ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContMDiffOn.neg
@[to_additive]
theorem ContMDiffOn.inv {f : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s) :
    ContMDiffOn I' I n (fun x => (f x)⁻¹) 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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s)
Goal: ContMDiffOn I' I n (fun x => -f x) s
-- ContMDiff.neg
@[to_additive]
theorem ContMDiff.inv {f : M → G} (hf : ContMDiff I' I n f) : ContMDiff I' I n fun x => (f x)⁻¹ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f : M → G} (hf : ContMDiff I' I n f)
Goal: ContMDiff I' I n fun x => -f x
-- ContMDiff.inv
@[to_additive]
theorem ContMDiff.inv {f : M → G} (hf : ContMDiff I' I n f) : ContMDiff I' I n fun x => (f x)⁻¹ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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] {n : ℕ∞} {f : M → G} (hf : ContMDiff I' I n f)
Goal: ContMDiff I' I n fun x => (f x)⁻¹
-- SmoothWithinAt.neg
@[to_additive]
nonrec theorem SmoothWithinAt.inv {f : M → G} {s : Set M} {x₀ : M}
    (hf : SmoothWithinAt I' I f s x₀) : SmoothWithinAt I' I (fun x => (f x)⁻¹) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 : M → G} {s : Set M} {x₀ : M} (hf : SmoothWithinAt I' I f s x₀)
Goal: SmoothWithinAt I' I (fun x => -f x) s x₀
-- SmoothWithinAt.inv
@[to_additive]
nonrec theorem SmoothWithinAt.inv {f : M → G} {s : Set M} {x₀ : M}
    (hf : SmoothWithinAt I' I f s x₀) : SmoothWithinAt I' I (fun x => (f x)⁻¹) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 : M → G} {s : Set M} {x₀ : M} (hf : SmoothWithinAt I' I f s x₀)
Goal: SmoothWithinAt I' I (fun x => (f x)⁻¹) s x₀
-- SmoothAt.inv
@[to_additive]
nonrec theorem SmoothAt.inv {f : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀) :
    SmoothAt I' I (fun x => (f x)⁻¹) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀)
Goal: SmoothAt I' I (fun x => (f x)⁻¹) x₀
-- SmoothAt.neg
@[to_additive]
nonrec theorem SmoothAt.inv {f : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀) :
    SmoothAt I' I (fun x => (f x)⁻¹) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀)
Goal: SmoothAt I' I (fun x => -f x) x₀
-- SmoothOn.neg
@[to_additive]
nonrec theorem SmoothOn.inv {f : M → G} {s : Set M} (hf : SmoothOn I' I f s) :
    SmoothOn I' I (fun x => (f x)⁻¹) 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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 : M → G} {s : Set M} (hf : SmoothOn I' I f s)
Goal: SmoothOn I' I (fun x => -f x) s
-- SmoothOn.inv
@[to_additive]
nonrec theorem SmoothOn.inv {f : M → G} {s : Set M} (hf : SmoothOn I' I f s) :
    SmoothOn I' I (fun x => (f x)⁻¹) 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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 : M → G} {s : Set M} (hf : SmoothOn I' I f s)
Goal: SmoothOn I' I (fun x => (f x)⁻¹) s
-- Smooth.neg
@[to_additive]
nonrec theorem Smooth.inv {f : M → G} (hf : Smooth I' I f) : Smooth I' I fun x => (f x)⁻¹ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 : M → G} (hf : Smooth I' I f)
Goal: Smooth I' I fun x => -f x
-- Smooth.inv
@[to_additive]
nonrec theorem Smooth.inv {f : M → G} (hf : Smooth I' I f) : Smooth I' I fun x => (f x)⁻¹ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 : M → G} (hf : Smooth I' I f)
Goal: Smooth I' I fun x => (f x)⁻¹
-- ContMDiffWithinAt.div
@[to_additive]
theorem ContMDiffWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
    (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
    ContMDiffWithinAt I' I n (fun x => f x / g x) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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] {n : ℕ∞} {f g : M → G} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => f x / g x) s x₀
-- ContMDiffWithinAt.sub
@[to_additive]
theorem ContMDiffWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
    (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
    ContMDiffWithinAt I' I n (fun x => f x / g x) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f g : M → G} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => f x - g x) s x₀
-- ContMDiffAt.div
@[to_additive]
theorem ContMDiffAt.div {f g : M → G} {x₀ : M} (hf : ContMDiffAt I' I n f x₀)
    (hg : ContMDiffAt I' I n g x₀) : ContMDiffAt I' I n (fun x => f x / g x) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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] {n : ℕ∞} {f g : M → G} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀)
Goal: ContMDiffAt I' I n (fun x => f x / g x) x₀
-- ContMDiffAt.sub
@[to_additive]
theorem ContMDiffAt.div {f g : M → G} {x₀ : M} (hf : ContMDiffAt I' I n f x₀)
    (hg : ContMDiffAt I' I n g x₀) : ContMDiffAt I' I n (fun x => f x / g x) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f g : M → G} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀)
Goal: ContMDiffAt I' I n (fun x => f x - g x) x₀
-- ContMDiffOn.div
@[to_additive]
theorem ContMDiffOn.div {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s)
    (hg : ContMDiffOn I' I n g s) : ContMDiffOn I' I n (fun x => f x / g x) 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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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] {n : ℕ∞} {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s)
Goal: ContMDiffOn I' I n (fun x => f x / g x) s
-- ContMDiffOn.sub
@[to_additive]
theorem ContMDiffOn.div {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s)
    (hg : ContMDiffOn I' I n g s) : ContMDiffOn I' I n (fun x => f x / g x) 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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s)
Goal: ContMDiffOn I' I n (fun x => f x - g x) s
-- ContMDiff.div
@[to_additive]
theorem ContMDiff.div {f g : M → G} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
    ContMDiff I' I n fun x => f x / g 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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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] {n : ℕ∞} {f g : M → G} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g)
Goal: ContMDiff I' I n fun x => f x / g x
-- ContMDiff.sub
@[to_additive]
theorem ContMDiff.div {f g : M → G} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
    ContMDiff I' I n fun x => f x / g 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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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] {n : ℕ∞} {f g : M → G} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g)
Goal: ContMDiff I' I n fun x => f x - g x
-- SmoothWithinAt.sub
@[to_additive]
nonrec theorem SmoothWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
    (hf : SmoothWithinAt I' I f s x₀) (hg : SmoothWithinAt I' I g s x₀) :
    SmoothWithinAt I' I (fun x => f x / g x) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 g : M → G} {s : Set M} {x₀ : M} (hf : SmoothWithinAt I' I f s x₀) (hg : SmoothWithinAt I' I g s x₀)
Goal: SmoothWithinAt I' I (fun x => f x - g x) s x₀
-- SmoothWithinAt.div
@[to_additive]
nonrec theorem SmoothWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
    (hf : SmoothWithinAt I' I f s x₀) (hg : SmoothWithinAt I' I g s x₀) :
    SmoothWithinAt I' I (fun x => f x / g x) s x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 g : M → G} {s : Set M} {x₀ : M} (hf : SmoothWithinAt I' I f s x₀) (hg : SmoothWithinAt I' I g s x₀)
Goal: SmoothWithinAt I' I (fun x => f x / g x) s x₀
-- SmoothAt.div
@[to_additive]
nonrec theorem SmoothAt.div {f g : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀)
    (hg : SmoothAt I' I g x₀) : SmoothAt I' I (fun x => f x / g x) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 g : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀) (hg : SmoothAt I' I g x₀)
Goal: SmoothAt I' I (fun x => f x / g x) x₀
-- SmoothAt.sub
@[to_additive]
nonrec theorem SmoothAt.div {f g : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀)
    (hg : SmoothAt I' I g x₀) : SmoothAt I' I (fun x => f x / g x) x₀ 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 g : M → G} {x₀ : M} (hf : SmoothAt I' I f x₀) (hg : SmoothAt I' I g x₀)
Goal: SmoothAt I' I (fun x => f x - g x) x₀
-- SmoothOn.sub
@[to_additive]
nonrec theorem SmoothOn.div {f g : M → G} {s : Set M} (hf : SmoothOn I' I f s)
    (hg : SmoothOn I' I g s) : SmoothOn I' I (f / g) 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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 g : M → G} {s : Set M} (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s)
Goal: SmoothOn I' I (f - g) s
-- SmoothOn.div
@[to_additive]
nonrec theorem SmoothOn.div {f g : M → G} {s : Set M} (hf : SmoothOn I' I f s)
    (hg : SmoothOn I' I g s) : SmoothOn I' I (f / g) 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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 g : M → G} {s : Set M} (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s)
Goal: SmoothOn I' I (f / g) s
-- Smooth.div
@[to_additive]
nonrec theorem Smooth.div {f g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g) :
    Smooth I' I (f / g) 
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_5} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup 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 g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g)
Goal: Smooth I' I (f / g)
-- Smooth.sub
@[to_additive]
nonrec theorem Smooth.div {f g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g) :
    Smooth I' I (f / g) 
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_5} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup 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 g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g)
Goal: Smooth I' I (f - g)
-- SmoothInv₀.smoothAt_inv₀
class SmoothInv₀ {𝕜 : Type*} [NontriviallyNormedField 𝕜] {H : Type*} [TopologicalSpace H]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (I : ModelWithCorners 𝕜 E H) (G : Type*)
    [Inv G] [Zero G] [TopologicalSpace G] [ChartedSpace H G] : Prop where
    smoothAt_inv₀ : ∀ ⦃x : G⦄, x ≠ 0 → SmoothAt I I (fun y ↦ y⁻¹) 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} [Inv G] [Zero G] [TopologicalSpace G] [ChartedSpace H G] [self : SmoothInv₀ I G] ⦃x : G⦄
Goal: x ≠ 0 → SmoothAt I I (fun y => y⁻¹) x
-- smoothAt_inv₀
theorem smoothAt_inv₀ {x : G} (hx : x ≠ 0) : SmoothAt I I (fun y ↦ y⁻¹) 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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G] {x : G} (hx : x ≠ 0)
Goal: SmoothAt I I (fun y => y⁻¹) x
-- hasContinuousInv₀_of_hasSmoothInv₀
theorem hasContinuousInv₀_of_hasSmoothInv₀ : HasContinuousInv₀ G 
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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G]
Goal: HasContinuousInv₀ G
-- SmoothOn_inv₀
theorem SmoothOn_inv₀ : SmoothOn I I (Inv.inv : G → G) {0}ᶜ 
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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G]
Goal: SmoothOn I I Inv.inv {0}ᶜ
-- ContMDiffWithinAt.inv₀
theorem ContMDiffWithinAt.inv₀ (hf : ContMDiffWithinAt I' I n f s a) (ha : f a ≠ 0) :
    ContMDiffWithinAt I' I n (fun x => (f x)⁻¹) s a 
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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ 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] {n : ℕ∞} {f : M → G} {s : Set M} {a : M} (hf : ContMDiffWithinAt I' I n f s a) (ha : f a ≠ 0)
Goal: ContMDiffWithinAt I' I n (fun x => (f x)⁻¹) s a
-- ContMDiffAt.inv₀
theorem ContMDiffAt.inv₀ (hf : ContMDiffAt I' I n f a) (ha : f a ≠ 0) :
    ContMDiffAt I' I n (fun x ↦ (f x)⁻¹) a 
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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ 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] {n : ℕ∞} {f : M → G} {a : M} (hf : ContMDiffAt I' I n f a) (ha : f a ≠ 0)
Goal: ContMDiffAt I' I n (fun x => (f x)⁻¹) a
-- ContMDiff.inv₀
theorem ContMDiff.inv₀ (hf : ContMDiff I' I n f) (h0 : ∀ x, f x ≠ 0) :
    ContMDiff I' I n (fun x ↦ (f x)⁻¹) 
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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ 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] {n : ℕ∞} {f : M → G} (hf : ContMDiff I' I n f) (h0 : ∀ (x : M), f x ≠ 0)
Goal: ContMDiff I' I n fun x => (f x)⁻¹
-- ContMDiffOn.inv₀
theorem ContMDiffOn.inv₀ (hf : ContMDiffOn I' I n f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
    ContMDiffOn I' I n (fun x => (f x)⁻¹) 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} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ 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] {n : ℕ∞} {f : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s) (h0 : ∀ x ∈ s, f x ≠ 0)
Goal: ContMDiffOn I' I n (fun x => (f x)⁻¹) s