ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- contMDiffWithinAt_id
theorem contMDiffWithinAt_id : ContMDiffWithinAt I I n (id : M → M) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} {x : M} {n : ℕ∞}
Goal: ContMDiffWithinAt I I n id s x
-- smoothWithinAt_id
theorem smoothWithinAt_id : SmoothWithinAt I I (id : M → M) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} {x : M}
Goal: SmoothWithinAt I I id s x
-- contMDiff_const
theorem contMDiff_const : ContMDiff I I' n fun _ : M => c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} {c : M'}
Goal: ContMDiff I I' n fun x => c
-- contMDiff_zero
@[to_additive]
theorem contMDiff_one [One M'] : ContMDiff I I' n (1 : M → M') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} [Zero M']
Goal: ContMDiff I I' n 0
-- contMDiff_one
@[to_additive]
theorem contMDiff_one [One M'] : ContMDiff I I' n (1 : M → M') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} [One M']
Goal: ContMDiff I I' n 1
-- smooth_const
theorem smooth_const : Smooth I I' fun _ : M => c 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {c : M'}
Goal: Smooth I I' fun x => c
-- smooth_one
@[to_additive]
theorem smooth_one [One M'] : Smooth I I' (1 : M → M') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] [One M']
Goal: Smooth I I' 1
-- smooth_zero
@[to_additive]
theorem smooth_one [One M'] : Smooth I I' (1 : M → M') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] [Zero M']
Goal: Smooth I I' 0
-- contMDiffOn_const
theorem contMDiffOn_const : ContMDiffOn I I' n (fun _ : M => c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {n : ℕ∞} {c : M'}
Goal: ContMDiffOn I I' n (fun x => c) s
-- contMDiffOn_one
@[to_additive]
theorem contMDiffOn_one [One M'] : ContMDiffOn I I' n (1 : M → M') s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {n : ℕ∞} [One M']
Goal: ContMDiffOn I I' n 1 s
-- contMDiffOn_zero
@[to_additive]
theorem contMDiffOn_one [One M'] : ContMDiffOn I I' n (1 : M → M') s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {n : ℕ∞} [Zero M']
Goal: ContMDiffOn I I' n 0 s
-- smoothOn_const
theorem smoothOn_const : SmoothOn I I' (fun _ : M => c) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {c : M'}
Goal: SmoothOn I I' (fun x => c) s
-- smoothOn_zero
@[to_additive]
theorem smoothOn_one [One M'] : SmoothOn I I' (1 : M → M') s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} [Zero M']
Goal: SmoothOn I I' 0 s
-- smoothOn_one
@[to_additive]
theorem smoothOn_one [One M'] : SmoothOn I I' (1 : M → M') s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} [One M']
Goal: SmoothOn I I' 1 s
-- contMDiffAt_const
theorem contMDiffAt_const : ContMDiffAt I I' n (fun _ : M => c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} {n : ℕ∞} {c : M'}
Goal: ContMDiffAt I I' n (fun x => c) x
-- contMDiffAt_zero
@[to_additive]
theorem contMDiffAt_one [One M'] : ContMDiffAt I I' n (1 : M → M') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} {n : ℕ∞} [Zero M']
Goal: ContMDiffAt I I' n 0 x
-- contMDiffAt_one
@[to_additive]
theorem contMDiffAt_one [One M'] : ContMDiffAt I I' n (1 : M → M') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} {n : ℕ∞} [One M']
Goal: ContMDiffAt I I' n 1 x
-- smoothAt_const
theorem smoothAt_const : SmoothAt I I' (fun _ : M => c) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} {c : M'}
Goal: SmoothAt I I' (fun x => c) x
-- smoothAt_zero
@[to_additive]
theorem smoothAt_one [One M'] : SmoothAt I I' (1 : M → M') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} [Zero M']
Goal: SmoothAt I I' 0 x
-- smoothAt_one
@[to_additive]
theorem smoothAt_one [One M'] : SmoothAt I I' (1 : M → M') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {x : M} [One M']
Goal: SmoothAt I I' 1 x
-- contMDiffWithinAt_const
theorem contMDiffWithinAt_const : ContMDiffWithinAt I I' n (fun _ : M => c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} {n : ℕ∞} {c : M'}
Goal: ContMDiffWithinAt I I' n (fun x => c) s x
-- contMDiffWithinAt_zero
@[to_additive]
theorem contMDiffWithinAt_one [One M'] : ContMDiffWithinAt I I' n (1 : M → M') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} {n : ℕ∞} [Zero M']
Goal: ContMDiffWithinAt I I' n 0 s x
-- contMDiffWithinAt_one
@[to_additive]
theorem contMDiffWithinAt_one [One M'] : ContMDiffWithinAt I I' n (1 : M → M') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} {n : ℕ∞} [One M']
Goal: ContMDiffWithinAt I I' n 1 s x
-- smoothWithinAt_const
theorem smoothWithinAt_const : SmoothWithinAt I I' (fun _ : M => c) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} {c : M'}
Goal: SmoothWithinAt I I' (fun x => c) s x
-- smoothWithinAt_one
@[to_additive]
theorem smoothWithinAt_one [One M'] : SmoothWithinAt I I' (1 : M → M') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} [One M']
Goal: SmoothWithinAt I I' 1 s x
-- smoothWithinAt_zero
@[to_additive]
theorem smoothWithinAt_one [One M'] : SmoothWithinAt I I' (1 : M → M') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {s : Set M} {x : M} [Zero M']
Goal: SmoothWithinAt I I' 0 s x
-- contMDiff_of_tsupport
@[to_additive "`f` is continuously differentiable if it is continuously
differentiable at each `x ∈ tsupport f`."]
theorem contMDiff_of_mulTSupport [One M'] {f : M → M'}
    (hf : ∀ x ∈ mulTSupport f, ContMDiffAt I I' n f x) : ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} [Zero M'] {f : M → M'} (hf : ∀ x ∈ tsupport f, ContMDiffAt I I' n f x)
Goal: ContMDiff I I' n f
-- contMDiff_of_mulTSupport
@[to_additive "`f` is continuously differentiable if it is continuously
differentiable at each `x ∈ tsupport f`."]
theorem contMDiff_of_mulTSupport [One M'] {f : M → M'}
    (hf : ∀ x ∈ mulTSupport f, ContMDiffAt I I' n f x) : ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} [One M'] {f : M → M'} (hf : ∀ x ∈ mulTSupport f, ContMDiffAt I I' n f x)
Goal: ContMDiff I I' n f
-- contMDiffWithinAt_of_not_mem_mulTSupport
@[to_additive contMDiffWithinAt_of_not_mem]
theorem contMDiffWithinAt_of_not_mem_mulTSupport {f : M → M'} [One M'] {x : M}
    (hx : x ∉ mulTSupport f) (n : ℕ∞) (s : Set M) : ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 → M'} [One M'] {x : M} (hx : x ∉ mulTSupport f) (n : ℕ∞) (s : Set M)
Goal: ContMDiffWithinAt I I' n f s x
-- contMDiffWithinAt_of_not_mem
@[to_additive contMDiffWithinAt_of_not_mem]
theorem contMDiffWithinAt_of_not_mem_mulTSupport {f : M → M'} [One M'] {x : M}
    (hx : x ∉ mulTSupport f) (n : ℕ∞) (s : Set M) : ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 → M'} [Zero M'] {x : M} (hx : x ∉ tsupport f) (n : ℕ∞) (s : Set M)
Goal: ContMDiffWithinAt I I' n f s x
-- contMDiffAt_of_not_mem_mulTSupport
@[to_additive contMDiffAt_of_not_mem]
theorem contMDiffAt_of_not_mem_mulTSupport {f : M → M'} [One M'] {x : M}
    (hx : x ∉ mulTSupport f) (n : ℕ∞) : ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 → M'} [One M'] {x : M} (hx : x ∉ mulTSupport f) (n : ℕ∞)
Goal: ContMDiffAt I I' n f x
-- contMDiffAt_of_not_mem
@[to_additive contMDiffAt_of_not_mem]
theorem contMDiffAt_of_not_mem_mulTSupport {f : M → M'} [One M'] {x : M}
    (hx : x ∉ mulTSupport f) (n : ℕ∞) : ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 → M'} [Zero M'] {x : M} (hx : x ∉ tsupport f) (n : ℕ∞)
Goal: ContMDiffAt I I' n f x
-- contMdiffAt_subtype_iff
theorem contMdiffAt_subtype_iff {n : ℕ∞} {U : Opens M} {f : M → M'} {x : U} :
    ContMDiffAt I I' n (fun x : U ↦ f x) x ↔ ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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∞} {U : TopologicalSpace.Opens M} {f : M → M'} {x : ↥U}
Goal: ContMDiffAt I I' n (fun x => f ↑x) x ↔ ContMDiffAt I I' n f ↑x
-- contMDiff_subtype_val
theorem contMDiff_subtype_val {n : ℕ∞} {U : Opens M} : ContMDiff I I n (Subtype.val : U → M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {n : ℕ∞} {U : TopologicalSpace.Opens M}
Goal: ContMDiff I I n Subtype.val
-- ContMDiff.extend_one
@[to_additive]
theorem ContMDiff.extend_one [T2Space M] [One M'] {n : ℕ∞} {U : Opens M} {f : U → M'}
    (supp : HasCompactMulSupport f) (diff : ContMDiff I I' n f) :
    ContMDiff I I' n (Subtype.val.extend f 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] [T2Space M] [One M'] {n : ℕ∞} {U : TopologicalSpace.Opens M} {f : ↥U → M'} (supp : HasCompactMulSupport f) (diff : ContMDiff I I' n f)
Goal: ContMDiff I I' n (Function.extend Subtype.val f 1)
-- ContMDiff.extend_zero
@[to_additive]
theorem ContMDiff.extend_one [T2Space M] [One M'] {n : ℕ∞} {U : Opens M} {f : U → M'}
    (supp : HasCompactMulSupport f) (diff : ContMDiff I I' n f) :
    ContMDiff I I' n (Subtype.val.extend f 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] [T2Space M] [Zero M'] {n : ℕ∞} {U : TopologicalSpace.Opens M} {f : ↥U → M'} (supp : HasCompactSupport f) (diff : ContMDiff I I' n f)
Goal: ContMDiff I I' n (Function.extend Subtype.val f 0)
-- contMDiff_inclusion
theorem contMDiff_inclusion {n : ℕ∞} {U V : Opens M} (h : U ≤ V) :
    ContMDiff I I n (Set.inclusion h : U → V) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {n : ℕ∞} {U V : TopologicalSpace.Opens M} (h : U ≤ V)
Goal: ContMDiff I I n (Set.inclusion h)
-- smooth_subtype_iff
theorem smooth_subtype_iff {U : Opens M} {f : M → M'} {x : U} :
    SmoothAt I I' (fun x : U ↦ f x) x ↔ SmoothAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] {U : TopologicalSpace.Opens M} {f : M → M'} {x : ↥U}
Goal: SmoothAt I I' (fun x => f ↑x) x ↔ SmoothAt I I' f ↑x
-- smooth_subtype_val
theorem smooth_subtype_val {U : Opens M} : Smooth I I (Subtype.val : U → M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {U : TopologicalSpace.Opens M}
Goal: Smooth I I Subtype.val
-- Smooth.extend_zero
@[to_additive]
theorem Smooth.extend_one [T2Space M] [One M'] {U : Opens M} {f : U → M'}
    (supp : HasCompactMulSupport f) (diff : Smooth I I' f) : Smooth I I' (Subtype.val.extend f 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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'] [T2Space M] [Zero M'] {U : TopologicalSpace.Opens M} {f : ↥U → M'} (supp : HasCompactSupport f) (diff : Smooth I I' f)
Goal: Smooth I I' (Function.extend Subtype.val f 0)