.DominatedFinMeasAdditive.add_measure_left
theorem add_measure_left {_ : MeasurableSpace α} (μ ν : Measure α)
    (hT : DominatedFinMeasAdditive ν T C) (hC : 0 ≤ C) : DominatedFinMeasAdditive (μ + ν) T C 
Arguments: {α : Type u_1} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ}
Goal: ∀ {x : MeasurableSpace α} (μ ν : MeasureTheory.Measure α),
  MeasureTheory.DominatedFinMeasAdditive ν T C → 0 ≤ C → MeasureTheory.DominatedFinMeasAdditive (μ + ν) T C
-- MeasureTheory.DominatedFinMeasAdditive.of_smul_measure
theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : DominatedFinMeasAdditive (c • μ) T C) :
    DominatedFinMeasAdditive μ T (c.toReal * C) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ} (c : ℝ≥0∞) (hc_ne_top : c ≠ ⊤) (hT : MeasureTheory.DominatedFinMeasAdditive (c • μ) T C)
Goal: MeasureTheory.DominatedFinMeasAdditive μ T (c.toReal * C)
-- MeasureTheory.DominatedFinMeasAdditive.of_measure_le_smul
theorem of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (h : μ ≤ c • μ')
    (hT : DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C) :
    DominatedFinMeasAdditive μ' T (c.toReal * C) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [SeminormedAddCommGroup β] {T : Set α → β} {C : ℝ} {μ' : MeasureTheory.Measure α} (c : ℝ≥0∞) (hc : c ≠ ⊤) (h : μ ≤ c • μ') (hT : MeasureTheory.DominatedFinMeasAdditive μ T C) (hC : 0 ≤ C)
Goal: MeasureTheory.DominatedFinMeasAdditive μ' T (c.toReal * C)
-- MeasureTheory.SimpleFunc.setToSimpleFunc_zero
@[simp]
theorem setToSimpleFunc_zero {m : MeasurableSpace α} (f : α →ₛ F) :
    setToSimpleFunc (0 : Set α → F →L[ℝ] F') f = 0 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α F)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc 0 f = 0
-- MeasureTheory.SimpleFunc.setToSimpleFunc_zero'
theorem setToSimpleFunc_zero' {T : Set α → E →L[ℝ] F'}
    (h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →ₛ E) (hf : Integrable f μ) :
    setToSimpleFunc T f = 0 
Arguments: {α : Type u_1} {E : Type u_2} {F' : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {T : Set α → E →L[ℝ] F'} (h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0) (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f = 0
-- MeasureTheory.SimpleFunc.setToSimpleFunc_zero_apply
@[simp]
theorem setToSimpleFunc_zero_apply {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') :
    setToSimpleFunc T (0 : α →ₛ F) = 0 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F')
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T 0 = 0
-- MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter
theorem setToSimpleFunc_eq_sum_filter {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F')
    (f : α →ₛ F) :
    setToSimpleFunc T f = ∑ x ∈ f.range.filter fun x => x ≠ 0, (T (f ⁻¹' {x})) x 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (f : MeasureTheory.SimpleFunc α F)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f = ∑ x ∈ Finset.filter (fun x => x ≠ 0) f.range, (T (↑f ⁻¹' {x})) x
-- MeasureTheory.SimpleFunc.map_setToSimpleFunc
theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G}
    (hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) :
    (f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x) 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} {G : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] [NormedAddCommGroup G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → F →L[ℝ] F') (h_add : MeasureTheory.FinMeasAdditive μ T) {f : MeasureTheory.SimpleFunc α G} (hf : MeasureTheory.Integrable (↑f) μ) {g : G → F} (hg : g 0 = 0)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (MeasureTheory.SimpleFunc.map g f) = ∑ x ∈ f.range, (T (↑f ⁻¹' {x})) (g x)
-- MeasureTheory.SimpleFunc.setToSimpleFunc_congr'
theorem setToSimpleFunc_congr' (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
    (hf : Integrable f μ) (hg : Integrable g μ)
    (h : Pairwise fun x y => T (f ⁻¹' {x} ∩ g ⁻¹' {y}) = 0) :
    f.setToSimpleFunc T = g.setToSimpleFunc T 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (hg : MeasureTheory.Integrable (↑g) μ) (h : Pairwise fun x y => T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f = MeasureTheory.SimpleFunc.setToSimpleFunc T g
-- MeasureTheory.SimpleFunc.setToSimpleFunc_congr
theorem setToSimpleFunc_congr (T : Set α → E →L[ℝ] F)
    (h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
    (hf : Integrable f μ) (h : f =ᵐ[μ] g) : f.setToSimpleFunc T = g.setToSimpleFunc T 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0) (h_add : MeasureTheory.FinMeasAdditive μ T) {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (h : ↑f =ᵐ[μ] ↑g)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f = MeasureTheory.SimpleFunc.setToSimpleFunc T g
-- MeasureTheory.SimpleFunc.setToSimpleFunc_congr_left
theorem setToSimpleFunc_congr_left (T T' : Set α → E →L[ℝ] F)
    (h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →ₛ E) (hf : Integrable f μ) :
    setToSimpleFunc T f = setToSimpleFunc T' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T T' : Set α → E →L[ℝ] F) (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = T' s) (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f = MeasureTheory.SimpleFunc.setToSimpleFunc T' f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_add_left
theorem setToSimpleFunc_add_left {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] F') {f : α →ₛ F} :
    setToSimpleFunc (T + T') f = setToSimpleFunc T f + setToSimpleFunc T' f 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] F') {f : MeasureTheory.SimpleFunc α F}
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc (T + T') f =
  MeasureTheory.SimpleFunc.setToSimpleFunc T f + MeasureTheory.SimpleFunc.setToSimpleFunc T' f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_add_left'
theorem setToSimpleFunc_add_left' (T T' T'' : Set α → E →L[ℝ] F)
    (h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) {f : α →ₛ E}
    (hf : Integrable f μ) : setToSimpleFunc T'' f = setToSimpleFunc T f + setToSimpleFunc T' f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T T' T'' : Set α → E →L[ℝ] F) (h_add : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T'' s = T s + T' s) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T'' f =
  MeasureTheory.SimpleFunc.setToSimpleFunc T f + MeasureTheory.SimpleFunc.setToSimpleFunc T' f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left
theorem setToSimpleFunc_smul_left {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (c : ℝ)
    (f : α →ₛ F) : setToSimpleFunc (fun s => c • T s) f = c • setToSimpleFunc T f 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (c : ℝ) (f : MeasureTheory.SimpleFunc α F)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc (fun s => c • T s) f = c • MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left'
theorem setToSimpleFunc_smul_left' (T T' : Set α → E →L[ℝ] F') (c : ℝ)
    (h_smul : ∀ s, MeasurableSet s → μ s < ∞ → T' s = c • T s) {f : α →ₛ E} (hf : Integrable f μ) :
    setToSimpleFunc T' f = c • setToSimpleFunc T f 
Arguments: {α : Type u_1} {E : Type u_2} {F' : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T T' : Set α → E →L[ℝ] F') (c : ℝ) (h_smul : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T' s = c • T s) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T' f = c • MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_add
theorem setToSimpleFunc_add (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
    (hf : Integrable f μ) (hg : Integrable g μ) :
    setToSimpleFunc T (f + g) = setToSimpleFunc T f + setToSimpleFunc T g 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (hg : MeasureTheory.Integrable (↑g) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (f + g) =
  MeasureTheory.SimpleFunc.setToSimpleFunc T f + MeasureTheory.SimpleFunc.setToSimpleFunc T g
-- MeasureTheory.SimpleFunc.setToSimpleFunc_neg
theorem setToSimpleFunc_neg (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f : α →ₛ E}
    (hf : Integrable f μ) : setToSimpleFunc T (-f) = -setToSimpleFunc T f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (-f) = -MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_sub
theorem setToSimpleFunc_sub (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
    (hf : Integrable f μ) (hg : Integrable g μ) :
    setToSimpleFunc T (f - g) = setToSimpleFunc T f - setToSimpleFunc T g 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (hg : MeasureTheory.Integrable (↑g) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (f - g) =
  MeasureTheory.SimpleFunc.setToSimpleFunc T f - MeasureTheory.SimpleFunc.setToSimpleFunc T g
-- MeasureTheory.SimpleFunc.setToSimpleFunc_smul_real
theorem setToSimpleFunc_smul_real (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) (c : ℝ)
    {f : α →ₛ E} (hf : Integrable f μ) : setToSimpleFunc T (c • f) = c • setToSimpleFunc T f 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) (c : ℝ) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (c • f) = c • MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_smul
theorem setToSimpleFunc_smul {E} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E]
    [NormedSpace ℝ E] [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T)
    (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) (c : 𝕜) {f : α →ₛ E} (hf : Integrable f μ) :
    setToSimpleFunc T (c • f) = c • setToSimpleFunc T f 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_6} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_7} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [NormedSpace 𝕜 F] (T : Set α → E →L[ℝ] F) (h_add : MeasureTheory.FinMeasAdditive μ T) (h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x) (c : 𝕜) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T (c • f) = c • MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left
theorem setToSimpleFunc_mono_left {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] G'')
    (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →ₛ F) : setToSimpleFunc T f ≤ setToSimpleFunc T' f 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] {m : MeasurableSpace α} (T T' : Set α → F →L[ℝ] G'') (hTT' : ∀ (s : Set α) (x : F), (T s) x ≤ (T' s) x) (f : MeasureTheory.SimpleFunc α F)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T' f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left'
theorem setToSimpleFunc_mono_left' (T T' : Set α → E →L[ℝ] G'')
    (hTT' : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, T s x ≤ T' s x) (f : α →ₛ E)
    (hf : Integrable f μ) : setToSimpleFunc T f ≤ setToSimpleFunc T' f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] (T T' : Set α → E →L[ℝ] G'') (hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x) (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T' f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg
theorem setToSimpleFunc_nonneg {m : MeasurableSpace α} (T : Set α → G' →L[ℝ] G'')
    (hT_nonneg : ∀ s x, 0 ≤ x → 0 ≤ T s x) (f : α →ₛ G') (hf : 0 ≤ f) :
    0 ≤ setToSimpleFunc T f 
Arguments: {α : Type u_1} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {m : MeasurableSpace α} (T : Set α → G' →L[ℝ] G'') (hT_nonneg : ∀ (s : Set α) (x : G'), 0 ≤ x → 0 ≤ (T s) x) (f : MeasureTheory.SimpleFunc α G') (hf : 0 ≤ f)
Goal: 0 ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg'
theorem setToSimpleFunc_nonneg' (T : Set α → G' →L[ℝ] G'')
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) (f : α →ₛ G') (hf : 0 ≤ f)
    (hfi : Integrable f μ) : 0 ≤ setToSimpleFunc T f 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] (T : Set α → G' →L[ℝ] G'') (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) (f : MeasureTheory.SimpleFunc α G') (hf : 0 ≤ f) (hfi : MeasureTheory.Integrable (↑f) μ)
Goal: 0 ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T f
-- MeasureTheory.SimpleFunc.setToSimpleFunc_mono
theorem setToSimpleFunc_mono {T : Set α → G' →L[ℝ] G''} (h_add : FinMeasAdditive μ T)
    (hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) {f g : α →ₛ G'}
    (hfi : Integrable f μ) (hgi : Integrable g μ) (hfg : f ≤ g) :
    setToSimpleFunc T f ≤ setToSimpleFunc T g 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {G' : Type u_7} {G'' : Type u_8} [NormedLatticeAddCommGroup G''] [NormedSpace ℝ G''] [NormedLatticeAddCommGroup G'] [NormedSpace ℝ G'] {T : Set α → G' →L[ℝ] G''} (h_add : MeasureTheory.FinMeasAdditive μ T) (hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x) {f g : MeasureTheory.SimpleFunc α G'} (hfi : MeasureTheory.Integrable (↑f) μ) (hgi : MeasureTheory.Integrable (↑g) μ) (hfg : f ≤ g)
Goal: MeasureTheory.SimpleFunc.setToSimpleFunc T f ≤ MeasureTheory.SimpleFunc.setToSimpleFunc T g
-- MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_sum_opNorm
theorem norm_setToSimpleFunc_le_sum_opNorm {m : MeasurableSpace α} (T : Set α → F' →L[ℝ] F)
    (f : α →ₛ F') : ‖f.setToSimpleFunc T‖ ≤ ∑ x ∈ f.range, ‖T (f ⁻¹' {x})‖ * ‖x‖ 
Arguments: {α : Type u_1} {F : Type u_3} {F' : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] {m : MeasurableSpace α} (T : Set α → F' →L[ℝ] F) (f : MeasureTheory.SimpleFunc α F')
Goal: ‖MeasureTheory.SimpleFunc.setToSimpleFunc T f‖ ≤ ∑ x ∈ f.range, ‖T (↑f ⁻¹' {x})‖ * ‖x‖