mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ‖x‖ / ((-o).oangle y (x + y)).tan = ‖y‖

Tactic:
add_comm

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ‖x‖ / ((-o).oangle y (y + x)).tan = ‖y‖
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ((-o).oangle x (x - y)).sin = ‖y‖ / ‖x - y‖

Tactic:
exact (-o).sin_oangle_sub_right_of_oangle_eq_pi_div_two h

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
⊢ ‖x‖ / ((-o).oangle x (x - y)).cos = ‖x - y‖

Tactic:
exact (-o).norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two h

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : 0 < r
⊢ o.oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)

Tactic:
o.oangle_smul_right_of_pos _ _ hr,

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : 0 < r
⊢ o.oangle x ((o.rotation ↑(π / 2)) x) = ↑(π / 2)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
⊢ (-o).oangle (r • ((-o).rotation (-↑(π / 2))) x) (x + r • ((-o).rotation (-↑(π / 2))) x) = ↑(Real.arctan r⁻¹)

Tactic:
add_comm

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
⊢ (-o).oangle (r • ((-o).rotation (-↑(π / 2))) x) (r • ((-o).rotation (-↑(π / 2))) x + x) = ↑(Real.arctan r⁻¹)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)
⊢ o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x - x) = ↑(Real.arctan r⁻¹)

Tactic:
rw [sub_eq_add_neg, hx, o.oangle_add_right_smul_rotation_pi_div_two]

Goal state after:
case neg.h
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)
⊢ r • (o.rotation ↑(π / 2)) x ≠ 0
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : x = r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))
⊢ o.oangle (x + -(r • (o.rotation ↑(π / 2)) x)) x = ↑(Real.arctan r)

Tactic:
add_comm

Goal state after:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : ¬r = 0
hx : x = r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))
⊢ o.oangle (-(r • (o.rotation ↑(π / 2)) x) + x) x = ↑(Real.arctan r)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r₁ r₂ r₃ r₄ : ℝ
hr₁ : ¬r₁ = 0
⊢ (o.oangle (r₁ • x + r₂ • y) ((-r₃ / r₁ * r₂) • y + r₄ • y)).sign =
    SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign

Tactic:
← add_smul,

Goal state after:
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r₁ r₂ r₃ r₄ : ℝ
hr₁ : ¬r₁ = 0
⊢ (o.oangle (r₁ • x + r₂ • y) ((-r₃ / r₁ * r₂ + r₄) • y)).sign = SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign
Module: Mathlib.Geometry.Euclidean.SignedDist

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
v : V
p : P
⊢ (↑((signedDist v) p)).linear = (innerₗ V) (NormedSpace.normalize v)

Tactic:
by
  change (innerₗ V (normalize v)).comp (LinearMap.id - 0) = _
  simp

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

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
ι : Type u_3
s : Finset ι
w₁ w₂ : ι → ℝ
p : ι → P
h₁ : ∑ i ∈ s, w₁ i = 1
h₂ : ∑ i ∈ s, w₂ i = 1
⊢ ∑ i ∈ s, (w₁ - w₂) i = 0

Tactic:
simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, h₁, h₂, sub_self]

Goal state after:
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
⊢ s.center ∈ s.orthRadius p ↔ p = s.center

Tactic:
rw [mem_orthRadius_iff_inner_left, ← neg_vsub_eq_vsub_rev, inner_neg_left]

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
⊢ -⟪p -ᵥ s.center, p -ᵥ s.center⟫ = 0 ↔ p = s.center
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

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 : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
⊢ ∀ (i : Fin 3), i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫

Tactic:
intro 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
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hf0 : f 0 = 0
hfi : Function.Injective f
hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
hfn0 : ∀ (i : Fin 3), i ≠ 0 → f i ≠ 0
i : Fin 3
hi : i ≠ 0
⊢ f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
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 q : P
h : s.orthRadius p = s.orthRadius q
hpq : p = q ∨ q = s.center
⊢ p = q

Tactic:
have hqp := orthRadius_le_orthRadius_iff.1 h.symm.le

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 q : P
h : s.orthRadius p = s.orthRadius q
hpq : p = q ∨ q = s.center
hqp : q = p ∨ p = s.center
⊢ p = q
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 q : P
⊢ ‖q -ᵥ p + (p -ᵥ s.center)‖ * ‖q -ᵥ p + (p -ᵥ s.center)‖ = ‖p -ᵥ s.center‖ * ‖p -ᵥ s.center‖ + ‖q -ᵥ p‖ * ‖q -ᵥ p‖ ↔
    q ∈ s.orthRadius p

Tactic:
add_comm

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 q : P
⊢ ‖q -ᵥ p + (p -ᵥ s.center)‖ * ‖q -ᵥ p + (p -ᵥ s.center)‖ = ‖q -ᵥ p‖ * ‖q -ᵥ p‖ + ‖p -ᵥ s.center‖ * ‖p -ᵥ s.center‖ ↔
    q ∈ s.orthRadius p
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: case neg.refine_2.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
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖p₁ -ᵥ s.center‖
⊢ ¬‖p₁ -ᵥ s.center‖ • (p₂ -ᵥ p₁) = 0

Tactic:
← Ne,

Goal state after:
case neg.refine_2.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
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖p₁ -ᵥ s.center‖
⊢ ‖p₁ -ᵥ s.center‖ • (p₂ -ᵥ p₁) ≠ 0
Module: Mathlib.Geometry.Euclidean.Sphere.OrthRadius

Goal state before:
pp: case inr.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
hf2 : Fact (Module.finrank ℝ V = 2)
s : Sphere P
p : P
hpc : p ≠ s.center
h : s.radius < dist p s.center
⊢ (Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)).ncard ≤ 2

Tactic:
simp [inter_orthRadius_eq_empty_of_radius_lt_dist h]

Goal state after:
Module: Mathlib.Geometry.Euclidean.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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
⊢ ∀ (i : Fin (n + 1)),
    i ≠ 0 → ∑ j, ⟪s.points i -ᵥ s.points 0, (s.height j)⁻¹ ^ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫ = 0

Tactic:
intro i hi

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hi : i ≠ 0
⊢ ∑ j, ⟪s.points i -ᵥ s.points 0, (s.height j)⁻¹ ^ 2 • (s.points j -ᵥ s.altitudeFoot j)⟫ = 0
Module: Mathlib.Geometry.Euclidean.Angle.Sphere

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 q : P
as : AffineSubspace ℝ P
h : s.IsTangentAt p as
hq_mem : q ∈ as
h1 : ⟪q -ᵥ p, p -ᵥ s.center⟫_ℝ = 0
⊢ ∠ q p s.center = π / 2

Tactic:
rw [inner_eq_zero_iff_angle_eq_pi_div_two] at h1

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 q : P
as : AffineSubspace ℝ P
h : s.IsTangentAt p as
hq_mem : q ∈ as
h1 : InnerProductGeometry.angle (q -ᵥ p) (p -ᵥ s.center) = π / 2
⊢ ∠ q p s.center = π / 2
Module: Mathlib.Geometry.Euclidean.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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hp : 1 ≤ s.excenterWeights ∅ i
j : Fin (n + 1)
hj : j ≠ i
⊢ 1 < s.excenterWeights ∅ j + s.excenterWeights ∅ i + ∑ i ∈ {j, i}ᶜ, s.excenterWeights ∅ i

Tactic:
add_comm

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
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
hp : 1 ≤ s.excenterWeights ∅ i
j : Fin (n + 1)
hj : j ≠ i
⊢ 1 < s.excenterWeights ∅ i + s.excenterWeights ∅ j + ∑ i ∈ {j, i}ᶜ, s.excenterWeights ∅ i
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case intro.succ.succ
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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = m + 1 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius

Tactic:
have ha2 : AffineIndependent ℝ fun i2 : ι2 => p i2 := ha.subtype _

Goal state after:
case intro.succ.succ
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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hm :
  ∀ {ι : Type u_3} [hne : Nonempty ι] [Finite ι] {p : ι → P},
    AffineIndependent ℝ p →
      ∀ (val : Fintype ι),
        Fintype.card ι = m + 1 →
          ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max 0 u_3) := { x // x ≠ i }
hc : Fintype.card ι2 = m + 1
this : Nonempty ι2
ha2 : AffineIndependent ℝ fun i2 ↦ p ↑i2
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
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
s : Simplex ℝ P 0
i : Fin 1
h : s.circumcenter ∈ affineSpan ℝ (Set.range s.points)
a : Fin 1
⊢ a = default

Tactic:
by simp only [Fin.eq_zero]

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

Goal state before:
pp: case inr
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)
p₁ p₂ p₃ p₄ : P
h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄
hc : Collinear ℝ {p₁, p₂, p₃, p₄}
⊢ Concyclic {p₁, p₂, p₃, p₄} ∨ Collinear ℝ {p₁, p₂, p₃, p₄}

Tactic:
· exact Or.inr hc

Goal state after:
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
⊢ 0 ≤ ⟪x, ortho y x⟫

Tactic:
by
  rw [ortho_eq_sub_inner x hy, inner_sub_right,
    inner_self_eq_one_of_norm_eq_one hx, real_inner_smul_right, real_inner_comm, sub_nonneg]
  grw [← sq, sq_le_one_iff_abs_le_one, abs_real_inner_le_norm, hx, hy, one_mul]

Goal state after:
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
⊢ (‖x - ⟪y, x⟫ • y‖⁻¹ * (⟪x, x⟫ - ⟪y, x⟫ * ⟪x, y⟫)) ^ 2 + ⟪x, y⟫ ^ 2 = 1

Tactic:
by_cases h₁ : x = y

Goal state after:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
h₁ : x = y
⊢ (‖x - ⟪y, x⟫ • y‖⁻¹ * (⟪x, x⟫ - ⟪y, x⟫ * ⟪x, y⟫)) ^ 2 + ⟪x, y⟫ ^ 2 = 1

case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
h₁ : ¬x = y
⊢ (‖x - ⟪y, x⟫ • y‖⁻¹ * (⟪x, x⟫ - ⟪y, x⟫ * ⟪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✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((if x ∈ {i₁, i₂} then
            (↑(#{i₁, i₂}))⁻¹ • (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) +
              (↑(#{i₁, i₂}))⁻¹ • (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯)
          else 0) -
          if x = i₁ ∨ x = i₂ then 1 • (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) else 0) +
      (0 - 1 • (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) -
        -1 • (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯)) =
    0

Tactic:
← add_smul,

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
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
h_faces : ↑((orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)
⊢ ∑ x,
        ((if x ∈ {i₁, i₂} then
            ((↑(#{i₁, i₂}))⁻¹ + (↑(#{i₁, i₂}))⁻¹) • (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯)
          else 0) -
          if x = i₁ ∨ x = i₂ then 1 • (s.pointsWithCircumcenter (pointIndex x) -ᵥ Classical.choice ⋯) else 0) +
      (0 - 1 • (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯) -
        -1 • (s.pointsWithCircumcenter circumcenterIndex -ᵥ Classical.choice ⋯)) =
    0
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
hxy : ¬x - ⟪x, y⟫ • y = 0
H : 1 - ⟪x, y⟫ ^ 2 ≠ 0
⊢ ⟪x - ⟪x, y⟫ • y, x - ⟪x, y⟫ • y⟫ = 1 - ⟪x, y⟫ ^ 2

Tactic:
rw [inner_sub_left, inner_sub_right, inner_sub_right, inner_self_eq_one_of_norm_eq_one hx,
      real_inner_smul_right, ← sq, real_inner_smul_left, real_inner_smul_left,
      real_inner_smul_right, inner_self_eq_one_of_norm_eq_one hy, real_inner_comm y x]

Goal state after:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hxy : ¬x - ⟪x, y⟫ • y = 0
H : 1 - ⟪x, y⟫ ^ 2 ≠ 0
⊢ 1 - ⟪y, x⟫ ^ 2 - (⟪y, x⟫ * ⟪y, x⟫ - ⟪y, x⟫ * (⟪y, x⟫ * 1)) = 1 - ⟪y, x⟫ ^ 2
Module: Mathlib.Topology.MetricSpace.Similarity

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
P₁ : Type u_3
P₂ : Type u_4
inst✝¹ : PseudoEMetricSpace P₁
inst✝ : PseudoEMetricSpace P₂
f : ι' ≃ ι
v₁ : ι → P₁
v₂ : ι → P₂
r : ℝ≥0
hr : r ≠ 0
h : ∀ (i₁ i₂ : ι'), edist ((v₁ ∘ ⇑f) i₁) ((v₁ ∘ ⇑f) i₂) = ↑r * edist ((v₂ ∘ ⇑f) i₁) ((v₂ ∘ ⇑f) i₂)
i₁ i₂ : ι
⊢ edist (v₁ i₁) (v₁ i₂) = ↑r * edist (v₂ i₁) (v₂ i₂)

Tactic:
simpa [f.right_inv i₁, f.right_inv i₂] using h (f.symm i₁) (f.symm i₂)

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)
⊢ (↑n + 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) = ∑ i, (s.points i -ᵥ s.circumcenter)

Tactic:
have h : Invertible (n + 2 + 1 : ℝ) := by norm_cast; apply invertibleOfPos

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)
h : Invertible (↑n + 2 + 1)
⊢ (↑n + 2 + 1) • (Finset.centroid ℝ univ s.points -ᵥ s.circumcenter) = ∑ i, (s.points i -ᵥ s.circumcenter)
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 ∉ fs, i ≠ i₁ ∧ i ≠ i₂

Tactic:
intro 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}
i : Fin (n + 3)
hi : i ∉ fs
⊢ i ≠ i₁ ∧ i ≠ i₂
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
p : P
v : V
hp : s.secondInter p v = p
⊢ ⟪v, p -ᵥ s.center⟫ = 0

pp: case refine_2
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
hp : ⟪v, p -ᵥ s.center⟫ = 0
⊢ s.secondInter p v = p

Tactic:
· by_cases hv : v = 0
    · simp [hv]
    rwa [Sphere.secondInter, eq_comm, eq_vadd_iff_vsub_eq, vsub_self, eq_comm, smul_eq_zero,
      or_iff_left hv, div_eq_zero_iff, inner_self_eq_zero, or_iff_left hv, mul_eq_zero,
      or_iff_right (by simp : (-2 : ℝ) ≠ 0)] at hp

Goal state after:
case refine_2
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
hp : ⟪v, p -ᵥ s.center⟫ = 0
⊢ s.secondInter p v = p
Module: Mathlib.Geometry.Euclidean.Sphere.SecondInter

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
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ (-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v = 0

Tactic:
← add_smul,

Goal state after:
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
s : Sphere P
p : P
v : V
hv : ¬v = 0
hv' : ⟪v, v⟫ ≠ 0
⊢ (-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫ + -2 * ⟪v, p -ᵥ s.center⟫ / ⟪v, v⟫) • v = 0
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case pos.inl
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
r : ℝ
hr_pos : 0 < r
hdist :
  ∀ (i₁ i₂ : Fin (Nat.succ 0).succ.succ),
    dist (![a, b, c] i₁) (![a, b, c] i₂) = r * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
h_ab : dist a b = r * dist a' b'
h_cb : dist c b = r * dist c' b'
h_ac : dist a c = r * dist a' c'
h_cos : dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c') = dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c)
h_cos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
h1 : a' = b'
h_dist_ab : a = b
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
simp_rw [h_dist_ab, h1, angle_self_left]

Goal state after: