ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothMap.coe_pow
@[to_additive (attr := simp)]
theorem coe_pow {G : Type*} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f : C^∞⟮I, N; I', G⟯) (n : ℕ) :
    ⇑(f ^ n) = (f : N → G) ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G] (f : C^⊤⟮I, N; I', G⟯) (n : ℕ)
Goal: ⇑(f ^ n) = ⇑f ^ n
-- SmoothMap.coe_nsmul
@[to_additive (attr := simp)]
theorem coe_pow {G : Type*} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f : C^∞⟮I, N; I', G⟯) (n : ℕ) :
    ⇑(f ^ n) = (f : N → G) ^ n 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [AddMonoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothAdd I' G] (f : C^⊤⟮I, N; I', G⟯) (n : ℕ)
Goal: ⇑(n • f) = n • ⇑f
-- SmoothMap.coeFnAddMonoidHom_apply
@[to_additive (attr := simps) "Coercion to a function as an `AddMonoidHom`.
  Similar to `AddMonoidHom.coeFn`."]
def coeFnMonoidHom {G : Type*} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G]
    [SmoothMul I' G] : C^∞⟮I, N; I', G⟯ →* N → G 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [AddMonoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothAdd I' G]
Goal: ∀ (a : C^⊤⟮I, N; I', G⟯) (a_1 : N), SmoothMap.coeFnAddMonoidHom a a_1 = a a_1
-- SmoothMap.coeFnMonoidHom_apply
@[to_additive (attr := simps) "Coercion to a function as an `AddMonoidHom`.
  Similar to `AddMonoidHom.coeFn`."]
def coeFnMonoidHom {G : Type*} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G]
    [SmoothMul I' G] : C^∞⟮I, N; I', G⟯ →* N → G 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Monoid G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
Goal: ∀ (a : C^⊤⟮I, N; I', G⟯) (a_1 : N), SmoothMap.coeFnMonoidHom a a_1 = a a_1
-- SmoothMap.coe_neg
@[to_additive (attr := simp)]
theorem coe_inv {G : Type*} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G]
    (f : C^∞⟮I, N; I', G⟯) : ⇑f⁻¹ = (⇑f)⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [AddGroup G] [TopologicalSpace G] [ChartedSpace H' G] [LieAddGroup I' G] (f : C^⊤⟮I, N; I', G⟯)
Goal: ⇑(-f) = -⇑f
-- SmoothMap.coe_inv
@[to_additive (attr := simp)]
theorem coe_inv {G : Type*} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G]
    (f : C^∞⟮I, N; I', G⟯) : ⇑f⁻¹ = (⇑f)⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G] (f : C^⊤⟮I, N; I', G⟯)
Goal: ⇑f⁻¹ = (⇑f)⁻¹
-- SmoothMap.coe_sub
@[to_additive (attr := simp)]
theorem coe_div {G : Type*} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G]
    (f g : C^∞⟮I, N; I', G⟯) : ⇑(f / g) = f / g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [AddGroup G] [TopologicalSpace G] [ChartedSpace H' G] [LieAddGroup I' G] (f g : C^⊤⟮I, N; I', G⟯)
Goal: ⇑(f - g) = ⇑f - ⇑g
-- SmoothMap.coe_div
@[to_additive (attr := simp)]
theorem coe_div {G : Type*} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G]
    (f g : C^∞⟮I, N; I', G⟯) : ⇑(f / g) = f / g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Group G] [TopologicalSpace G] [ChartedSpace H' G] [LieGroup I' G] (f g : C^⊤⟮I, N; I', G⟯)
Goal: ⇑(f / g) = ⇑f / ⇑g
-- SmoothMap.coeFnRingHom_apply
@[simps]
def coeFnRingHom {R : Type*} [CommRing R] [TopologicalSpace R] [ChartedSpace H' R]
    [SmoothRing I' R] : C^∞⟮I, N; I', R⟯ →+* N → R 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {R : Type u_10} [CommRing R] [TopologicalSpace R] [ChartedSpace H' R] [SmoothRing I' R]
Goal: ∀ (a : C^⊤⟮I, N; I', R⟯) (a_1 : N), SmoothMap.coeFnRingHom a a_1 = a a_1
-- SmoothMap.coe_smul
@[simp]
theorem coe_smul {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (r : 𝕜)
    (f : C^∞⟮I, N; 𝓘(𝕜, V), V⟯) : ⇑(r • f) = r • ⇑f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {V : Type u_10} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (r : 𝕜) (f : C^⊤⟮I, N; 𝓘(𝕜, V), V⟯)
Goal: ⇑(r • f) = r • ⇑f
-- SmoothMap.smul_comp
@[simp]
theorem smul_comp {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (r : 𝕜)
    (g : C^∞⟮I'', N'; 𝓘(𝕜, V), V⟯) (h : C^∞⟮I, N; I'', N'⟯) : (r • g).comp h = r • g.comp h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {E'' : Type u_7} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_8} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {N' : Type u_9} [TopologicalSpace N'] [ChartedSpace H'' N'] {V : Type u_10} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (r : 𝕜) (g : C^⊤⟮I'', N'; 𝓘(𝕜, V), V⟯) (h : C^⊤⟮I, N; I'', N'⟯)
Goal: (r • g).comp h = r • g.comp h
-- SmoothMap.coeFnLinearMap_apply
@[simps]
def coeFnLinearMap {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V] :
    C^∞⟮I, N; 𝓘(𝕜, V), V⟯ →ₗ[𝕜] N → V 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {V : Type u_10} [NormedAddCommGroup V] [NormedSpace 𝕜 V]
Goal: ∀ (a : C^⊤⟮I, N; 𝓘(𝕜, V), V⟯) (a_1 : N), SmoothMap.coeFnLinearMap a a_1 = a a_1
-- SmoothMap.coeFnAlgHom_apply
@[simps]
def coeFnAlgHom : C^∞⟮I, N; 𝓘(𝕜, A), A⟯ →ₐ[𝕜] N → A 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {A : Type u_10} [NormedRing A] [NormedAlgebra 𝕜 A] [SmoothRing 𝓘(𝕜, A) A]
Goal: ∀ (a : C^⊤⟮I, N; 𝓘(𝕜, A), A⟯) (a_1 : N), SmoothMap.coeFnAlgHom a a_1 = a a_1
-- SmoothMap.smul_comp'
@[simp]
theorem smul_comp' {V : Type*} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (f : C^∞⟮I'', N'; 𝕜⟯)
    (g : C^∞⟮I'', N'; 𝓘(𝕜, V), V⟯) (h : C^∞⟮I, N; I'', N'⟯) :
    (f • g).comp h = f.comp h • g.comp h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {E'' : Type u_7} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_8} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {N' : Type u_9} [TopologicalSpace N'] [ChartedSpace H'' N'] {V : Type u_10} [NormedAddCommGroup V] [NormedSpace 𝕜 V] (f : C^⊤⟮I'', N'; 𝓘(𝕜, 𝕜), 𝕜⟯) (g : C^⊤⟮I'', N'; 𝓘(𝕜, V), V⟯) (h : C^⊤⟮I, N; I'', N'⟯)
Goal: (f • g).comp h = f.comp h • g.comp h
-- ContinuousLinearMap.hasMFDerivWithinAt
protected theorem hasMFDerivWithinAt : HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {s : Set E} {x : E}
Goal: HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x f
-- ContinuousLinearMap.hasMFDerivAt
protected theorem hasMFDerivAt : HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {x : E}
Goal: HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x f
-- ContinuousLinearMap.mdifferentiableWithinAt
protected theorem mdifferentiableWithinAt : MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {s : Set E} {x : E}
Goal: MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x
-- ContinuousLinearMap.mdifferentiableOn
protected theorem mdifferentiableOn : MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {s : Set E}
Goal: MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s
-- ContinuousLinearMap.mdifferentiableAt
protected theorem mdifferentiableAt : MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {x : E}
Goal: MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x
-- ContinuousLinearMap.mdifferentiable
protected theorem mdifferentiable : MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E')
Goal: MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') ⇑f
-- ContinuousLinearMap.mfderiv_eq
theorem mfderiv_eq : mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') f x = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {x : E}
Goal: mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x = f
-- ContinuousLinearMap.mfderivWithin_eq
theorem mfderivWithin_eq (hs : UniqueMDiffWithinAt 𝓘(𝕜, E) s x) :
    mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') f s x = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {s : Set E} {x : E} (hs : UniqueMDiffWithinAt 𝓘(𝕜, E) s x)
Goal: mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x = f
-- ContinuousLinearEquiv.hasMFDerivWithinAt
protected theorem hasMFDerivWithinAt : HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x (f : E →L[𝕜] E') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {s : Set E} {x : E}
Goal: HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x ↑f
-- ContinuousLinearEquiv.hasMFDerivAt
protected theorem hasMFDerivAt : HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x (f : E →L[𝕜] E') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {x : E}
Goal: HasMFDerivAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x ↑f
-- ContinuousLinearEquiv.mdifferentiableWithinAt
protected theorem mdifferentiableWithinAt : MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {s : Set E} {x : E}
Goal: MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x
-- ContinuousLinearEquiv.mdifferentiableOn
protected theorem mdifferentiableOn : MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {s : Set E}
Goal: MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s
-- ContinuousLinearEquiv.mdifferentiableAt
protected theorem mdifferentiableAt : MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {x : E}
Goal: MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x
-- ContinuousLinearEquiv.mdifferentiable
protected theorem mdifferentiable : MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E')
Goal: MDifferentiable 𝓘(𝕜, E) 𝓘(𝕜, E') ⇑f
-- ContinuousLinearEquiv.mfderiv_eq
theorem mfderiv_eq : mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') f x = (f : E →L[𝕜] E') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {x : E}
Goal: mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) x = ↑f
-- ContinuousLinearEquiv.mfderivWithin_eq
theorem mfderivWithin_eq (hs : UniqueMDiffWithinAt 𝓘(𝕜, E) s x) :
    mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') f s x = (f : E →L[𝕜] E') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {s : Set E} {x : E} (hs : UniqueMDiffWithinAt 𝓘(𝕜, E) s x)
Goal: mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') (⇑f) s x = ↑f
-- hasMFDerivAt_id
theorem hasMFDerivAt_id (x : M) :
    HasMFDerivAt I I (@id M) x (ContinuousLinearMap.id 𝕜 (TangentSpace I 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] [SmoothManifoldWithCorners I M] (x : M)
Goal: HasMFDerivAt I I id x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))
-- hasMFDerivWithinAt_id
theorem hasMFDerivWithinAt_id (s : Set M) (x : M) :
    HasMFDerivWithinAt I I (@id M) s x (ContinuousLinearMap.id 𝕜 (TangentSpace I 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] [SmoothManifoldWithCorners I M] (s : Set M) (x : M)
Goal: HasMFDerivWithinAt I I id s x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))
-- mdifferentiableAt_id
theorem mdifferentiableAt_id : MDifferentiableAt I I (@id 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] [SmoothManifoldWithCorners I M] {x : M}
Goal: MDifferentiableAt I I id x
-- mdifferentiableWithinAt_id
theorem mdifferentiableWithinAt_id : MDifferentiableWithinAt I I (@id 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] [SmoothManifoldWithCorners I M] {s : Set M} {x : M}
Goal: MDifferentiableWithinAt I I id s x
-- mdifferentiable_id
theorem mdifferentiable_id : MDifferentiable I I (@id 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] [SmoothManifoldWithCorners I M]
Goal: MDifferentiable I I id
-- mdifferentiableOn_id
theorem mdifferentiableOn_id : MDifferentiableOn I I (@id 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] [SmoothManifoldWithCorners I M] {s : Set M}
Goal: MDifferentiableOn I I id s
-- mfderiv_id
@[simp, mfld_simps]
theorem mfderiv_id : mfderiv I I (@id M) x = ContinuousLinearMap.id 𝕜 (TangentSpace I 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] [SmoothManifoldWithCorners I M] {x : M}
Goal: mfderiv I I id x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)
-- mfderivWithin_id
theorem mfderivWithin_id (hxs : UniqueMDiffWithinAt I s x) :
    mfderivWithin I I (@id M) s x = ContinuousLinearMap.id 𝕜 (TangentSpace I 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] [SmoothManifoldWithCorners I M] {s : Set M} {x : M} (hxs : UniqueMDiffWithinAt I s x)
Goal: mfderivWithin I I id s x = ContinuousLinearMap.id 𝕜 (TangentSpace I x)
-- tangentMap_id
@[simp, mfld_simps]
theorem tangentMap_id : tangentMap I I (id : M → M) = id 
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] [SmoothManifoldWithCorners I M]
Goal: tangentMap I I id = id
-- tangentMapWithin_id
theorem tangentMapWithin_id {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.proj) :
    tangentMapWithin I I (id : M → M) s p = 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] [SmoothManifoldWithCorners I M] {s : Set M} {p : TangentBundle I M} (hs : UniqueMDiffWithinAt I s p.proj)
Goal: tangentMapWithin I I id s p = p
-- hasMFDerivAt_const
theorem hasMFDerivAt_const (c : M') (x : M) :
    HasMFDerivAt I I' (fun _ : M => c) x (0 : TangentSpace I x →L[𝕜] TangentSpace I' 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] (c : M') (x : M)
Goal: HasMFDerivAt I I' (fun x => c) x 0
-- hasMFDerivWithinAt_const
theorem hasMFDerivWithinAt_const (c : M') (s : Set M) (x : M) :
    HasMFDerivWithinAt I I' (fun _ : M => c) s x (0 : TangentSpace I x →L[𝕜] TangentSpace I' 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] (c : M') (s : Set M) (x : M)
Goal: HasMFDerivWithinAt I I' (fun x => c) s x 0
-- mdifferentiableAt_const
theorem mdifferentiableAt_const : MDifferentiableAt 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {x : M} {c : M'}
Goal: MDifferentiableAt I I' (fun x => c) x
-- mdifferentiableWithinAt_const
theorem mdifferentiableWithinAt_const : MDifferentiableWithinAt 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {s : Set M} {x : M} {c : M'}
Goal: MDifferentiableWithinAt I I' (fun x => c) s x
-- mdifferentiable_const
theorem mdifferentiable_const : MDifferentiable 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {c : M'}
Goal: MDifferentiable I I' fun x => c
-- mdifferentiableOn_const
theorem mdifferentiableOn_const : MDifferentiableOn 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {s : Set M} {c : M'}
Goal: MDifferentiableOn I I' (fun x => c) s