awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos :
  dist a c * dist a c =
    k * dist a' b' * (k * dist a' b') + k * dist c' b' * (k * dist c' b') -
      2 * (k * dist a' b') * (k * dist c' b') * Real.cos (∠ a b c)
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos₁ : HPow.hPow (β := ℕ) (dist a c) 2 =
  HPow.hPow (β := ℕ) k 2 *
    (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
      dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c))
Tactic: field_simp at hcos
Cleared hypotheses: hcos
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos :
  HPow.hPow (β := ℕ) (dist a c) 2 =
    HPow.hPow (β := ℕ) k 2 *
      (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
        dist a' b' * dist c' b' * 2 * Real.cos (∠ a b c))
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos₁ : HPow.hPow (β := ℕ) (dist a c) 2 =
  HPow.hPow (β := ℕ) k 2 *
    (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
      dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c'))
Tactic: rw [h] at hcos
Cleared hypotheses: hcos
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos :
  HPow.hPow (β := ℕ) (dist a c) 2 =
    HPow.hPow (β := ℕ) k 2 *
      (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
        dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c'))
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos' : dist a' c' * dist a' c' =
  dist a' b' * dist a' b' + dist c' b' * dist c' b' - 2 * dist a' b' * dist c' b' * Real.cos (∠ a' b' c')
Tactic: have hcos' := law_cos a' b' c'
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos :
  HPow.hPow (β := ℕ) (dist a c) 2 =
    HPow.hPow (β := ℕ) k 2 *
      (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
        dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c'))
hcos' :
  dist a' c' * dist a' c' =
    dist a' b' * dist a' b' + dist c' b' * dist c' b' - 2 * dist a' b' * dist c' b' * Real.cos (∠ a' b' c')
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos'₁ : HPow.hPow (β := ℕ) (dist a' c') 2 =
  HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
    dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
Tactic: field_simp at hcos'
Cleared hypotheses: hcos'
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos :
  HPow.hPow (β := ℕ) (dist a c) 2 =
    HPow.hPow (β := ℕ) k 2 *
      (HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
        dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c'))
hcos' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
      dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos₁ : HPow.hPow (β := ℕ) (dist a c) 2 = HPow.hPow (β := ℕ) (k * dist a' c') 2
Tactic: rw [← hcos', ← mul_pow] at hcos
Cleared hypotheses: hcos, hcos'
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos : HPow.hPow (β := ℕ) (dist a c) 2 = HPow.hPow (β := ℕ) (k * dist a' c') 2
hcos' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
      dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
⊢ ![a, b, c] ∼ ![a', b', c']
have dist_ac_pos : 0 < dist a c
Tactic: have dist_ac_pos : 0 < dist a c := by grind [dist_pos, ne₁₃_of_not_collinear]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos : HPow.hPow (β := ℕ) (dist a c) 2 = HPow.hPow (β := ℕ) (k * dist a' c') 2
hcos' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (dist c' b') 2 -
      dist a' b' * dist c' b' * 2 * Real.cos (∠ a' b' c')
dist_ac_pos : 0 < dist a c
⊢ ![a, b, c] ∼ ![a', b', c']
have k_dist_a'c' : 0 ≤ k * dist a' c'
Tactic: have k_dist_a'c' : 0 ≤ k * dist a' c' := by positivity
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos : HPow.hPow (β := ℕ) (dist a c) 2 = HPow.hPow (β := ℕ) (k * dist a' c') 2
hcos' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
⊢ ![a, b, c] ∼ ![a', b', c']
have hcos₁ : dist c a = k * dist c' a'
Tactic: rw [pow_left_inj₀ (le_of_lt dist_ac_pos) k_dist_a'c' (by norm_num), dist_comm a _,
    dist_comm a' _] at hcos
Cleared hypotheses: hcos, hcos', dist_ac_pos, k_dist_a'c'
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (hypothesis)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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 c b = k * dist c' b'
hcos : dist c a = k * dist c' a'
hcos' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
⊢ ![a, b, c] ∼ ![a', b', c']
have h_bc₁ : dist b c = k * dist b' c'
Tactic: rw [dist_comm c _, dist_comm c' _] at h_bc
Cleared hypotheses: h_bc, hcos, hcos', dist_ac_pos, k_dist_a'c'
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
⊢ ![a, b, c] ∼ ![a', b', c']
have rw : ∃ (r : ℝ),
  0 < r ∧
    Pairwise fun (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)
Tactic: rw [similar_iff_exists_pos_pairwise_dist_eq]
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
⊢ ∃ (r : ℝ),
    0 < r ∧
      Pairwise fun (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)
have h₁ : 0 < k ∧
  Pairwise fun (i₁ i₂ : Fin (Nat.succ 0).succ.succ) ↦
    dist (![a, b, c] i₁) (![a, b, c] i₂) = k * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
Tactic: use k
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
⊢ 0 < k ∧
    Pairwise fun (i₁ i₂ : Fin (Nat.succ 0).succ.succ) ↦
      dist (![a, b, c] i₁) (![a, b, c] i₂) = k * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
have h₁ : Pairwise fun (i₁ i₂ : Fin (Nat.succ 0).succ.succ) ↦
  dist (![a, b, c] i₁) (![a, b, c] i₂) = k * dist (![a', b', c'] i₁) (![a', b', c'] i₂)
Tactic: refine ⟨k_pos, ?_⟩
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
i j : Fin (Nat.succ 0).succ.succ
hij : i ≠ j
⊢ dist (![a, b, c] i) (![a, b, c] j) = k * dist (![a', b', c'] i) (![a', b', c'] j)
have h₁ : ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, sorry⟩ ≠ j),
  dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ))) (![a, b, c] j) =
    k *
      dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a', b', c'] j)
Tactic: fin_cases i
Cleared hypotheses: i, hij
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
i j : Fin (Nat.succ 0).succ.succ
hij : i ≠ j
h₁ :
  ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, ⋯⟩ ≠ j),
    dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a, b, c] j) =
      k *
        dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
          (![a', b', c'] j)
⊢ dist (![a, b, c] i) (![a, b, c] j) = k * dist (![a', b', c'] i) (![a', b', c'] j)
have h₂ : ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨1, sorry⟩ ≠ j),
  dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨1, sorry⟩ : Fin (Nat.succ 0).succ.succ))) (![a, b, c] j) =
    k *
      dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨1, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a', b', c'] j)
Tactic: fin_cases i
Cleared hypotheses: i, hij
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
i j : Fin (Nat.succ 0).succ.succ
hij : i ≠ j
h₁ :
  ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, ⋯⟩ ≠ j),
    dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a, b, c] j) =
      k *
        dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
          (![a', b', c'] j)
h₂ :
  ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨1, ⋯⟩ ≠ j),
    dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨1, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a, b, c] j) =
      k *
        dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨1, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
          (![a', b', c'] j)
⊢ dist (![a, b, c] i) (![a, b, c] j) = k * dist (![a', b', c'] i) (![a', b', c'] j)
have h₃ : ∀ (hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨2, sorry⟩ ≠ j),
  dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨2, sorry⟩ : Fin (Nat.succ 0).succ.succ))) (![a, b, c] j) =
    k *
      dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨2, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a', b', c'] j)
Tactic: fin_cases i
Cleared hypotheses: i, hij
-- Lean 4 proof step from Mathlib.Geometry.Euclidean.Similarity (goal)
Goal:
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst : NormedAddCommGroup V₁
inst_1 : NormedAddCommGroup V₂
inst_2 : InnerProductSpace ℝ V₁
inst_3 : InnerProductSpace ℝ V₂
inst_4 : MetricSpace P₁
inst_5 : MetricSpace P₂
inst_6 : NormedAddTorsor V₁ P₁
inst_7 : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h_not_col : ¬Collinear ℝ (P := P₁) {a, b, c}
h_not_col' : ¬Collinear ℝ (P := P₂) {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' :
  HPow.hPow (β := ℕ) (dist a' c') 2 =
    HPow.hPow (β := ℕ) (dist a' b') 2 + HPow.hPow (β := ℕ) (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'
j : Fin (Nat.succ 0).succ.succ
hij : (fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, ⋯⟩ ≠ j
⊢ dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ))) (![a, b, c] j) =
    k *
      dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, ⋯⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a', b', c'] j)
have h₁ : ∀
  (hij :
    Ne (α := Fin (Nat.succ 0).succ.succ) ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, sorry⟩)
      ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) ⟨0, sorry⟩)),
  dist (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
      (![a, b, c] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ))) =
    k *
      dist (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
        (![a', b', c'] ((fun (i : Fin (Nat.succ 0).succ.succ) ↦ i) (⟨0, sorry⟩ : Fin (Nat.succ 0).succ.succ)))
Tactic: fin_cases j
Cleared hypotheses: j, hij