ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Trivialization.smoothAt_iff
theorem Trivialization.smoothAt_iff {f : M → TotalSpace F E} {x₀ : M} (he : f x₀ ∈ e.source) :
    SmoothAt IM (IB.prod 𝓘(𝕜, F)) f x₀ ↔
      SmoothAt IM IB (fun x => (f x).proj) x₀ ∧ SmoothAt IM 𝓘(𝕜, F) (fun x ↦ (e (f x)).2) x₀ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {x₀ : M} (he : f x₀ ∈ e.source)
Goal: SmoothAt IM (IB.prod 𝓘(𝕜, F)) f x₀ ↔
  SmoothAt IM IB (fun x => (f x).proj) x₀ ∧ SmoothAt IM 𝓘(𝕜, F) (fun x => (↑e (f x)).2) x₀
-- Trivialization.smoothOn_iff
theorem Trivialization.smoothOn_iff {f : M → TotalSpace F E} {s : Set M}
    (he : MapsTo f s e.source) :
    SmoothOn IM (IB.prod 𝓘(𝕜, F)) f s ↔
      SmoothOn IM IB (fun x => (f x).proj) s ∧ SmoothOn IM 𝓘(𝕜, F) (fun x ↦ (e (f x)).2) s 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} {s : Set M} (he : Set.MapsTo f s e.source)
Goal: SmoothOn IM (IB.prod 𝓘(𝕜, F)) f s ↔
  SmoothOn IM IB (fun x => (f x).proj) s ∧ SmoothOn IM 𝓘(𝕜, F) (fun x => (↑e (f x)).2) s
-- Trivialization.smooth_iff
theorem Trivialization.smooth_iff {f : M → TotalSpace F E} (he : ∀ x, f x ∈ e.source) :
    Smooth IM (IB.prod 𝓘(𝕜, F)) f ↔
      Smooth IM IB (fun x => (f x).proj) ∧ Smooth IM 𝓘(𝕜, F) (fun x ↦ (e (f x)).2) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {M : Type u_5} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] {e : Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {f : M → Bundle.TotalSpace F E} (he : ∀ (x : M), f x ∈ e.source)
Goal: Smooth IM (IB.prod 𝓘(𝕜, F)) f ↔ (Smooth IM IB fun x => (f x).proj) ∧ Smooth IM 𝓘(𝕜, F) fun x => (↑e (f x)).2
-- Trivialization.smoothOn
theorem Trivialization.smoothOn (e : Trivialization F (π F E)) [MemTrivializationAtlas e] :
    SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) e e.source 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e]
Goal: SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e) e.source
-- Trivialization.smoothOn_symm
theorem Trivialization.smoothOn_symm (e : Trivialization F (π F E)) [MemTrivializationAtlas e] :
    SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) e.toPartialHomeomorph.symm e.target 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [FiberBundle F E] [VectorBundle 𝕜 F E] [SmoothVectorBundle F E IB] (e : Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e]
Goal: SmoothOn (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) (↑e.symm) e.target
-- VectorBundleCore.IsSmooth.smoothOn_coordChange
class IsSmooth (IB : ModelWithCorners 𝕜 EB HB) : Prop where
  smoothOn_coordChange :
    ∀ i j, SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (Z.coordChange i j) (Z.baseSet i ∩ Z.baseSet j)
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_11} {Z : VectorBundleCore 𝕜 B F ι} {IB : ModelWithCorners 𝕜 EB HB} [self : Z.IsSmooth IB] (i j : ι)
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (Z.coordChange i j) (Z.baseSet i ∩ Z.baseSet j)
-- VectorBundleCore.smoothOn_coordChange
theorem smoothOn_coordChange (IB : ModelWithCorners 𝕜 EB HB) [h : Z.IsSmooth IB] (i j : ι) :
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (Z.coordChange i j) (Z.baseSet i ∩ Z.baseSet j) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] [TopologicalSpace B] [ChartedSpace HB B] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ι : Type u_11} (Z : VectorBundleCore 𝕜 B F ι) (IB : ModelWithCorners 𝕜 EB HB) [h : Z.IsSmooth IB] (i j : ι)
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (Z.coordChange i j) (Z.baseSet i ∩ Z.baseSet j)
-- VectorPrebundle.IsSmooth.exists_smoothCoordChange
class IsSmooth (a : VectorPrebundle 𝕜 F E) : Prop where
  exists_smoothCoordChange :
    ∀ᵉ (e ∈ a.pretrivializationAtlas) (e' ∈ a.pretrivializationAtlas),
      ∃ f : B → F →L[𝕜] F,
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) f (e.baseSet ∩ e'.baseSet) ∧
          ∀ (b : B) (_ : b ∈ e.baseSet ∩ e'.baseSet) (v : F),
            f b v = (e' ⟨b, e.symm b v⟩).2
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [(x : B) → TopologicalSpace (E x)] {a : VectorPrebundle 𝕜 F E} [self : VectorPrebundle.IsSmooth IB a] (e : Pretrivialization F Bundle.TotalSpace.proj)
Goal: e ∈ a.pretrivializationAtlas →
  ∀ e' ∈ a.pretrivializationAtlas,
    ∃ f,
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) f (e.baseSet ∩ e'.baseSet) ∧
        ∀ b ∈ e.baseSet ∩ e'.baseSet, ∀ (v : F), (f b) v = (↑e' { proj := b, snd := e.symm b v }).2
-- VectorPrebundle.smoothOn_smoothCoordChange
theorem smoothOn_smoothCoordChange (he : e ∈ a.pretrivializationAtlas)
    (he' : e' ∈ a.pretrivializationAtlas) :
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (a.smoothCoordChange IB he he') (e.baseSet ∩ e'.baseSet) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle 𝕜 F E) [ha : VectorPrebundle.IsSmooth IB a] {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas)
Goal: SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (VectorPrebundle.smoothCoordChange IB a he he') (e.baseSet ∩ e'.baseSet)
-- VectorPrebundle.smoothCoordChange_apply
theorem smoothCoordChange_apply (he : e ∈ a.pretrivializationAtlas)
    (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
    a.smoothCoordChange IB he he' b v = (e' ⟨b, e.symm b v⟩).2 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle 𝕜 F E) [ha : VectorPrebundle.IsSmooth IB a] {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F)
Goal: (VectorPrebundle.smoothCoordChange IB a he he' b) v = (↑e' { proj := b, snd := e.symm b v }).2
-- VectorPrebundle.mk_smoothCoordChange
theorem mk_smoothCoordChange (he : e ∈ a.pretrivializationAtlas)
    (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F) :
    (b, a.smoothCoordChange IB he he' b v) = e' ⟨b, e.symm b v⟩ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle 𝕜 F E) [ha : VectorPrebundle.IsSmooth IB a] {e e' : Pretrivialization F Bundle.TotalSpace.proj} (he : e ∈ a.pretrivializationAtlas) (he' : e' ∈ a.pretrivializationAtlas) {b : B} (hb : b ∈ e.baseSet ∩ e'.baseSet) (v : F)
Goal: (b, (VectorPrebundle.smoothCoordChange IB a he he' b) v) = ↑e' { proj := b, snd := e.symm b v }
-- VectorPrebundle.smoothVectorBundle
theorem smoothVectorBundle : @SmoothVectorBundle
    _ _ F E _ _ _ _ _ _ IB _ _ _ _ _ _ a.totalSpaceTopology _ a.toFiberBundle a.toVectorBundle 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_4} {E : B → Type u_6} [NontriviallyNormedField 𝕜] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_8} [TopologicalSpace HB] (IB : ModelWithCorners 𝕜 EB HB) [TopologicalSpace B] [ChartedSpace HB B] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [(x : B) → TopologicalSpace (E x)] (a : VectorPrebundle 𝕜 F E) [ha : VectorPrebundle.IsSmooth IB a]
Goal: SmoothVectorBundle F E IB
-- SmoothBumpCovering.eventuallyEq_one'
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure SmoothBumpCovering (s : Set M := univ) where
    c : ι → M
    toFun : ∀ i, SmoothBumpFunction I (c i)
    c_mem' : ∀ i, c i ∈ s
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothBumpCovering ι I M s) (x : M)
Goal: x ∈ s → ∃ i, ↑(self.toFun i) =ᶠ[𝓝 x] 1
-- SmoothBumpCovering.c_mem'
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure SmoothBumpCovering (s : Set M := univ) where
    c : ι → M
    toFun : ∀ i, SmoothBumpFunction I (c i)
    c_mem' : ∀ i, c i ∈ s
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothBumpCovering ι I M s) (i : ι)
Goal: self.c i ∈ s
-- SmoothBumpCovering.locallyFinite'
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure SmoothBumpCovering (s : Set M := univ) where
    c : ι → M
    toFun : ∀ i, SmoothBumpFunction I (c i)
    c_mem' : ∀ i, c i ∈ s
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    eventuallyEq_one' : ∀ x ∈ s, ∃ i, toFun i =ᶠ[𝓝 x] 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothBumpCovering ι I M s)
Goal: LocallyFinite fun i => Function.support ↑(self.toFun i)
-- SmoothPartitionOfUnity.sum_le_one'
structure SmoothPartitionOfUnity (s : Set M := univ) where
    toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    nonneg' : ∀ i x, 0 ≤ toFun i x
    sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
    sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothPartitionOfUnity ι I M s) (x : M)
Goal: ∑ᶠ (i : ι), (self.toFun i) x ≤ 1
-- SmoothPartitionOfUnity.sum_eq_one'
structure SmoothPartitionOfUnity (s : Set M := univ) where
    toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    nonneg' : ∀ i x, 0 ≤ toFun i x
    sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
    sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothPartitionOfUnity ι I M s) (x : M)
Goal: x ∈ s → ∑ᶠ (i : ι), (self.toFun i) x = 1
-- SmoothPartitionOfUnity.locallyFinite'
structure SmoothPartitionOfUnity (s : Set M := univ) where
    toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    nonneg' : ∀ i x, 0 ≤ toFun i x
    sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
    sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothPartitionOfUnity ι I M s)
Goal: LocallyFinite fun i => Function.support ⇑(self.toFun i)
-- SmoothPartitionOfUnity.nonneg'
structure SmoothPartitionOfUnity (s : Set M := univ) where
    toFun : ι → C^∞⟮I, M; 𝓘(ℝ), ℝ⟯
    locallyFinite' : LocallyFinite fun i => support (toFun i)
    nonneg' : ∀ i x, 0 ≤ toFun i x
    sum_eq_one' : ∀ x ∈ s, ∑ᶠ i, toFun i x = 1
    sum_le_one' : ∀ x, ∑ᶠ i, toFun i x ≤ 1
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothPartitionOfUnity ι I M s) (i : ι) (x : M)
Goal: 0 ≤ (self.toFun i) x
-- SmoothPartitionOfUnity.locallyFinite
protected theorem locallyFinite : LocallyFinite fun i => support (f i) 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s)
Goal: LocallyFinite fun i => Function.support ⇑(f i)
-- SmoothPartitionOfUnity.nonneg
theorem nonneg (i : ι) (x : M) : 0 ≤ f i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) (i : ι) (x : M)
Goal: 0 ≤ (f i) x
-- SmoothPartitionOfUnity.sum_eq_one
theorem sum_eq_one {x} (hx : x ∈ s) : ∑ᶠ i, f i x = 1 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {x : M} (hx : x ∈ s)
Goal: ∑ᶠ (i : ι), (f i) x = 1
-- SmoothPartitionOfUnity.exists_pos_of_mem
theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {x : M} (hx : x ∈ s)
Goal: ∃ i, 0 < (f i) x
-- SmoothPartitionOfUnity.sum_le_one
theorem sum_le_one (x : M) : ∑ᶠ i, f i x ≤ 1 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) (x : M)
Goal: ∑ᶠ (i : ι), (f i) x ≤ 1
-- SmoothPartitionOfUnity.toPartitionOfUnity_toFun
@[simps]
def toPartitionOfUnity : PartitionOfUnity ι M s 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) (i : ι)
Goal: f.toPartitionOfUnity i = ↑(f i)
-- SmoothPartitionOfUnity.smooth_sum
theorem smooth_sum : Smooth I 𝓘(ℝ) fun x => ∑ᶠ i, f i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s)
Goal: Smooth I 𝓘(ℝ, ℝ) fun x => ∑ᶠ (i : ι), (f i) x
-- SmoothPartitionOfUnity.le_one
theorem le_one (i : ι) (x : M) : f i x ≤ 1 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) (i : ι) (x : M)
Goal: (f i) x ≤ 1
-- SmoothPartitionOfUnity.sum_nonneg
theorem sum_nonneg (x : M) : 0 ≤ ∑ᶠ i, f i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) (x : M)
Goal: 0 ≤ ∑ᶠ (i : ι), (f i) x
-- SmoothPartitionOfUnity.contMDiff_smul
theorem contMDiff_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n g x) :
    ContMDiff I 𝓘(ℝ, F) n fun x => f i x • g x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞} {g : M → F} {i : ι} (hg : ∀ x ∈ tsupport ⇑(f i), ContMDiffAt I 𝓘(ℝ, F) n g x)
Goal: ContMDiff I 𝓘(ℝ, F) n fun x => (f i) x • g x
-- SmoothPartitionOfUnity.smooth_smul
theorem smooth_smul {g : M → F} {i} (hg : ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) g x) :
    Smooth I 𝓘(ℝ, F) fun x => f i x • g x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {g : M → F} {i : ι} (hg : ∀ x ∈ tsupport ⇑(f i), SmoothAt I 𝓘(ℝ, F) g x)
Goal: Smooth I 𝓘(ℝ, F) fun x => (f i) x • g x
-- SmoothPartitionOfUnity.contMDiff_finsum_smul
theorem contMDiff_finsum_smul {g : ι → M → F}
    (hg : ∀ (i), ∀ x ∈ tsupport (f i), ContMDiffAt I 𝓘(ℝ, F) n (g i) x) :
    ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ i, f i x • g i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞} {g : ι → M → F} (hg : ∀ (i : ι), ∀ x ∈ tsupport ⇑(f i), ContMDiffAt I 𝓘(ℝ, F) n (g i) x)
Goal: ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- SmoothPartitionOfUnity.smooth_finsum_smul
theorem smooth_finsum_smul {g : ι → M → F}
    (hg : ∀ (i), ∀ x ∈ tsupport (f i), SmoothAt I 𝓘(ℝ, F) (g i) x) :
    Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ i, f i x • g i x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {g : ι → M → F} (hg : ∀ (i : ι), ∀ x ∈ tsupport ⇑(f i), SmoothAt I 𝓘(ℝ, F) (g i) x)
Goal: Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- SmoothPartitionOfUnity.contMDiffAt_finsum
theorem contMDiffAt_finsum {x₀ : M} {g : ι → M → F}
    (hφ : ∀ i, x₀ ∈ tsupport (f i) → ContMDiffAt I 𝓘(ℝ, F) n (g i) x₀) :
    ContMDiffAt I 𝓘(ℝ, F) n (fun x ↦ ∑ᶠ i, f i x • g i x) x₀ 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {n : ℕ∞} {x₀ : M} {g : ι → M → F} (hφ : ∀ (i : ι), x₀ ∈ tsupport ⇑(f i) → ContMDiffAt I 𝓘(ℝ, F) n (g i) x₀)
Goal: ContMDiffAt I 𝓘(ℝ, F) n (fun x => ∑ᶠ (i : ι), (f i) x • g i x) x₀
-- SmoothPartitionOfUnity.contDiffAt_finsum
theorem contDiffAt_finsum {s : Set E} (f : SmoothPartitionOfUnity ι 𝓘(ℝ, E) E s) {x₀ : E}
    {g : ι → E → F} (hφ : ∀ i, x₀ ∈ tsupport (f i) → ContDiffAt ℝ n (g i) x₀) :
    ContDiffAt ℝ n (fun x ↦ ∑ᶠ i, f i x • g i x) x₀ 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {n : ℕ∞} {s : Set E} (f : SmoothPartitionOfUnity ι 𝓘(ℝ, E) E s) {x₀ : E} {g : ι → E → F} (hφ : ∀ (i : ι), x₀ ∈ tsupport ⇑(f i) → ContDiffAt ℝ n (g i) x₀)
Goal: ContDiffAt ℝ n (fun x => ∑ᶠ (i : ι), (f i) x • g i x) x₀
-- SmoothPartitionOfUnity.finsum_smul_mem_convex
theorem finsum_smul_mem_convex {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s)
    (hg : ∀ i, f i x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t) : ∑ᶠ i, f i x • g i x ∈ t 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace ℝ F] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) {g : ι → M → F} {t : Set F} {x : M} (hx : x ∈ s) (hg : ∀ (i : ι), (f i) x ≠ 0 → g i x ∈ t) (ht : Convex ℝ t)
Goal: ∑ᶠ (i : ι), (f i) x • g i x ∈ t