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

Goal state before:
pp: case «1».«0».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«0».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«0».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«1».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₁₃ h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«1».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«1».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₁₃ h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«2».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«2».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «1».«2».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«0».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«0».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«0».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«1».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«1».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«1».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
h₁₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₂₃ : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«2».«0»
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
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₁₃ h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨0, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«2».«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
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
h₁₃ h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨1, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

Goal state before:
pp: case «2».«2».«2»
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
x✝ :
  ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2 ∧
    ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2 ∧ ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h012 : ∠ (t.points 0) (t.points 1) (t.points 2) < π / 2
h120 : ∠ (t.points 1) (t.points 2) (t.points 0) < π / 2
h201 : ∠ (t.points 2) (t.points 0) (t.points 1) < π / 2
h210 : ∠ (t.points 2) (t.points 1) (t.points 0) < π / 2
h021 : ∠ (t.points 0) (t.points 2) (t.points 1) < π / 2
h102 : ∠ (t.points 1) (t.points 0) (t.points 2) < π / 2
h₁₂ h₁₃ h₂₃ : (fun i ↦ i) ⟨2, ⋯⟩ ≠ (fun i ↦ i) ⟨2, ⋯⟩
⊢ ∠ (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) (t.points ((fun i ↦ i) ⟨2, ⋯⟩)) < π / 2

Tactic:
simp [*] at *

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

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 : 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
⊢ 0 < r

Tactic:
simp [inv_pos, sub_pos, r, (hw01 i).2]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : ¬Disjoint (ModelWithCorners.interior M) (ModelWithCorners.boundary M)
x : M
h1 : x ∈ ModelWithCorners.interior M
h2 : x ∈ ModelWithCorners.boundary M
⊢ ↑(extChartAt I x) x ∈ interior ?m.77 ∧ ↑(extChartAt I x) x ∈ frontier ?m.77

pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
h : ¬Disjoint (ModelWithCorners.interior M) (ModelWithCorners.boundary M)
x : M
h1 : x ∈ ModelWithCorners.interior M
h2 : x ∈ ModelWithCorners.boundary M
⊢ Set E

Tactic:
exact ⟨h1, h2⟩

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

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝³ : I.Boundaryless
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDiffAt f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖

Tactic:
have hI : range I = univ := ModelWithCorners.Boundaryless.range_eq_univ

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℂ E
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℂ F
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝³ : I.Boundaryless
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDiffAt f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
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
e e' : OpenPartialHomeomorph M H
x : M
hn : n ≠ 0
he : e ∈ atlas H M
he' : e' ∈ atlas H M
hex : x ∈ e.source
hex' : x ∈ e'.source
hx : ↑(e.extend I) x ∈ interior (e.extend I).target
⊢ ↑(e'.extend I) x ∈ interior (e'.extend I).target

Tactic:
let φ := I.extendCoordChange e e'
  have hφ : ContDiffOn 𝕜 n φ φ.source := contDiffOn_extendCoordChange
    (IsManifold.subset_maximalAtlas he) (IsManifold.subset_maximalAtlas he')
  suffices h : Function.Surjective (fderivWithin 𝕜 φ φ.source (e.extend I x)) →
      e'.extend I x ∈ interior (range I) by
    refine e'.mem_interior_extend_target (by simp [hex']) <| h ?_
    exact (isInvertible_fderivWithin_extendCoordChange hn (IsManifold.subset_maximalAtlas he)
      (IsManifold.subset_maximalAtlas he') <| by simp [hex, hex']).surjective
  intro hφx'
  /- Reduce the situation to the real case, then apply
  `DifferentiableAt.mem_interior_convex_of_surjective_fderiv`. -/
  wlog _ : IsRCLikeNormedField 𝕜
  · simp [I.range_eq_univ_of_not_isRCLikeNormedField ‹_›]
  let _ := IsRCLikeNormedField.rclike 𝕜
  let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
  have hφx : φ.source ∈ 𝓝 (e.extend I x) := by
    simp_rw [φ, extendCoordChange, PartialEquiv.trans_source, PartialEquiv.symm_source,
      Filter.inter_mem_iff, mem_interior_iff_mem_nhds.1 hx, true_and, e'.extend_source]
    exact e.extend_preimage_mem_nhds hex <| e'.open_source.mem_nhds hex'
  rw [← ContinuousLinearMap.coe_restrictScalars' (R := ℝ),
    (hφ.differentiableOn hn _ (by simp [φ, hex, hex'])).restrictScalars_fderivWithin (𝕜 := ℝ)
      (uniqueDiffWithinAt_of_mem_nhds hφx), fderivWithin_of_mem_nhds <| hφx] at hφx'
  rw [show e'.extend I x = φ (e.extend I x) by simp [φ, hex]]
  replace hφ := ((hφ.restrict_scalars ℝ).differentiableOn hn).differentiableAt hφx
  exact hφ.mem_interior_convex_of_surjective_fderiv hφx I.convex_range I.isClosed_range
    I.nonempty_interior (φ.mapsTo.mono_right <| by simp [φ, inter_assoc]) hφx'

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
n : ℕ∞ω
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
G : Type u_4
inst✝³ : TopologicalSpace G
inst✝² : ChartedSpace H G
inst✝¹ : Monoid G
inst✝ : ContMDiffMul I ∞ G
g✝ h : G
r : 𝕜
X✝ Y✝ : LeftInvariantDerivation I G
f f' : C^∞⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯
X Y : LeftInvariantDerivation I G
g : G
⊢ (𝒅ₕ ⋯) ((Derivation.evalAt 1) ⁅↑X, ↑Y⁆) = (Derivation.evalAt g) ⁅↑X, ↑Y⁆

Tactic:
ext f
      have hX := Derivation.congr_fun (left_invariant' g X) (Y f)
      have hY := Derivation.congr_fun (left_invariant' g Y) (X f)
      rw [hfdifferential_apply, fdifferential_apply, Derivation.evalAt_apply] at hX hY ⊢
      rw [comp_L] at hX hY
      rw [Derivation.commutator_apply, ContMDiffMap.coe_sub, Pi.sub_apply, coe_derivation]
      rw [coe_derivation] at hX hY ⊢
      rw [hX, hY]
      rfl

Goal state after: