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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E₁ : Type u_2
inst✝⁵ : NormedAddCommGroup E₁
inst✝⁴ : NormedSpace 𝕜 E₁
H : Type u_6
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E₁ H
M : Type u_10
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
s : TopologicalSpace.Opens M
⊢ IsSmoothEmbedding I I n Subtype.val

Tactic:
rw [isSmoothEmbedding_iff]
  exact ⟨IsImmersion.of_opens s, IsEmbedding.subtypeVal⟩

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

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
A : (mfderiv[Icc x y] (Subtype.val ∘ projIcc x y ⋯) ↑z) 1 = (mfderiv[Icc x y] id ↑z) 1
⊢ (mfderiv% Subtype.val z) 1 = 1

Tactic:
← mfderivWithin_comp_projIcc_one,

Goal state after:
x y : ℝ
h : Fact (x < y)
z : ↑(Icc x y)
A : (mfderiv[Icc x y] (Subtype.val ∘ projIcc x y ⋯) ↑z) 1 = (mfderiv[Icc x y] id ↑z) 1
⊢ (mfderiv[Icc x y] (Subtype.val ∘ projIcc x y ⋯) ↑z) 1 = 1
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: case e'_3.e'_6
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
hvy' : ⟪a • v, ↑y⟫_ℝ = 0
⊢ ‖y‖ ^ 2 = ‖↑y‖ * ‖↑y‖

Tactic:
exact sq _

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

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveOn γ v s
t : ℝ
ht : t ∈ s
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t)) =
    ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))

Tactic:
rw [ContinuousLinearMap.ext_iff]

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
s : Set ℝ
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveOn γ v s
t : ℝ
ht : t ∈ s
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ ∀ (x : TangentSpace 𝓘(ℝ, ℝ) t),
    (mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t))) x =
      (ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))) x
Module: Mathlib.Geometry.Manifold.IntegralCurve.Basic

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt% γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t)) =
    ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))

Tactic:
rw [ContinuousLinearMap.ext_iff]

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
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : IsManifold I 1 M
hγ : IsMIntegralCurveAt γ v t₀
t : ℝ
ht1 : γ ⁻¹' (extChartAt I (γ t₀)).source ∈ 𝓝 t
ht2 : HasMFDerivAt% γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))
hsrc : γ t ∈ (chartAt H (γ t₀)).source
⊢ ∀ (x : TangentSpace 𝓘(ℝ, ℝ) t),
    (mfderiv% ↑(chartAt H (γ t₀)) (γ t) ∘SL ContinuousLinearMap.smulRight 1 (v (γ t))) x =
      (ContinuousLinearMap.toSpanSingleton ℝ ((tangentCoordChange I (γ t) (γ t₀) (γ t)) (v (γ t)))) x
Module: Mathlib.MeasureTheory.Constructions.UnitInterval

Goal state before:
pp: x y : ↑I
⊢ volume (uIcc x y) = edist y x

Tactic:
by
  simp only [uIcc, volume_apply, image_subtype_val_Icc, Icc.coe_inf, Icc.coe_sup, Real.volume_Icc,
    max_sub_min_eq_abs, edist_dist, Subtype.dist_eq, Real.dist_eq]

Goal state after:
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 : ℝ
h : Fact (a < b)
γ : ↑(Icc a b) → M
this : MeasurePreserving Subtype.val volume (volume.restrict (Icc a b))
⊢ ∫⁻ (t : ↑(Icc a b)), ‖(mfderiv[Icc a b] (γ ∘ projIcc a b ⋯) ↑t) 1‖ₑ =
    ∫⁻ (t : ℝ) in Icc a b, ‖(mfderiv[Icc a b] (γ ∘ projIcc a b ⋯) t) 1‖ₑ

Tactic:
rw [← MeasurePreserving.lintegral_comp_emb this
    (MeasurableEmbedding.subtype_coe measurableSet_Icc)]

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 : ℝ
h : Fact (a < b)
γ : ↑(Icc a b) → M
this : MeasurePreserving Subtype.val volume (volume.restrict (Icc a b))
⊢ ∫⁻ (t : ↑(Icc a b)), ‖(mfderiv[Icc a b] (γ ∘ projIcc a b ⋯) ↑t) 1‖ₑ =
    ∫⁻ (a_1 : { x // x ∈ Icc a b }), ‖(mfderiv[Icc a b] (γ ∘ projIcc a b ⋯) ↑a_1) 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 : M
r : ℝ≥0∞
a b : ℝ
hr : riemannianEDist I x y < r
hab : a < b
⊢ ∃ γ,
    γ a = x ∧
      γ b = y ∧ ContMDiff 𝓘(ℝ, ℝ) I 1 γ ∧ pathELength I γ a b < r ∧ (γ =ᶠ[𝓝 a] fun x_1 ↦ x) ∧ γ =ᶠ[𝓝 b] fun x ↦ y

Tactic:
by
  /- We start from a path from `x` to `y` defined on `[0, 1]` with length `< r`. Then, we
  reparameterize it using a smooth monotone map `η` from `[a, b]` to `[0, 1]` which is moreover
  locally constant around `a` and `b`.
  Such a map is easy to build with `Real.smoothTransition`.

  Note that this is a very standard construction in differential topology.
  TODO: refactor once we have more differential topology in Mathlib and this gets duplicated. -/
  rcases exists_lt_of_riemannianEDist_lt hr with ⟨γ, hγx, hγy, γ_smooth, hγ⟩
  rcases exists_between hab with ⟨a', haa', ha'b⟩
  rcases exists_between ha'b with ⟨b', ha'b', hb'b⟩
  let η (t : ℝ) : ℝ := Real.smoothTransition ((b' - a')⁻¹ * (t - a'))
  have A (t) (ht : t < a') : η t = 0 := by
    simp only [η, Real.smoothTransition.zero_iff_nonpos]
    apply mul_nonpos_of_nonneg_of_nonpos
    · simpa using ha'b'.le
    · linarith
  have A' (t) (ht : t < a') : (γ ∘ η) t = x := by simp [A t ht, hγx]
  have B (t) (ht : b' < t) : η t = 1 := by
    simp only [η, Real.smoothTransition.eq_one_iff_one_le, inv_mul_eq_div]
    rw [one_le_div₀] <;> linarith
  have B' (t) (ht : b' < t) : (γ ∘ η) t = y := by simp [B t ht, hγy]
  refine ⟨γ ∘ η, A' _ haa', B' _ hb'b, ?_, ?_, ?_, ?_⟩
  · rw [← contMDiffOn_univ]
    apply γ_smooth.comp
    · rw [contMDiffOn_univ, contMDiff_iff_contDiff]
      fun_prop
    · intro t ht
      exact ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
  · convert! hγ using 1
    rw [← A a haa', ← B b hb'b]
    apply pathELength_comp_of_monotoneOn hab.le
    · apply Monotone.monotoneOn
      apply Real.smoothTransition.monotone.comp
      intro t u htu
      dsimp only
      gcongr
    · simp only [η]
      apply (ContDiff.contDiffOn _).differentiableOn one_ne_zero
      fun_prop
    · rw [A a haa', B b hb'b]
      apply γ_smooth.mdifferentiableOn one_ne_zero
  · filter_upwards [Iio_mem_nhds haa'] with t ht using A' t ht
  · filter_upwards [Ioi_mem_nhds hb'b] with t ht using B' t ht

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

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, 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
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
      r

Tactic:
let g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦
    inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)

Goal state after:
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, 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
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) x ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) y‖ <
      r
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
m n : ℕ∞
inst✝ : IsManifold I (↑n + 1) M
U V : (x : M) → TangentSpace I x
x : M
hU : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, U x⟩) x
hV : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, V x⟩) x
hmn : minSmoothness 𝕜 (↑m + 1) ≤ ↑n
⊢ ContMDiffAt I I.tangent (↑m) (fun x₀ ↦ ⟨x₀, mlieBracket I U V x₀⟩) x

Tactic:
by
  simp only [← contMDiffWithinAt_univ, ← mlieBracketWithin_univ] at hU hV ⊢
  exact hU.mlieBracketWithin_vectorField hV uniqueMDiffOn_univ (mem_univ _) hmn

Goal state after:
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
⊢ ↑(h.codChart.extend J) ∘ f ∘ ↑(h.domChart.extend I).symm '' (h.domChart.extend I).target ⊆
    (h.codChart.extend J).target

Tactic:
Set.image_comp,

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
⊢ ↑(h.codChart.extend J) '' f ∘ ↑(h.domChart.extend I).symm '' (h.domChart.extend I).target ⊆
    (h.codChart.extend J).target
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
⊢ ↑(h.codChart.extend J) '' f ∘ ↑(h.domChart.extend I).symm '' (h.domChart.extend I).target ⊆
    (h.codChart.extend J).target

Tactic:
Set.image_comp,

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
⊢ ↑(h.codChart.extend J) '' f '' ↑(h.domChart.extend I).symm '' (h.domChart.extend I).target ⊆
    (h.codChart.extend J).target
Module: Mathlib.Geometry.Manifold.VectorBundle.LocalFrame

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → Module 𝕜 (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : VectorBundle 𝕜 F V
inst✝¹ : ContMDiffVectorBundle n F V I
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
i : ι
⊢ ContMDiffOn I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, e.localFrame b i x⟩) e.baseSet

Tactic:
rw [e.contMDiffOn_section_baseSet_iff]

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
F : Type u_5
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁷ : TopologicalSpace (TotalSpace F V)
inst✝⁶ : (x : M) → AddCommGroup (V x)
inst✝⁵ : (x : M) → Module 𝕜 (V x)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : VectorBundle 𝕜 F V
inst✝¹ : ContMDiffVectorBundle n F V I
ι : Type u_7
e : Trivialization F TotalSpace.proj
inst✝ : MemTrivializationAtlas e
b : Basis ι 𝕜 F
i : ι
⊢ ContMDiffOn I 𝓘(𝕜, F) n (fun x ↦ (↑e ⟨x, e.localFrame b i x⟩).2) e.baseSet
Module: Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov cov' : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
hcov' : IsCovariantDerivativeOn F cov' s
x : M
hx : x ∈ s
σ✝ σ'✝ : (x : M) → V x
hσ : MDiffAt (T% σ✝) x
hσ' : MDiffAt (T% σ'✝) x
⊢ differenceAux cov cov' (σ✝ + σ'✝) x = differenceAux cov cov' σ✝ x + differenceAux cov cov' σ'✝ x

Tactic:
simp [differenceAux, hcov.add hσ hσ', hcov'.add hσ hσ']

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
F : Type u_5
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
V : M → Type u_6
inst✝⁶ : TopologicalSpace (TotalSpace F V)
inst✝⁵ : (x : M) → AddCommGroup (V x)
inst✝⁴ : (x : M) → Module 𝕜 (V x)
inst✝³ : (x : M) → TopologicalSpace (V x)
inst✝² : ∀ (x : M), IsTopologicalAddGroup (V x)
inst✝¹ : ∀ (x : M), ContinuousSMul 𝕜 (V x)
inst✝ : FiberBundle F V
cov cov' : ((x : M) → V x) → (x : M) → TangentSpace I x →L[𝕜] V x
s : Set M
hcov : IsCovariantDerivativeOn F cov s
hcov' : IsCovariantDerivativeOn F cov' s
x : M
hx : x ∈ s
σ✝ σ'✝ : (x : M) → V x
hσ : MDiffAt (T% σ✝) x
hσ' : MDiffAt (T% σ'✝) x
⊢ cov σ✝ x + cov σ'✝ x - (cov' σ✝ x + cov' σ'✝ x) = cov σ✝ x - cov' σ✝ x + (cov σ'✝ x - cov' σ'✝ x)
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

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

Tactic:
rintro (a | a)
    · exact congr_arg r (zero_add a)
    · exact congr_arg sr (sub_zero a)

Goal state after:
Module: Mathlib.GroupTheory.Coxeter.Basic

Goal state before:
pp: case mul_left
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
p : W → Prop
w : W
one : p 1
mul_simple_left : ∀ (w : W) (i : B), p w → p (cs.simple i * w)
p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop := fun w x ↦ p w
i : W
mi : ∃ y, cs.simple y = i
y : W
my : y ∈ Submonoid.closure (Set.range cs.simple)
ih : p y
⊢ p (i * y)

Tactic:
exact mi.choose_spec ▸ mul_simple_left _ _ ih

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

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, 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
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
      r

Tactic:
let g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦
    inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)

Goal state after:
B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
h'δ : 1 + δ * C < r' ^ 2
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, 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
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
⊢ ∀ᶠ (y : B) in 𝓝 x,
    ‖Trivialization.symmL ℝ (trivializationAt F E x) y ∘SL
          Trivialization.continuousLinearMapAt ℝ (trivializationAt F E x) x‖ <
      r
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
w : Word M
hnil : ¬w.toList = []
m₂ m₁ : M (w.toList.head hnil).fst
hw : ⟨(w.toList.head hnil).fst, m₁⟩ ∉ w.toList.tail
hm1 : ¬m₁ = 1
hm' : (∀ (x : M (w.toList.head hnil).fst), ¬w.toList.head? = some ⟨(w.toList.head hnil).fst, x⟩) ∧ m₁ = m₂
⊢ w.toList.head? = some ⟨(w.toList.head hnil).fst, (w.toList.head hnil).snd⟩

Tactic:
rw [List.head?_eq_some_head]

Goal state after:
Module: Mathlib.GroupTheory.Coxeter.Inversion

Goal state before:
pp: B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
t : W
ht : t ∈ cs.rightInvSeq ω.reverse
⊢ cs.IsReflection t

Tactic:
exact cs.isReflection_of_mem_rightInvSeq ω.reverse ht

Goal state after:
Module: Mathlib.GroupTheory.FiniteAbelian.Duality

Goal state before:
pp: G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
ι : Type
w✝ : Fintype ι
n : ι → ℕ
h₁ : ∀ (i : ι), 1 < n i
h₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h₂.some
e' : (((i : ι) → Multiplicative (ZMod (n i))) →* Mˣ) ≃* ((i : ι) → Multiplicative (ZMod (n i)) →* Mˣ) :=
  Pi.monoidHomMulEquiv (fun i ↦ Multiplicative (ZMod (n i))) Mˣ
this : ∀ (i : ι), NeZero (n i)
inst : ∀ (i : ι), HasEnoughRootsOfUnity M (Nat.card (Multiplicative (ZMod (n i))))
⊢ Nonempty ((G →* Mˣ) ≃* G)

Tactic:
let E i := (IsCyclic.monoidHom_equiv_self (Multiplicative (ZMod (n i))) M).some

Goal state after:
G : Type u_1
M : Type u_2
inst✝² : CommGroup G
inst✝¹ : Finite G
inst✝ : CommMonoid M
hM : HasEnoughRootsOfUnity M (Monoid.exponent G)
ι : Type
w✝ : Fintype ι
n : ι → ℕ
h₁ : ∀ (i : ι), 1 < n i
h₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))
e : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) := h₂.some
e' : (((i : ι) → Multiplicative (ZMod (n i))) →* Mˣ) ≃* ((i : ι) → Multiplicative (ZMod (n i)) →* Mˣ) :=
  Pi.monoidHomMulEquiv (fun i ↦ Multiplicative (ZMod (n i))) Mˣ
this : ∀ (i : ι), NeZero (n i)
inst : ∀ (i : ι), HasEnoughRootsOfUnity M (Nat.card (Multiplicative (ZMod (n i))))
E : (i : ι) → (Multiplicative (ZMod (n i)) →* Mˣ) ≃* Multiplicative (ZMod (n i)) := fun i ↦ ⋯.some
⊢ Nonempty ((G →* Mˣ) ≃* G)