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

Goal state before:
pp: case refine_1
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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
hu : Set.univ = {j₁, j₂, j₃}
⊢ {t₁.orthocenter, t₁.points i₂, t₂.points j₃} ⊆ ↑(affineSpan ℝ (Set.range t₁.points))

Tactic:
h₃,

Goal state after:
case refine_1
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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
hu : Set.univ = {j₁, j₂, j₃}
⊢ {t₁.orthocenter, t₁.points i₂, t₁.points i₃} ⊆ ↑(affineSpan ℝ (Set.range t₁.points))
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case right
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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points)
hu : {j₂}ᶜ = {j₁, j₃}
hle : (altitude t₁ i₃).directionᗮ ≤ line[ℝ, t₁.orthocenter, t₁.points i₃].directionᗮ
hui : {i₃}ᶜ = {i₁, i₂}
⊢ t₁.points i₁ -ᵥ t₁.points i₂ ∈ vectorSpan ℝ (t₁.points '' {i₃}ᶜ)

Tactic:
rw [hui, Set.image_insert_eq, Set.image_singleton]

Goal state after:
case right
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
t₁ t₂ : Triangle ℝ P
i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3
hi₁₂ : i₁ ≠ i₂
hi₁₃ : i₁ ≠ i₃
hi₂₃ : i₂ ≠ i₃
hj₁₂ : j₁ ≠ j₂
hj₁₃ : j₁ ≠ j₃
hj₂₃ : j₂ ≠ j₃
h₁ : t₂.points j₁ = t₁.orthocenter
h₂ : t₂.points j₂ = t₁.points i₂
h₃ : t₂.points j₃ = t₁.points i₃
he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points)
hu : {j₂}ᶜ = {j₁, j₃}
hle : (altitude t₁ i₃).directionᗮ ≤ line[ℝ, t₁.orthocenter, t₁.points i₃].directionᗮ
hui : {i₃}ᶜ = {i₁, i₂}
⊢ t₁.points i₁ -ᵥ t₁.points i₂ ∈ vectorSpan ℝ {t₁.points i₁, t₁.points i₂}
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : WithTop ℕ∞
inst✝ : IsManifold I n M
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
φ : PartialEquiv E E := extendCoordChange e e'
hφ : ContDiffOn 𝕜 n (↑φ) φ.source
h : Surjective ⇑(fderivWithin 𝕜 (↑φ) φ.source (↑(e.extend I) x)) → ↑(e'.extend I) x ∈ interior (range ↑I)
⊢ ↑(e'.symm.restr I.source).symm x ∈ e'.target

Tactic:
by simp [hex']

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
⊢ EqOn (norm ∘ f) (const M ‖f c‖) U

Tactic:
set V := {z ∈ U | ‖f z‖ = ‖f c‖}

Goal state after:
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‖}
⊢ EqOn (norm ∘ f) (const M ‖f c‖) U
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
⊢ EqOn (norm ∘ f) (const M ‖f c‖) U

Tactic:
by
  set V := {z ∈ U | ‖f z‖ = ‖f c‖}
  suffices U ⊆ V from fun x hx ↦ (this hx).2
  have hVo : IsOpen V := by
    refine isOpen_iff_mem_nhds.2 fun x hx ↦ inter_mem (ho.mem_nhds hx.1) ?_
    replace hm : IsLocalMax (‖f ·‖) x :=
      mem_of_superset (ho.mem_nhds hx.1) fun z hz ↦ (hm hz).out.trans_eq hx.2.symm
    replace hd : ∀ᶠ y in 𝓝 x, MDiffAt f y :=
      (eventually_mem_nhds_iff.2 (ho.mem_nhds hx.1)).mono fun z ↦ hd.mdifferentiableAt
    exact (Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax hd hm).mono fun _ ↦
      (Eq.trans · hx.2)
  have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, rfl⟩
  set W := U ∩ {z | ‖f z‖ = ‖f c‖}ᶜ
  have hWo : IsOpen W := hd.continuousOn.norm.isOpen_inter_preimage ho isOpen_ne
  have hdVW : Disjoint V W := disjoint_compl_right.mono inf_le_right inf_le_right
  have hUVW : U ⊆ V ∪ W := fun x hx => (eq_or_ne ‖f x‖ ‖f c‖).imp (.intro hx) (.intro hx)
  exact hc.subset_left_of_subset_union hVo hWo hdVW hUVW hVne

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: 𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : TopologicalSpace (TotalSpace F E)
inst✝⁶ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁵ : NormedAddCommGroup EB
inst✝⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝² : TopologicalSpace B
inst✝¹ : ChartedSpace HB B
inst✝ : FiberBundle F E
s : (b : B) → E b
u : Set B
b₀ : B
⊢ MDiffAt[u] (T% s) b₀ ↔ (MDiffAt[u] fun b ↦ (↑(trivializationAt F E b₀) ⟨b, s b⟩).2) b₀

Tactic:
rw [mdifferentiableWithinAt_totalSpace]
  change MDifferentiableWithinAt _ _ id _ _ ∧ _ ↔ _
  simp [mdifferentiableWithinAt_id]

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable

Goal state before:
pp: case refine_1.h
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace (TotalSpace F E)
inst✝¹¹ : (x : B) → TopologicalSpace (E x)
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : FiberBundle F E
inst✝⁶ : (x : B) → AddCommGroup (E x)
inst✝⁵ : (x : B) → Module 𝕜 (E x)
inst✝⁴ : VectorBundle 𝕜 F E
HB : Type u_7
inst✝³ : TopologicalSpace HB
inst✝² : ChartedSpace HB B
EB : Type u_8
inst✝¹ : NormedAddCommGroup EB
inst✝ : NormedSpace 𝕜 EB
I : ModelWithCorners 𝕜 EB HB
f : B → 𝕜
s : (x : B) → E x
u : Set B
x₀ : B
hf : MDiffAt[u] f x₀
e : Trivialization F TotalSpace.proj := ⋯
hs : (MDiffAt[u] fun b ↦ (↑e ⟨b, s b⟩).2) x₀
⊢ ∀ x ∈ e.baseSet, (fun b ↦ (↑e ⟨b, (f • s) b⟩).2) x = (fun p ↦ f p • (↑e ⟨p, s p⟩).2) x

Tactic:
exact fun x hx ↦ (e.linear 𝕜 hx).2 ..

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : M
⊢ MDiffAt 0 x

Tactic:
by exact mdifferentiable_const ..

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

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝¹ : EuclideanHalfSpace n
xval : EuclideanSpace ℝ (Fin n)
xprop : 0 ≤ xval.ofLp 0
x✝ : ⟨xval, xprop⟩ ∈ univ
⊢ ⟨toLp 2 (update (↑⟨xval, xprop⟩).ofLp 0 (max ((↑⟨xval, xprop⟩).ofLp 0) 0)), ⋯⟩ = ⟨xval, xprop⟩

Tactic:
Subtype.mk_eq_mk,

Goal state after:
n : ℕ
inst✝ : NeZero n
x✝¹ : EuclideanHalfSpace n
xval : EuclideanSpace ℝ (Fin n)
xprop : 0 ≤ xval.ofLp 0
x✝ : ⟨xval, xprop⟩ ∈ univ
⊢ toLp 2 (update (↑⟨xval, xprop⟩).ofLp 0 (max ((↑⟨xval, xprop⟩).ofLp 0) 0)) = xval
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ℝ
hz : z ∈ Icc x y
h'z : ↑(projIcc x y ⋯ z) < y
this :
  ContDiff ℝ n fun w ↦
    have this := toLp 2 fun x_1 ↦ w - x;
    this
w : ℝ
hw : w ∈ Icc x y
i : Fin 1
⊢ x ≤ min y w

Tactic:
simp [hw.1, h.out.le]

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

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
n : WithTop ℕ∞
z : ℝ
hz : z ∈ Icc x y
h'z : ¬↑(projIcc x y ⋯ z) < y
this :
  ContDiff ℝ n fun w ↦
    have this := toLp 2 fun x ↦ y - w;
    this
w : ℝ
hw : w ∈ Icc x y
i : Fin 1
⊢ x ≤ min y w

Tactic:
simp [hw.1, h.out.le]

Goal state after:
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
w : E
hw : w ∈ (ℝ ∙ v)ᗮ
h₁ : 0 < ‖w‖ ^ 2 + 4
this : ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ ^ 2 = (‖w‖ ^ 2 + 4) ^ 2
⊢ ‖4 • w + (‖w‖ ^ 2 - 4) • v‖ = ‖w‖ ^ 2 + 4

Tactic:
by
    simpa only [sq_eq_sq_iff_abs_eq_abs, abs_norm, abs_of_pos h₁] using this

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

Goal state before:
pp: case mpr
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
⊢ ∃ v_1 ∈ 𝓝 t₀, ∀ y ∈ v_1, HasMFDerivAt% γ y (ContinuousLinearMap.smulRight 1 (v (γ y)))

Tactic:
refine ⟨s', h2.mem_nhds h3, ?_⟩

Goal state after:
case mpr
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
s : Set ℝ
hs : s ∈ 𝓝 t₀
h : IsMIntegralCurveOn γ v s
s' : Set ℝ
h1 : s' ⊆ s
h2 : IsOpen s'
h3 : t₀ ∈ s'
⊢ ∀ y ∈ s', HasMFDerivAt% γ y (ContinuousLinearMap.smulRight 1 (v (γ y)))
Module: Mathlib.Geometry.Manifold.IntegralCurve.Transform

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
γ : ℝ → M
v : (x : M) → TangentSpace I x
t₀ a : ℝ
ha : a ≠ 0
⊢ IsMIntegralCurveAt (γ ∘ fun x ↦ x * a) (a • v) (t₀ / a) ↔ IsMIntegralCurveAt γ v t₀

Tactic:
by
  refine ⟨fun hγ ↦ ?_, fun hγ ↦ hγ.comp_mul_ne_zero ha⟩
  convert! hγ.comp_mul_ne_zero (inv_ne_zero ha)
  · ext t
    simp only [Function.comp_apply, mul_assoc, inv_mul_eq_div, div_self ha, mul_one]
  · simp only [smul_smul, inv_mul_eq_div, div_self ha, one_smul]
  · simp only [div_inv_eq_mul, div_mul_cancel₀ _ ha]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f s
hmn : m + 1 ≤ n
hs : UniqueMDiff[s]
x₀ : TangentBundle I M
hx₀ : x₀ ∈ TotalSpace.proj ⁻¹' s
s' : Set (TangentBundle I M) := TotalSpace.proj ⁻¹' s
b₁ : TangentBundle I M → M := fun p ↦ p.proj
v : (y : TangentBundle I M) → TangentSpace I (b₁ y) := fun y ↦ y.snd
hv : ContMDiffWithinAt I.tangent I.tangent m (fun y ↦ ⟨b₁ y, v y⟩) s' x₀
b₂ : TangentBundle I M → M' := f ∘ b₁
⊢ ContMDiffWithinAt I.tangent I'.tangent m (tangentMapWithin I I' f s) (TotalSpace.proj ⁻¹' s) x₀

Tactic:
have hb₂ : CMDiffAt[s'] m b₂ x₀ :=
    ((hf (b₁ x₀) hx₀).of_le (le_self_add.trans hmn)).comp _
      (contMDiffWithinAt_proj (TangentSpace I)) (fun x h ↦ h)

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
s : Set M
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f s
hmn : m + 1 ≤ n
hs : UniqueMDiff[s]
x₀ : TangentBundle I M
hx₀ : x₀ ∈ TotalSpace.proj ⁻¹' s
s' : Set (TangentBundle I M) := TotalSpace.proj ⁻¹' s
b₁ : TangentBundle I M → M := fun p ↦ p.proj
v : (y : TangentBundle I M) → TangentSpace I (b₁ y) := fun y ↦ y.snd
hv : ContMDiffWithinAt I.tangent I.tangent m (fun y ↦ ⟨b₁ y, v y⟩) s' x₀
b₂ : TangentBundle I M → M' := f ∘ b₁
hb₂ : ContMDiffWithinAt I.tangent I' m b₂ s' x₀
⊢ ContMDiffWithinAt I.tangent I'.tangent m (tangentMapWithin I I' f s) (TotalSpace.proj ⁻¹' s) x₀
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case e'_22
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : m + 1 ≤ n
⊢ tangentMap I I' f = tangentMapWithin I I' f univ

Tactic:
tangentMapWithin_univ

Goal state after:
case e'_22
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : m + 1 ≤ n
⊢ tangentMap I I' f = tangentMap I I' f
Module: Mathlib.Geometry.Manifold.ContMDiffMFDeriv

Goal state before:
pp: case e'_5
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : 1 ≤ n
⊢ tangentMap I I' f = tangentMapWithin I I' f univ

Tactic:
tangentMapWithin_univ

Goal state after:
case e'_5
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
f : M → M'
Is : IsManifold I 1 M
I's : IsManifold I' 1 M'
hf : ContMDiffOn I I' n f univ
hmn : 1 ≤ n
⊢ tangentMap I I' f = tangentMap I I' f
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
n : ℕ∞ω
inst✝ : IsManifold I (n + 1) M
m : ℕ∞ω
U V : (x : M) → TangentSpace I x
s : Set M
x : M
hU : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, V x⟩) s x
hs : UniqueMDiff[s]
hx : x ∈ s
hmn : minSmoothness 𝕜 (m + 1) ≤ n
⊢ minSmoothness 𝕜 2 ≤ minSmoothness 𝕜 (m + (1 + 1))

Tactic:
exact minSmoothness_monotone le_add_self

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

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 : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
c : ℝ≥0∞
hc : 0 < c
⊢ ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y < c

Tactic:
by
  rcases eventually_riemannianEDist_le_edist_extChartAt I x with ⟨C, C_pos, hC⟩
  have : (extChartAt I x) ⁻¹' (Metric.eball (extChartAt I x x) (c / C)) ∈ 𝓝 x := by
    apply (continuousAt_extChartAt x).preimage_mem_nhds
    exact Metric.eball_mem_nhds _ (ENNReal.div_pos hc.ne' (by simp))
  filter_upwards [this, hC] with y hy h'y
  apply h'y.trans_lt
  have : edist (extChartAt I x x) (extChartAt I x y) < c / C := by
    simpa only [mem_preimage, Metric.mem_eball'] using hy
  rwa [ENNReal.lt_div_iff_mul_lt, mul_comm] at this
  · exact Or.inl (mod_cast C_pos.ne')
  · simp

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
⊢ (noncenter G).toFinsetᶜ.card = (Finset.univ \ (noncenter G).toFinset).card

Tactic:
Finset.compl_eq_univ_sdiff

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
val✝ : Fintype G
⊢ (Finset.univ \ (noncenter G).toFinset).card = (Finset.univ \ (noncenter G).toFinset).card
Module: Mathlib.GroupTheory.Commutator.Finite

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : Finite ↑(commutatorSet G)
inst✝ : Group.FG G
⊢ (center G).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank G

Tactic:
by
  obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
  rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]
  exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)

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

Goal state before:
pp: case intro
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
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance
this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance
eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯
x : M
⊢ Injective ⇑(mfderiv% (⇑eEF ∘ ⇑f.embeddingPiTangent) x)

Tactic:
rw [mfderiv_comp _ eEF.differentiableAt.mdifferentiableAt
      (f.embeddingPiTangent.contMDiff.mdifferentiableAt (by simp)),
    eEF.mfderiv_eq]

Goal state after:
case intro
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
ι : Type u_1
inst✝ : Finite ι
f : SmoothBumpCovering ι I M
val✝ : Fintype ι
F : Type := EuclideanSpace ℝ (Fin (finrank ℝ (ι → E × ℝ)))
this✝ : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.mpr inferInstance
this : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.mp inferInstance
eEF : (ι → E × ℝ) ≃L[ℝ] F := ContinuousLinearEquiv.ofFinrankEq ⋯
x : M
⊢ Injective ⇑(↑eEF ∘SL mfderiv% ⇑f.embeddingPiTangent x)
Module: Mathlib.GroupTheory.SpecificGroups.Dihedral

Goal state before:
pp: n : ℕ
x✝¹ : n + 3 ≠ 1
x✝ : n + 3 ≠ 2
h' : IsMulCommutative (DihedralGroup (n + 3))
⊢ False

Tactic:
have := h'.is_comm.comm (r 1) (sr 0)
    rw [r_mul_sr, zero_sub, sr_mul_r, zero_add, sr.injEq, neg_eq_iff_add_eq_zero,
      one_add_one_eq_two, ← ZMod.val_eq_zero, ZMod.val_two_eq_two_mod] at this
    simpa using Nat.le_of_dvd Nat.zero_lt_two <| Nat.dvd_of_mod_eq_zero this

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: ι : Type u_1
G : ι → Type u_4
inst✝¹ : (i : ι) → Group (G i)
N : Type u_5
inst✝ : Group N
f : (i : ι) → G i →* N
i : ι
x : G i
⊢ (lift f) (of x) = (f i) x

Tactic:
by simp only [lift_of]

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

Goal state before:
pp: B : Type u_1
B' : Type u_2
e : B ≃ B'
W : Type u_3
H : Type u_4
inst✝¹ : Group W
inst✝ : Group H
M : CoxeterMatrix B
cs : CoxeterSystem M W
i j : B
p k : ℕ
hk : k < p
⊢ k < (alternatingWord i j p).length

Tactic:
simp [hk]

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

Goal state before:
pp: case neg
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
ih :
  cs.wordProd (alternatingWord i i' m) = (if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)
hm : ¬Even m
h₁ : Even (m + 1)
h₂ : (m + 1) / 2 = m / 2 + 1
⊢ cs.simple (if Even m then i' else i) *
      ((if Even m then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ (m / 2)) =
    (if Even (m + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((m + 1) / 2)

Tactic:
simp [hm, h₁, h₂, ← pow_succ', ← mul_assoc]

Goal state after:
Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case inr
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)
i j : ι
w : Word M
m : M i
h : i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩
⊢ ⟨i, m⟩ ∈ w.toList

Tactic:
· revert h; cases w.toList <;> simp +contextual

Goal state after:
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
⊢ (if Even (2 * k + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((2 * k + 1) / 2) =
    (if Even (↑(M.M i i') * 2 - (2 * k + 1)) then 1 else cs.simple i) *
      (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') * 2 - (2 * k + 1)) / 2)

Tactic:
have : ¬Even (2 * k + 1) := Int.not_even_iff_odd.2 ⟨k, rfl⟩
    rw [if_neg this]
    have : ¬Even (↑(M i i') * 2 - (2 * k + 1)) :=
      Int.not_even_iff_odd.2 ⟨↑(M i i') - k - 1, by ring⟩
    rw [if_neg this]
    rw [(by ring : ↑(M i i') * 2 - (2 * k + 1) = -1 + (-k + ↑(M i i')) * 2),
      (by ring : 2 * k + 1 = 1 + k * 2)]
    repeat rw [Int.add_mul_ediv_right _ _ (by simp)]
    simp [zpow_add, simple_mul_simple_pow', ← inv_zpow, ← mul_assoc]

Goal state after: