mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsImmersionAtOfComplement F I J n f x
this : f '' h.domChart.source ⊆ h.codChart.source
⊢ ↑(h.codChart.extend J) '' f '' h.domChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source

Tactic:
this,

Goal state after:
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsImmersionAtOfComplement F I J n f x
this : f '' h.domChart.source ⊆ h.codChart.source
⊢ ↑(h.codChart.extend J) '' h.codChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
⊢ ↑((IccRightChart x y).extend (𝓡∂ 1)) ⊤ ∈ frontier (range ↑(𝓡∂ 1))

Tactic:
IccRightChart_extend_top,

Goal state after:
x y : ℝ
hxy : Fact (x < y)
⊢ 0 ∈ frontier (range ↑(𝓡∂ 1))
Module: Mathlib.Geometry.Manifold.Instances.UnitsOfNormedAlgebra

Goal state before:
pp: R : Type u_1
inst✝⁸ : NormedRing R
inst✝⁷ : CompleteSpace R
n : ℕ∞ω
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra 𝕜 R
H : Type u_3
inst✝⁴ : TopologicalSpace H
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_5
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
this : (val ∘ fun x ↦ x⁻¹) = Ring.inverse ∘ val
⊢ ContMDiff 𝓘(𝕜, R) 𝓘(𝕜, R) n (val ∘ fun a ↦ a⁻¹)

Tactic:
this,

Goal state after:
R : Type u_1
inst✝⁸ : NormedRing R
inst✝⁷ : CompleteSpace R
n : ℕ∞ω
𝕜 : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAlgebra 𝕜 R
H : Type u_3
inst✝⁴ : TopologicalSpace H
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_5
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
this : (val ∘ fun x ↦ x⁻¹) = Ring.inverse ∘ val
⊢ ContMDiff 𝓘(𝕜, R) 𝓘(𝕜, R) n (Ring.inverse ∘ val)
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : ↑x ≠ v
a : ℝ := ((innerSL ℝ) v) ↑x
y : ↥(ℝ ∙ v)ᗮ := (ℝ ∙ v)ᗮ.orthogonalProjectionOnto ↑x
split : ↑x = a • v + ↑y
hvy : ⟪v, ↑y⟫_ℝ = 0
pythag : 1 = a ^ 2 + ‖y‖ ^ 2
ha : 0 < 1 - a
⊢ ((|2| / (1 - a) * ‖y‖) ^ 2 + 4)⁻¹ * (4 * (2 / (1 - a) * 1)) = 1

Tactic:
field_simp
    linear_combination 4 * pythag

Goal state after:
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ EqOn (γ a') (γ a) (Ioo (-a') a')

Tactic:
apply isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless _ hv
    (hγ a' (by positivity)) ((hγ a (lt_of_lt_of_le hpos hle)).mono _)
    (by rw [hγx a, hγx a'])

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ 0 ∈ Ioo (-a') a'

E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ Ioo (-a') a' ⊆ Ioo (-a) a
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b c : ℝ
γ : ℝ → M
h : a ≤ b
h' : b ≤ c
this : Icc a c = Icc a b ∪ Ioc b c
⊢ ∫⁻ (t : ℝ) in Icc a c, ‖(mfderiv% γ t) 1‖ₑ = pathELength I γ a b + pathELength I γ b c

Tactic:
this,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type u_2
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : (x : M) → ENorm (TangentSpace I x)
a b c : ℝ
γ : ℝ → M
h : a ≤ b
h' : b ≤ c
this : Icc a c = Icc a b ∪ Ioc b c
⊢ ∫⁻ (t : ℝ) in Icc a b ∪ Ioc b c, ‖(mfderiv% γ t) 1‖ₑ = pathELength I γ a b + pathELength I γ b c
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case neg
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'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
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 : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
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
hfi : ¬(mfderiv[s] f x₀).IsInvertible
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullbackWithin I I' f V s) (mpullbackWithin I I' f W s) s x₀

Tactic:
simp only [mlieBracketWithin_apply, mpullbackWithin_apply,
      ContinuousLinearMap.inverse_of_not_isInvertible hfi, zero_apply]

Goal state after:
case neg
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'
inst✝² : IsManifold I 2 M
inst✝¹ : IsManifold I' 2 M'
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 : ContMDiffWithinAt I I' 2 f s x₀
hx₀ : x₀ ∈ s
hst : f ⁻¹' t ∈ 𝓝[s] x₀
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
hfi : ¬(mfderiv[s] f x₀).IsInvertible
⊢ 0 =
    (mfderiv% ↑(extChartAt I x₀) x₀).inverse
      (lieBracketWithin 𝕜 (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm) (mpullbackWithin I I' f V s) (range ↑I))
        (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x₀).symm) (mpullbackWithin I I' f W s) (range ↑I))
        (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀))
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
this✝ : mfderiv[Icc a b] (γ ∘ f) t = mfderiv[Icc (f b) (f a)] γ (f t) ∘SL mfderiv[Icc a b] f t
this : (mfderiv[Icc a b] f t) 1 = derivWithin f (Icc a b) t • 1
⊢ ‖(mfderiv[Icc (f b) (f a)] γ (f t)) (derivWithin f (Icc a b) t • 1)‖ₑ =
    ENNReal.ofReal (-derivWithin f (Icc a b) t) * ‖(mfderiv[f '' Icc a b] γ (f t)) 1‖ₑ

Tactic:
have : 0 ≤ -derivWithin f (Icc a b) t := by simp [hf.derivWithin_nonpos]

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
a b : ℝ
γ : ℝ → M
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
f : ℝ → ℝ
h✝ : a ≤ b
hf : AntitoneOn f (Icc a b)
h'f : DifferentiableOn ℝ f (Icc a b)
hγ : MDiff[Icc (f b) (f a)] γ
h : a < b
f_im : f '' Icc a b = Icc (f b) (f a)
B : ∀ t ∈ Icc a b, HasDerivWithinAt f (derivWithin f (Icc a b) t) (Icc a b) t
t : ℝ
ht : t ∈ Icc a b
this✝¹ : mfderiv[Icc a b] (γ ∘ f) t = mfderiv[Icc (f b) (f a)] γ (f t) ∘SL mfderiv[Icc a b] f t
this✝ : (mfderiv[Icc a b] f t) 1 = derivWithin f (Icc a b) t • 1
this : 0 ≤ -derivWithin f (Icc a b) t
⊢ ‖(mfderiv[Icc (f b) (f a)] γ (f t)) (derivWithin f (Icc a b) t • 1)‖ₑ =
    ENNReal.ofReal (-derivWithin f (Icc a b) t) * ‖(mfderiv[f '' Icc a b] γ (f t)) 1‖ₑ
Module: Mathlib.Geometry.Manifold.Riemannian.PathELength

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun x ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun x ↦ y
γ : ℝ → M := (Iic 1).piecewise γ₁ γ₂
this : riemannianEDist I x z ≤ pathELength I γ 0 2
⊢ pathELength I γ 0 1 = pathELength I γ₁ 0 1

Tactic:
apply pathELength_congr

Goal state after:
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : (x : M) → ENorm (TangentSpace I x)
inst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)
x y z : M
r : ℝ≥0∞
hr : riemannianEDist I x y + riemannianEDist I y z < r
u : ℝ≥0∞
hu : u > riemannianEDist I x y
v : ℝ≥0∞
hv : v > riemannianEDist I y z
huv : u + v < r
γ₁ : ℝ → M
hγ₁0 : γ₁ 0 = x
hγ₁1 : γ₁ 1 = y
hγ₁_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₁
hγ₁ : pathELength I γ₁ 0 1 < u
hγ₁_const : γ₁ =ᶠ[𝓝 1] fun x ↦ y
γ₂ : ℝ → M
hγ₂1 : γ₂ 1 = y
hγ₂2 : γ₂ 2 = z
hγ₂_smooth : ContMDiff 𝓘(ℝ, ℝ) I 1 γ₂
hγ₂ : pathELength I γ₂ 1 2 < v
hγ₂_const : γ₂ =ᶠ[𝓝 1] fun x ↦ y
γ : ℝ → M := ⋯
this : riemannianEDist I x z ≤ pathELength I γ 0 2
⊢ EqOn γ γ₁ (Icc 0 1)
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

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'
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : IsManifold I' (minSmoothness 𝕜 2) M'
inst✝ : CompleteSpace E
n : ℕ∞ω
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 : ContMDiffAt I I' n f x₀
hx₀ : x₀ ∈ s
hn : minSmoothness 𝕜 2 ≤ n
hst : f ⁻¹' t ∈ 𝓝[s] x₀
this : mpullback I I' f (mlieBracketWithin I' V W t) x₀ = mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀
⊢ mpullback I I' f (mlieBracketWithin I' V W t) x₀ = mlieBracketWithin I (mpullback I I' f V) (mpullback I I' f W) s x₀

Tactic:
this,

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'
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : IsManifold I' (minSmoothness 𝕜 2) M'
inst✝ : CompleteSpace E
n : ℕ∞ω
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 : ContMDiffAt I I' n f x₀
hx₀ : x₀ ∈ s
hn : minSmoothness 𝕜 2 ≤ n
hst : f ⁻¹' t ∈ 𝓝[s] x₀
this : mpullback I I' f (mlieBracketWithin I' V W t) x₀ = mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀
⊢ mpullbackWithin I I' f (mlieBracketWithin I' V W t) s x₀ =
    mlieBracketWithin I (mpullback I I' f V) (mpullback I I' f W) s x₀
Module: Mathlib.Geometry.Manifold.Submersion

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
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 E'' G
M : Type u_11
N : Type u_13
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsSubmersionAtOfComplement F I J n f x
this : f '' h.domChart.source ⊆ h.codChart.source
⊢ ↑(h.codChart.extend J) '' f '' h.domChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source

Tactic:
this,

Goal state after:
𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
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 E'' G
M : Type u_11
N : Type u_13
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsSubmersionAtOfComplement F I J n f x
this : f '' h.domChart.source ⊆ h.codChart.source
⊢ ↑(h.codChart.extend J) '' h.codChart.source ⊆ ↑(h.codChart.extend J) '' (h.codChart.extend J).source
Module: Mathlib.Geometry.Manifold.WhitneyEmbedding

Goal state before:
pp: ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
y : M
a✝ : y ∈ s
h :
  ∀ (x_1 : ι),
    (↑(f.toFun x_1) x • ↑(extChartAt I (f.c x_1)) x, ↑(f.toFun x_1) x) =
      (↑(f.toFun x_1) y • ↑(extChartAt I (f.c x_1)) y, ↑(f.toFun x_1) y)
h₁ :
  ↑(f.toFun (f.ind x hx)) x • ↑(extChartAt I (f.c (f.ind x hx))) x =
    ↑(f.toFun (f.ind x hx)) y • ↑(extChartAt I (f.c (f.ind x hx))) y
h₂ : ↑(f.toFun (f.ind x hx)) x = ↑(f.toFun (f.ind x hx)) y
⊢ x = y

Tactic:
rw [f.apply_ind x hx] at h₂

Goal state after:
ι : Type uι
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I ∞ M
inst✝¹ : T2Space M
inst✝ : Fintype ι
s : Set M
f : SmoothBumpCovering ι I M s
x : M
hx : x ∈ s
y : M
a✝ : y ∈ s
h :
  ∀ (x_1 : ι),
    (↑(f.toFun x_1) x • ↑(extChartAt I (f.c x_1)) x, ↑(f.toFun x_1) x) =
      (↑(f.toFun x_1) y • ↑(extChartAt I (f.c x_1)) y, ↑(f.toFun x_1) y)
h₁ :
  ↑(f.toFun (f.ind x hx)) x • ↑(extChartAt I (f.c (f.ind x hx))) x =
    ↑(f.toFun (f.ind x hx)) y • ↑(extChartAt I (f.c (f.ind x hx))) y
h₂ : 1 = ↑(f.toFun (f.ind x hx)) y
⊢ x = y
Module: Mathlib.GroupTheory.GroupAction.CardCommute

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
this :
  ∀ (ω : Quotient (orbitRel α β)),
    Fintype.card α / Fintype.card ↥(stabilizer α (φ ω)) = Fintype.card (α ⧸ stabilizer α (φ ω))
⊢ Fintype.card β = ∑ ω, Fintype.card α / Fintype.card ↥(stabilizer α (φ ω))

Tactic:
this,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : Group α
inst✝⁴ : MulAction α β
inst✝³ : Fintype α
inst✝² : Fintype β
inst✝¹ : Fintype (Quotient (orbitRel α β))
inst✝ : (b : β) → Fintype ↥(stabilizer α b)
φ : Quotient (orbitRel α β) → β
hφ : LeftInverse Quotient.mk'' φ
this :
  ∀ (ω : Quotient (orbitRel α β)),
    Fintype.card α / Fintype.card ↥(stabilizer α (φ ω)) = Fintype.card (α ⧸ stabilizer α (φ ω))
⊢ Fintype.card β = ∑ x, Fintype.card (α ⧸ stabilizer α (φ x))
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
x : M
hU : ContMDiffAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) x
hV : ContMDiffAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) x
hW : ContMDiffAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) x
⊢ mlieBracket I U (mlieBracket I V W) x = mlieBracket I (mlieBracket I U V) W x + mlieBracket I V (mlieBracket I U W) x

Tactic:
simp only [← mlieBracketWithin_univ, ← contMDiffWithinAt_univ] at hU hV hW ⊢

Goal state after:
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
x : M
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) univ x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) univ x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) univ x
⊢ mlieBracketWithin I U (mlieBracketWithin I V W univ) univ x =
    mlieBracketWithin I (mlieBracketWithin I U V univ) W univ x +
      mlieBracketWithin I V (mlieBracketWithin I U W univ) univ x
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
⊢ ∀ (a : DihedralGroup n), 1 * a = a

Tactic:
rintro (a | a)

Goal state after:
case r
n : ℕ
a : ZMod n
⊢ 1 * r a = r a

case sr
n : ℕ
a : ZMod n
⊢ 1 * sr a = sr a
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
⊢ ∀ (a : DihedralGroup n), a * 1 = a

Tactic:
rintro (a | a)

Goal state after:
case r
n : ℕ
a : ZMod n
⊢ r a * 1 = r a

case sr
n : ℕ
a : ZMod n
⊢ sr a * 1 = sr a
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
⊢ ∀ (a : DihedralGroup n), a.inv * a = 1

Tactic:
rintro (a | a)

Goal state after:
case r
n : ℕ
a : ZMod n
⊢ (r a).inv * r a = 1

case sr
n : ℕ
a : ZMod n
⊢ (sr a).inv * sr a = 1
Module: Mathlib.GroupTheory.PresentedGroup

Goal state before:
pp: α : Type u_1
rels : Set (FreeGroup α)
this : of = ⇑(QuotientGroup.mk' (Subgroup.normalClosure rels)) ∘ FreeGroup.of
⊢ Subgroup.closure (Set.range of) = ⊤

Tactic:
this,

Goal state after:
α : Type u_1
rels : Set (FreeGroup α)
this : of = ⇑(QuotientGroup.mk' (Subgroup.normalClosure rels)) ∘ FreeGroup.of
⊢ Subgroup.closure (Set.range (⇑(QuotientGroup.mk' (Subgroup.normalClosure rels)) ∘ FreeGroup.of)) = ⊤
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : cs.simple = ⇑cs.mulEquiv.symm ∘ PresentedGroup.of
⊢ Subgroup.closure (Set.range cs.simple) = ⊤

Tactic:
this,

Goal state after:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : cs.simple = ⇑cs.mulEquiv.symm ∘ PresentedGroup.of
⊢ Subgroup.closure (Set.range (⇑cs.mulEquiv.symm ∘ PresentedGroup.of)) = ⊤
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : Set.range cs.simple = Set.range cs.simple ∪ (Set.range cs.simple)⁻¹
⊢ Submonoid.closure (Set.range cs.simple) = ⊤

Tactic:
this,

Goal state after:
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
this : Set.range cs.simple = Set.range cs.simple ∪ (Set.range cs.simple)⁻¹
⊢ Submonoid.closure (Set.range cs.simple ∪ (Set.range cs.simple)⁻¹) = ⊤