ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<VAR>hgg' : g =ᶠ[ae μ] g'<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<VAR>hgg' : g =ᶠ[ae μ] g'<VAR>hg'_meas : StronglyMeasurable g'<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<VAR>hgg' : g =ᶠ[ae μ] g'<VAR>hg'_meas : StronglyMeasurable g'<VAR>hg'_Lp : Memℒp g' p μ<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj✝ : Disjoint (Function.support f) (Function.support g)<VAR>hPf : P (Memℒp.toLp f hf)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<VAR>hgg' : g =ᶠ[ae μ] g'<VAR>hg'_meas : StronglyMeasurable g'<VAR>hg'_Lp : Memℒp g' p μ<VAR>h_disj : Disjoint (Function.support f') (Function.support g')<GOAL>P (Memℒp.toLp f hf + Memℒp.toLp g hg)
<VAR>α : Type u_1<VAR>E' : Type u_2<VAR>F : Type u_3<VAR>F' : Type u_4<VAR>𝕜 : Type u_5<VAR>p : ℝ≥0∞<VAR>inst✝¹² : RCLike 𝕜<VAR>inst✝¹¹ : NormedAddCommGroup E'<VAR>inst✝¹⁰ : InnerProductSpace 𝕜 E'<VAR>inst✝⁹ : CompleteSpace E'<VAR>inst✝⁸ : NormedSpace ℝ E'<VAR>inst✝⁷ : NormedAddCommGroup F<VAR>inst✝⁶ : NormedSpace 𝕜 F<VAR>inst✝⁵ : NormedAddCommGroup F'<VAR>inst✝⁴ : NormedSpace 𝕜 F'<VAR>inst✝³ : NormedSpace ℝ F'<VAR>inst✝² : CompleteSpace F'<VAR>m m0 : MeasurableSpace α<VAR>μ : Measure α<VAR>inst✝¹ : Fact (1 ≤ p)<VAR>inst✝ : NormedSpace ℝ F<VAR>hm : m ≤ m0<VAR>hp_ne_top : p ≠ ⊤<VAR>P : ↥(Lp F p μ) → Prop<VAR>h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ⊤), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)<VAR>h_add : ∀ ⦃f g : α → F⦄ (hf : Memℒp f p μ) (hg : Memℒp g p μ), StronglyMeasurable f → StronglyMeasurable g → Disjoint (Function.support f) (Function.support g) → P (Memℒp.toLp f hf) → P (Memℒp.toLp g hg) → P (Memℒp.toLp f hf + Memℒp.toLp g hg)<VAR>h_closed : IsClosed {f | P ↑f}<VAR>f✝ : ↥(Lp F p μ)<VAR>hf✝ : AEStronglyMeasurable' m (↑↑f✝) μ<VAR>f g : α → F<VAR>hf : Memℒp f p μ<VAR>hg : Memℒp g p μ<VAR>hfm : AEStronglyMeasurable' m f μ<VAR>hgm : AEStronglyMeasurable' m g μ<VAR>h_disj✝ : Disjoint (Function.support f) (Function.support g)<VAR>hPg : P (Memℒp.toLp g hg)<VAR>s_f : Set α := Function.support (AEStronglyMeasurable'.mk f hfm)<VAR>hs_f : MeasurableSet s_f<VAR>hs_f_eq : s_f =ᶠ[ae μ] Function.support f<VAR>s_g : Set α := Function.support (AEStronglyMeasurable'.mk g hgm)<VAR>hs_g : MeasurableSet s_g<VAR>hs_g_eq : s_g =ᶠ[ae μ] Function.support g<VAR>h_inter_empty : s_f ∩ s_g =ᶠ[ae μ] ∅<VAR>f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable'.mk f hfm)<VAR>hff' : f =ᶠ[ae μ] f'<VAR>hf'_meas : StronglyMeasurable f'<VAR>hf'_Lp : Memℒp f' p μ<VAR>hPf : P (Memℒp.toLp f' hf'_Lp)<VAR>g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable'.mk g hgm)<VAR>hgg' : g =ᶠ[ae μ] g'<VAR>hg'_meas : StronglyMeasurable g'<VAR>hg'_Lp : Memℒp g' p μ<VAR>h_disj : Disjoint (Function.support f') (Function.support g')<GOAL>P (Memℒp.toLp f' hf'_Lp + Memℒp.toLp g hg)