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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
as : AffineSubspace ℝ P
p : P
hp : p ∈ as
x : P
h : s.IsTangentAt x as
⊢ s.radius ≤ dist p s.center

Tactic:
refine le_of_sq_le_sq ?_ dist_nonneg

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
s : Sphere P
as : AffineSubspace ℝ P
p : P
hp : p ∈ as
x : P
h : s.IsTangentAt x as
⊢ s.radius ^ 2 ≤ dist p s.center ^ 2
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
hpc : p ≠ s.center
hp : dist p s.center ≠ s.radius
hf : Module.finrank ℝ V = 1
p' : P
hr : 0 ≤ s.radius
⊢ p' -ᵥ p ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ → ¬‖p' -ᵥ p‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2

Tactic:
hpo

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
s : Sphere P
p : P
hpc : p ≠ s.center
hp : dist p s.center ≠ s.radius
hf : Module.finrank ℝ V = 1
p' : P
hr : 0 ≤ s.radius
hpo : p' -ᵥ p ∈ (ℝ ∙ (p -ᵥ s.center))ᗮ
⊢ ¬‖p' -ᵥ p‖ ^ 2 = s.radius ^ 2 - dist p s.center ^ 2
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ ∡ (touchpoint t {i₁} i₃) (t.points i₂) (excenter t {i₁}) + ↑π =
    ∡ (excenter t {i₁}) (t.points i₂) (touchpoint t {i₁} i₁) + ↑π

Tactic:
add_left_inj

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ ∡ (touchpoint t {i₁} i₃) (t.points i₂) (excenter t {i₁}) = ∡ (excenter t {i₁}) (t.points i₂) (touchpoint t {i₁} i₁)
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: case refine_2
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hj : j ∈ {k | k ≠ i}
⊢ -(⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫ / (s.height i * s.height j)) * (s.height j)⁻¹ <
    (s.height j)⁻¹

Tactic:
refine mul_lt_of_lt_one_left ?_ ?_

Goal state after:
case refine_2.refine_1
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hj : j ∈ {k | k ≠ i}
⊢ 0 < (s.height j)⁻¹

case refine_2.refine_2
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hj : j ∈ {k | k ≠ i}
⊢ -(⟪s.points i -ᵥ s.altitudeFoot i, s.points j -ᵥ s.altitudeFoot j⟫ / (s.height i * s.height j)) < 1
Module: Mathlib.Geometry.Euclidean.Triangle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
h : ⟪x, x - y⟫ * ‖x‖⁻¹ = ⟪y, y - x⟫ * ‖y‖⁻¹
⊢ ‖x‖ = ‖y‖

Tactic:
rw [inner_sub_right, inner_sub_right, real_inner_comm x y, real_inner_self_eq_norm_mul_norm,
      real_inner_self_eq_norm_mul_norm, mul_sub_right_distrib, mul_sub_right_distrib,
      mul_self_mul_inv, mul_self_mul_inv, sub_eq_sub_iff_sub_eq_sub, ← mul_sub_left_distrib] at h

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hpi : angle x y ≠ π
hxy : ¬x = y
h : ‖x‖ - ‖y‖ = ⟪x, y⟫ * (‖x‖⁻¹ - ‖y‖⁻¹)
⊢ ‖x‖ = ‖y‖
Module: Mathlib.Geometry.Euclidean.Circumcenter

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
fs : Finset (Fin (n + 1))
⊢ ∑ i ∈ fs, centroidWeights ℝ fs i • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯ =
    ∑ i, (if i ∈ fs then (↑(#fs))⁻¹ else 0) • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯

Tactic:
centroidWeights,

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
fs : Finset (Fin (n + 1))
⊢ ∑ i ∈ fs, Function.const (Fin (n + 1)) (↑(#fs))⁻¹ i • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯ =
    ∑ i, (if i ∈ fs then (↑(#fs))⁻¹ else 0) • (s.points i -ᵥ Classical.choice ⋯) +ᵥ Classical.choice ⋯
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ (‖ortho y x‖⁻¹ * ⟪x, ortho y x⟫) ^ 2 + ⟪x, y⟫ ^ 2 = 1

Tactic:
ortho_eq_sub_inner _ hy,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ (‖x - ⟪y, x⟫ • y‖⁻¹ * ⟪x, x - ⟪y, x⟫ • y⟫) ^ 2 + ⟪x, y⟫ ^ 2 = 1
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
n : ℕ
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = n
hc : Cospherical ps
r : ℝ
hr : ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumradius = r
⊢ ∃ c, ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumsphere = c

Tactic:
obtain ⟨c, hc⟩ := exists_circumcenter_eq_of_cospherical_subset h hd hc

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
ps : Set P
h : ps ⊆ ↑s
inst✝¹ : Nonempty ↥s
n : ℕ
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = n
hc✝ : Cospherical ps
r : ℝ
hr : ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumradius = r
c : P
hc : ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumcenter = c
⊢ ∃ c, ∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumsphere = c
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: case mp
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
i : Fin (n + 1)
w : Fin (n + 1) → ℝ
hw : ∑ j, w j = 1
⊢ (Finset.affineCombination ℝ Finset.univ s.points) w = s.touchpoint signs i → w = s.touchpointWeights signs i

pp: case mpr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
i : Fin (n + 1)
w : Fin (n + 1) → ℝ
hw : ∑ j, w j = 1
⊢ w = s.touchpointWeights signs i → (Finset.affineCombination ℝ Finset.univ s.points) w = s.touchpoint signs i

Tactic:
· rw [← s.affineCombination_touchpointWeights]
    exact fun h ↦ (affineIndependent_iff_eq_of_fintype_affineCombination_eq ℝ s.points).1
      s.independent _ _ hw (s.sum_touchpointWeights _ _) h

Goal state after:
case mpr
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
i : Fin (n + 1)
w : Fin (n + 1) → ℝ
hw : ∑ j, w j = 1
⊢ w = s.touchpointWeights signs i → (Finset.affineCombination ℝ Finset.univ s.points) w = s.touchpoint signs i
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝¹ x✝ : ι
⊢ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝) ↔ dist (v₁ x✝¹) (v₁ x✝) = dist (v₂ x✝¹) (v₂ x✝)

Tactic:
rw [dist_nndist, dist_nndist]; norm_cast

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ ∀ (i_1 : Fin (n + 1)), ¬i_1 = i → 0 < s.touchpointWeights ∅ i i_1

Tactic:
intro j hj

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hj : ¬j = i
⊢ 0 < s.touchpointWeights ∅ i j
Module: Mathlib.Geometry.Euclidean.Incenter

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ ∀ (i_1 : Fin (n + 1)), ¬i_1 = i → 0 < s.touchpointWeights {i} i i_1

Tactic:
intro j hj

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hj : ¬j = i
⊢ 0 < s.touchpointWeights {i} i j
Module: Mathlib.Geometry.Euclidean.Incenter

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ line[ℝ, t.points i₂, t.points i₃].SSameSide (t.points i₁) (Simplex.incenter t)

Tactic:
convert! t.sSameSide_point_incenter i₁

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ {t.points i₂, t.points i₃} = Set.range (Simplex.faceOpposite t i₁).points
Module: Mathlib.Geometry.Euclidean.Incenter

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hw : ∑ j, Simplex.touchpointWeights t {i₁} i₂ j = 1
⊢ Finset.univ = {i₁, i₂, i₃}

Tactic:
clear hw

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ Finset.univ = {i₁, i₂, i₃}
Module: Mathlib.Geometry.Euclidean.Incenter

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
hw : Simplex.touchpointWeights t {i₁} i₂ i₁ + Simplex.touchpointWeights t {i₁} i₂ i₃ = 1
i : Fin (2 + 1)
⊢ i = i₁ ∨ i = i₂ ∨ i = i₃

Tactic:
clear hw

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
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
i : Fin (2 + 1)
⊢ i = i₁ ∨ i = i₂ ∨ i = i₃
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_not_col : ¬Collinear ℝ {a, b, c}
h_not_col' : ¬Collinear ℝ {a', b', c'}
h : ∠ a b c = ∠ a' b' c'
hd : dist a b * dist b' c' = dist b c * dist a' b'
dist_a'b' : dist a' b' ≠ 0
⊢ dist b' c' ≠ 0

Tactic:
simp [ne₂₃_of_not_collinear h_not_col']

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

Goal state before:
pp: case e'_3.e'_6
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)
e : Fin (n + 3) ≃ Fin (n + 3)
i₁ i₂ : Fin (n + 3)
⊢ centroidWeights ℝ {e.symm i₁, e.symm i₂}ᶜ = centroidWeights ℝ {i₁, i₂}ᶜ ∘ ⇑e.toEmbedding

Tactic:
centroidWeights,

Goal state after:
case e'_3.e'_6
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)
e : Fin (n + 3) ≃ Fin (n + 3)
i₁ i₂ : Fin (n + 3)
⊢ Function.const (Fin (n + 3)) (↑(#{e.symm i₁, e.symm i₂}ᶜ))⁻¹ =
    Function.const (Fin (n + 3)) (↑(#{i₁, i₂}ᶜ))⁻¹ ∘ ⇑e.toEmbedding
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:
rw [mongePlane_def, direction_inf_of_mem_inf s.mongePoint_mem_mongePlane, direction_mk',
    direction_affineSpan]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
⊢ s.secondInter (s.secondInter p v) v = p

Tactic:
by
  by_cases hv : v = 0; · simp [hv]
  have hv' : ⟪v, v⟫ ≠ 0 := inner_self_ne_zero.2 hv
  simp only [Sphere.secondInter, vadd_vsub_assoc, vadd_vadd, inner_add_right, inner_smul_right,
    div_mul_cancel₀ _ hv']
  rw [← @vsub_eq_zero_iff_eq V, vadd_vsub, ← add_smul, ← add_div]
  convert! zero_smul ℝ _
  convert! zero_div (G₀ := ℝ) _
  ring

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

Goal state before:
pp: P : Type u_2
inst✝ : MetricSpace P
s : Sphere P
p : P
hr : 0 ≤ s.radius
⊢ 0 < s.power p ↔ s.radius < dist p s.center

Tactic:
by
  rw [power, sub_pos, pow_lt_pow_iff_left₀ hr dist_nonneg two_ne_zero]

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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hq : dist a s.center = dist b s.center
⊢ |dist b s.center ^ 2 - dist p s.center ^ 2| = |s.power p|

Tactic:
mem_sphere.mp hb,

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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hq : dist a s.center = dist b s.center
⊢ |s.radius ^ 2 - dist p s.center ^ 2| = |s.power p|
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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ |dist b s.center ^ 2 - dist p s.center ^ 2| = 0

Tactic:
mem_sphere.mp hb,

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
s : Sphere P
p a b : P
hp : p ∈ line[ℝ, a, b]
ha : a ∈ s
hb : b ∈ s
hp_on : p ∈ s
hq : dist a s.center = dist b s.center
⊢ |s.radius ^ 2 - dist p s.center ^ 2| = 0
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ n * X ^ 1
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
d : ℕ
hd : X ^ (n + d) = X ^ n
⊢ X ^ (n + (d + 1)) = X ^ n

Tactic:
pow_one

Goal state after:
case succ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : n ≠ 0
hXn : X ^ n = X ^ n * X
hn₁ : ¬n = 1
this :
  ∀ {G : Type u_1} [inst : Group G] [inst_1 : DecidableEq G] {X : Finset G} {n : ℕ},
    1 ∈ X → X.Nontrivial → n ≠ 0 → X ^ n = X ^ (n + 1) → n = 1 → ↑X = ↑(closure ↑X)
d : ℕ
hd : X ^ (n + d) = X ^ n
⊢ X ^ (n + (d + 1)) = X ^ n
Module: Mathlib.Geometry.Group.Growth.LinearLowerBound

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
⊢ StrictMonoOn (fun n ↦ X ^ n) {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}

Tactic:
refine strictMonoOn_of_lt_add_one ⟨?_⟩ fun n _ _ hn ↦
    pow_ssubset_pow_succ_of_pow_ne_closure hX₁ hX hn

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
⊢ ∀ ⦃x : ℕ⦄,
    x ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)} →
      ∀ ⦃y : ℕ⦄, y ∈ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)} → Set.Icc x y ⊆ {n | ↑X ^ (n - 1) ≠ ↑(closure ↑X)}
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✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁸ : NormedAddCommGroup EB
inst✝⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
⊢ (MDiffAt fun x ↦ ⟨x, 0⟩) x

Tactic:
mdifferentiableAt_section

Goal state after:
𝕜 : Type u_1
B : Type u_2
F : Type u_4
E : B → Type u_6
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
inst✝¹⁰ : TopologicalSpace (TotalSpace F E)
inst✝⁹ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁸ : NormedAddCommGroup EB
inst✝⁷ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁶ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝⁵ : TopologicalSpace B
inst✝⁴ : ChartedSpace HB B
inst✝³ : FiberBundle F E
inst✝² : (x : B) → AddCommMonoid (E x)
inst✝¹ : (x : B) → Module 𝕜 (E x)
inst✝ : VectorBundle 𝕜 F E
x : B
⊢ (MDiffAt fun b ↦ (↑(trivializationAt F E x) ⟨b, 0⟩).2) x
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✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
inst✝⁸ : TopologicalSpace (TotalSpace F E)
inst✝⁷ : (x : B) → TopologicalSpace (E x)
EB : Type u_7
inst✝⁶ : NormedAddCommGroup EB
inst✝⁵ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝⁴ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝³ : TopologicalSpace B
inst✝² : ChartedSpace HB B
inst✝¹ : FiberBundle F E
s : (x : B) → E x
x : B
inst✝ : Subsingleton F
⊢ MDiffAt[univ] (T% s) x

Tactic:
apply mdifferentiableWithinAt_section_of_subsingleton

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

Goal state before:
pp: 𝕜₁ : Type u_1
inst✝²³ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝²² : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
B : Type u_3
F₁ : Type u_4
inst✝²¹ : NormedAddCommGroup F₁
inst✝²⁰ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁹ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁸ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁷ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁶ : NormedAddCommGroup F₂
inst✝¹⁵ : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹⁴ : (x : B) → AddCommGroup (E₂ x)
inst✝¹³ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝¹² : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹¹ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
inst✝⁶ : RingHomIsometric σ
inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁
inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
⊢ ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
by
  have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous
  have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous
  have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁
  have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂'
  refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_continuousOn (h₃.mono ?_))).congr ?_
  · mfld_set_tac
  · mfld_set_tac
  · intro b _
    ext L v
    dsimp [continuousLinearMapCoordChange]

Goal state after: