awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
x : M
s : Set M
f' f₁' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h' : f' = f₁'
⊢ HasMFDerivWithinAt I I' f s x f₁'

Tactic:
h' ▸ h

Premises used: Eq.rec
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
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'

Tactic:
⟨ContinuousWithinAt.congr_of_eventuallyEq h.1 h₁ hx, ?_⟩

Premises used: ContinuousWithinAt.congr_of_eventuallyEq, And.left, And.intro, HasMFDerivWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
refine :
  HasFDerivWithinAt (writtenInExtChartAt I I' x f₁) f' ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E))
    ((↑(extChartAt I x) : M → E) x)
⊢ HasMFDerivWithinAt I I' f₁ s x f'

Tactic:
refine ⟨ContinuousWithinAt.congr_of_eventuallyEq h.1 h₁ hx, ?_⟩
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
h₂ :
  writtenInExtChartAt I I' x
      f₁ =ᶠ[𝓝[(↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x) : M → E) x]
    writtenInExtChartAt I I' x f
hx₁ :
  writtenInExtChartAt I I' x f₁ ((↑(extChartAt I x) : M → E) x) =
    writtenInExtChartAt I I' x f ((↑(extChartAt I x) : M → E) x)
⊢ HasFDerivWithinAt (writtenInExtChartAt I I' x f₁) f' ((↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E))
    ((↑(extChartAt I x) : M → E) x)

Tactic:
apply HasFDerivWithinAt.congr_of_eventuallyEq h.2

Premises used: HasFDerivWithinAt.congr_of_eventuallyEq, And.right
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ (↑(extChartAt I x).symm : E → M) ⁻¹' {y : M | f₁ y = f y} ∈
    𝓝[(↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x) : M → E) x

Tactic:
have :
      (extChartAt I x).symm ⁻¹' {y | f₁ y = f y} ∈
        𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x :=
      extChartAt_preimage_mem_nhdsWithin h₁
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
this :
  (↑(extChartAt I x).symm : E → M) ⁻¹' {y : M | f₁ y = f y} ∈
    𝓝[(↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x) : M → E) x
apply :
  ∀ y ∈ (↑(extChartAt I x).symm : E → M) ⁻¹' {y : M | f₁ y = f y},
    Membership.mem (γ := Set E)
      {x_1 : E | (fun (x_2 : E) ↦ writtenInExtChartAt I I' x f₁ x_2 = writtenInExtChartAt I I' x f x_2) x_1} y
⊢ writtenInExtChartAt I I' x
      f₁ =ᶠ[𝓝[(↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x) : M → E) x]
    writtenInExtChartAt I I' x f

Tactic:
apply Filter.mem_of_superset this fun y => _

Premises used: Filter.mem_of_superset
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
this :
  (↑(extChartAt I x).symm : E → M) ⁻¹' {y : M | f₁ y = f y} ∈
    𝓝[(↑(extChartAt I x).symm : E → M) ⁻¹' s ∩ range (↑I : H → E)] (↑(extChartAt I x) : M → E) x
⊢ ∀ y ∈ (↑(extChartAt I x).symm : E → M) ⁻¹' {y : M | f₁ y = f y},
    Membership.mem (γ := Set E)
      {x_1 : E | (fun (x_2 : E) ↦ writtenInExtChartAt I I' x f₁ x_2 = writtenInExtChartAt I I' x f x_2) x_1} y

Tactic:
simp +contextual only [hx, mfld_simps]
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f s x f'
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ writtenInExtChartAt I I' x f₁ ((↑(extChartAt I x) : M → E) x) =
    writtenInExtChartAt I I' x f ((↑(extChartAt I x) : M → E) x)

Tactic:
simp only [hx, mfld_simps]
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s t : Set M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (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
⊢ HasMFDerivWithinAt I I' f₁ t x f'

Tactic:
(h.mono h₁).congr_of_eventuallyEq (Filter.mem_inf_of_right ht) hx

Premises used: Filter.mem_inf_of_right, HasMFDerivWithinAt.congr_of_eventuallyEq, HasMFDerivWithinAt.mono
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivAt I I' f x f'
h₁ : f₁ =ᶠ[𝓝 x] f
rw : ∀ (h : HasMFDerivWithinAt I I' f univ x f') (h₁ : f₁ =ᶠ[𝓝 x] f), HasMFDerivWithinAt I I' f₁ univ x f'
⊢ HasMFDerivAt I I' f₁ x f'

Tactic:
rw [← hasMFDerivWithinAt_univ] at h ⊢

Premises used: hasMFDerivWithinAt_univ
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f univ x f'
h₁ : f₁ =ᶠ[𝓝 x] f
apply : f₁ =ᶠ[𝓝[univ] x] f
⊢ HasMFDerivWithinAt I I' f₁ univ x f'

Tactic:
apply h.congr_of_eventuallyEq _ (mem_of_mem_nhds h₁ :)

Premises used: mem_of_mem_nhds, HasMFDerivWithinAt.congr_of_eventuallyEq
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
f' : ContinuousLinearMap.{u_1, u_1, u_2, u_5} (RingHom.id 𝕜) (TangentSpace I x) (TangentSpace I' (f x))
h : HasMFDerivWithinAt I I' f univ x f'
h₁ rwa rwa₁ rwa₂ rwa₃ : f₁ =ᶠ[𝓝 x] f
⊢ f₁ =ᶠ[𝓝[univ] x] f

Tactic:
rwa [nhdsWithin_univ]

Premises used: nhdsWithin_univ
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h₁ : ∀ y ∈ s, f₁ y = f y
hx : f₁ x = f x
⊢ MDifferentiableWithinAt I I' f₁ s x ↔ MDifferentiableWithinAt I I' f s x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropWithinAt_congr_iff h₁ hx

Premises used: StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_iff, differentiableWithinAt_localInvariantProp, MDifferentiableWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h : MDifferentiableWithinAt I I' f s x
h₁ : ∀ y ∈ s, f₁ y = f y
hx : x ∈ s
⊢ MDifferentiableWithinAt I I' f₁ s x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropWithinAt_congr_of_mem h h₁ hx

Premises used: differentiableWithinAt_localInvariantProp, StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_of_mem, MDifferentiableWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h₁ : ∀ y ∈ s, f₁ y = f y
hx : x ∈ s
⊢ MDifferentiableWithinAt I I' f₁ s x ↔ MDifferentiableWithinAt I I' f s x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropWithinAt_congr_iff_of_mem h₁ hx

Premises used: StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_iff_of_mem, differentiableWithinAt_localInvariantProp, MDifferentiableWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ MDifferentiableWithinAt I I' f₁ s x ↔ MDifferentiableWithinAt I I' f s x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropWithinAt_congr_iff_of_eventuallyEq h₁ hx

Premises used: StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_iff_of_eventuallyEq, differentiableWithinAt_localInvariantProp, MDifferentiableWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
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

Tactic:
(h.hasMFDerivWithinAt.congr_of_eventuallyEq h₁ hx).mdifferentiableWithinAt

Premises used: MDifferentiableWithinAt.hasMFDerivWithinAt, HasMFDerivWithinAt.mdifferentiableWithinAt, HasMFDerivWithinAt.congr_of_eventuallyEq
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h : MDifferentiableWithinAt I I' f s x
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : x ∈ s
⊢ MDifferentiableWithinAt I I' f₁ s x

Tactic:
h.congr_of_eventuallyEq h₁ (mem_of_mem_nhdsWithin hx h₁ :)

Premises used: MDifferentiableWithinAt.congr_of_eventuallyEq, mem_of_mem_nhdsWithin
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h : MDifferentiableWithinAt I I' f s x
h₁ : f₁ =ᶠ[𝓝[insert x s] x] f
⊢ MDifferentiableWithinAt I I' f₁ s x

Tactic:
(h.insert.congr_of_eventuallyEq_of_mem h₁ (mem_insert x s)).of_insert

Premises used: MDifferentiableWithinAt.of_insert, MDifferentiableWithinAt.insert, MDifferentiableWithinAt.congr_of_eventuallyEq_of_mem, Set.mem_insert
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
h₁ : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ MDifferentiableWithinAt I I' f s x ↔ MDifferentiableWithinAt I I' f₁ s x

Tactic:
mdifferentiablefWithinAt_iff h₁.symm hx.symm

Premises used: Filter.EventuallyEq.mdifferentiablefWithinAt_iff, Eq.symm, Filter.EventuallyEq.symm
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s t : Set M
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

Tactic:
(HasMFDerivWithinAt.congr_mono h.hasMFDerivWithinAt ht hx h₁).mdifferentiableWithinAt

Premises used: MDifferentiableWithinAt.hasMFDerivWithinAt, HasMFDerivWithinAt.mdifferentiableWithinAt, HasMFDerivWithinAt.congr_mono
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s : Set M
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

Tactic:
(HasMFDerivWithinAt.congr_mono h.hasMFDerivWithinAt ht hx (Subset.refl _)).mdifferentiableWithinAt

Premises used: MDifferentiableWithinAt.hasMFDerivWithinAt, Set.Subset.refl, HasMFDerivWithinAt.mdifferentiableWithinAt, HasMFDerivWithinAt.congr_mono
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
s t : Set M
h : MDifferentiableWithinAt I I' f s x
ht : ∀ x ∈ t, f₁ x = f x
hst : s ⊆ t
hxt : x ∈ t
⊢ MDifferentiableWithinAt I I' f₁ s x

Tactic:
h.congr (fun _y hy ↦ ht _y (hst hy)) (ht x hxt)

Premises used: MDifferentiableWithinAt.congr
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
x : M
h₁ : f₁ =ᶠ[𝓝 x] f
⊢ MDifferentiableAt I I' f₁ x ↔ MDifferentiableAt I I' f x

Tactic:
differentiableWithinAt_localInvariantProp.liftPropAt_congr_iff_of_eventuallyEq h₁

Premises used: MDifferentiableAt, StructureGroupoid.LocalInvariantProp.liftPropAt_congr_iff_of_eventuallyEq, differentiableWithinAt_localInvariantProp
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
s : Set M
h : MDifferentiableOn I I' f s
h₁ : ∀ y ∈ s, f₁ y = f y
⊢ MDifferentiableOn I I' f₁ s

Tactic:
differentiableWithinAt_localInvariantProp.liftPropOn_congr h h₁

Premises used: ChartedSpace.LiftPropOn, StructureGroupoid.LocalInvariantProp.liftPropOn_congr, MDifferentiableOn, differentiableWithinAt_localInvariantProp, MDifferentiableWithinAt
-- Module: Mathlib.Geometry.Manifold.MFDeriv.Basic
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
E' : Type u_5
inst_6 : NormedAddCommGroup E'
inst_7 : NormedSpace 𝕜 E'
H' : Type u_6
inst_8 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f f₁ : M → M'
s : Set M
h₁ : ∀ y ∈ s, f₁ y = f y
⊢ MDifferentiableOn I I' f₁ s ↔ MDifferentiableOn I I' f s

Tactic:
differentiableWithinAt_localInvariantProp.liftPropOn_congr_iff h₁

Premises used: ChartedSpace.LiftPropOn, MDifferentiableOn, differentiableWithinAt_localInvariantProp, MDifferentiableWithinAt, StructureGroupoid.LocalInvariantProp.liftPropOn_congr_iff