ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasMFDerivAt.const_smul
theorem HasMFDerivAt.const_smul (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f') (s : 𝕜) :
    HasMFDerivAt I 𝓘(𝕜, E') (s • f) z (s • 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] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f : M → E'} {f' : TangentSpace I z →L[𝕜] E'} (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f') (s : 𝕜)
Goal: HasMFDerivAt I 𝓘(𝕜, E') (s • f) z (s • f')
-- MDifferentiableAt.const_smul
theorem MDifferentiableAt.const_smul (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (s : 𝕜) :
    MDifferentiableAt I 𝓘(𝕜, E') (s • f) z 
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'] {z : M} {f : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (s : 𝕜)
Goal: MDifferentiableAt I 𝓘(𝕜, E') (s • f) z
-- MDifferentiable.const_smul
theorem MDifferentiable.const_smul (s : 𝕜) (hf : MDifferentiable I 𝓘(𝕜, E') f) :
    MDifferentiable I 𝓘(𝕜, E') (s • 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] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : M → E'} (s : 𝕜) (hf : MDifferentiable I 𝓘(𝕜, E') f)
Goal: MDifferentiable I 𝓘(𝕜, E') (s • f)
-- const_smul_mfderiv
theorem const_smul_mfderiv (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (s : 𝕜) :
    (mfderiv I 𝓘(𝕜, E') (s • f) z : TangentSpace I z →L[𝕜] E') =
      (s • mfderiv I 𝓘(𝕜, E') f z : TangentSpace I z →L[𝕜] E') 
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'] {z : M} {f : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (s : 𝕜)
Goal: mfderiv I 𝓘(𝕜, E') (s • f) z = s • mfderiv I 𝓘(𝕜, E') f z
-- HasMFDerivAt.neg
theorem HasMFDerivAt.neg (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f') :
    HasMFDerivAt I 𝓘(𝕜, E') (-f) z (-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] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f : M → E'} {f' : TangentSpace I z →L[𝕜] E'} (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f')
Goal: HasMFDerivAt I 𝓘(𝕜, E') (-f) z (-f')
-- hasMFDerivAt_neg
theorem hasMFDerivAt_neg : HasMFDerivAt I 𝓘(𝕜, E') (-f) z (-f') ↔ HasMFDerivAt I 𝓘(𝕜, E') f z 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] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {z : M} {f : M → E'} {f' : TangentSpace I z →L[𝕜] E'}
Goal: HasMFDerivAt I 𝓘(𝕜, E') (-f) z (-f') ↔ HasMFDerivAt I 𝓘(𝕜, E') f z f'
-- MDifferentiableAt.neg
theorem MDifferentiableAt.neg (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) :
    MDifferentiableAt I 𝓘(𝕜, E') (-f) z 
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'] {z : M} {f : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z)
Goal: MDifferentiableAt I 𝓘(𝕜, E') (-f) z
-- mdifferentiableAt_neg
theorem mdifferentiableAt_neg :
    MDifferentiableAt I 𝓘(𝕜, E') (-f) z ↔ MDifferentiableAt I 𝓘(𝕜, E') f z 
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'] {z : M} {f : M → E'}
Goal: MDifferentiableAt I 𝓘(𝕜, E') (-f) z ↔ MDifferentiableAt I 𝓘(𝕜, E') f z
-- MDifferentiable.neg
theorem MDifferentiable.neg (hf : MDifferentiable I 𝓘(𝕜, E') f) : MDifferentiable I 𝓘(𝕜, E') (-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] [SmoothManifoldWithCorners I M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : M → E'} (hf : MDifferentiable I 𝓘(𝕜, E') f)
Goal: MDifferentiable I 𝓘(𝕜, E') (-f)
-- mfderiv_neg
theorem mfderiv_neg (f : M → E') (x : M) :
    (mfderiv I 𝓘(𝕜, E') (-f) x : TangentSpace I x →L[𝕜] E') =
      (-mfderiv I 𝓘(𝕜, E') f x : TangentSpace I x →L[𝕜] E') 
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'] (f : M → E') (x : M)
Goal: mfderiv I 𝓘(𝕜, E') (-f) x = -mfderiv I 𝓘(𝕜, E') f x
-- HasMFDerivAt.sub
theorem HasMFDerivAt.sub (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f')
    (hg : HasMFDerivAt I 𝓘(𝕜, E') g z g') : HasMFDerivAt I 𝓘(𝕜, E') (f - g) z (f' - g') 
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'] {z : M} {f g : M → E'} {f' g' : TangentSpace I z →L[𝕜] E'} (hf : HasMFDerivAt I 𝓘(𝕜, E') f z f') (hg : HasMFDerivAt I 𝓘(𝕜, E') g z g')
Goal: HasMFDerivAt I 𝓘(𝕜, E') (f - g) z (f' - g')
-- MDifferentiableAt.sub
theorem MDifferentiableAt.sub (hf : MDifferentiableAt I 𝓘(𝕜, E') f z)
    (hg : MDifferentiableAt I 𝓘(𝕜, E') g z) : MDifferentiableAt I 𝓘(𝕜, E') (f - g) z 
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'] {z : M} {f g : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (hg : MDifferentiableAt I 𝓘(𝕜, E') g z)
Goal: MDifferentiableAt I 𝓘(𝕜, E') (f - g) z
-- MDifferentiable.sub
theorem MDifferentiable.sub (hf : MDifferentiable I 𝓘(𝕜, E') f)
    (hg : MDifferentiable I 𝓘(𝕜, E') g) : MDifferentiable I 𝓘(𝕜, E') (f - g) 
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'] {f g : M → E'} (hf : MDifferentiable I 𝓘(𝕜, E') f) (hg : MDifferentiable I 𝓘(𝕜, E') g)
Goal: MDifferentiable I 𝓘(𝕜, E') (f - g)
-- mfderiv_sub
theorem mfderiv_sub (hf : MDifferentiableAt I 𝓘(𝕜, E') f z)
    (hg : MDifferentiableAt I 𝓘(𝕜, E') g z) :
    (by exact mfderiv I 𝓘(𝕜, E') (f - g) z : TangentSpace I z →L[𝕜] E') =
      (by exact mfderiv I 𝓘(𝕜, E') f z) - (by exact mfderiv I 𝓘(𝕜, E') g z) 
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'] {z : M} {f g : M → E'} (hf : MDifferentiableAt I 𝓘(𝕜, E') f z) (hg : MDifferentiableAt I 𝓘(𝕜, E') g z)
Goal: mfderiv I 𝓘(𝕜, E') (f - g) z = mfderiv I 𝓘(𝕜, E') f z - mfderiv I 𝓘(𝕜, E') g z
-- HasMFDerivWithinAt.mul'
theorem HasMFDerivWithinAt.mul' (hp : HasMFDerivWithinAt I 𝓘(𝕜, F') p s z p')
    (hq : HasMFDerivWithinAt I 𝓘(𝕜, F') q s z q') :
    HasMFDerivWithinAt I 𝓘(𝕜, F') (p * q) s z (p z • q' + p'.smulRight (q z) : E →L[𝕜] 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] [SmoothManifoldWithCorners I M] {s : Set M} {z : M} {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} {p' q' : TangentSpace I z →L[𝕜] F'} (hp : HasMFDerivWithinAt I 𝓘(𝕜, F') p s z p') (hq : HasMFDerivWithinAt I 𝓘(𝕜, F') q s z q')
Goal: HasMFDerivWithinAt I 𝓘(𝕜, F') (p * q) s z (p z • q' + p'.smulRight (q z))
-- HasMFDerivAt.mul'
theorem HasMFDerivAt.mul' (hp : HasMFDerivAt I 𝓘(𝕜, F') p z p')
    (hq : HasMFDerivAt I 𝓘(𝕜, F') q z q') :
    HasMFDerivAt I 𝓘(𝕜, F') (p * q) z (p z • q' + p'.smulRight (q z) : E →L[𝕜] 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] [SmoothManifoldWithCorners I M] {z : M} {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} {p' q' : TangentSpace I z →L[𝕜] F'} (hp : HasMFDerivAt I 𝓘(𝕜, F') p z p') (hq : HasMFDerivAt I 𝓘(𝕜, F') q z q')
Goal: HasMFDerivAt I 𝓘(𝕜, F') (p * q) z (p z • q' + p'.smulRight (q z))
-- MDifferentiableWithinAt.mul
theorem MDifferentiableWithinAt.mul (hp : MDifferentiableWithinAt I 𝓘(𝕜, F') p s z)
    (hq : MDifferentiableWithinAt I 𝓘(𝕜, F') q s z) :
    MDifferentiableWithinAt I 𝓘(𝕜, F') (p * q) s z 
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} {z : M} {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDifferentiableWithinAt I 𝓘(𝕜, F') p s z) (hq : MDifferentiableWithinAt I 𝓘(𝕜, F') q s z)
Goal: MDifferentiableWithinAt I 𝓘(𝕜, F') (p * q) s z
-- MDifferentiableAt.mul
theorem MDifferentiableAt.mul (hp : MDifferentiableAt I 𝓘(𝕜, F') p z)
    (hq : MDifferentiableAt I 𝓘(𝕜, F') q z) : MDifferentiableAt I 𝓘(𝕜, F') (p * q) z 
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] {z : M} {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDifferentiableAt I 𝓘(𝕜, F') p z) (hq : MDifferentiableAt I 𝓘(𝕜, F') q z)
Goal: MDifferentiableAt I 𝓘(𝕜, F') (p * q) z
-- MDifferentiableOn.mul
theorem MDifferentiableOn.mul (hp : MDifferentiableOn I 𝓘(𝕜, F') p s)
    (hq : MDifferentiableOn I 𝓘(𝕜, F') q s) : MDifferentiableOn I 𝓘(𝕜, F') (p * q) 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} {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDifferentiableOn I 𝓘(𝕜, F') p s) (hq : MDifferentiableOn I 𝓘(𝕜, F') q s)
Goal: MDifferentiableOn I 𝓘(𝕜, F') (p * q) s
-- MDifferentiable.mul
theorem MDifferentiable.mul (hp : MDifferentiable I 𝓘(𝕜, F') p)
    (hq : MDifferentiable I 𝓘(𝕜, F') q) : MDifferentiable I 𝓘(𝕜, F') (p * q) 
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] {F' : Type u_11} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDifferentiable I 𝓘(𝕜, F') p) (hq : MDifferentiable I 𝓘(𝕜, F') q)
Goal: MDifferentiable I 𝓘(𝕜, F') (p * q)
-- HasMFDerivWithinAt.mul
theorem HasMFDerivWithinAt.mul (hp : HasMFDerivWithinAt I 𝓘(𝕜, F') p s z p')
    (hq : HasMFDerivWithinAt I 𝓘(𝕜, F') q s z q') :
    HasMFDerivWithinAt I 𝓘(𝕜, F') (p * q) s z (p z • q' + q z • p' : E →L[𝕜] 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] [SmoothManifoldWithCorners I M] {s : Set M} {z : M} {F' : Type u_11} [NormedCommRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} {p' q' : TangentSpace I z →L[𝕜] F'} (hp : HasMFDerivWithinAt I 𝓘(𝕜, F') p s z p') (hq : HasMFDerivWithinAt I 𝓘(𝕜, F') q s z q')
Goal: HasMFDerivWithinAt I 𝓘(𝕜, F') (p * q) s z (p z • q' + q z • p')
-- HasMFDerivAt.mul
theorem HasMFDerivAt.mul (hp : HasMFDerivAt I 𝓘(𝕜, F') p z p')
    (hq : HasMFDerivAt I 𝓘(𝕜, F') q z q') :
    HasMFDerivAt I 𝓘(𝕜, F') (p * q) z (p z • q' + q z • p' : E →L[𝕜] 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] [SmoothManifoldWithCorners I M] {z : M} {F' : Type u_11} [NormedCommRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} {p' q' : TangentSpace I z →L[𝕜] F'} (hp : HasMFDerivAt I 𝓘(𝕜, F') p z p') (hq : HasMFDerivAt I 𝓘(𝕜, F') q z q')
Goal: HasMFDerivAt I 𝓘(𝕜, F') (p * q) z (p z • q' + q z • p')
-- smoothSheaf.obj_eq
lemma smoothSheaf.obj_eq (U : (Opens (TopCat.of M))ᵒᵖ) :
    (smoothSheaf IM I M N).presheaf.obj U = C^∞⟮IM, (unop U : Opens M); I, N⟯ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] (N : Type u) [TopologicalSpace N] [ChartedSpace H N] (U : (TopologicalSpace.Opens ↑(TopCat.of M))ᵒᵖ)
Goal: (smoothSheaf IM I M N).presheaf.obj U = ContMDiffMap IM I (↥(Opposite.unop U)) N ⊤
-- smoothSheaf.ι_evalHom
@[simp, reassoc, elementwise] lemma smoothSheaf.ι_evalHom (x : TopCat.of M) (U) :
    colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheaf IM I M N).val) U ≫
    smoothSheaf.evalHom IM I N x =
    smoothSheaf.evalAt _ _ _ _ _ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] (N : Type u) [TopologicalSpace N] [ChartedSpace H N] (x : ↑(TopCat.of M)) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Limits.colimit.ι ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheaf IM I M N).val) U)
    (smoothSheaf.evalHom IM I N x) =
  smoothSheaf.evalAt IM I N x (Opposite.unop U)
-- smoothSheaf.eval_surjective
lemma smoothSheaf.eval_surjective (x : M) : Function.Surjective (smoothSheaf.eval IM I N x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] (N : Type u) [TopologicalSpace N] [ChartedSpace H N] (x : M)
Goal: Function.Surjective (smoothSheaf.eval IM I N x)
-- smoothSheaf.eval_germ
@[simp] lemma smoothSheaf.eval_germ (U : Opens M) (x : U)
    (f : (smoothSheaf IM I M N).presheaf.obj (op U)) :
    smoothSheaf.eval IM I N (x : M) ((smoothSheaf IM I M N).presheaf.germ x f) = f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] {N : Type u} [TopologicalSpace N] [ChartedSpace H N] (U : TopologicalSpace.Opens M) (x : ↥U) (f : (smoothSheaf IM I M N).presheaf.obj (Opposite.op U))
Goal: smoothSheaf.eval IM I N (↑x) ((smoothSheaf IM I M N).presheaf.germ x f) = ↑f x
-- smoothSheaf.smooth_section
lemma smoothSheaf.smooth_section {U : (Opens (TopCat.of M))ᵒᵖ}
    (f : (smoothSheaf IM I M N).presheaf.obj U) :
    Smooth IM I f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u} [TopologicalSpace M] [ChartedSpace HM M] {N : Type u} [TopologicalSpace N] [ChartedSpace H N] {U : (TopologicalSpace.Opens ↑(TopCat.of M))ᵒᵖ} (f : (smoothSheaf IM I M N).presheaf.obj U)
Goal: Smooth IM I ↑f
-- smoothSheafCommRing.ι_forgetStalk_hom
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.ι_forgetStalk_hom (x : TopCat.of M) (U) :
    CategoryStruct.comp
      (Z := (smoothSheaf IM I M R).presheaf.stalk x)
      (DFunLike.coe
        (α := ((forget CommRingCat).obj ((smoothSheafCommRing IM I M R).presheaf.obj
          (op ((OpenNhds.inclusion x).obj U.unop)))))
        (colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U))
      (forgetStalk IM I M R x).hom =
    colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheaf IM I M R).presheaf) U 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : ↑(TopCat.of M)) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ)
Goal: CategoryTheory.CategoryStruct.comp
    (⇑(CategoryTheory.Limits.colimit.ι
        ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheafCommRing IM I M R).presheaf) U))
    (smoothSheafCommRing.forgetStalk IM I M R x).hom =
  CategoryTheory.Limits.colimit.ι ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheaf IM I M R).presheaf) U
-- smoothSheafCommRing.ι_forgetStalk_inv
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.ι_forgetStalk_inv (x : TopCat.of M) (U) :
    colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheaf IM I M R).presheaf) U ≫
    (smoothSheafCommRing.forgetStalk IM I M R x).inv =
    (forget CommRingCat).map
      (colimit.ι ((OpenNhds.inclusion x).op ⋙ (smoothSheafCommRing IM I M R).presheaf) U) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : ↑(TopCat.of M)) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Limits.colimit.ι ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheaf IM I M R).presheaf)
      U)
    (smoothSheafCommRing.forgetStalk IM I M R x).inv =
  (CategoryTheory.forget CommRingCat).map
    (CategoryTheory.Limits.colimit.ι
      ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheafCommRing IM I M R).presheaf) U)
-- smoothSheafCommRing.ι_evalHom
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.ι_evalHom (x : TopCat.of M) (U) :
    colimit.ι ((OpenNhds.inclusion x).op ⋙ _) U ≫ smoothSheafCommRing.evalHom IM I M R x =
    smoothSheafCommRing.evalAt _ _ _ _ _ _ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : ↑(TopCat.of M)) (U : (TopologicalSpace.OpenNhds x)ᵒᵖ)
Goal: CategoryTheory.CategoryStruct.comp
    (CategoryTheory.Limits.colimit.ι
      ((TopologicalSpace.OpenNhds.inclusion x).op.comp (smoothSheafCommRing IM I M R).presheaf) U)
    (smoothSheafCommRing.evalHom IM I M R x) =
  smoothSheafCommRing.evalAt IM I M R x (Opposite.unop U)
-- smoothSheafCommRing.evalHom_germ
@[simp] lemma smoothSheafCommRing.evalHom_germ (U : Opens (TopCat.of M)) (x : U)
    (f : (smoothSheafCommRing IM I M R).presheaf.obj (op U)) :
    smoothSheafCommRing.evalHom IM I M R (x : TopCat.of M)
      ((smoothSheafCommRing IM I M R).presheaf.germ x f)
    = f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (U : TopologicalSpace.Opens ↑(TopCat.of M)) (x : ↥U) (f : ↑((smoothSheafCommRing IM I M R).presheaf.obj (Opposite.op U)))
Goal: (smoothSheafCommRing.evalHom IM I M R ↑x) (((smoothSheafCommRing IM I M R).presheaf.germ x) f) = ↑f x
-- smoothSheafCommRing.forgetStalk_inv_comp_eval
@[simp, reassoc, elementwise] lemma smoothSheafCommRing.forgetStalk_inv_comp_eval
    (x : TopCat.of M) :
    (smoothSheafCommRing.forgetStalk IM I M R x).inv ≫
     (DFunLike.coe (smoothSheafCommRing.evalHom IM I M R x)) =
    smoothSheaf.evalHom _ _ _ _ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {EM : Type u_2} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_3} [TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM) {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u) [TopologicalSpace M] [ChartedSpace HM M] (R : Type u) [TopologicalSpace R] [ChartedSpace H R] [CommRing R] [SmoothRing I R] (x : ↑(TopCat.of M))
Goal: CategoryTheory.CategoryStruct.comp (smoothSheafCommRing.forgetStalk IM I M R x).inv
    ⇑(smoothSheafCommRing.evalHom IM I M R x) =
  smoothSheaf.evalHom IM I R x