ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- InnerProductGeometry.angle_add_le_pi_div_two_of_inner_eq_zero
theorem angle_add_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x + y) ≤ π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: InnerProductGeometry.angle x (x + y) ≤ π / 2
-- InnerProductGeometry.angle_add_lt_pi_div_two_of_inner_eq_zero
theorem angle_add_lt_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x + y) < π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0)
Goal: InnerProductGeometry.angle x (x + y) < π / 2
-- InnerProductGeometry.cos_angle_add_of_inner_eq_zero
theorem cos_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x + y)) = ‖x‖ / ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x‖ / ‖x + y‖
-- InnerProductGeometry.sin_angle_add_of_inner_eq_zero
theorem sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y ≠ 0)
Goal: Real.sin (InnerProductGeometry.angle x (x + y)) = ‖y‖ / ‖x + y‖
-- InnerProductGeometry.tan_angle_add_of_inner_eq_zero
theorem tan_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.tan (InnerProductGeometry.angle x (x + y)) = ‖y‖ / ‖x‖
-- InnerProductGeometry.cos_angle_add_mul_norm_of_inner_eq_zero
theorem cos_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x + y)) * ‖x + y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.cos (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖x‖
-- InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
theorem sin_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.sin (angle x (x + y)) * ‖x + y‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.sin (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖y‖
-- InnerProductGeometry.tan_angle_add_mul_norm_of_inner_eq_zero
theorem tan_angle_add_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    Real.tan (angle x (x + y)) * ‖x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y = 0)
Goal: Real.tan (InnerProductGeometry.angle x (x + y)) * ‖x‖ = ‖y‖
-- InnerProductGeometry.norm_div_cos_angle_add_of_inner_eq_zero
theorem norm_div_cos_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    ‖x‖ / Real.cos (angle x (x + y)) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y = 0)
Goal: ‖x‖ / Real.cos (InnerProductGeometry.angle x (x + y)) = ‖x + y‖
-- InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero
theorem norm_div_sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.sin (angle x (x + y)) = ‖x + y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x = 0 ∨ y ≠ 0)
Goal: ‖y‖ / Real.sin (InnerProductGeometry.angle x (x + y)) = ‖x + y‖
-- InnerProductGeometry.norm_div_tan_angle_add_of_inner_eq_zero
theorem norm_div_tan_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.tan (angle x (x + y)) = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x = 0 ∨ y ≠ 0)
Goal: ‖y‖ / Real.tan (InnerProductGeometry.angle x (x + y)) = ‖x‖
-- InnerProductGeometry.angle_sub_eq_arccos_of_inner_eq_zero
theorem angle_sub_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: InnerProductGeometry.angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖)
-- InnerProductGeometry.angle_sub_eq_arcsin_of_inner_eq_zero
theorem angle_sub_eq_arcsin_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y ≠ 0)
Goal: InnerProductGeometry.angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)
-- InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero
theorem angle_sub_eq_arctan_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x - y) = Real.arctan (‖y‖ / ‖x‖) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0)
Goal: InnerProductGeometry.angle x (x - y) = Real.arctan (‖y‖ / ‖x‖)
-- InnerProductGeometry.angle_sub_pos_of_inner_eq_zero
theorem angle_sub_pos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    0 < angle x (x - y) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x = 0 ∨ y ≠ 0)
Goal: 0 < InnerProductGeometry.angle x (x - y)
-- InnerProductGeometry.angle_sub_le_pi_div_two_of_inner_eq_zero
theorem angle_sub_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    angle x (x - y) ≤ π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: InnerProductGeometry.angle x (x - y) ≤ π / 2
-- InnerProductGeometry.angle_sub_lt_pi_div_two_of_inner_eq_zero
theorem angle_sub_lt_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
    angle x (x - y) < π / 2 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0)
Goal: InnerProductGeometry.angle x (x - y) < π / 2
-- InnerProductGeometry.cos_angle_sub_of_inner_eq_zero
theorem cos_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.cos (InnerProductGeometry.angle x (x - y)) = ‖x‖ / ‖x - y‖
-- InnerProductGeometry.sin_angle_sub_of_inner_eq_zero
theorem sin_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y ≠ 0) :
    Real.sin (angle x (x - y)) = ‖y‖ / ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y ≠ 0)
Goal: Real.sin (InnerProductGeometry.angle x (x - y)) = ‖y‖ / ‖x - y‖
-- InnerProductGeometry.tan_angle_sub_of_inner_eq_zero
theorem tan_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.tan (angle x (x - y)) = ‖y‖ / ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.tan (InnerProductGeometry.angle x (x - y)) = ‖y‖ / ‖x‖
-- InnerProductGeometry.cos_angle_sub_mul_norm_of_inner_eq_zero
theorem cos_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.cos (angle x (x - y)) * ‖x - y‖ = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.cos (InnerProductGeometry.angle x (x - y)) * ‖x - y‖ = ‖x‖
-- InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero
theorem sin_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
    Real.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0)
Goal: Real.sin (InnerProductGeometry.angle x (x - y)) * ‖x - y‖ = ‖y‖
-- InnerProductGeometry.tan_angle_sub_mul_norm_of_inner_eq_zero
theorem tan_angle_sub_mul_norm_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    Real.tan (angle x (x - y)) * ‖x‖ = ‖y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y = 0)
Goal: Real.tan (InnerProductGeometry.angle x (x - y)) * ‖x‖ = ‖y‖
-- InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero
theorem norm_div_cos_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0 ∨ y = 0) :
    ‖x‖ / Real.cos (angle x (x - y)) = ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x ≠ 0 ∨ y = 0)
Goal: ‖x‖ / Real.cos (InnerProductGeometry.angle x (x - y)) = ‖x - y‖
-- InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero
theorem norm_div_sin_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x = 0 ∨ y ≠ 0)
Goal: ‖y‖ / Real.sin (InnerProductGeometry.angle x (x - y)) = ‖x - y‖
-- InnerProductGeometry.norm_div_tan_angle_sub_of_inner_eq_zero
theorem norm_div_tan_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
    ‖y‖ / Real.tan (angle x (x - y)) = ‖x‖ 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y : V} (h : ⟪x, y⟫_ℝ = 0) (h0 : x = 0 ∨ y ≠ 0)
Goal: ‖y‖ / Real.tan (InnerProductGeometry.angle x (x - y)) = ‖x‖
-- EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two
theorem dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two (p1 p2 p3 : P) :
    dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 ↔
      ∠ p1 p2 p3 = π / 2 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p1 p2 p3 : P)
Goal: dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 ↔ ∠ p1 p2 p3 = π / 2
-- EuclideanGeometry.angle_eq_arccos_of_angle_eq_pi_div_two
theorem angle_eq_arccos_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    ∠ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: ∠ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃)
-- EuclideanGeometry.angle_eq_arcsin_of_angle_eq_pi_div_two
theorem angle_eq_arcsin_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂) : ∠ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂)
Goal: ∠ p₂ p₃ p₁ = Real.arcsin (dist p₁ p₂ / dist p₁ p₃)
-- EuclideanGeometry.angle_eq_arctan_of_angle_eq_pi_div_two
theorem angle_eq_arctan_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₃ ≠ p₂) : ∠ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂) 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₃ ≠ p₂)
Goal: ∠ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂)
-- EuclideanGeometry.angle_pos_of_angle_eq_pi_div_two
theorem angle_pos_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : 0 < ∠ p₂ p₃ p₁ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ ≠ p₂ ∨ p₃ = p₂)
Goal: 0 < ∠ p₂ p₃ p₁
-- EuclideanGeometry.angle_le_pi_div_two_of_angle_eq_pi_div_two
theorem angle_le_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    ∠ p₂ p₃ p₁ ≤ π / 2 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: ∠ p₂ p₃ p₁ ≤ π / 2
-- EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two
theorem angle_lt_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₃ ≠ p₂) : ∠ p₂ p₃ p₁ < π / 2 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₃ ≠ p₂)
Goal: ∠ p₂ p₃ p₁ < π / 2
-- EuclideanGeometry.cos_angle_of_angle_eq_pi_div_two
theorem cos_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: Real.cos (∠ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃
-- EuclideanGeometry.sin_angle_of_angle_eq_pi_div_two
theorem sin_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂) : Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ ≠ p₂ ∨ p₃ ≠ p₂)
Goal: Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃
-- EuclideanGeometry.tan_angle_of_angle_eq_pi_div_two
theorem tan_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂
-- EuclideanGeometry.cos_angle_mul_dist_of_angle_eq_pi_div_two
theorem cos_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.cos (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: Real.cos (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₃ p₂
-- EuclideanGeometry.sin_angle_mul_dist_of_angle_eq_pi_div_two
theorem sin_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
    Real.sin (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
Goal: Real.sin (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂
-- EuclideanGeometry.tan_angle_mul_dist_of_angle_eq_pi_div_two
theorem tan_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ = p₂ ∨ p₃ ≠ p₂) : Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ = p₂ ∨ p₃ ≠ p₂)
Goal: Real.tan (∠ p₂ p₃ p₁) * dist p₃ p₂ = dist p₁ p₂
-- EuclideanGeometry.dist_div_cos_angle_of_angle_eq_pi_div_two
theorem dist_div_cos_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ = p₂ ∨ p₃ ≠ p₂) : dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ = p₂ ∨ p₃ ≠ p₂)
Goal: dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃
-- EuclideanGeometry.dist_div_sin_angle_of_angle_eq_pi_div_two
theorem dist_div_sin_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ ≠ p₂ ∨ p₃ = p₂)
Goal: dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃
-- EuclideanGeometry.dist_div_tan_angle_of_angle_eq_pi_div_two
theorem dist_div_tan_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
    (h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂ 
Arguments: {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) (h0 : p₁ ≠ p₂ ∨ p₃ = p₂)
Goal: dist p₁ p₂ / Real.tan (∠ p₂ p₃ p₁) = dist p₃ p₂
-- NFA.mem_stepSet
theorem mem_stepSet (s : σ) (S : Set σ) (a : α) : s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (s : σ) (S : Set σ) (a : α)
Goal: s ∈ M.stepSet S a ↔ ∃ t ∈ S, s ∈ M.step t a
-- NFA.stepSet_empty
@[simp]
theorem stepSet_empty (a : α) : M.stepSet ∅ a = ∅ 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (a : α)
Goal: M.stepSet ∅ a = ∅
-- NFA.evalFrom_nil
@[simp]
theorem evalFrom_nil (S : Set σ) : M.evalFrom S [] = S 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (S : Set σ)
Goal: M.evalFrom S [] = S
-- NFA.evalFrom_singleton
@[simp]
theorem evalFrom_singleton (S : Set σ) (a : α) : M.evalFrom S [a] = M.stepSet S a 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (S : Set σ) (a : α)
Goal: M.evalFrom S [a] = M.stepSet S a
-- NFA.evalFrom_append_singleton
@[simp]
theorem evalFrom_append_singleton (S : Set σ) (x : List α) (a : α) :
    M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (S : Set σ) (x : List α) (a : α)
Goal: M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a
-- NFA.eval_nil
@[simp]
theorem eval_nil : M.eval [] = M.start 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ)
Goal: M.eval [] = M.start
-- NFA.eval_singleton
@[simp]
theorem eval_singleton (a : α) : M.eval [a] = M.stepSet M.start a 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (a : α)
Goal: M.eval [a] = M.stepSet M.start a
-- NFA.eval_append_singleton
@[simp]
theorem eval_append_singleton (x : List α) (a : α) : M.eval (x ++ [a]) = M.stepSet (M.eval x) a 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) (x : List α) (a : α)
Goal: M.eval (x ++ [a]) = M.stepSet (M.eval x) a
-- NFA.mem_accepts
theorem mem_accepts {x : List α} : x ∈ M.accepts ↔ ∃ S ∈ M.accept, S ∈ M.evalFrom M.start x 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) {x : List α}
Goal: x ∈ M.accepts ↔ ∃ S ∈ M.accept, S ∈ M.evalFrom M.start x
-- NFA.toDFA_correct
@[simp]
theorem toDFA_correct : M.toDFA.accepts = M.accepts 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ)
Goal: M.toDFA.accepts = M.accepts
-- NFA.pumping_lemma
theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
    (hlen : Fintype.card (Set σ) ≤ List.length x) :
    ∃ a b c,
      x = a ++ b ++ c ∧
        a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts 
Arguments: {α : Type u} {σ : Type v} (M : NFA α σ) [Fintype σ] {x : List α} (hx : x ∈ M.accepts) (hlen : Fintype.card (Set σ) ≤ x.length)
Goal: ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts
-- DFA.toNFA_step
@[simps] def toNFA (M : DFA α σ') : NFA α σ' 
Arguments: {α : Type u} {σ' : Type v} (M : DFA α σ') (s : σ') (a : α)
Goal: M.toNFA.step s a = {M.step s a}
-- DFA.toNFA_accept
@[simps] def toNFA (M : DFA α σ') : NFA α σ' 
Arguments: {α : Type u} {σ' : Type v} (M : DFA α σ')
Goal: M.toNFA.accept = M.accept
-- DFA.toNFA_start
@[simps] def toNFA (M : DFA α σ') : NFA α σ' 
Arguments: {α : Type u} {σ' : Type v} (M : DFA α σ')
Goal: M.toNFA.start = {M.start}
-- DFA.toNFA_evalFrom_match
@[simp]
theorem toNFA_evalFrom_match (M : DFA α σ) (start : σ) (s : List α) :
    M.toNFA.evalFrom {start} s = {M.evalFrom start s} 
Arguments: {α : Type u} {σ : Type v} (M : DFA α σ) (start : σ) (s : List α)
Goal: M.toNFA.evalFrom {start} s = {M.evalFrom start s}
-- DFA.toNFA_correct
@[simp]
theorem toNFA_correct (M : DFA α σ) : M.toNFA.accepts = M.accepts 
Arguments: {α : Type u} {σ : Type v} (M : DFA α σ)
Goal: M.toNFA.accepts = M.accepts
-- contMDiffWithinAt_iff_contDiffWithinAt
theorem contMDiffWithinAt_iff_contDiffWithinAt {f : E → E'} {s : Set E} {x : E} :
    ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {n : ℕ∞} {f : E → E'} {s : Set E} {x : E}
Goal: ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x
-- contMDiffAt_iff_contDiffAt
theorem contMDiffAt_iff_contDiffAt {f : E → E'} {x : E} :
    ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f x ↔ ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {n : ℕ∞} {f : E → E'} {x : E}
Goal: ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f x ↔ ContDiffAt 𝕜 n f x
-- contMDiffOn_iff_contDiffOn
theorem contMDiffOn_iff_contDiffOn {f : E → E'} {s : Set E} :
    ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, E') n f s ↔ ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {n : ℕ∞} {f : E → E'} {s : Set E}
Goal: ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, E') n f s ↔ ContDiffOn 𝕜 n f s
-- contMDiff_iff_contDiff
theorem contMDiff_iff_contDiff {f : E → E'} : ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f ↔ ContDiff 𝕜 n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {n : ℕ∞} {f : E → E'}
Goal: ContMDiff 𝓘(𝕜, E) 𝓘(𝕜, E') n f ↔ ContDiff 𝕜 n f
-- ContDiffWithinAt.comp_contMDiffWithinAt
theorem ContDiffWithinAt.comp_contMDiffWithinAt {g : F → F'} {f : M → F} {s : Set M} {t : Set F}
    {x : M} (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x)
    (h : s ⊆ f ⁻¹' t) : ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} {s : Set M} {t : Set F} {x : M} (hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x) (h : s ⊆ f ⁻¹' t)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x
-- ContDiffAt.comp_contMDiffWithinAt
theorem ContDiffAt.comp_contMDiffWithinAt {g : F → F'} {f : M → F} {s : Set M}
    {x : M} (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x) :
    ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} {s : Set M} {x : M} (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContMDiffWithinAt I 𝓘(𝕜, F) n f s x)
Goal: ContMDiffWithinAt I 𝓘(𝕜, F') n (g ∘ f) s x
-- ContDiffAt.comp_contMDiffAt
theorem ContDiffAt.comp_contMDiffAt {g : F → F'} {f : M → F} {x : M} (hg : ContDiffAt 𝕜 n g (f x))
    (hf : ContMDiffAt I 𝓘(𝕜, F) n f x) : ContMDiffAt I 𝓘(𝕜, F') n (g ∘ f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {F' : Type u_11} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {n : ℕ∞} {g : F → F'} {f : M → F} {x : M} (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContMDiffAt I 𝓘(𝕜, F) n f x)
Goal: ContMDiffAt I 𝓘(𝕜, F') n (g ∘ f) x