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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
this : p = p₁
⊢ 2 • ∡ p₂ p₁ p = 2 • ∡ p p₁ p₃

Tactic:
by

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

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = s.direction
⊢ ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
⊢ 1 ≠ 0

Tactic:
by

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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
this : {p₁} = ![p₁, p₂, p₃] '' {0}
⊢ {p₁} = ![p₁, p₂, p₃] '' ({0, 1} ∩ {0, 2})

Tactic:
by

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

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = s.direction
hv : ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
t₁ t₂ : ℝ
hop : t₁ = 0
hps : t₂ • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ s
hpc₁ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁
hpc₂ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₂ = r₂
⊢ p₂ -ᵥ p₁ ≠ 0

Tactic:
by

Goal state after:
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
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
⊢ 2 ≠ 0

Tactic:
by

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

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = s.direction
hv : ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
t₁ t₂ : ℝ
hop : t₁ = 0
hps : t₂ • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ s
hpc₁ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁
hpc₂ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₂ = r₂
hp' : p₂ -ᵥ p₁ ≠ 0
⊢ dist (1 • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁

Tactic:
by

Goal state after:
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
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
⊢ f 1 = f 2

Tactic:
by

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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
⊢ line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}

Tactic:
by

Goal state after:
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
hfn0' : ∀ (i : Fin 3), i ≠ 0 → f i = -2 * ⟪v, p 0 -ᵥ c⟫ / ⟪v, v⟫
hf12 : f 1 = f 2
⊢ 1 ≠ 2

Tactic:
by

Goal state after:
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
hs : Cospherical s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
⊢ AffineIndependent ℝ p

Tactic:
by

Goal state after:
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
hs : Cospherical s
p₁ p₂ p₃ : P
h₁ : p₁ ∈ s
h₂ : p₂ ∈ s
h₃ : p₃ ∈ s
h₁₂ : p₁ ≠ p₂
h₁₃ : p₁ ≠ p₃
h₂₃ : p₂ ≠ p₃
⊢ AffineIndependent ℝ ![p₁, p₂, p₃]

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
h : 0 = t
⊢ p = p₁

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
h : t = 1
⊢ p = p₂

Tactic:
by

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

Goal state before:
pp: case refine_1.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 q : P
h : s.orthRadius p ≤ s.orthRadius q
h' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ
r : ℝ
hr : r • (p -ᵥ s.center) = q -ᵥ s.center
hp : ⟪p -ᵥ s.center, p -ᵥ s.center⟫ = 0
⊢ p = q ∨ q = s.center

Tactic:
simp only [inner_self_eq_zero, vsub_eq_zero_iff_eq] at hp
      rw [hp, vsub_self, smul_zero, eq_comm, vsub_eq_zero_iff_eq] at hr
      exact .inr hr

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

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
⊢ p = p₁ ∨ p = p₂

Tactic:
by

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

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
inst✝ : FiniteDimensional ℝ V
hd : finrank ℝ V = 2
c₁ c₂ p₁ p₂ p : P
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
⊢ finrank ℝ ↥⊤.direction = 2

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
h : u = v
⊢ p = p₁

Tactic:
by

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 q : P
⊢ s.orthRadius p ≤ s.orthRadius q ↔ p = q ∨ q = s.center

Tactic:
by

Goal state after:
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
p q : P
as : AffineSubspace ℝ P
hp : s.IsTangentAt p as
hq : s.IsTangentAt q as
⊢ p = q

Tactic:
by

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 q : P
⊢ s.orthRadius p = s.orthRadius q ↔ p = q

Tactic:
by

Goal state after:
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
⊢ s.IsTangentAt s.center as ↔ s.radius = 0 ∧ s.center ∈ as

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
⊢ p -ᵥ o = (1 - t) • u + t • v

Tactic:
by

Goal state after:
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
p q : P
as : AffineSubspace ℝ P
h : s.IsTangentAt p as
hq : q ∈ as
⊢ dist q s.center ^ 2 = s.radius ^ 2 + dist q p ^ 2

Tactic:
by

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

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
inst✝ : FiniteDimensional ℝ V
s : Sphere P
p : P
hp : p ≠ s.center
⊢ Module.finrank ℝ ↥(s.orthRadius p).direction + 1 = Module.finrank ℝ V

Tactic:
by

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

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 q : P
as : AffineSubspace ℝ P
h : s.IsTangentAt p as
x✝ : q ∈ s ∧ q ∈ as
hs : q ∈ s
has : q ∈ as
⊢ q = p

Tactic:
have hd := h.dist_sq_eq_of_mem has
    rw [hs] at hd
    simpa using hd

Goal state after:
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
p q : P
as : AffineSubspace ℝ P
h : s.IsTangentAt p as
⊢ q ∈ s ∧ q ∈ as ↔ q = p

Tactic:
by

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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}
⊢ line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤

Tactic:
by

Goal state after:
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
p₁ p₂ q : P
as₁ as₂ : AffineSubspace ℝ P
h₁ : s.IsTangentAt p₁ as₁
h₂ : s.IsTangentAt p₂ as₂
hq_mem₁ : q ∈ as₁
hq_mem₂ : q ∈ as₂
⊢ dist q p₁ = dist q p₂

Tactic:
by

Goal state after:
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 q : P
h : s.IsTangentAt p as
hq : q ∈ as
hqp : q ≠ p
this : s.radius ^ 2 < dist q s.center ^ 2
⊢ s.radius < dist q s.center

Tactic:
by

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
f : P ≃ᵃⁱ[ℝ] P
h : f s.center = s.center
⊢ map (↑f.toAffineEquiv) (s.orthRadius p) = s.orthRadius (f p)

Tactic:
by

Goal state after:
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 q : P
h : s.IsTangentAt p as
hq : q ∈ as
hqp : q ≠ p
⊢ s.radius < dist q s.center

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
hpo : p -ᵥ o = (1 - t) • u + t • v
⊢ u ∈ Metric.closedBall 0 s.radius

Tactic:
by

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 q : P
⊢ (s.orthRadius p).direction ≤ (s.orthRadius q).direction ↔ ∃ r, q -ᵥ s.center = r • (p -ᵥ s.center)

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
o : P := s.center
t : ℝ
hpt : (AffineMap.lineMap p₁ p₂) t = p
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
hne₁ : p ≠ p₁
hne₂ : p ≠ p₂
ht₀' : 0 < t
ht₁' : t < 1
u : V := p₁ -ᵥ o
v : V := p₂ -ᵥ o
hu : ‖u‖ = s.radius
hv : ‖v‖ = s.radius
huv : u ≠ v
hpo : p -ᵥ o = (1 - t) • u + t • v
⊢ v ∈ Metric.closedBall 0 s.radius

Tactic:
by

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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}
hsup : line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤
this : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) ∈ affineSpan ℝ {p₁}
⊢ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁

Tactic:
by

Goal state after:
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
ht : s.IsTangentAt p as
hr : s.radius ≠ 0
hfr : Module.finrank ℝ ↥as.direction + 1 = Module.finrank ℝ V
⊢ as = s.orthRadius p

Tactic:
by

Goal state after:
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
p : P
⊢ s.IsTangent (s.orthRadius p) ↔ p ∈ s

Tactic:
by

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 q : P
⊢ s.orthRadius p ∥ s.orthRadius q ↔ ∃ r, r ≠ 0 ∧ q -ᵥ s.center = r • (p -ᵥ s.center)

Tactic:
by

Goal state after:
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
p₁ p₂ p : P
s : Sphere P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp : Sbtw ℝ p₁ p p₂
⊢ dist s.center p < s.radius

Tactic:
by

Goal state after:
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
h : s.IsTangent as
p : P
hp : p ∈ as
⊢ s.radius ≤ dist p s.center

Tactic:
by

Goal state after:
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
h : s.IsTangent as
p : P
hp : dist p s.center < s.radius
⊢ p ∉ as

Tactic:
by

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 q : P
⊢ dist q s.center ^ 2 = dist p s.center ^ 2 + dist q p ^ 2 ↔ q ∈ s.orthRadius p

Tactic:
by

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

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
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p p₁ p₂ p₃ : P
ha : AffineIndependent ℝ ![p₁, p₂, p₃]
h : dist p ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = dist p ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
ho : ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = ↑((orthogonalProjection line[ℝ, p₁, p₃]) p)
hs : ↑((orthogonalProjection (line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃])) p) = ↑((orthogonalProjection line[ℝ, p₁, p₂]) p)
hinf : line[ℝ, p₁, p₂] ⊓ line[ℝ, p₁, p₃] = affineSpan ℝ {p₁}
hsup : line[ℝ, p₁, p₂] ⊔ line[ℝ, p₁, p₃] = ⊤
⊢ ↑((orthogonalProjection line[ℝ, p₁, p₂]) p) = p₁

Tactic:
by

Goal state after:
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
h : s.IsTangent as
⊢ Metric.infDist s.center ↑as = s.radius

Tactic:
by

Goal state after: