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

Goal state before:
pp: case h.«2».«0»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
dist_self,

Goal state after:
case h.«2».«0»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«2».«1»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
dist_self,

Goal state after:
case h.«2».«1»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩))
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
h_cospherical' : Cospherical {p₁', p₂', p₃', p₄'}
⊢ Cospherical {p₁, p₂, p₃, p₄}

Tactic:
#adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
    (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal.
    It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in the new
    canonicalizer; a minimization would help. The original proof was:
    `grind [Set.image_insert_eq, Set.image_singleton]` -/

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
h_cospherical' : Cospherical {p₁', p₂', p₃', p₄'}
⊢ Cospherical {p₁, p₂, p₃, p₄}
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«2».«2»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
dist_self,

Goal state after:
case h.«2».«2»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ 0 = k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«2».«2»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ 0 = k * dist (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
dist_self,

Goal state after:
case h.«2».«2»
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'
dist_a'b' : dist a' b' ≠ 0
dist_b'c' : dist b' c' ≠ 0
k : ℝ := dist a b / dist a' b'
hd : k = dist b c / dist b' c'
hk : k = dist a b / dist a' b'
k_pos : 0 < k
h_ab : dist a b = k * dist a' b'
h_bc : dist b c = k * dist b' c'
hcos : dist c a = k * dist c' a'
hcos' : dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
k_dist_a'c' : 0 ≤ k * dist a' c'
hij : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ 0 = k * 0
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 : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
i₂ : Fin (n + 3)
hne : i₁ ≠ i₂
⊢ p -ᵥ s.mongePoint ∈ (ℝ ∙ (s.points i₁ -ᵥ s.points i₂))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)

Tactic:
← s.direction_mongePlane,

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 : Fin (n + 3)
p : P
h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂
i₂ : Fin (n + 3)
hne : i₁ ≠ i₂
⊢ p -ᵥ s.mongePoint ∈ (s.mongePlane i₁ i₂).direction
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
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' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - 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')
⊢ ∠ a b c = ∠ a' b' c'

Tactic:
rw [h_cos', sub_right_inj] at h_cos

Goal state after:
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')
⊢ ∠ a b c = ∠ a' b' c'
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}
hab : a ≠ b
hcd : c ≠ d
hapb : Collinear ℝ {a, p, b}
hcpd : Collinear ℝ {c, p, d}
⊢ dist a p * dist b p = dist c p * dist d p

Tactic:
exact mul_dist_eq_mul_dist_of_cospherical h
    (hapb.mem_affineSpan_of_mem_of_ne (by simp) (by simp) (by simp) hab)
    (hcpd.mem_affineSpan_of_mem_of_ne (by simp) (by simp) (by simp) hcd)

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

Goal state before:
pp: case inl
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ 0 ≠ ↑(closure ↑X)
⊢ X ^ 0 ⊂ X ^ (0 + 1)

pp: case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ n ≠ ↑(closure ↑X)
hn : n ≠ 0
⊢ X ^ n ⊂ X ^ (n + 1)

Tactic:
· simpa [ssubset_iff_subset_not_subset, hX₁, -Finset.subset_singleton_iff]
      using! hX.not_subset_singleton

Goal state after:
case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ n ≠ ↑(closure ↑X)
hn : n ≠ 0
⊢ X ^ n ⊂ X ^ (n + 1)
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
hAsymm : A⁻¹ = A
⊢ #A ≤ #(image (⇑(QuotientGroup.mk' H)) A) * #({x ∈ A ^ 2 | x ∈ H})

Tactic:
set π := QuotientGroup.mk' H
  rw [card_eq_sum_card_image π]
  refine sum_le_card_nsmul _ _ _ <| forall_mem_image.2 fun a ha ↦ ?_
  calc
    #{a' ∈ A | π a' = π a}
    _ ≤ #({a' ∈ A | π a' = π a}⁻¹ * {a' ∈ A | π a' = π a}) :=
      card_le_card_mul_left ⟨a⁻¹, by simpa⟩
    _ ≤ #{x ∈ A⁻¹ * A | x ∈ H} := by
      gcongr
      simp only [mul_subset_iff, mem_inv', map_inv, mem_filter, and_imp]
      rintro x hx hxa y hy hya
      refine ⟨mul_mem_mul (by simpa) hy, (QuotientGroup.eq_one_iff _).1 (?_ : π _ = _)⟩
      simp [hya, ← hxa]
    _ = #{x ∈ A ^ 2 | x ∈ H} := by simp [hAsymm, sq]

Goal state after:
Module: Mathlib.Geometry.Manifold.Algebra.SmoothFunctions

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
E' : Type u_3
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedSpace 𝕜 E'
H : Type u_4
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
N : Type u_6
inst✝⁸ : TopologicalSpace N
inst✝⁷ : ChartedSpace H N
E'' : Type u_7
inst✝⁶ : NormedAddCommGroup E''
inst✝⁵ : NormedSpace 𝕜 E''
H'' : Type u_8
inst✝⁴ : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
N' : Type u_9
inst✝³ : TopologicalSpace N'
inst✝² : ChartedSpace H'' N'
n : ℕ∞ω
V : Type u_10
inst✝¹ : NormedAddCommGroup V
inst✝ : NormedSpace 𝕜 V
f : C^n⟮I, N; 𝓘(𝕜, V), V⟯
⊢ 1 • f = f

Tactic:
by ext x; exact one_smul 𝕜 (f x)

Goal state after:
Module: Mathlib.Geometry.Manifold.LocalDiffeomorph

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
F' : Type u_4
inst✝¹⁰ : NormedAddCommGroup F'
inst✝⁹ : NormedSpace 𝕜 F'
H₁ : Type u_5
inst✝⁸ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁷ : TopologicalSpace H₂
H₃ : Type u_7
inst✝⁶ : TopologicalSpace H₃
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
K : ModelWithCorners 𝕜 F' H₃
M : Type u_8
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H₁ M
N : Type u_9
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H₂ N
P : Type u_10
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace H₃ P
n : WithTop ℕ∞
f : M → N
s : Set M
x : M
I' : ?m.25
hf : IsLocalDiffeomorphAt I J n f x
hn : n ≠ 0
⊢ Function.LeftInverse ⇑(mfderiv% ↑hf.localInverse.toPartialEquiv (f x)) ⇑(mfderiv% f x)

Tactic:
apply ContinuousLinearMap.leftInverse_of_comp

Goal state after:
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹² : NormedAddCommGroup F
inst✝¹¹ : NormedSpace 𝕜 F
F' : Type u_4
inst✝¹⁰ : NormedAddCommGroup F'
inst✝⁹ : NormedSpace 𝕜 F'
H₁ : Type u_5
inst✝⁸ : TopologicalSpace H₁
H₂ : Type u_6
inst✝⁷ : TopologicalSpace H₂
H₃ : Type u_7
inst✝⁶ : TopologicalSpace H₃
I : ModelWithCorners 𝕜 E H₁
J : ModelWithCorners 𝕜 F H₂
K : ModelWithCorners 𝕜 F' H₃
M : Type u_8
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H₁ M
N : Type u_9
inst✝³ : TopologicalSpace N
inst✝² : ChartedSpace H₂ N
P : Type u_10
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace H₃ P
n : WithTop ℕ∞
f : M → N
s : Set M
x : M
I' : ?m.25
hf : IsLocalDiffeomorphAt I J n f x
hn : n ≠ 0
⊢ mfderiv% ↑hf.localInverse.toPartialEquiv (f x) ∘SL mfderiv% f x = ContinuousLinearMap.id 𝕜 (TangentSpace I 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 : 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
⊢ (MDiffAt[u] fun x ↦ ⟨x, ∑' (i : ι), t i x⟩) x₀

Tactic:
obtain ⟨u', hu', hfin⟩ := ht x₀
  -- All sections `t i` but a finite set `s` vanish near `x₀`: choose a neighbourhood `u` of `x₀`
  -- and a finite set `s` of sections which don't vanish.
  let s := {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ u').Nonempty}
  have := hfin.fintype
  have : MDiffAt[u ∩ u'] (T% (fun x ↦ ∑ i ∈ s, (t i x))) x₀ :=
     .sum_section fun i _ ↦ ((ht' i).mono inter_subset_left)
  apply (mdifferentiableWithinAt_inter hu').mp
  apply this.congr' (fun y hy ↦ ?_) inter_subset_right (mem_of_mem_nhds hu')
  rw [TotalSpace.mk_inj, tsum_eq_sum']
  refine support_subset_iff'.mpr fun i hi ↦ ?_
  by_contra! h
  have : i ∈ s.toFinset := by
    refine Set.mem_toFinset.mpr ?_
    simp only [s, ne_eq, Set.mem_ofPred_eq]
    use y
    simp [h, hy]
  exact hi this

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
F : Type u_8
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : M
s : Set M
hs : UniqueMDiffAt[s] x
⊢ MDiffAt[s] 0 x

Tactic:
by exact mdifferentiableWithinAt_const

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁴ : TopologicalSpace H
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
s : Set M
x : M
V W : (x : M) → TangentSpace I x
h : s ∈ 𝓝 x
⊢ mlieBracketWithin I V W s x = mlieBracket I V W x

Tactic:
by
  rw [← mlieBracketWithin_univ, ← univ_inter s, mlieBracketWithin_inter h]

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Icc

Goal state before:
pp: case neg
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
H : Type u_2
inst✝² : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
x y : ℝ
h : Fact (x < y)
f : ↑(Icc x y) → M
w : ↑(Icc x y)
hw : ¬MDiffAt f w
⊢ ¬MDiffAt[Icc x y] (f ∘ projIcc x y ⋯) ↑w

Tactic:
rwa [mdifferentiableWithinAt_comp_projIcc_iff]

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(ℝ ∙ ↑(-v))ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
⊢ Injective
    ⇑(fderiv ℝ (↑(ChartedSpace.chartAt ↑v) ∘ Subtype.val ∘ ↑(ChartedSpace.chartAt v).symm)
        (↑(ChartedSpace.chartAt v) v))

Tactic:
suffices Injective (fderiv ℝ ((stereoInvFunAux (-v : E) ∘ (↑)) ∘ U.symm) 0) by
    convert! this using 3
    congr 2
    apply stereographic'_neg (v := v)

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
n : ℕ
inst✝ : Fact (finrank ℝ E = n + 1)
v : ↑(sphere 0 1)
U : ↥(ℝ ∙ ↑(-v))ᗮ ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin n) := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr
⊢ Injective ⇑(fderiv ℝ ((stereoInvFunAux (-↑v) ∘ Subtype.val) ∘ ⇑U.symm) 0)
Module: Mathlib.Geometry.Manifold.IntegralCurve.UniformTime

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ γ a' ?m.104 = γ a ?m.104

Tactic:
hγx a,

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I 1 M
inst✝¹ : T2Space M
v : (x : M) → TangentSpace I x
inst✝ : BoundarylessManifold I M
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
x : M
γ : ℝ → ℝ → M
hγx : ∀ (a : ℝ), γ a 0 = x
hγ : ∀ a > 0, IsMIntegralCurveOn (γ a) v (Ioo (-a) a)
a a' : ℝ
hpos : 0 < a'
hle : a' ≤ a
⊢ γ a' 0 = x