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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ s.mongePoint = (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)

Tactic:
mongePoint_eq_smul_vsub_vadd_circumcenter,

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ (↑(n + 2 + 1) / ↑(n + 2 - 1)) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) +ᵥ s.circumcenter =
    (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6.pointIndex
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
n : ℕ
s : Simplex ℝ P (n + 2)
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) * ((if a✝ ∈ univ then (↑(#univ))⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹

Tactic:
rw [add_tsub_assoc_of_le (by decide : 1 ≤ 2), (by decide : 2 - 1 = 1)]

Goal state after:
case e_6.pointIndex
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
n : ℕ
s : Simplex ℝ P (n + 2)
a✝ : Fin (n + 2 + 1)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * ((if a✝ ∈ univ then (↑(#univ))⁻¹ else 0) - 0) + 0 = (↑(n + 1))⁻¹
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case e_6.circumcenterIndex
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 2 - 1) * (0 - 1) + 1 = -2 / ↑(n + 1)

Tactic:
rw [add_tsub_assoc_of_le (by decide : 1 ≤ 2), (by decide : 2 - 1 = 1)]

Goal state after:
case e_6.circumcenterIndex
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
n : ℕ
s : Simplex ℝ P (n + 2)
⊢ ↑(n + 2 + 1) / ↑(n + 1) * (0 - 1) + 1 = -2 / ↑(n + 1)
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: n : ℕ
i₁ i₂ : Fin (n + 3)
h : i₁ ≠ i₂
⊢ ∑ i, mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ i = 0

Tactic:
by
  rw [mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]
  simp_rw [Pi.sub_apply, sum_sub_distrib, sum_mongePointWeightsWithCircumcenter]
  rw [sum_centroidWeightsWithCircumcenter, sub_self]
  simp [← card_pos, card_compl, h]

Goal state after:
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
i : Fin (n + 3)
hi : i ∉ fs
⊢ i ≠ i₁ ∧ i ≠ i₂

Tactic:
constructor <;> · intro hj; simp [fs, ← hj] at hi

Goal state after:
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
⊢ Cospherical {p₁, p₂, p₃, p₄}

Tactic:
have hindep : AffineIndependent ℝ ![p₁, p, p₃] := affineIndependent_iff_not_collinear_set.mpr hn

Goal state after:
V : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
P : Type u_2
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ p : P
h : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p
hp₁p₂ : ∠ p₁ p p₂ = π
hp₃p₄ : ∠ p₃ p p₄ = π
hn : ¬Collinear ℝ {p₁, p, p₃}
hp₁p₂_sbtw : Sbtw ℝ p₁ p p₂
hp₃p₄_sbtw : Sbtw ℝ p₃ p p₄
hindep : AffineIndependent ℝ ![p₁, p, p₃]
⊢ Cospherical {p₁, p₂, p₃, p₄}
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
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
⊢ (s.mongePlane i₁ i₂).direction = (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)

Tactic:
by
  rw [mongePlane_def, direction_inf_of_mem_inf s.mongePoint_mem_mongePlane, direction_mk',
    direction_affineSpan]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ![a, b, c] ∼ ![a', b', c']
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
rw [similar_iff_exists_pos_dist_eq] at h
  rcases h with ⟨r, hr_pos, hdist⟩
  have h_ab : dist a b = r * dist a' b' := hdist 0 1
  have h_cb : dist c b = r * dist c' b' := hdist 2 1
  have h_ac : dist a c = r * dist a' c' := hdist 0 2
  have h_cos := law_cos a b c
  rw [h_ab, h_cb, h_ac] at h_cos
  field_simp at h_cos
  have h_cos' := law_cos a' b' c'
  field_simp at h_cos'
  rw [h_cos', sub_right_inj] at h_cos
  by_cases heq : dist a' b' * dist c' b' * 2 = 0
  · rw [mul_eq_zero_iff_right (by norm_num), mul_eq_zero] at heq
    rcases heq with h1 | h2
    · have h_dist_ab : dist a b = 0 := by grind
      rw [dist_eq_zero] at h_dist_ab h1
      simp_rw [h_dist_ab, h1, angle_self_left]
    · have h_dist_cb : dist c b = 0 := by grind
      rw [dist_eq_zero] at h_dist_cb h2
      simp_rw [h_dist_cb, h2, angle_self_right]
  rw [mul_right_inj' heq] at h_cos
  apply Real.injOn_cos at h_cos
  repeat grind [angle_nonneg, angle_le_pi]

Goal state after:
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ n ≠ ↑(closure ↑X)
⊢ X ^ n ⊂ X ^ (n + 1)

Tactic:
obtain rfl | hn := eq_or_ne n 0
  · simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]
      using! hX.not_subset_singleton
  refine (pow_subset_pow_right hX₁ <| n.le_add_right _).ssubset_of_ne ?_
  contrapose hXclosure with hXn
  rw [← closure_pow (mod_cast hX₁) hn]
  wlog hn₁ : n = 1
  · simp +contextual only [pow_one] at this
    replace hXn d : X ^ (n + d) = X ^ n := by
      induction d with
      | zero => rw [add_zero]
      | succ d hd =>
        rw [pow_add, pow_one] at hXn
        rw [← add_assoc, pow_add, pow_one, hd, ← hXn]
    exact mod_cast this (one_mem_pow hX₁) (hX.pow hn) one_ne_zero
      (by simp [hXn, ← pow_mul, mul_two]) (by simp)
  subst hn₁
  simp only [ne_eq, one_ne_zero, not_false_eq_true, Nat.reduceAdd, pow_one] at *
  let Xgp : Subgroup G :=
  { carrier := X
    mul_mem' := fun {x y} hx hy ↦ by
      norm_cast at *
      simpa [← hXn, ← sq] using! mul_mem_mul hx hy
    one_mem' := hX₁
    inv_mem' := fun {x} hx ↦ by
      norm_cast at *
      have : x • X ⊆ X := by
        simpa [← hXn, add_assoc, ← sq] using! smul_finset_subset_mul (t := X) hx
      have : x • X = X := eq_of_subset_of_card_le this (card_smul_finset ..).ge
      rw [← eq_inv_smul_iff] at this
      rw [this]
      simpa [mem_inv_smul_finset_iff] }
  exact subset_closure.antisymm <| (closure_le Xgp).2 subset_rfl

Goal state after:
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
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
⊢ t₁.points i₁ ≠ t₂.points j₂ ∧
    t₁.points i₁ ∈ affineSpan ℝ (Set.range t₂.points) ∧
      t₁.points i₁ -ᵥ t₂.points j₂ ∈ (affineSpan ℝ (t₂.points '' {j₂}ᶜ)).directionᗮ

Tactic:
h₂

Goal state after:
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
⊢ t₁.points i₁ ≠ t₁.points i₂ ∧
    t₁.points i₁ ∈ affineSpan ℝ (Set.range t₂.points) ∧
      t₁.points i₁ -ᵥ t₁.points i₂ ∈ (affineSpan ℝ (t₂.points '' {j₂}ᶜ)).directionᗮ
Module: Mathlib.Geometry.Manifold.LocalDiffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H₁ : Type u_5
inst✝⁵ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁴ : TopologicalSpace H₂
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
M : Type u_8
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H₁ M
N : Type u_9
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H₂ N
n : WithTop ℕ∞
f : M → N
hf : IsLocalDiffeomorph I J n f
⊢ IsLocalHomeomorph f

Tactic:
isLocalHomeomorph_iff_isLocalHomeomorphOn_univ

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H₁ : Type u_5
inst✝⁵ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁴ : TopologicalSpace H₂
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
M : Type u_8
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H₁ M
N : Type u_9
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace H₂ N
n : WithTop ℕ∞
f : M → N
hf : IsLocalDiffeomorph I J n f
⊢ IsLocalHomeomorphOn f univ
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
hn : n ≠ 0
x : M
hx : x ∈ ModelWithCorners.interior M
⊢ (chartAt H x).source ∩ ↑(extChartAt I x) ⁻¹' interior (extChartAt I x).target ⊆ ModelWithCorners.interior M

Tactic:
exact fun y hy ↦ (I.isInteriorPoint_iff_of_mem_atlas hn (chart_mem_atlas H x) hy.1).2 hy.2

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

Goal state before:
pp: 𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst✝³¹ : NontriviallyNormedField 𝕜
inst✝³⁰ : (x : B₁) → AddCommGroup (E₁ x)
inst✝²⁹ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝²⁸ : NormedAddCommGroup F₁
inst✝²⁷ : NormedSpace 𝕜 F₁
inst✝²⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁵ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝²⁴ : (x : B₂) → AddCommGroup (E₂ x)
inst✝²³ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝²² : NormedAddCommGroup F₂
inst✝²¹ : NormedSpace 𝕜 F₂
inst✝²⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁹ : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst✝¹⁸ : NormedAddCommGroup EB₁
inst✝¹⁷ : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst✝¹⁶ : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst✝¹⁵ : TopologicalSpace B₁
inst✝¹⁴ : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst✝¹³ : NormedAddCommGroup EB₂
inst✝¹² : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst✝¹¹ : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst✝¹⁰ : TopologicalSpace B₂
inst✝⁹ : ChartedSpace HB₂ B₂
EM : Type u_13
inst✝⁸ : NormedAddCommGroup EM
inst✝⁷ : NormedSpace 𝕜 EM
HM : Type u_14
inst✝⁶ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace HM M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : (MDiffAt[s] fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) m₀
hv :
  (MDiffAt[s] fun x ↦ ⟨b₁ x, v x⟩.proj) m₀ ∧
    (MDiffAt[s] fun x ↦ (↑(trivializationAt F₁ E₁ ⟨b₁ m₀, v m₀⟩.proj) ⟨b₁ x, v x⟩).2) m₀
hb₂ : MDiffAt[s] b₂ m₀
⊢ (fun x ↦ ⟨b₁ x, v x⟩.proj) m₀ ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet

Tactic:
exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)

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

Goal state before:
pp: 𝕜 : Type u_1
F₁ : Type u_2
F₂ : Type u_3
B₁ : Type u_4
B₂ : Type u_5
M : Type u_6
E₁ : B₁ → Type u_7
E₂ : B₂ → Type u_8
inst✝³¹ : NontriviallyNormedField 𝕜
inst✝³⁰ : (x : B₁) → AddCommGroup (E₁ x)
inst✝²⁹ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝²⁸ : NormedAddCommGroup F₁
inst✝²⁷ : NormedSpace 𝕜 F₁
inst✝²⁶ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝²⁵ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝²⁴ : (x : B₂) → AddCommGroup (E₂ x)
inst✝²³ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝²² : NormedAddCommGroup F₂
inst✝²¹ : NormedSpace 𝕜 F₂
inst✝²⁰ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁹ : (x : B₂) → TopologicalSpace (E₂ x)
EB₁ : Type u_9
inst✝¹⁸ : NormedAddCommGroup EB₁
inst✝¹⁷ : NormedSpace 𝕜 EB₁
HB₁ : Type u_10
inst✝¹⁶ : TopologicalSpace HB₁
IB₁ : ModelWithCorners 𝕜 EB₁ HB₁
inst✝¹⁵ : TopologicalSpace B₁
inst✝¹⁴ : ChartedSpace HB₁ B₁
EB₂ : Type u_11
inst✝¹³ : NormedAddCommGroup EB₂
inst✝¹² : NormedSpace 𝕜 EB₂
HB₂ : Type u_12
inst✝¹¹ : TopologicalSpace HB₂
IB₂ : ModelWithCorners 𝕜 EB₂ HB₂
inst✝¹⁰ : TopologicalSpace B₂
inst✝⁹ : ChartedSpace HB₂ B₂
EM : Type u_13
inst✝⁸ : NormedAddCommGroup EM
inst✝⁷ : NormedSpace 𝕜 EM
HM : Type u_14
inst✝⁶ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace HM M
n : WithTop ℕ∞
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ :
  ContMDiffWithinAt IM 𝓘(𝕜, F₁ →L[𝕜] F₂) n (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m))
    (insert m₀ s) m₀
hv :
  ContMDiffWithinAt IM IB₁ n (fun x ↦ ⟨b₁ x, v x⟩.proj) (insert m₀ s) m₀ ∧
    ContMDiffWithinAt IM 𝓘(𝕜, F₁) n (fun x ↦ (↑(trivializationAt F₁ E₁ ⟨b₁ m₀, v m₀⟩.proj) ⟨b₁ x, v x⟩).2) (insert m₀ s)
      m₀
hb₂ : ContMDiffWithinAt IM IB₂ n b₂ (insert m₀ s) m₀
⊢ (fun x ↦ ⟨b₁ x, v x⟩.proj) m₀ ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet

Tactic:
exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)

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

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
⊢ (fun x ↦ { proj := b₁ x, snd := v x }.proj) m₀ ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet

Tactic:
exact FiberBundle.mem_baseSet_trivializationAt' (b₁ m₀)

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

Goal state before:
pp: 𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
⊢ ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₂ m ∈ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet

Tactic:
apply hb₂

Goal state after:
𝕜 : Type u_8
F₁ : Type u_9
F₂ : Type u_10
B₁ : Type u_11
B₂ : Type u_12
M : Type u_13
E₁ : B₁ → Type u_14
E₂ : B₂ → Type u_15
inst✝¹⁹ : NontriviallyNormedField 𝕜
inst✝¹⁸ : (x : B₁) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B₁) → Module 𝕜 (E₁ x)
inst✝¹⁶ : NormedAddCommGroup F₁
inst✝¹⁵ : NormedSpace 𝕜 F₁
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
inst✝¹³ : (x : B₁) → TopologicalSpace (E₁ x)
inst✝¹² : (x : B₂) → AddCommGroup (E₂ x)
inst✝¹¹ : (x : B₂) → Module 𝕜 (E₂ x)
inst✝¹⁰ : NormedAddCommGroup F₂
inst✝⁹ : NormedSpace 𝕜 F₂
inst✝⁸ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁷ : (x : B₂) → TopologicalSpace (E₂ x)
inst✝⁶ : TopologicalSpace B₁
inst✝⁵ : TopologicalSpace B₂
inst✝⁴ : TopologicalSpace M
inst✝³ : FiberBundle F₁ E₁
inst✝² : VectorBundle 𝕜 F₁ E₁
inst✝¹ : FiberBundle F₂ E₂
inst✝ : VectorBundle 𝕜 F₂ E₂
b₁ : M → B₁
b₂ : M → B₂
m₀ : M
ϕ : (m : M) → E₁ (b₁ m) →L[𝕜] E₂ (b₂ m)
v : (m : M) → E₁ (b₁ m)
s : Set M
hϕ : ContinuousWithinAt (fun m ↦ inCoordinates F₁ E₁ F₂ E₂ (b₁ m₀) (b₁ m) (b₂ m₀) (b₂ m) (ϕ m)) (insert m₀ s) m₀
hv :
  ContinuousWithinAt (fun x ↦ { proj := b₁ x, snd := v x }.proj) (insert m₀ s) m₀ ∧
    ContinuousWithinAt
      (fun x ↦ (↑(trivializationAt F₁ E₁ { proj := b₁ m₀, snd := v m₀ }.proj) { proj := b₁ x, snd := v x }).2)
      (insert m₀ s) m₀
hb₂ : ContinuousWithinAt b₂ (insert m₀ s) m₀
A : ∀ᶠ (m : M) in 𝓝[insert m₀ s] m₀, b₁ m ∈ (trivializationAt F₁ E₁ (b₁ m₀)).baseSet
⊢ (trivializationAt F₂ E₂ (b₂ m₀)).baseSet ∈ 𝓝 (b₂ m₀)
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 := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := fun x ↦ (mfderiv[s] f x).inverse
⊢ ContMDiffWithinAt I I.tangent m (fun x ↦ ⟨x, mpullbackWithin I I' f V s x⟩) (s ∩ f ⁻¹' t) x₀

Tactic:
have hv : ContMDiffWithinAt I I'.tangent m
      (fun x ↦ (v x : TangentBundle I' M')) (s ∩ f ⁻¹' t) x₀ := by
    apply hV.comp x₀ ((hf.of_le (le_trans (le_self_add) hmn)).mono inter_subset_left)
    exact MapsTo.mono_left (mapsTo_preimage _ _) inter_subset_right

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 := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → TangentSpace I' (f x) →L[𝕜] TangentSpace I x := fun x ↦ (mfderiv[s] f x).inverse
hv : ContMDiffWithinAt I I'.tangent m (fun x ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀
⊢ ContMDiffWithinAt I I.tangent m (fun x ↦ ⟨x, mpullbackWithin I I' f V s x⟩) (s ∩ f ⁻¹' t) x₀
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : x✝⁶ ∈ {x | 0 ≤ x.ofLp 0}
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : x✝⁵ ∈ {x | 0 ≤ x.ofLp 0}
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
⊢ x✝⁴ • x✝⁶ + x✝³ • x✝⁵ ∈ {x | 0 ≤ x.ofLp 0}

Tactic:
dsimp at hx hy ⊢

Goal state after:
n : ℕ
inst✝ : NeZero n
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : 0 ≤ x✝⁶.ofLp 0
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : 0 ≤ x✝⁵.ofLp 0
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
⊢ 0 ≤ x✝⁴ * x✝⁶.ofLp 0 + x✝³ * x✝⁵.ofLp 0
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : x✝⁶ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : x✝⁵ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
i : Fin n
⊢ 0 ≤ (x✝⁴ • x✝⁶ + x✝³ • x✝⁵).ofLp i

Tactic:
dsimp at hx hy ⊢

Goal state after:
n : ℕ
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : ∀ (i : Fin n), 0 ≤ x✝⁶.ofLp i
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : ∀ (i : Fin n), 0 ≤ x✝⁵.ofLp i
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
i : Fin n
⊢ 0 ≤ x✝⁴ * x✝⁶.ofLp i + x✝³ * x✝⁵.ofLp i
Module: Mathlib.Geometry.Manifold.Immersion

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_7
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_11
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : ℕ∞ω
inst✝ : IsManifold I n M
⊢ IsImmersionOfComplement PUnit.{u_15 + 1} I I n id

Tactic:
intro x
  apply IsImmersionAtOfComplement.mk_of_continuousAt (continuousAt_id) (.prodUnique 𝕜 E _)
    (chartAt H x) (chartAt H x) (mem_chart_source H x) (mem_chart_source H x)
    (chart_mem_maximalAtlas x) (chart_mem_maximalAtlas x)
  intro y hy
  have : I ((chartAt H x) ((chartAt H x).symm (I.symm y))) = y := by
    rw [(chartAt H x).right_inv (by simp_all), I.right_inv (by simp_all)]
  simpa

Goal state after: