ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- contMDiff_finprod
@[to_additive]
theorem contMDiff_finprod (h : ∀ i, ContMDiff I' I n (f i))
    (hfin : LocallyFinite fun i => mulSupport (f i)) : ContMDiff I' I n fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {n : ℕ∞} (h : ∀ (i : ι), ContMDiff I' I n (f i)) (hfin : LocallyFinite fun i => Function.mulSupport (f i))
Goal: ContMDiff I' I n fun x => ∏ᶠ (i : ι), f i x
-- contMDiff_finsum
@[to_additive]
theorem contMDiff_finprod (h : ∀ i, ContMDiff I' I n (f i))
    (hfin : LocallyFinite fun i => mulSupport (f i)) : ContMDiff I' I n fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {n : ℕ∞} (h : ∀ (i : ι), ContMDiff I' I n (f i)) (hfin : LocallyFinite fun i => Function.support (f i))
Goal: ContMDiff I' I n fun x => ∑ᶠ (i : ι), f i x
-- contMDiff_finsum_cond
@[to_additive]
theorem contMDiff_finprod_cond (hc : ∀ i, p i → ContMDiff I' I n (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    ContMDiff I' I n fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {n : ℕ∞} {p : ι → Prop} (hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)) (hf : LocallyFinite fun i => Function.support (f i))
Goal: ContMDiff I' I n fun x => ∑ᶠ (i : ι) (_ : p i), f i x
-- contMDiff_finprod_cond
@[to_additive]
theorem contMDiff_finprod_cond (hc : ∀ i, p i → ContMDiff I' I n (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    ContMDiff I' I n fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {n : ℕ∞} {p : ι → Prop} (hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i))
Goal: ContMDiff I' I n fun x => ∏ᶠ (i : ι) (_ : p i), f i x
-- smoothAt_finsum
@[to_additive]
theorem smoothAt_finprod
    (lf : LocallyFinite fun i ↦ mulSupport <| f i) (h : ∀ i, SmoothAt I' I (f i) x₀) :
    SmoothAt I' I (fun x ↦ ∏ᶠ i, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.support (f i)) (h : ∀ (i : ι), SmoothAt I' I (f i) x₀)
Goal: SmoothAt I' I (fun x => ∑ᶠ (i : ι), f i x) x₀
-- smoothAt_finprod
@[to_additive]
theorem smoothAt_finprod
    (lf : LocallyFinite fun i ↦ mulSupport <| f i) (h : ∀ i, SmoothAt I' I (f i) x₀) :
    SmoothAt I' I (fun x ↦ ∏ᶠ i, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.mulSupport (f i)) (h : ∀ (i : ι), SmoothAt I' I (f i) x₀)
Goal: SmoothAt I' I (fun x => ∏ᶠ (i : ι), f i x) x₀
-- smoothWithinAt_finset_prod'
@[to_additive]
theorem smoothWithinAt_finset_prod' (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x) :
    SmoothWithinAt I' I (∏ i ∈ t, f i) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x)
Goal: SmoothWithinAt I' I (∏ i ∈ t, f i) s x
-- smoothWithinAt_finset_sum'
@[to_additive]
theorem smoothWithinAt_finset_prod' (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x) :
    SmoothWithinAt I' I (∏ i ∈ t, f i) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x)
Goal: SmoothWithinAt I' I (∑ i ∈ t, f i) s x
-- smoothWithinAt_finset_sum
@[to_additive]
theorem smoothWithinAt_finset_prod (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x) :
    SmoothWithinAt I' I (fun x => ∏ i ∈ t, f i x) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x)
Goal: SmoothWithinAt I' I (fun x => ∑ i ∈ t, f i x) s x
-- smoothWithinAt_finset_prod
@[to_additive]
theorem smoothWithinAt_finset_prod (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x) :
    SmoothWithinAt I' I (fun x => ∏ i ∈ t, f i x) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x)
Goal: SmoothWithinAt I' I (fun x => ∏ i ∈ t, f i x) s x
-- smoothAt_finset_prod'
@[to_additive]
theorem smoothAt_finset_prod' (h : ∀ i ∈ t, SmoothAt I' I (f i) x) :
    SmoothAt I' I (∏ i ∈ t, f i) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothAt I' I (f i) x)
Goal: SmoothAt I' I (∏ i ∈ t, f i) x
-- smoothAt_finset_sum'
@[to_additive]
theorem smoothAt_finset_prod' (h : ∀ i ∈ t, SmoothAt I' I (f i) x) :
    SmoothAt I' I (∏ i ∈ t, f i) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothAt I' I (f i) x)
Goal: SmoothAt I' I (∑ i ∈ t, f i) x
-- smoothAt_finset_prod
@[to_additive]
theorem smoothAt_finset_prod (h : ∀ i ∈ t, SmoothAt I' I (f i) x) :
    SmoothAt I' I (fun x => ∏ i ∈ t, f i x) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothAt I' I (f i) x)
Goal: SmoothAt I' I (fun x => ∏ i ∈ t, f i x) x
-- smoothAt_finset_sum
@[to_additive]
theorem smoothAt_finset_prod (h : ∀ i ∈ t, SmoothAt I' I (f i) x) :
    SmoothAt I' I (fun x => ∏ i ∈ t, f i x) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothAt I' I (f i) x)
Goal: SmoothAt I' I (fun x => ∑ i ∈ t, f i x) x
-- smoothOn_finset_prod'
@[to_additive]
theorem smoothOn_finset_prod' (h : ∀ i ∈ t, SmoothOn I' I (f i) s) :
    SmoothOn I' I (∏ i ∈ t, f i) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothOn I' I (f i) s)
Goal: SmoothOn I' I (∏ i ∈ t, f i) s
-- smoothOn_finset_sum'
@[to_additive]
theorem smoothOn_finset_prod' (h : ∀ i ∈ t, SmoothOn I' I (f i) s) :
    SmoothOn I' I (∏ i ∈ t, f i) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothOn I' I (f i) s)
Goal: SmoothOn I' I (∑ i ∈ t, f i) s
-- smoothOn_finset_sum
@[to_additive]
theorem smoothOn_finset_prod (h : ∀ i ∈ t, SmoothOn I' I (f i) s) :
    SmoothOn I' I (fun x => ∏ i ∈ t, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothOn I' I (f i) s)
Goal: SmoothOn I' I (fun x => ∑ i ∈ t, f i x) s
-- smoothOn_finset_prod
@[to_additive]
theorem smoothOn_finset_prod (h : ∀ i ∈ t, SmoothOn I' I (f i) s) :
    SmoothOn I' I (fun x => ∏ i ∈ t, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, SmoothOn I' I (f i) s)
Goal: SmoothOn I' I (fun x => ∏ i ∈ t, f i x) s
-- smooth_finset_sum'
@[to_additive]
theorem smooth_finset_prod' (h : ∀ i ∈ t, Smooth I' I (f i)) : Smooth I' I (∏ i ∈ t, f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, Smooth I' I (f i))
Goal: Smooth I' I (∑ i ∈ t, f i)
-- smooth_finset_prod'
@[to_additive]
theorem smooth_finset_prod' (h : ∀ i ∈ t, Smooth I' I (f i)) : Smooth I' I (∏ i ∈ t, f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, Smooth I' I (f i))
Goal: Smooth I' I (∏ i ∈ t, f i)
-- smooth_finset_prod
@[to_additive]
theorem smooth_finset_prod (h : ∀ i ∈ t, Smooth I' I (f i)) :
    Smooth I' I fun x => ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, Smooth I' I (f i))
Goal: Smooth I' I fun x => ∏ i ∈ t, f i x
-- smooth_finset_sum
@[to_additive]
theorem smooth_finset_prod (h : ∀ i ∈ t, Smooth I' I (f i)) :
    Smooth I' I fun x => ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} (h : ∀ i ∈ t, Smooth I' I (f i))
Goal: Smooth I' I fun x => ∑ i ∈ t, f i x
-- smooth_finprod
@[to_additive]
theorem smooth_finprod (h : ∀ i, Smooth I' I (f i))
    (hfin : LocallyFinite fun i => mulSupport (f i)) : Smooth I' I fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} (h : ∀ (i : ι), Smooth I' I (f i)) (hfin : LocallyFinite fun i => Function.mulSupport (f i))
Goal: Smooth I' I fun x => ∏ᶠ (i : ι), f i x
-- smooth_finsum
@[to_additive]
theorem smooth_finprod (h : ∀ i, Smooth I' I (f i))
    (hfin : LocallyFinite fun i => mulSupport (f i)) : Smooth I' I fun x => ∏ᶠ i, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} (h : ∀ (i : ι), Smooth I' I (f i)) (hfin : LocallyFinite fun i => Function.support (f i))
Goal: Smooth I' I fun x => ∑ᶠ (i : ι), f i x
-- smooth_finprod_cond
@[to_additive]
theorem smooth_finprod_cond (hc : ∀ i, p i → Smooth I' I (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    Smooth I' I fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {p : ι → Prop} (hc : ∀ (i : ι), p i → Smooth I' I (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i))
Goal: Smooth I' I fun x => ∏ᶠ (i : ι) (_ : p i), f i x
-- smooth_finsum_cond
@[to_additive]
theorem smooth_finprod_cond (hc : ∀ i, p i → Smooth I' I (f i))
    (hf : LocallyFinite fun i => mulSupport (f i)) :
    Smooth I' I fun x => ∏ᶠ (i) (_ : p i), f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {p : ι → Prop} (hc : ∀ (i : ι), p i → Smooth I' I (f i)) (hf : LocallyFinite fun i => Function.support (f i))
Goal: Smooth I' I fun x => ∑ᶠ (i : ι) (_ : p i), f i x
-- ContMDiffWithinAt.div_const
@[to_additive]
theorem ContMDiffWithinAt.div_const (hf : ContMDiffWithinAt I' I n f s x) :
    ContMDiffWithinAt I' I n (fun x ↦ f x / c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f : M → G} {s : Set M} {x : M} {n : ℕ∞} (c : G) (hf : ContMDiffWithinAt I' I n f s x)
Goal: ContMDiffWithinAt I' I n (fun x => f x / c) s x
-- ContMDiffWithinAt.sub_const
@[to_additive]
theorem ContMDiffWithinAt.div_const (hf : ContMDiffWithinAt I' I n f s x) :
    ContMDiffWithinAt I' I n (fun x ↦ f x / c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f : M → G} {s : Set M} {x : M} {n : ℕ∞} (c : G) (hf : ContMDiffWithinAt I' I n f s x)
Goal: ContMDiffWithinAt I' I n (fun x => f x - c) s x
-- ContMDiffAt.div_const
@[to_additive]
nonrec theorem ContMDiffAt.div_const (hf : ContMDiffAt I' I n f x) :
    ContMDiffAt I' I n (fun x ↦ f x / c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [DivInvMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f : M → G} {x : M} {n : ℕ∞} (c : G) (hf : ContMDiffAt I' I n f x)
Goal: ContMDiffAt I' I n (fun x => f x / c) x
-- ContMDiffAt.sub_const
@[to_additive]
nonrec theorem ContMDiffAt.div_const (hf : ContMDiffAt I' I n f x) :
    ContMDiffAt I' I n (fun x ↦ f x / c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f : M → G} {x : M} {n : ℕ∞} (c : G) (hf : ContMDiffAt I' I n f x)
Goal: ContMDiffAt I' I n (fun x => f x - c) x
-- ContMDiffOn.sub_const
@[to_additive]
theorem ContMDiffOn.div_const (hf : ContMDiffOn I' I n f s) :
    ContMDiffOn I' I n (fun x ↦ f x / c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [SubNegMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f : M → G} {s : Set M} {n : ℕ∞} (c : G) (hf : ContMDiffOn I' I n f s)
Goal: ContMDiffOn I' I n (fun x => f x - c) s