ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothPartitionOfUnity.mem_finsupport
@[simp]
theorem mem_finsupport {i : ι} : i ∈ ρ.finsupport x₀ ↔ i ∈ support fun i ↦ ρ i x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M) {i : ι}
Goal: i ∈ ρ.finsupport x₀ ↔ i ∈ Function.support fun i => (ρ i) x₀
-- SmoothPartitionOfUnity.coe_finsupport
@[simp]
theorem coe_finsupport : (ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
Goal: ↑(ρ.finsupport x₀) = Function.support fun i => (ρ i) x₀
-- SmoothPartitionOfUnity.sum_finsupport
theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i ∈ ρ.finsupport x₀, ρ i x₀ = 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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M) (hx₀ : x₀ ∈ s)
Goal: ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ = 1
-- SmoothPartitionOfUnity.sum_finsupport'
theorem sum_finsupport' (hx₀ : x₀ ∈ s) {I : Finset ι} (hI : ρ.finsupport x₀ ⊆ I) :
    ∑ i ∈ I, ρ i x₀ = 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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M) (hx₀ : x₀ ∈ s)
Goal: ∀ {I_1 : Finset ι}, ρ.finsupport x₀ ⊆ I_1 → ∑ i ∈ I_1, (ρ i) x₀ = 1
-- SmoothPartitionOfUnity.sum_finsupport_smul_eq_finsum
theorem sum_finsupport_smul_eq_finsum {A : Type*} [AddCommGroup A] [Module ℝ A] (φ : ι → M → A) :
    ∑ i ∈ ρ.finsupport x₀, ρ i x₀ • φ i x₀ = ∑ᶠ i, ρ i x₀ • φ i x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M) {A : Type u_1} [AddCommGroup A] [Module ℝ A] (φ : ι → M → A)
Goal: ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀
-- SmoothPartitionOfUnity.finite_tsupport
theorem finite_tsupport : {i | x₀ ∈ tsupport (ρ i)}.Finite 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
Goal: {i | x₀ ∈ tsupport ⇑(ρ i)}.Finite
-- SmoothPartitionOfUnity.mem_fintsupport_iff
theorem mem_fintsupport_iff (i : ι) : i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport (ρ 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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M) (i : ι)
Goal: i ∈ ρ.fintsupport x₀ ↔ x₀ ∈ tsupport ⇑(ρ i)
-- SmoothPartitionOfUnity.eventually_fintsupport_subset
theorem eventually_fintsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
Goal: ∀ᶠ (y : M) in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀
-- SmoothPartitionOfUnity.finsupport_subset_fintsupport
theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
Goal: ρ.finsupport x₀ ⊆ ρ.fintsupport x₀
-- SmoothPartitionOfUnity.eventually_finsupport_subset
theorem eventually_finsupport_subset : ∀ᶠ y in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀ 
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} (ρ : SmoothPartitionOfUnity ι I M s) (x₀ : M)
Goal: ∀ᶠ (y : M) in 𝓝 x₀, ρ.finsupport y ⊆ ρ.fintsupport x₀
-- SmoothPartitionOfUnity.isSubordinate_toPartitionOfUnity
@[simp]
theorem isSubordinate_toPartitionOfUnity :
    f.toPartitionOfUnity.IsSubordinate U ↔ f.IsSubordinate U 
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} {U : ι → Set M}
Goal: f.toPartitionOfUnity.IsSubordinate U ↔ f.IsSubordinate U
-- SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul
theorem IsSubordinate.contMDiff_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
    (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, ContMDiffOn I 𝓘(ℝ, F) n (g i) (U i)) :
    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∞} {U : ι → Set M} {g : ι → M → F} (hf : f.IsSubordinate U) (ho : ∀ (i : ι), IsOpen (U i)) (hg : ∀ (i : ι), ContMDiffOn I 𝓘(ℝ, F) n (g i) (U i))
Goal: ContMDiff I 𝓘(ℝ, F) n fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- SmoothPartitionOfUnity.IsSubordinate.smooth_finsum_smul
theorem IsSubordinate.smooth_finsum_smul {g : ι → M → F} (hf : f.IsSubordinate U)
    (ho : ∀ i, IsOpen (U i)) (hg : ∀ i, SmoothOn I 𝓘(ℝ, F) (g i) (U i)) :
    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} {U : ι → Set M} {g : ι → M → F} (hf : f.IsSubordinate U) (ho : ∀ (i : ι), IsOpen (U i)) (hg : ∀ (i : ι), SmoothOn I 𝓘(ℝ, F) (g i) (U i))
Goal: Smooth I 𝓘(ℝ, F) fun x => ∑ᶠ (i : ι), (f i) x • g i x
-- BumpCovering.smooth_toPartitionOfUnity
theorem smooth_toPartitionOfUnity {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 : BumpCovering ι M s)
    (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) (i : ι) : Smooth I 𝓘(ℝ) (f.toPartitionOfUnity 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 : BumpCovering ι M s) (hf : ∀ (i : ι), Smooth I 𝓘(ℝ, ℝ) ⇑(f i)) (i : ι)
Goal: Smooth I 𝓘(ℝ, ℝ) ⇑(f.toPartitionOfUnity i)
-- BumpCovering.toSmoothPartitionOfUnity_toPartitionOfUnity
@[simp]
theorem toSmoothPartitionOfUnity_toPartitionOfUnity (f : BumpCovering ι M s)
    (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) :
    (f.toSmoothPartitionOfUnity hf).toPartitionOfUnity = f.toPartitionOfUnity 
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 : BumpCovering ι M s) (hf : ∀ (i : ι), Smooth I 𝓘(ℝ, ℝ) ⇑(f i))
Goal: (f.toSmoothPartitionOfUnity hf).toPartitionOfUnity = f.toPartitionOfUnity
-- BumpCovering.coe_toSmoothPartitionOfUnity
@[simp]
theorem coe_toSmoothPartitionOfUnity (f : BumpCovering ι M s) (hf : ∀ i, Smooth I 𝓘(ℝ) (f i))
    (i : ι) : ⇑(f.toSmoothPartitionOfUnity hf i) = f.toPartitionOfUnity 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 : BumpCovering ι M s) (hf : ∀ (i : ι), Smooth I 𝓘(ℝ, ℝ) ⇑(f i)) (i : ι)
Goal: ⇑((f.toSmoothPartitionOfUnity hf) i) = ⇑(f.toPartitionOfUnity i)
-- BumpCovering.IsSubordinate.toSmoothPartitionOfUnity
theorem IsSubordinate.toSmoothPartitionOfUnity {f : BumpCovering ι M s} {U : ι → Set M}
    (h : f.IsSubordinate U) (hf : ∀ i, Smooth I 𝓘(ℝ) (f i)) :
    (f.toSmoothPartitionOfUnity hf).IsSubordinate U 
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 : BumpCovering ι M s} {U : ι → Set M} (h : f.IsSubordinate U) (hf : ∀ (i : ι), Smooth I 𝓘(ℝ, ℝ) ⇑(f i))
Goal: (f.toSmoothPartitionOfUnity hf).IsSubordinate U
-- SmoothBumpCovering.IsSubordinate.support_subset
theorem IsSubordinate.support_subset {fs : SmoothBumpCovering ι I M s} {U : M → Set M}
    (h : fs.IsSubordinate U) (i : ι) : support (fs i) ⊆ U (fs.c i) 
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 : Set M} {fs : SmoothBumpCovering ι I M s} {U : M → Set M} (h : fs.IsSubordinate U) (i : ι)
Goal: Function.support ↑(fs.toFun i) ⊆ U (fs.c i)
-- SmoothBumpCovering.exists_isSubordinate
theorem exists_isSubordinate [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s)
    (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
    ∃ (ι : Type uM) (f : SmoothBumpCovering ι I M s), f.IsSubordinate U 
Arguments: {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 : Set M} {U : M → Set M} [T2Space M] [SigmaCompactSpace M] (hs : IsClosed s) (hU : ∀ x ∈ s, U x ∈ 𝓝 x)
Goal: ∃ ι f, f.IsSubordinate U
-- SmoothBumpCovering.locallyFinite
protected theorem locallyFinite : LocallyFinite fun i => support (fs i) 
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 : Set M} (fs : SmoothBumpCovering ι I M s)
Goal: LocallyFinite fun i => Function.support ↑(fs.toFun i)
-- SmoothBumpCovering.point_finite
protected theorem point_finite (x : M) : {i | fs i x ≠ 0}.Finite 
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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M)
Goal: {i | ↑(fs.toFun i) x ≠ 0}.Finite
-- SmoothBumpCovering.mem_chartAt_source_of_eq_one
theorem mem_chartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
    x ∈ (chartAt H (fs.c i)).source 
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 : Set M} (fs : SmoothBumpCovering ι I M s) {i : ι} {x : M} (h : ↑(fs.toFun i) x = 1)
Goal: x ∈ (chartAt H (fs.c i)).source
-- SmoothBumpCovering.mem_extChartAt_source_of_eq_one
theorem mem_extChartAt_source_of_eq_one {i : ι} {x : M} (h : fs i x = 1) :
    x ∈ (extChartAt I (fs.c i)).source 
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 : Set M} (fs : SmoothBumpCovering ι I M s) {i : ι} {x : M} (h : ↑(fs.toFun i) x = 1)
Goal: x ∈ (extChartAt I (fs.c i)).source
-- SmoothBumpCovering.eventuallyEq_one
theorem eventuallyEq_one (x : M) (hx : x ∈ s) : fs (fs.ind x hx) =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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: ↑(fs.toFun (fs.ind x hx)) =ᶠ[𝓝 x] 1
-- SmoothBumpCovering.apply_ind
theorem apply_ind (x : M) (hx : x ∈ s) : fs (fs.ind x hx) 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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: ↑(fs.toFun (fs.ind x hx)) x = 1
-- SmoothBumpCovering.mem_support_ind
theorem mem_support_ind (x : M) (hx : x ∈ s) : x ∈ support (fs <| fs.ind x hx) 
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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: x ∈ Function.support ↑(fs.toFun (fs.ind x hx))
-- SmoothBumpCovering.mem_chartAt_ind_source
theorem mem_chartAt_ind_source (x : M) (hx : x ∈ s) : x ∈ (chartAt H (fs.c (fs.ind x hx))).source 
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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: x ∈ (chartAt H (fs.c (fs.ind x hx))).source
-- SmoothBumpCovering.mem_extChartAt_ind_source
theorem mem_extChartAt_ind_source (x : M) (hx : x ∈ s) :
    x ∈ (extChartAt I (fs.c (fs.ind x hx))).source 
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 : Set M} (fs : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: x ∈ (extChartAt I (fs.c (fs.ind x hx))).source
-- SmoothBumpCovering.isSubordinate_toBumpCovering
@[simp, nolint simpNF]
theorem isSubordinate_toBumpCovering {f : SmoothBumpCovering ι I M s} {U : M → Set M} :
    (f.toBumpCovering.IsSubordinate fun i => U (f.c i)) ↔ f.IsSubordinate U 
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] [SmoothManifoldWithCorners I M] {s : Set M} [T2Space M] {f : SmoothBumpCovering ι I M s} {U : M → Set M}
Goal: (f.toBumpCovering.IsSubordinate fun i => U (f.c i)) ↔ f.IsSubordinate U
-- SmoothBumpCovering.toSmoothPartitionOfUnity_apply
theorem toSmoothPartitionOfUnity_apply (i : ι) (x : M) :
    fs.toSmoothPartitionOfUnity i x = fs i x * ∏ᶠ (j) (_ : WellOrderingRel j i), (1 - fs j x) 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] (i : ι) (x : M)
Goal: (fs.toSmoothPartitionOfUnity i) x = ↑(fs.toFun i) x * ∏ᶠ (j : ι) (_ : WellOrderingRel j i), (1 - ↑(fs.toFun j) x)
-- SmoothBumpCovering.toSmoothPartitionOfUnity_eq_mul_prod
theorem toSmoothPartitionOfUnity_eq_mul_prod (i : ι) (x : M) (t : Finset ι)
    (ht : ∀ j, WellOrderingRel j i → fs j x ≠ 0 → j ∈ t) :
    fs.toSmoothPartitionOfUnity i x =
      fs i x * ∏ j ∈ t.filter fun j => WellOrderingRel j i, (1 - fs j x) 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] (i : ι) (x : M) (t : Finset ι) (ht : ∀ (j : ι), WellOrderingRel j i → ↑(fs.toFun j) x ≠ 0 → j ∈ t)
Goal: (fs.toSmoothPartitionOfUnity i) x =
  ↑(fs.toFun i) x * ∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - ↑(fs.toFun j) x)
-- SmoothBumpCovering.exists_finset_toSmoothPartitionOfUnity_eventuallyEq
theorem exists_finset_toSmoothPartitionOfUnity_eventuallyEq (i : ι) (x : M) :
    ∃ t : Finset ι,
      fs.toSmoothPartitionOfUnity i =ᶠ[𝓝 x]
        fs i * ∏ j ∈ t.filter fun j => WellOrderingRel j i, ((1 : M → ℝ) - fs j) 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] (i : ι) (x : M)
Goal: ∃ t,
  ⇑(fs.toSmoothPartitionOfUnity i) =ᶠ[𝓝 x]
    ↑(fs.toFun i) * ∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - ↑(fs.toFun j))
-- SmoothBumpCovering.toSmoothPartitionOfUnity_zero_of_zero
theorem toSmoothPartitionOfUnity_zero_of_zero {i : ι} {x : M} (h : fs i x = 0) :
    fs.toSmoothPartitionOfUnity i x = 0 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] {i : ι} {x : M} (h : ↑(fs.toFun i) x = 0)
Goal: (fs.toSmoothPartitionOfUnity i) x = 0
-- SmoothBumpCovering.support_toSmoothPartitionOfUnity_subset
theorem support_toSmoothPartitionOfUnity_subset (i : ι) :
    support (fs.toSmoothPartitionOfUnity i) ⊆ support (fs i) 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] (i : ι)
Goal: Function.support ⇑(fs.toSmoothPartitionOfUnity i) ⊆ Function.support ↑(fs.toFun i)
-- SmoothBumpCovering.IsSubordinate.toSmoothPartitionOfUnity
theorem IsSubordinate.toSmoothPartitionOfUnity {f : SmoothBumpCovering ι I M s} {U : M → Set M}
    (h : f.IsSubordinate U) : f.toSmoothPartitionOfUnity.IsSubordinate fun i => U (f.c i) 
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] [SmoothManifoldWithCorners I M] {s : Set M} [T2Space M] {f : SmoothBumpCovering ι I M s} {U : M → Set M} (h : f.IsSubordinate U)
Goal: f.toSmoothPartitionOfUnity.IsSubordinate fun i => U (f.c i)
-- SmoothBumpCovering.sum_toSmoothPartitionOfUnity_eq
theorem sum_toSmoothPartitionOfUnity_eq (x : M) :
    ∑ᶠ i, fs.toSmoothPartitionOfUnity i x = 1 - ∏ᶠ i, (1 - fs i x) 
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] [SmoothManifoldWithCorners I M] {s : Set M} (fs : SmoothBumpCovering ι I M s) [T2Space M] (x : M)
Goal: ∑ᶠ (i : ι), (fs.toSmoothPartitionOfUnity i) x = 1 - ∏ᶠ (i : ι), (1 - ↑(fs.toFun i) x)
-- exists_smooth_zero_one_of_isClosed
theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
    (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc 0 1 
Arguments: {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] [SmoothManifoldWithCorners I M] [T2Space M] [SigmaCompactSpace M] {s t : Set M} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : M), f x ∈ Set.Icc 0 1
-- exists_smooth_zero_one_nhds_of_isClosed
theorem exists_smooth_zero_one_nhds_of_isClosed [T2Space M] [NormalSpace M] [SigmaCompactSpace M]
    {s t : Set M} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, (∀ᶠ x in 𝓝ˢ s, f x = 0) ∧ (∀ᶠ x in 𝓝ˢ t, f x = 1) ∧
      ∀ x, f x ∈ Icc 0 1 
Arguments: {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] [SmoothManifoldWithCorners I M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] {s t : Set M} (hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t)
Goal: ∃ f, (∀ᶠ (x : M) in 𝓝ˢ s, f x = 0) ∧ (∀ᶠ (x : M) in 𝓝ˢ t, f x = 1) ∧ ∀ (x : M), f x ∈ Set.Icc 0 1
-- exists_smooth_one_nhds_of_subset_interior
theorem exists_smooth_one_nhds_of_subset_interior [T2Space M] [NormalSpace M] [SigmaCompactSpace M]
    {s t : Set M} (hs : IsClosed s) (hd : s ⊆ interior t) :
    ∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, (∀ᶠ x in 𝓝ˢ s, f x = 1) ∧ (∀ x ∉ t, f x = 0) ∧
      ∀ x, f x ∈ Icc 0 1 
Arguments: {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] [SmoothManifoldWithCorners I M] [T2Space M] [NormalSpace M] [SigmaCompactSpace M] {s t : Set M} (hs : IsClosed s) (hd : s ⊆ interior t)
Goal: ∃ f, (∀ᶠ (x : M) in 𝓝ˢ s, f x = 1) ∧ (∀ x ∉ t, f x = 0) ∧ ∀ (x : M), f x ∈ Set.Icc 0 1
-- SmoothPartitionOfUnity.exists_isSubordinate
theorem exists_isSubordinate {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ i, IsOpen (U i))
    (hU : s ⊆ ⋃ i, U i) : ∃ f : SmoothPartitionOfUnity ι I M s, f.IsSubordinate U 
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] [SmoothManifoldWithCorners I M] [T2Space M] [SigmaCompactSpace M] {s : Set M} (hs : IsClosed s) (U : ι → Set M) (ho : ∀ (i : ι), IsOpen (U i)) (hU : s ⊆ ⋃ i, U i)
Goal: ∃ f, f.IsSubordinate U
-- SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source_of_isClosed
theorem exists_isSubordinate_chartAt_source_of_isClosed {s : Set M} (hs : IsClosed s) :
    ∃ f : SmoothPartitionOfUnity s I M s,
      f.IsSubordinate (fun x ↦ (chartAt H (x : M)).source) 
Arguments: {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] [SmoothManifoldWithCorners I M] [T2Space M] [SigmaCompactSpace M] {s : Set M} (hs : IsClosed s)
Goal: ∃ f, f.IsSubordinate fun x => (chartAt H ↑x).source
-- SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source
theorem exists_isSubordinate_chartAt_source :
    ∃ f : SmoothPartitionOfUnity M I M univ, f.IsSubordinate (fun x ↦ (chartAt H x).source) 
Arguments: {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] [SmoothManifoldWithCorners I M] [T2Space M] [SigmaCompactSpace M]
Goal: ∃ f, f.IsSubordinate fun x => (chartAt H x).source