ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MDifferentiableOn.continuousOn
theorem MDifferentiableOn.continuousOn (h : MDifferentiableOn I I' f s) : ContinuousOn f 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'] {f : M → M'} {s : Set M} (h : MDifferentiableOn I I' f s)
Goal: ContinuousOn f s
-- MDifferentiable.continuous
theorem MDifferentiable.continuous (h : MDifferentiable I I' f) : Continuous 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'] {f : M → M'} (h : MDifferentiable I I' f)
Goal: Continuous f
-- tangentMapWithin_subset
theorem tangentMapWithin_subset {p : TangentBundle I M} (st : s ⊆ t)
    (hs : UniqueMDiffWithinAt I s p.1) (h : MDifferentiableWithinAt I I' f t p.1) :
    tangentMapWithin I I' f s p = tangentMapWithin I I' f t p 
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'} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {p : TangentBundle I M} (st : s ⊆ t) (hs : UniqueMDiffWithinAt I s p.proj) (h : MDifferentiableWithinAt I I' f t p.proj)
Goal: tangentMapWithin I I' f s p = tangentMapWithin I I' f t p
-- tangentMapWithin_univ
theorem tangentMapWithin_univ : tangentMapWithin I I' f univ = tangentMap I I' 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'] {f : M → M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M']
Goal: tangentMapWithin I I' f Set.univ = tangentMap I I' f
-- tangentMapWithin_eq_tangentMap
theorem tangentMapWithin_eq_tangentMap {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.1)
    (h : MDifferentiableAt I I' f p.1) : tangentMapWithin I I' f s p = tangentMap I I' f p 
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'} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.proj) (h : MDifferentiableAt I I' f p.proj)
Goal: tangentMapWithin I I' f s p = tangentMap I I' f p
-- tangentMapWithin_proj
@[simp, mfld_simps]
theorem tangentMapWithin_proj {p : TangentBundle I M} :
    (tangentMapWithin I I' f s p).proj = f p.proj 
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'} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {p : TangentBundle I M}
Goal: (tangentMapWithin I I' f s p).proj = f p.proj
-- tangentMap_proj
@[simp, mfld_simps]
theorem tangentMap_proj {p : TangentBundle I M} : (tangentMap I I' f p).proj = f p.proj 
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'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {p : TangentBundle I M}
Goal: (tangentMap I I' f p).proj = f p.proj
-- MDifferentiableWithinAt.prod_mk
theorem MDifferentiableWithinAt.prod_mk {f : M → M'} {g : M → M''}
    (hf : MDifferentiableWithinAt I I' f s x) (hg : MDifferentiableWithinAt I I'' g s x) :
    MDifferentiableWithinAt I (I'.prod I'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {x : M} {s : Set M} {f : M → M'} {g : M → M''} (hf : MDifferentiableWithinAt I I' f s x) (hg : MDifferentiableWithinAt I I'' g s x)
Goal: MDifferentiableWithinAt I (I'.prod I'') (fun x => (f x, g x)) s x
-- MDifferentiableAt.prod_mk
theorem MDifferentiableAt.prod_mk {f : M → M'} {g : M → M''} (hf : MDifferentiableAt I I' f x)
    (hg : MDifferentiableAt I I'' g x) :
    MDifferentiableAt I (I'.prod I'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {x : M} {f : M → M'} {g : M → M''} (hf : MDifferentiableAt I I' f x) (hg : MDifferentiableAt I I'' g x)
Goal: MDifferentiableAt I (I'.prod I'') (fun x => (f x, g x)) x
-- MDifferentiableOn.prod_mk
theorem MDifferentiableOn.prod_mk {f : M → M'} {g : M → M''} (hf : MDifferentiableOn I I' f s)
    (hg : MDifferentiableOn I I'' g s) :
    MDifferentiableOn I (I'.prod I'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {s : Set M} {f : M → M'} {g : M → M''} (hf : MDifferentiableOn I I' f s) (hg : MDifferentiableOn I I'' g s)
Goal: MDifferentiableOn I (I'.prod I'') (fun x => (f x, g x)) s
-- MDifferentiable.prod_mk
theorem MDifferentiable.prod_mk {f : M → M'} {g : M → M''} (hf : MDifferentiable I I' f)
    (hg : MDifferentiable I I'' g) : MDifferentiable I (I'.prod I'') fun x => (f x, g 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M → M''} (hf : MDifferentiable I I' f) (hg : MDifferentiable I I'' g)
Goal: MDifferentiable I (I'.prod I'') fun x => (f x, g x)
-- MDifferentiableWithinAt.prod_mk_space
theorem MDifferentiableWithinAt.prod_mk_space {f : M → E'} {g : M → E''}
    (hf : MDifferentiableWithinAt I 𝓘(𝕜, E') f s x)
    (hg : MDifferentiableWithinAt I 𝓘(𝕜, E'') g s x) :
    MDifferentiableWithinAt I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {x : M} {s : Set M} {f : M → E'} {g : M → E''} (hf : MDifferentiableWithinAt I 𝓘(𝕜, E') f s x) (hg : MDifferentiableWithinAt I 𝓘(𝕜, E'') g s x)
Goal: MDifferentiableWithinAt I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) s x
-- MDifferentiableAt.prod_mk_space
theorem MDifferentiableAt.prod_mk_space {f : M → E'} {g : M → E''}
    (hf : MDifferentiableAt I 𝓘(𝕜, E') f x) (hg : MDifferentiableAt I 𝓘(𝕜, E'') g x) :
    MDifferentiableAt I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {x : M} {f : M → E'} {g : M → E''} (hf : MDifferentiableAt I 𝓘(𝕜, E') f x) (hg : MDifferentiableAt I 𝓘(𝕜, E'') g x)
Goal: MDifferentiableAt I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) x
-- MDifferentiableOn.prod_mk_space
theorem MDifferentiableOn.prod_mk_space {f : M → E'} {g : M → E''}
    (hf : MDifferentiableOn I 𝓘(𝕜, E') f s) (hg : MDifferentiableOn I 𝓘(𝕜, E'') g s) :
    MDifferentiableOn I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {s : Set M} {f : M → E'} {g : M → E''} (hf : MDifferentiableOn I 𝓘(𝕜, E') f s) (hg : MDifferentiableOn I 𝓘(𝕜, E'') g s)
Goal: MDifferentiableOn I 𝓘(𝕜, E' × E'') (fun x => (f x, g x)) s
-- MDifferentiable.prod_mk_space
theorem MDifferentiable.prod_mk_space {f : M → E'} {g : M → E''} (hf : MDifferentiable I 𝓘(𝕜, E') f)
    (hg : MDifferentiable I 𝓘(𝕜, E'') g) : MDifferentiable I 𝓘(𝕜, E' × E'') fun x => (f x, g 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {f : M → E'} {g : M → E''} (hf : MDifferentiable I 𝓘(𝕜, E') f) (hg : MDifferentiable I 𝓘(𝕜, E'') g)
Goal: MDifferentiable I 𝓘(𝕜, E' × E'') fun x => (f x, g x)
-- HasMFDerivAt.congr_mfderiv
theorem HasMFDerivAt.congr_mfderiv (h : HasMFDerivAt I I' f x f') (h' : f' = f₁') :
    HasMFDerivAt I I' f x f₁' 
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'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivAt I I' f x f') (h' : f' = f₁')
Goal: HasMFDerivAt I I' f x f₁'
-- HasMFDerivWithinAt.congr_mfderiv
theorem HasMFDerivWithinAt.congr_mfderiv (h : HasMFDerivWithinAt I I' f s x f') (h' : f' = f₁') :
    HasMFDerivWithinAt I I' f s x f₁' 
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'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (h' : f' = f₁')
Goal: HasMFDerivWithinAt I I' f s x f₁'
-- HasMFDerivWithinAt.congr_of_eventuallyEq
theorem HasMFDerivWithinAt.congr_of_eventuallyEq (h : HasMFDerivWithinAt I I' f s x f')
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasMFDerivWithinAt I I' f₁ s x 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'] {f f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasMFDerivWithinAt I I' f₁ s x f'
-- HasMFDerivWithinAt.congr_mono
theorem HasMFDerivWithinAt.congr_mono (h : HasMFDerivWithinAt I I' f s x f')
    (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasMFDerivWithinAt I I' f₁ t x 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'] {f f₁ : M → M'} {x : M} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: HasMFDerivWithinAt I I' f₁ t x f'
-- HasMFDerivAt.congr_of_eventuallyEq
theorem HasMFDerivAt.congr_of_eventuallyEq (h : HasMFDerivAt I I' f x f') (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasMFDerivAt I I' f₁ x 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'] {f f₁ : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivAt I I' f x f') (h₁ : f₁ =ᶠ[𝓝 x] f)
Goal: HasMFDerivAt I I' f₁ x f'
-- MDifferentiableWithinAt.congr_of_eventuallyEq
theorem MDifferentiableWithinAt.congr_of_eventuallyEq (h : MDifferentiableWithinAt I I' f s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : MDifferentiableWithinAt I I' f₁ 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 f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: MDifferentiableWithinAt I I' f₁ s x
-- Filter.EventuallyEq.mdifferentiableWithinAt_iff
theorem Filter.EventuallyEq.mdifferentiableWithinAt_iff (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    MDifferentiableWithinAt I I' f s x ↔ MDifferentiableWithinAt I I' f₁ 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 f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: MDifferentiableWithinAt I I' f s x ↔ MDifferentiableWithinAt I I' f₁ s x
-- MDifferentiableWithinAt.congr_mono
theorem MDifferentiableWithinAt.congr_mono (h : MDifferentiableWithinAt I I' f s x)
    (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s) :
    MDifferentiableWithinAt I I' f₁ t 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 f₁ : M → M'} {x : M} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x) (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: MDifferentiableWithinAt I I' f₁ t x
-- MDifferentiableWithinAt.congr
theorem MDifferentiableWithinAt.congr (h : MDifferentiableWithinAt I I' f s x)
    (ht : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x) : MDifferentiableWithinAt I I' f₁ 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 f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x) (ht : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: MDifferentiableWithinAt I I' f₁ s x
-- MDifferentiableOn.congr_mono
theorem MDifferentiableOn.congr_mono (h : MDifferentiableOn I I' f s) (h' : ∀ x ∈ t, f₁ x = f x)
    (h₁ : t ⊆ s) : MDifferentiableOn I I' f₁ t 
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 f₁ : M → M'} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableOn I I' f s) (h' : ∀ x ∈ t, f₁ x = f x) (h₁ : t ⊆ s)
Goal: MDifferentiableOn I I' f₁ t
-- MDifferentiableAt.congr_of_eventuallyEq
theorem MDifferentiableAt.congr_of_eventuallyEq (h : MDifferentiableAt I I' f x)
    (hL : f₁ =ᶠ[𝓝 x] f) : MDifferentiableAt I I' f₁ 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 f₁ : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableAt I I' f x) (hL : f₁ =ᶠ[𝓝 x] f)
Goal: MDifferentiableAt I I' f₁ x
-- MDifferentiableWithinAt.mfderivWithin_congr_mono
theorem MDifferentiableWithinAt.mfderivWithin_congr_mono (h : MDifferentiableWithinAt I I' f s x)
    (hs : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (hxt : UniqueMDiffWithinAt I t x) (h₁ : t ⊆ s) :
    mfderivWithin I I' f₁ t x = (mfderivWithin I I' 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 f₁ : M → M'} {x : M} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x) (hs : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (hxt : UniqueMDiffWithinAt I t x) (h₁ : t ⊆ s)
Goal: mfderivWithin I I' f₁ t x = mfderivWithin I I' f s x
-- Filter.EventuallyEq.mfderivWithin_eq
theorem Filter.EventuallyEq.mfderivWithin_eq (hs : UniqueMDiffWithinAt I s x) (hL : f₁ =ᶠ[𝓝[s] x] f)
    (hx : f₁ x = f x) : mfderivWithin I I' f₁ s x = (mfderivWithin I I' 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 f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (hs : UniqueMDiffWithinAt I s x) (hL : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: mfderivWithin I I' f₁ s x = mfderivWithin I I' f s x