mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
t t₀ : Triangle ℝ P
ht : Set.range t.points ⊆ insert t₀.orthocenter (Set.range t₀.points)
ht₀o : t₀.orthocenter ∉ Set.range t₀.points
ht₀s : s = insert t₀.orthocenter (Set.range t₀.points)
i₁ i₂ i₃ j₂ j₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h₁₂₃ : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
h₁ : t.points i₁ = t₀.orthocenter
hj₂₃ : j₂ ≠ j₃
h₂ : t₀.points j₂ = t.points i₂
h₃ : t₀.points j₃ = t.points i₃
j₁ : Fin 3
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₁₂₃ : ∀ (j : Fin 3), j = j₁ ∨ j = j₂ ∨ j = j₃
h : t₀.points j₁ = t.orthocenter
hui : Set.univ = {i₁, i₂, i₃}
huj : Set.univ = {j₁, j₂, j₃}
⊢ {t₀.orthocenter, t₀.points j₁, t₀.points j₂, t₀.points j₃} =
    {t₀.points j₁, t₀.orthocenter, t₀.points j₂, t₀.points j₃}

Tactic:
rw [Set.insert_comm]

Goal state after:
Module: Mathlib.Geometry.Manifold.Complex

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝³ : I.Boundaryless
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
f : M → F
U : Set M
c : M
hd : MDiff[U] f
hc : IsPreconnected U
ho : IsOpen[inst✝²] U
hcU : c ∈ U
hm : IsMaxOn (norm ∘ f) U c
V : Set M := {z | z ∈ U ∧ ‖f z‖ = ‖f c‖}
hVo : IsOpen[inst✝²] V
hVne : (U ∩ V).Nonempty
W : Set M := U ∩ {z | ‖f z‖ = ‖f c‖}ᶜ
hWo : IsOpen[inst✝²] W
hdVW : Disjoint V W
hUVW : U ⊆ V ∪ W
⊢ U ⊆ V

Tactic:
exact hc.subset_left_of_subset_union hVo hWo hdVW hUVW hVne

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : 2 ≤ n
b₁ : M → M' := f
b₂ : M → M := ⋯
v : (x : M) → TangentSpace I' (f x) := ⋯
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := ⋯
hv : (MDiffAt[s ∩ f ⁻¹' t] fun x ↦ ⟨f x, v x⟩) x₀
this : (MDiffAt[s] fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) x₀
⊢ ContDiffAt 𝕜 1 inverse (inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x₀ (f x₀) (f x₀) (mfderiv[s] f x₀))

Tactic:
apply IsInvertible.contDiffAt_map_inverse

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
n : ℕ∞ω
t : Set M'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
inst✝ : CompleteSpace E
hV : MDiffAt[t] (T% V) (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : 2 ≤ n
b₁ : M → M' := f
b₂ : M → M := ⋯
v : (x : M) → TangentSpace I' (f x) := ⋯
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := ⋯
hv : (MDiffAt[s ∩ f ⁻¹' t] fun x ↦ ⟨f x, v x⟩) x₀
this : (MDiffAt[s] fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) x₀
⊢ (inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x₀ (f x₀) (f x₀) (mfderiv[s] f x₀)).IsInvertible
Module: Mathlib.Geometry.Manifold.VectorField.Pullback

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
m n : ℕ∞ω
t : Set M'
inst✝² : CompleteSpace E
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hV : ContMDiffWithinAt I' I'.tangent m (fun x ↦ ⟨x, V x⟩) t (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : m + 1 ≤ n
b₁ : M → M' := f
b₂ : M → M := ⋯
v : (x : M) → TangentSpace I' (f x) := ⋯
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := ⋯
hv : ContMDiffWithinAt I I'.tangent m (fun x ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀
this :
  ContMDiffWithinAt I 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) s x₀
⊢ ContDiffAt 𝕜 m inverse (inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x₀ (f x₀) (f x₀) (mfderiv[s] f x₀))

Tactic:
apply IsInvertible.contDiffAt_map_inverse

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
H' : Type u_5
inst✝⁷ : TopologicalSpace H'
E' : Type u_6
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁴ : TopologicalSpace M'
inst✝³ : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
m n : ℕ∞ω
t : Set M'
inst✝² : CompleteSpace E
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
hV : ContMDiffWithinAt I' I'.tangent m (fun x ↦ ⟨x, V x⟩) t (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : (mfderiv[s] f x₀).IsInvertible
hx₀ : x₀ ∈ s
hs : UniqueMDiff[s]
hmn : m + 1 ≤ n
b₁ : M → M' := f
b₂ : M → M := ⋯
v : (x : M) → TangentSpace I' (f x) := ⋯
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := ⋯
hv : ContMDiffWithinAt I I'.tangent m (fun x ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀
this :
  ContMDiffWithinAt I 𝓘(𝕜, E →L[𝕜] E') m
    (fun x ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderiv[s] f x)) s x₀
⊢ (inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x₀ (f x₀) (f x₀) (mfderiv[s] f x₀)).IsInvertible
Module: Mathlib.Geometry.Manifold.LocalSourceTargetProperty

Goal state before:
pp: case refine_5
𝕜 : Type u_1
E : Type u_2
F : Type u_4
H : Type u_6
G : Type u_8
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : TopologicalSpace H
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 F G
M : Type u_10
N : Type u_12
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f g : M → N
x : M
P : (M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop
hP : IsLocalSourceTargetProperty P
hf : LiftSourceTargetPropertyAt I J n f x P
h' : f =ᶠ[𝓝 x] g
s' : Set M
hxs' : s' ∈ 𝓝 x
hfg : EqOn f g s'
s : Set M
hss' : s ⊆ s'
hs : IsOpen[inst✝³] s
hxs : x ∈ s
⊢ P g (hf.domChart.restr s) hf.codChart

Tactic:
· exact hP.congr (hfg.mono hss' |>.mono (by grind)) <| hP.mono_source hs hf.property

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
x : M
V W : (x : M) → TangentSpace I x
c : 𝕜
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hW : MDiffAt (T% W) x
⊢ mlieBracket I V (c • W) x = c • mlieBracket I V W x

Tactic:
by
  simp only [← mlieBracketWithin_univ] at hW ⊢
  exact mlieBracketWithin_const_smul_right hW (uniqueMDiffWithinAt_univ _)

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ (fun f ↦ f.ofLp i) ⁻¹' interior (Ici a) = f ⁻¹' Ioi a

pp: case hf₂
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ Continuous fun f ↦ f.ofLp i

Tactic:
interior_Ici

Goal state after:
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ (fun f ↦ f.ofLp i) ⁻¹' Ioi a = f ⁻¹' Ioi a

case hf₂
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ Continuous fun f ↦ f.ofLp i
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ (fun f ↦ f.ofLp i) ⁻¹' closure (Ioi a) = f ⁻¹' Ici a

pp: case hfc
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ Continuous fun f ↦ f.ofLp i

Tactic:
closure_Ioi

Goal state after:
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ (fun f ↦ f.ofLp i) ⁻¹' Ici a = f ⁻¹' Ici a

case hfc
n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
f : (PiLp p fun x ↦ ℝ) → ℝ := fun x ↦ x.ofLp i
⊢ Continuous fun f ↦ f.ofLp i
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
h : {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a
h' : {y | ∀ (i : Fin n), a < y.ofLp i} = ⋂ i, f i ⁻¹' Ioi a
i : Fin n
⊢ (fun f ↦ f.ofLp i) ⁻¹' interior (Ici a) = f i ⁻¹' Ioi a

pp: case hf₂
n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
h : {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a
h' : {y | ∀ (i : Fin n), a < y.ofLp i} = ⋂ i, f i ⁻¹' Ioi a
i : Fin n
⊢ Continuous fun f ↦ f.ofLp i

Tactic:
interior_Ici

Goal state after:
n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
h : {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a
h' : {y | ∀ (i : Fin n), a < y.ofLp i} = ⋂ i, f i ⁻¹' Ioi a
i : Fin n
⊢ (fun f ↦ f.ofLp i) ⁻¹' Ioi a = f i ⁻¹' Ioi a

case hf₂
n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
h : {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a
h' : {y | ∀ (i : Fin n), a < y.ofLp i} = ⋂ i, f i ⁻¹' Ioi a
i : Fin n
⊢ Continuous fun f ↦ f.ofLp i
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
⊢ ModelWithCorners.boundary ↑(Icc x y) = {⊥, ⊤}

Tactic:
ext p
  rcases Set.eq_endpoints_or_mem_Ioo_of_mem_Icc p.2 with (hp | hp | hp)
  · have : p = ⊥ := SetCoe.ext hp
    rw [this]
    apply iff_of_true Icc_isBoundaryPoint_bot (mem_insert ⊥ {⊤})
  · have : p = ⊤ := SetCoe.ext hp
    rw [this]
    apply iff_of_true Icc_isBoundaryPoint_top (mem_insert_of_mem ⊥ rfl)
  · apply iff_of_false
    · simpa [← mem_compl_iff, ModelWithCorners.compl_boundary] using!
        Icc_isInteriorPoint_interior hp
    · rintro (rfl | rfl) <;> simp at hp

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case hg
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
A' : x₀ = ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)
h'f : MDiffAt[s] f x₀
this✝ : (mfderiv[range ↑I] ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)).inverse = mfderiv% ↑(extChartAt I x₀) x₀
V' : (x : E') → TangentSpace 𝓘(𝕜, E') x := mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) V (range ↑I')
hV' : V' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) V (range ↑I')
W' : (x : E') → TangentSpace 𝓘(𝕜, E') x := mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) W (range ↑I')
hW' : W' = mpullbackWithin 𝓘(𝕜, E') I' (↑(extChartAt I' (f x₀)).symm) W (range ↑I')
F : E → E' := (↑(extChartAt I' (f x₀)) ∘ f) ∘ ↑(extChartAt I x₀).symm
hF : F = (↑(extChartAt I' (f x₀)) ∘ f) ∘ ↑(extChartAt I x₀).symm
hFx₀ : ↑(extChartAt I' (f x₀)) (f x₀) = F (↑(extChartAt I x₀) x₀)
Y : (x : M') → TangentSpace I' x
this : (extChartAt I x₀).target ∈ 𝓝[↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x₀) x₀
y : E
hy : y ∈ ↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I
h'''y : y ∈ (extChartAt I x₀).target
h'y : f (↑(extChartAt I x₀).symm y) ∈ (extChartAt I' (f x₀)).source
h''y : f (↑(extChartAt I x₀).symm y) ∈ (chartAt H' (f x₀)).source
huy : UniqueMDiffAt[↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I] y
⊢ MDiffAt ↑(extChartAt I' (f x₀)) (f (↑(extChartAt I x₀).symm y))

Tactic:
exact mdifferentiableAt_extChartAt h''y

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁸ : TopologicalSpace B
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → NormedAddCommGroup (E x)
inst✝³ : (x : B) → InnerProductSpace ℝ (E x)
inst✝² : FiberBundle F E
inst✝¹ : VectorBundle ℝ F E
M : Type u_4
inst✝ : TopologicalSpace M
h : IsContinuousRiemannianBundle F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContinuousWithinAt (fun m ↦ { proj := b m, snd := v m }) s x
hw : ContinuousWithinAt (fun m ↦ { proj := b m, snd := w m }) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁸, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hf : ContinuousWithinAt b s x
this : ContinuousWithinAt (fun m ↦ TotalSpace.mk' ℝ (b m) (((g (b m)) (v m)) (w m))) s x
⊢ ContinuousWithinAt (fun m ↦ ((g (b m)) (v m)) (w m)) s x

Tactic:
simp only [FiberBundle.continuousWithinAt_totalSpace] at this

Goal state after:
B : Type u_1
inst✝⁸ : TopologicalSpace B
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁵ : TopologicalSpace (TotalSpace F E)
inst✝⁴ : (x : B) → NormedAddCommGroup (E x)
inst✝³ : (x : B) → InnerProductSpace ℝ (E x)
inst✝² : FiberBundle F E
inst✝¹ : VectorBundle ℝ F E
M : Type u_4
inst✝ : TopologicalSpace M
h : IsContinuousRiemannianBundle F E
b : M → B
v w : (x : M) → E (b x)
s : Set M
x : M
hv : ContinuousWithinAt (fun m ↦ { proj := b m, snd := v m }) s x
hw : ContinuousWithinAt (fun m ↦ { proj := b m, snd := w m }) s x
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁸, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
hf : ContinuousWithinAt b s x
this :
  ContinuousWithinAt (fun x ↦ b x) s x ∧
    ContinuousWithinAt
      (fun x_1 ↦
        (↑(trivializationAt ℝ (Trivial B ℝ) (b x)) (TotalSpace.mk' ℝ (b x_1) (((g (b x_1)) (v x_1)) (w x_1)))).2)
      s x
⊢ ContinuousWithinAt (fun m ↦ ((g (b m)) (v m)) (w m)) s x
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
f : ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x)
⊢ f ∈ nonunits ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x) ↔ f ∈ ↑(RingHom.ker (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x))

Tactic:
rw [mem_nonunits_iff, not_iff_comm, Iff.comm]

Goal state after:
𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝⁴ : NormedAddCommGroup EM
inst✝³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace HM M
x : M
f : ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x)
⊢ IsUnit f ↔ f ∉ ↑(RingHom.ker (eval IM 𝓘(𝕜, 𝕜) M 𝕜 x))
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
x : M
inst✝³ : IsManifold I 2 M
inst✝² : CompleteSpace E
cov : ((x : M) → TangentSpace I x) → (x : M) → TangentSpace I x →L[𝕜] TangentSpace I x
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
hcov : IsCovariantDerivativeOn E cov
X₀ : TangentSpace I x
⊢ ((hcov.torsion x) X₀) X₀ = 0

Tactic:
simp [torsion_apply_eq_extend]

Goal state after:
Module: Mathlib.GroupTheory.ClassEquation

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
val✝ : Fintype G
x✝ : ConjClasses G
g : G
hg : (carrier (Quot.mk (⇑(IsConj.setoid G)) g)).Subsingleton
⊢ ↑(carrier (Quot.mk (⇑(IsConj.setoid G)) g)).toFinset = ↑{g}

Tactic:
by simpa using hg.eq_singleton_of_mem mem_carrier_mk

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: case inl
n : ℕ
⊢ 0 < n → r 1 ^ n ≠ 1

Tactic:
hn

Goal state after:
case inl
n : ℕ
hn : 0 < n
⊢ r 1 ^ n ≠ 1
Module: Mathlib.GroupTheory.CommutingProbability

Goal state before:
pp: M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
⊢ commProb M ≤ 1

Tactic:
refine div_le_one_of_le₀ ?_ (sq_nonneg (Nat.card M : ℚ))

Goal state after:
M : Type u_1
inst✝¹ : Mul M
inst✝ : Finite M
⊢ ↑(Nat.card { p // Commute p.1 p.2 }) ≤ ↑(Nat.card M) ^ 2
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m m' : M i
w : Word M
⊢ (m * m') • w = m • m' • w

Tactic:
dsimp +instances [instHSMul]

Goal state after:
ι : Type u_1
M : ι → Type u_2
inst✝³ : (i : ι) → Monoid (M i)
N : Type u_3
inst✝² : Monoid N
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i : ι
m m' : M i
w : Word M
⊢ rcons { head := m * m' * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯ } =
    rcons
      {
        head :=
          m *
            ((equivPair i)
                (rcons { head := m' * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯ })).head,
        tail :=
          ((equivPair i)
              (rcons { head := m' * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯ })).tail,
        fstIdx_ne := ⋯ }
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case inr
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
this✝ : ¬Even (2 * k + 1)
this : ¬Even (↑(M.M i i') * 2 - (2 * k + 1))
⊢ cs.simple i' * (cs.simple i * cs.simple i') ^ ((1 + k * 2) / 2) =
    cs.simple i * (cs.simple i' * cs.simple i) ^ ((-1 + (-k + ↑(M.M i i')) * 2) / 2)

Tactic:
repeat rw [Int.add_mul_ediv_right _ _ (by simp)]

Goal state after:
case inr
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
this✝ : ¬Even (2 * k + 1)
this : ¬Even (↑(M.M i i') * 2 - (2 * k + 1))
⊢ cs.simple i' * (cs.simple i * cs.simple i') ^ (1 / 2 + k) =
    cs.simple i * (cs.simple i' * cs.simple i) ^ (-1 / 2 + (-k + ↑(M.M i i')))
Module: Mathlib.GroupTheory.Coxeter.Length

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
⊢ cs.length (cs.simple i * w) ≠ cs.length w

Tactic:
rw [← length_inv]
  simpa using cs.length_mul_simple_ne w⁻¹ i

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
i j : ι
w : NeWord M i j
⊢ w.toList ≠ []

Tactic:
induction w
  · rintro ⟨rfl⟩
  · apply List.append_ne_nil_of_left_ne_nil
    assumption

Goal state after:
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case inl
M : Type u_2
inst✝ : AddCommGroup M
a : ℚ
m : M
s : ℕ+
h : 0 ≤ a
⊢ (have this := ⟨a, h⟩;
      this) •
      mk m s =
    mk (a.num • m) (⟨a.den, ⋯⟩ * s)

Tactic:
nnqsmul_mk,

Goal state after:
case inl
M : Type u_2
inst✝ : AddCommGroup M
a : ℚ
m : M
s : ℕ+
h : 0 ≤ a
⊢ mk
      ((have this := ⟨a, h⟩;
          this).num •
        m)
      (⟨(have this := ⟨a, h⟩;
            this).den,
          ⋯⟩ *
        s) =
    mk (a.num • m) (⟨a.den, ⋯⟩ * s)
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: M✝ : Type u_1
inst✝¹ : AddCommMonoid M✝
M : Type u_2
inst✝ : AddCommGroup M
a : ℚ
x y : DivisibleHull M
⊢ a • (x + y) = a • x + a • y

Tactic:
simp [qsmul_def, smul_add]

Goal state after:
Module: Mathlib.GroupTheory.DoubleCoset

Goal state before:
pp: G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K (Quotient.out (mk H K g)) = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
⊢ ∃ h k, h ∈ H ∧ k ∈ K ∧ Quotient.out (mk H K g) = h * g * k

Tactic:
rw [out_eq'] at this

Goal state after:
G : Type u_1
inst✝ : Group G
H K : Subgroup G
g : G
this : mk H K g = mk H K g ↔ ∃ h ∈ H, ∃ k ∈ K, g = h * Quotient.out (mk H K g) * k
⊢ ∃ h k, h ∈ H ∧ k ∈ K ∧ Quotient.out (mk H K g) = h * g * k
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
hab : ↑sb • ma ≤ ↑sa • mb
mc : M
sc : ℕ+
hbc : ↑sc • mb ≤ ↑sb • mc
⊢ ↑sc • ↑sb • ma ≤ ↑sa • ↑sb • mc

Tactic:
rw [← nsmul_le_nsmul_iff_right (show sc.val ≠ 0 by simp), smul_comm _ _ mb] at hab

Goal state after:
case mk.mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
ma : M
sa : ℕ+
mb : M
sb : ℕ+
mc : M
sc : ℕ+
hab : ↑sc • ↑sb • ma ≤ ↑sa • ↑sc • mb
hbc : ↑sc • mb ≤ ↑sb • mc
⊢ ↑sc • ↑sb • ma ≤ ↑sa • ↑sb • mc
Module: Mathlib.GroupTheory.DivisibleHull

Goal state before:
pp: case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
mb : M
sb : ℕ+
mc : M
sc : ℕ+
h : ↑sc • mb < ↑sb • mc
⊢ a • mk mb sb < a • mk mc sc

Tactic:
nnqsmul_mk,

Goal state after:
case mk.mk
M✝ : Type u_1
inst✝³ : AddCommMonoid M✝
M : Type u_2
inst✝² : AddCommMonoid M
inst✝¹ : LinearOrder M
inst✝ : IsOrderedCancelAddMonoid M
a : ℚ≥0
ha : 0 < a
mb : M
sb : ℕ+
mc : M
sc : ℕ+
h : ↑sc • mb < ↑sb • mc
⊢ mk (a.num • mb) (⟨a.den, ⋯⟩ * sb) < mk (a.num • mc) (⟨a.den, ⋯⟩ * sc)