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

Goal state before:
pp: case h.«0».«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) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
try {rw [dist_self, dist_self, mul_zero]}

Goal state after:
case h.«0».«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) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«1».«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) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
try {rw [dist_self, dist_self, mul_zero]}

Goal state after:
case h.«1».«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) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨0, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨0, ⋯⟩))
Module: Mathlib.Geometry.Euclidean.Similarity

Goal state before:
pp: case h.«1».«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) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
try {rw [dist_self, dist_self, mul_zero]}

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

Goal state before:
pp: case h.«1».«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) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))

Tactic:
try {rw [dist_self, dist_self, mul_zero]}

Goal state after:
case h.«1».«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) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ dist (![a, b, c] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a, b, c] ((fun i ↦ i) ⟨2, ⋯⟩)) =
    k * dist (![a', b', c'] ((fun i ↦ i) ⟨1, ⋯⟩)) (![a', b', c'] ((fun i ↦ i) ⟨2, ⋯⟩))
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:
try {rw [dist_self, dist_self, mul_zero]}

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:
try {rw [dist_self, dist_self, mul_zero]}

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.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:
try {rw [dist_self, dist_self, mul_zero]}

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

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
v : V
r : ℝ
hr : r ≠ 0
⊢ s.secondInter p (r • v) = s.secondInter p v

Tactic:
simp_rw [Sphere.secondInter, real_inner_smul_left, inner_smul_right, smul_smul,
    div_mul_eq_div_div]
  rw [mul_comm, ← mul_div_assoc, ← mul_div_assoc, mul_div_cancel_left₀ _ hr, mul_comm, mul_assoc,
    mul_div_cancel_left₀ _ hr, mul_comm]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
p : P
h₁₂ : i₁ ≠ i₂
i₃ : Fin 3
h₂ : p ∈ mongePlane t i₃ i₁
h₁ : p ∈ mongePlane t i₃ i₂
h₂₃ : i₂ ≠ i₃
h₁₃ : i₁ ≠ i₃
ha : ∀ (i : Fin 3), i₃ ≠ i → p ∈ mongePlane t i₃ i
⊢ p = mongePoint t

Tactic:
exact eq_mongePoint_of_forall_mem_mongePlane ha

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

Goal state before:
pp: case hw
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 : Sphere P
n : ℕ
inst✝ : n.AtLeastTwo
sx : Affine.Simplex ℝ P n
i : Fin (n + 1)
hi : sx.points i ∈ s
hsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.center ≤ s.radius
w : Fin (n + 1) → ℝ
hw : ∑ i, w i = 1
hw01 : ∀ (i : Fin (n + 1)), w i ∈ Set.Ioo 0 1
r : ℝ := (1 - w i)⁻¹
hrpos : 0 < r
p' : P := (AffineMap.lineMap (sx.points i) ((Finset.affineCombination ℝ Finset.univ sx.points) w)) r
hp' : p' -ᵥ sx.points i = r • ((Finset.affineCombination ℝ Finset.univ sx.points) w -ᵥ sx.points i)
⊢ ∑ i_1, (r • (w - Pi.single i 1) + Pi.single i 1) i_1 = 1

Tactic:
simp [Finset.sum_add_distrib, ← Finset.mul_sum, hw]

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
s : Set P
ho : OrthocentricSystem s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
⊢ AffineIndependent ℝ p

Tactic:
rcases ho with ⟨t, hto, hst⟩
  rw [hst] at hps
  rcases exists_dist_eq_circumradius_of_subset_insert_orthocenter hto hps hpi with ⟨c, _, hc⟩
  exact Cospherical.affineIndependent ⟨c, t.circumradius, hc⟩ Set.Subset.rfl hpi

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : WithTop ℕ∞
inst✝ : IsManifold I n M
hn : n ≠ 0
⊢ IsClosed[inst✝²] (ModelWithCorners.boundary M)

Tactic:
rw [← I.compl_interior, isClosed_compl_iff]

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
n : WithTop ℕ∞
inst✝ : IsManifold I n M
hn : n ≠ 0
⊢ IsOpen[inst✝²] (ModelWithCorners.interior M)
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: case e'_1
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 V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) = x

Tactic:
exact extChartAt_to_inv x

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

Goal state before:
pp: case e'_2
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 V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
e_1✝ : TangentSpace I (↑(extChartAt I x).symm (↑(extChartAt I x) x)) = TangentSpace I x
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) = x

Tactic:
exact extChartAt_to_inv x

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

Goal state before:
pp: case e'_3
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 V₁ W₁ : (x : M) → TangentSpace I x
hV : V₁ =ᶠ[𝓝[s] x] V
hxV : V₁ x = V x
hW : W₁ =ᶠ[𝓝[s] x] W
hxW : W₁ x = W x
I1 : NormedAddCommGroup (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) :=
  { toNorm := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_1,
    toAddCommGroup := instAddCommGroupTangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x),
    toPseudoMetricSpace := EventuallyEq.mlieBracketWithin_vectorField_eq._aux_1_13, eq_of_dist_eq_zero := ⋯,
    dist_eq := ⋯ }
_I2 : NormedSpace 𝕜 (TangentSpace 𝓘(𝕜, E) (↑(extChartAt I x) x)) := inst✝²
e_1✝ : TangentSpace I (↑(extChartAt I x).symm (↑(extChartAt I x) x)) = TangentSpace I x
⊢ ↑(extChartAt I x).symm (↑(extChartAt I x) x) = x

Tactic:
exact extChartAt_to_inv x

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

Goal state before:
pp: case property
𝕜 : Type u_1
E : Type u_2
E' : Type u_3
F : Type u_4
F' : Type u_5
H : Type u_6
H' : Type u_7
G : Type u_8
G' : Type u_9
inst✝²⁴ : NontriviallyNormedField 𝕜
inst✝²³ : NormedAddCommGroup E
inst✝²² : NormedSpace 𝕜 E
inst✝²¹ : NormedAddCommGroup E'
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace 𝕜 F'
inst✝¹⁵ : TopologicalSpace H
inst✝¹⁴ : TopologicalSpace H'
inst✝¹³ : TopologicalSpace G
inst✝¹² : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 F G
J' : ModelWithCorners 𝕜 F' G'
M : Type u_10
M' : Type u_11
N : Type u_12
N' : Type u_13
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : ChartedSpace H' M'
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
n : ℕ∞ω
f : M → N
x : M
P : (M → N) → OpenPartialHomeomorph M H → OpenPartialHomeomorph N G → Prop
inst✝³ : IsManifold I n M
inst✝² : IsManifold I' n M'
inst✝¹ : IsManifold J n N
inst✝ : IsManifold J' n N'
Q : (M' → N') → OpenPartialHomeomorph M' H' → OpenPartialHomeomorph N' G' → Prop
R : (M × M' → N × N') → OpenPartialHomeomorph (M × M') (H × H') → OpenPartialHomeomorph (N × N') (G × G') → Prop
hf : LiftSourceTargetPropertyAt I J n f x P
g : M' → N'
x' : M'
hg : LiftSourceTargetPropertyAt I' J' n g x' Q
h :
  ∀ {f : M → N} {φ₁ : OpenPartialHomeomorph M H} {ψ₁ : OpenPartialHomeomorph N G} {g : M' → N'}
    {φ₂ : OpenPartialHomeomorph M' H'} {ψ₂ : OpenPartialHomeomorph N' G'},
    P f φ₁ ψ₁ → Q g φ₂ ψ₂ → R (Prod.map f g) (φ₁.prod φ₂) (ψ₁.prod ψ₂)
⊢ R (Prod.map f g) (hf.domChart.prod hg.domChart) (hf.codChart.prod hg.codChart)

Tactic:
· exact h hf.property hg.property

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

Goal state before:
pp: x y : ℝ
hxy : Fact (x < y)
z : EuclideanHalfSpace 1
hz : (↑z).ofLp 0 ≤ y - x
⊢ ↑(↑(IccLeftChart x y).symm z) = ↑⟨(↑z).ofLp 0 + x, ⋯⟩

Tactic:
simp only [IccLeftChart_symm_apply, inf_eq_left]

Goal state after:
x y : ℝ
hxy : Fact (x < y)
z : EuclideanHalfSpace 1
hz : (↑z).ofLp 0 ≤ y - x
⊢ (↑z).ofLp 0 + x ≤ y
Module: Mathlib.Geometry.Manifold.Instances.Sphere

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv : ‖v‖ = 1
x : ↑(sphere 0 1)
hx : x ∈ {⟨v, ⋯⟩}ᶜ
h : ↑x = v
⊢ x ∈ {⟨v, ⋯⟩}

Tactic:
rw [← h] at hv

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
hv✝ : ‖v‖ = 1
x : ↑(sphere 0 1)
hv : ‖↑x‖ = 1
hx : x ∈ {⟨v, ⋯⟩}ᶜ
h : ↑x = v
⊢ x ∈ {⟨v, ⋯⟩}
Module: Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
H : Type u_2
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
inst✝¹ : IsManifold I 1 M
γ γ' : ℝ → M
v : (x : M) → TangentSpace I x
t₀ : ℝ
inst✝ : T2Space M
a b : ℝ
ht₀ : t₀ ∈ Ioo a b
hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t)
hv : ContMDiff I I.tangent 1 fun x ↦ ⟨x, v x⟩
hγ : IsMIntegralCurveOn γ v (Ioo a b)
hγ' : IsMIntegralCurveOn γ' v (Ioo a b)
h : γ t₀ = γ' t₀
s : Set ℝ := ⋯
hs : s = {t | γ t = γ' t} ∩ Ioo a b
t₁ : ℝ
ht₁ : t₁ ∈ s
hmem : Ioo a b ∈ 𝓝 t₁
heq : γ =ᶠ[𝓝 t₁] γ'
⊢ ∀ (x : ℝ), γ x = γ' x ∧ a < x ∧ x < b → ({t | γ t = γ' t} ∩ Ioo a b) x

Tactic:
exact fun _ ht ↦ ht

Goal state after: