mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ edist (v₁ x✝¹) (v₁ x✝) = edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝)

Tactic:
rw [edist_nndist, edist_nndist]; norm_cast

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: case inl.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
hy : 0 • x + 0 • z ≠ 0
⊢ angle x z = angle x (0 • x + 0 • z) + angle (0 • x + 0 • z) z

pp: case inl.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kz : ℝ≥0
hy : 0 • x + kz • z ≠ 0
hkz : 0 < kz
⊢ angle x z = angle x (0 • x + kz • z) + angle (0 • x + kz • z) z

pp: case inr.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx : ℝ≥0
hkx : 0 < kx
hy : kx • x + 0 • z ≠ 0
⊢ angle x z = angle x (kx • x + 0 • z) + angle (kx • x + 0 • z) z

pp: case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx kz : ℝ≥0
hy : kx • x + kz • z ≠ 0
hkx : 0 < kx
hkz : 0 < kz
⊢ angle x z = angle x (kx • x + kz • z) + angle (kx • x + kz • z) z

Tactic:
· simp at hy

Goal state after:
case inl.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kz : ℝ≥0
hy : 0 • x + kz • z ≠ 0
hkz : 0 < kz
⊢ angle x z = angle x (0 • x + kz • z) + angle (0 • x + kz • z) z

case inr.inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx : ℝ≥0
hkx : 0 < kx
hy : kx • x + 0 • z ≠ 0
⊢ angle x z = angle x (kx • x + 0 • z) + angle (kx • x + 0 • z) z

case inr.inr
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx kz : ℝ≥0
hy : kx • x + kz • z ≠ 0
hkx : 0 < kx
hkz : 0 < kz
⊢ angle x z = angle x (kx • x + kz • z) + angle (kx • x + kz • z) z
Module: Mathlib.Geometry.Euclidean.Inversion.ImageHyperplane

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
c y : P
R : ℝ
hR : R ≠ 0
hy : y ≠ c
⊢ inversion c R '' sphere y (dist y c) = insert c ↑(perpBisector c (inversion c R y))

Tactic:
rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),
    preimage_inversion_sphere_dist_center hR hy]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
⊢ Similar ![a, b, c] ![a', b', c']

Tactic:
have hr_pos : 0 < r := by positivity

Goal state after:
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
a b c : P₁
a' b' c' : P₂
h_ne : dist a b ≠ 0
h_ne' : dist a' b' ≠ 0
heq1 : dist a b * dist b' c' = dist b c * dist a' b'
heq2 : dist a b * dist c' a' = dist c a * dist a' b'
r : ℝ := dist a b / dist a' b'
hr : r = dist a b / dist a' b'
hr_pos : 0 < r
⊢ Similar ![a, b, c] ![a', b', c']
Module: Mathlib.Geometry.Euclidean.Sphere.Power

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
h₃ : ‖z - y‖ = ‖z + y‖
r : ℝ
hr : x = r • y
⊢ ⟪z, y⟫ = 0

Tactic:
rwa [inner_eq_zero_iff_angle_eq_pi_div_two, ← norm_add_eq_norm_sub_iff_angle_eq_pi_div_two,
      eq_comm]

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

Goal state before:
pp: case pos
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
⊢ ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, s.points i₁ -ᵥ s.points i₂⟫ = 0

Tactic:
simp [h]

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
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : ∠ a p b = π
hcpd : ∠ c p d = π
⊢ dist a p * dist b p = dist c p * dist d p

Tactic:
rw [EuclideanGeometry.angle_eq_pi_iff_sbtw] at hapb hcpd

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
a b c d p : P
h : Cospherical {a, b, c, d}
hapb : Sbtw ℝ a p b
hcpd : Sbtw ℝ c p d
⊢ dist a p * dist b p = dist c p * dist d p
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case neg
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}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ -(∑ x ∈ fs,
            ∑ i₂_1,
              (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
                  (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
                (dist (s.points x) (s.pointsWithCircumcenter i₂_1) *
                  dist (s.points x) (s.pointsWithCircumcenter i₂_1)) +
          ∑ i₂_1,
            -2 / ↑(n + 1) * (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
              (dist s.circumcenter (s.pointsWithCircumcenter i₂_1) *
                dist s.circumcenter (s.pointsWithCircumcenter i₂_1))) /
      2 =
    0

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}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ ∀ x ∈ univ,
    x ∉ fs →
      ∑ i₂_1,
          (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
            (dist (s.points x) (s.pointsWithCircumcenter i₂_1) * dist (s.points x) (s.pointsWithCircumcenter i₂_1)) =
        0

Tactic:
· simp_rw [sum_pointsWithCircumcenter, pointsWithCircumcenter_eq_circumcenter,
      pointsWithCircumcenter_point, Pi.sub_apply, pointWeightsWithCircumcenter]
    rw [← sum_subset fs.subset_univ _]
    · simp_rw [fs, sum_insert (notMem_singleton.2 h), sum_singleton]
      repeat rw [← sum_subset fs.subset_univ _]
      · simp_rw [fs, sum_insert (notMem_singleton.2 h), sum_singleton]
        simp [h, Ne.symm h, dist_comm (s.points i₁)]
      all_goals intro i _ hi; simp [hfs i hi]
    · intro i _ hi
      simp [hfs i hi]

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)
i₁ i₂ : Fin (n + 3)
h : ¬i₁ = i₂
hs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = 0
fs : Finset (Fin (n + 3)) := {i₁, i₂}
hfs : ∀ i ∉ fs, i ≠ i₁ ∧ i ≠ i₂
⊢ ∀ x ∈ univ,
    x ∉ fs →
      ∑ i₂_1,
          (if x = i₁ ∨ x = i₂ then (↑(n + 1))⁻¹ else 0) *
              (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i₂_1 *
            (dist (s.points x) (s.pointsWithCircumcenter i₂_1) * dist (s.points x) (s.pointsWithCircumcenter i₂_1)) =
        0
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
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hmem : p₂ ∈ line[ℝ, p₁, p]
⊢ {p₁, p} ⊆ Set.range t.points

Tactic:
simp [ht]

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]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hmem : p₂ ∈ line[ℝ, p₁, p]
⊢ {p₁, p} ⊆ {p₃, p, p₁}
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
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hmem : p₄ ∈ line[ℝ, p₃, p]
⊢ {p₃, p} ⊆ Set.range t.points

Tactic:
simp [ht]

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]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hmem : p₄ ∈ line[ℝ, p₃, p]
⊢ {p₃, p} ⊆ {p₃, p, p₁}
Module: Mathlib.Geometry.Euclidean.MongePoint

Goal state before:
pp: case pos
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)
h : i₁ = i₂
⊢ #{e.symm i₁, e.symm i₂} = #{i₁, i₂}

Tactic:
simp [h]

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

Goal state before:
pp: case neg
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)
h : ¬i₁ = i₂
⊢ #{e.symm i₁, e.symm i₂} = #{i₁, i₂}

Tactic:
simp [h]

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
hindep : AffineIndependent ℝ ![p₁, p, p₃]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
⊢ Cospherical {p₁', p₂', p₃', p₄'}

Tactic:
have hf2 : Fact (finrank ℝ S.direction = 2) := ⟨by
    rw [hS, direction_affineSpan, t.independent.finrank_vectorSpan]
    simp⟩

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]
t : Affine.Triangle ℝ P := { points := ![p₁, p, p₃], independent := hindep }
ht : t = { points := ![p₁, p, p₃], independent := hindep }
S : AffineSubspace ℝ P := affineSpan ℝ (Set.range t.points)
hS : S = affineSpan ℝ (Set.range t.points)
hp₂ : p₂ ∈ S
hp₄ : p₄ ∈ S
s_isom : ↥S →ᵃⁱ[ℝ] P := S.subtypeₐᵢ
p₁' : ↥S := ⟨p₁, ⋯⟩
p' : ↥S := ⟨p, ⋯⟩
p₃' : ↥S := ⟨p₃, ⋯⟩
p₂' : ↥S := ⟨p₂, hp₂⟩
p₄' : ↥S := ⟨p₄, hp₄⟩
h_dist' : dist p₁' p' * dist p₂' p' = dist p₃' p' * dist p₄' p'
hp₁'p₂' : ∠ p₁' p' p₂' = π
hp₃'p₄' : ∠ p₃' p' p₄' = π
hf2 : Fact (finrank ℝ ↥S.direction = 2)
⊢ Cospherical {p₁', p₂', p₃', p₄'}
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

Goal state before:
pp: case neg.inl
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
hp : dist p s.center = s.radius
v : V
r : ℝ
hv : ¬v = 0
h : r = 0
⊢ r • v +ᵥ p = p ∨ r • v +ᵥ p = (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v +ᵥ p

Tactic:
simp [h]

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

Goal state before:
pp: case neg.inr
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
hp : dist p s.center = s.radius
v : V
r : ℝ
hv : ¬v = 0
h : r = -2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫
⊢ r • v +ᵥ p = p ∨ r • v +ᵥ p = (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v +ᵥ p

Tactic:
simp [h]

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

Goal state before:
pp: case pos
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' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : p' = p
⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))

Tactic:
simp [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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ dist t.orthocenter
        ((affineCombination ℝ univ (pointsWithCircumcenter t)) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)) *
      dist t.orthocenter
        ((affineCombination ℝ univ (pointsWithCircumcenter t)) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)) =
    circumradius t * circumradius t

Tactic:
t.orthocenter_eq_mongePoint,

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 : Fin 3
h : i₁ ≠ i₂
⊢ dist (mongePoint t)
        ((affineCombination ℝ univ (pointsWithCircumcenter t)) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)) *
      dist (mongePoint t)
        ((affineCombination ℝ univ (pointsWithCircumcenter t)) (reflectionCircumcenterWeightsWithCircumcenter i₁ i₂)) =
    circumradius t * circumradius t
Module: Mathlib.Geometry.Group.Growth.QuotientInter

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
a : G ⧸ H
ha : a ∈ ⇑π '' ↑A ^ m
⊢ ∃ a ∈ ?m.112, ?m.113 a = ?m.114

Tactic:
by simpa using ha

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

Goal state before:
pp: G : Type u_1
inst✝³ : Group G
inst✝² : DecidableEq G
H : Subgroup G
inst✝¹ : DecidablePred fun x ↦ x ∈ H
inst✝ : H.Normal
A : Finset G
m n : ℕ
π : G →* G ⧸ H := QuotientGroup.mk' H
φ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)
hφ : Set.InjOn φ (⇑π '' ↑A ^ m)
hφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m → φ a ∈ A ^ m
a : G ⧸ H
ha : a ∈ ⇑π '' ↑A ^ m
⊢ ∃ a_1 ∈ ↑A ^ m, π a_1 = a

Tactic:
by simpa using ha

Goal state after:
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
x : M
hf : IsLocalDiffeomorphAt I J n f x
⊢ ↑(Exists.choose hf).toPartialEquiv x ∈ hf.localInverse.source

Tactic:
exact (Classical.choose hf).map_source hf.choose_spec.1

Goal state after:
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
x : M
hf : IsLocalDiffeomorphAt I J n f x
y : N
hy : y ∈ hf.localInverse.source
⊢ f (↑hf.localInverse.toPartialEquiv y) = y

Tactic:
by
  have : hf.localInverse y ∈ hf.choose.source := by
    rw [← hf.choose.symm_target]
    exact hf.choose.symm.map_source hy
  rw [hf.choose_spec.2 this]
  exact hf.choose.right_inv hy

Goal state after:
Module: Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_7
inst✝² : TopologicalSpace N
inst✝¹ : ChartedSpace H' N
n : WithTop ℕ∞
hn : n ≠ 0
f : M → N
hf : IsLocalDiffeomorph I I' n f
inst✝ : BoundarylessManifold I' N
⊢ BoundarylessManifold I M

Tactic:
by
  simp [← Boundaryless.iff_boundary_eq_empty, ← hf.preimage_boundary hn,
    Boundaryless.boundary_eq_empty]

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✝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
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
u' : Set B
hu' : u' ∈ 𝓝 x₀
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}.Finite
s : Set ι := {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this✝ : Fintype ↑{i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this : (MDiffAt[u ∩ u'] fun x ↦ ⟨x, ∑ i ∈ s.toFinset, t i x⟩) x₀
y : B
hy : y ∈ u'
i : ι
hi : i ∉ ↑s.toFinset
h : t i y ≠ 0
⊢ ({x | ¬t i x = 0} ∩ u').Nonempty

Tactic:
use y

Goal state after:
case 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
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
u' : Set B
hu' : u' ∈ 𝓝 x₀
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}.Finite
s : Set ι := ⋯
this✝ : Fintype ↑{i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
this : (MDiffAt[u ∩ u'] fun x ↦ ⟨x, ∑ i ∈ s.toFinset, t i x⟩) x₀
y : B
hy : y ∈ u'
i : ι
hi : i ∉ ↑s.toFinset
h : t i y ≠ 0
⊢ y ∈ {x | ¬t i x = 0} ∩ u'
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 : 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
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
y : B
hy : y ∈ univ
U : Set B
hu : U ∈ 𝓝 y
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ U).Nonempty}.Finite
x : ι
hx : t x y ≠ 0
⊢ ((fun i ↦ {x | t i x ≠ 0}) x ∩ U).Nonempty

Tactic:
use y

Goal state after:
case 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
u : Set B
x₀ : B
ι : Type u_9
t : ι → (x : B) → E x
ht : LocallyFinite fun i ↦ {x | t i x ≠ 0}
ht' : ∀ (i : ι), MDiffAt[u] (T% t) x₀
y : B
hy : y ∈ univ
U : Set B
hu : U ∈ 𝓝 y
hfin : {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ U).Nonempty}.Finite
x : ι
hx : t x y ≠ 0
⊢ y ∈ (fun i ↦ {x | t i x ≠ 0}) x ∩ U