ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- eq_rec_inj
@[simp]
theorem eq_rec_inj {a a' : α} (h : a = a') {C : α → Type*} (x y : C a) :
    (Eq.ndrec x h : C a') = Eq.ndrec y h ↔ x = y 
Arguments: {α : Sort u_1} {a a' : α} (h : a = a') {C : α → Type u_3} (x y : C a)
Goal: h ▸ x = h ▸ y ↔ x = y
-- cast_inj
@[simp]
theorem cast_inj {α β : Type u} (h : α = β) {x y : α} : cast h x = cast h y ↔ x = y 
Arguments: {α β : Type u} (h : α = β) {x y : α}
Goal: cast h x = cast h y ↔ x = y
-- Function.LeftInverse.eq_rec_eq
theorem Function.LeftInverse.eq_rec_eq {γ : β → Sort v} {f : α → β} {g : β → α}
    (h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
    -- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
    @Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a 
Arguments: {α : Sort u_1} {β : Sort u_2} {γ : β → Sort v} {f : α → β} {g : β → α} (h : Function.LeftInverse g f) (C : (a : α) → γ (f a)) (a : α)
Goal: ⋯ ▸ C (g (f a)) = C a
-- Function.LeftInverse.eq_rec_on_eq
theorem Function.LeftInverse.eq_rec_on_eq {γ : β → Sort v} {f : α → β} {g : β → α}
    (h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
    -- TODO: mathlib3 uses `(congr_arg f (h a)).recOn (C (g (f a)))` for LHS
    @Eq.recOn β (f (g (f a))) (fun x _ ↦ γ x) (f a) (congr_arg f (h a)) (C (g (f a))) = C a 
Arguments: {α : Sort u_1} {β : Sort u_2} {γ : β → Sort v} {f : α → β} {g : β → α} (h : Function.LeftInverse g f) (C : (a : α) → γ (f a)) (a : α)
Goal: Eq.recOn ⋯ (C (g (f a))) = C a
-- Function.LeftInverse.cast_eq
theorem Function.LeftInverse.cast_eq {γ : β → Sort v} {f : α → β} {g : β → α}
    (h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
    cast (congr_arg (fun a ↦ γ (f a)) (h a)) (C (g (f a))) = C a 
Arguments: {α : Sort u_1} {β : Sort u_2} {γ : β → Sort v} {f : α → β} {g : β → α} (h : Function.LeftInverse g f) (C : (a : α) → γ (f a)) (a : α)
Goal: cast ⋯ (C (g (f a))) = C a
-- IsSymmOp.flip_eq
theorem IsSymmOp.flip_eq (op) [IsSymmOp α β op] : flip op = op 
Arguments: {α : Sort u_1} {β : Sort u_2} (op : α → α → β) [IsSymmOp α β op]
Goal: flip op = op
-- InvImage.equivalence
theorem InvImage.equivalence {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β)
    (h : Equivalence r) : Equivalence (InvImage r f) 
Arguments: {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β) (h : Equivalence r)
Goal: Equivalence (InvImage r f)
-- small_subset
theorem small_subset {α : Type v} {s t : Set α} (hts : t ⊆ s) [Small.{u} s] : Small.{u} t 
Arguments: {α : Type v} {s t : Set α} (hts : t ⊆ s) [Small.{u, v} ↑s]
Goal: Small.{u, v} ↑t
-- univLE_max
theorem univLE_max : UnivLE.{u, max u v} 
Goal: UnivLE.{u, max u v}
-- Small.trans_univLE
theorem Small.trans_univLE.{u, v} (α : Type w) [hα : Small.{u} α] [h : UnivLE.{u, v}] :
    Small.{v} α 
Arguments: (α : Type w) [hα : Small.{u, w} α] [h : UnivLE.{u, v}]
Goal: Small.{v, w} α
-- UnivLE.trans
theorem UnivLE.trans [UnivLE.{u, v}] [UnivLE.{v, w}] : UnivLE.{u, w} 
Arguments: [UnivLE.{u, v}] [UnivLE.{v, w}]
Goal: UnivLE.{u, w}
-- equivShrink_symm_zero
@[simp]
lemma equivShrink_symm_zero [Zero α] [Small α] : (equivShrink α).symm 0 = 0 
Arguments: {α : Type u_1} [Zero α] [Small.{u_2, u_1} α]
Goal: (equivShrink α).symm 0 = 0
-- equivShrink_symm_one
@[to_additive existing (attr := simp)]
lemma equivShrink_symm_one [One α] [Small α] : (equivShrink α).symm 1 = 1 
Arguments: {α : Type u_1} [One α] [Small.{u_2, u_1} α]
Goal: (equivShrink α).symm 1 = 1
-- equivShrink_symm_add
@[simp]
lemma equivShrink_symm_add [Add α] [Small α] (x y : Shrink α) :
    (equivShrink α).symm (x + y) = (equivShrink α).symm x + (equivShrink α).symm y 
Arguments: {α : Type u_1} [Add α] [Small.{u_2, u_1} α] (x y : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm (x + y) = (equivShrink α).symm x + (equivShrink α).symm y
-- equivShrink_add
@[simp]
lemma equivShrink_add [Add α] [Small α] (x y : α) :
    equivShrink α (x + y) = equivShrink α x + equivShrink α y 
Arguments: {α : Type u_1} [Add α] [Small.{u_2, u_1} α] (x y : α)
Goal: (equivShrink α) (x + y) = (equivShrink α) x + (equivShrink α) y
-- equivShrink_symm_mul
@[to_additive existing (attr := simp)]
lemma equivShrink_symm_mul [Mul α] [Small α] (x y : Shrink α) :
    (equivShrink α).symm (x * y) = (equivShrink α).symm x * (equivShrink α).symm y 
Arguments: {α : Type u_1} [Mul α] [Small.{u_2, u_1} α] (x y : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm (x * y) = (equivShrink α).symm x * (equivShrink α).symm y
-- equivShrink_mul
@[to_additive existing (attr := simp)]
lemma equivShrink_mul [Mul α] [Small α] (x y : α) :
    equivShrink α (x * y) = equivShrink α x * equivShrink α y 
Arguments: {α : Type u_1} [Mul α] [Small.{u_2, u_1} α] (x y : α)
Goal: (equivShrink α) (x * y) = (equivShrink α) x * (equivShrink α) y
-- equivShrink_symm_sub
@[simp]
lemma equivShrink_symm_sub [Sub α] [Small α] (x y : Shrink α) :
    (equivShrink α).symm (x - y) = (equivShrink α).symm x - (equivShrink α).symm y 
Arguments: {α : Type u_1} [Sub α] [Small.{u_2, u_1} α] (x y : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm (x - y) = (equivShrink α).symm x - (equivShrink α).symm y
-- equivShrink_sub
@[simp]
lemma equivShrink_sub [Sub α] [Small α] (x y : α) :
    equivShrink α (x - y) = equivShrink α x - equivShrink α y 
Arguments: {α : Type u_1} [Sub α] [Small.{u_2, u_1} α] (x y : α)
Goal: (equivShrink α) (x - y) = (equivShrink α) x - (equivShrink α) y
-- equivShrink_symm_div
@[to_additive existing (attr := simp)]
lemma equivShrink_symm_div [Div α] [Small α] (x y : Shrink α) :
    (equivShrink α).symm (x / y) = (equivShrink α).symm x / (equivShrink α).symm y 
Arguments: {α : Type u_1} [Div α] [Small.{u_2, u_1} α] (x y : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm (x / y) = (equivShrink α).symm x / (equivShrink α).symm y
-- equivShrink_div
@[to_additive existing (attr := simp)]
lemma equivShrink_div [Div α] [Small α] (x y : α) :
    equivShrink α (x / y) = equivShrink α x / equivShrink α y 
Arguments: {α : Type u_1} [Div α] [Small.{u_2, u_1} α] (x y : α)
Goal: (equivShrink α) (x / y) = (equivShrink α) x / (equivShrink α) y
-- equivShrink_symm_neg
@[simp]
lemma equivShrink_symm_neg [Neg α] [Small α] (x : Shrink α) :
    (equivShrink α).symm (-x) = -(equivShrink α).symm x 
Arguments: {α : Type u_1} [Neg α] [Small.{u_2, u_1} α] (x : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm (-x) = -(equivShrink α).symm x
-- equivShrink_neg
@[simp]
lemma equivShrink_neg [Neg α] [Small α] (x : α) :
    equivShrink α (-x) = -equivShrink α x 
Arguments: {α : Type u_1} [Neg α] [Small.{u_2, u_1} α] (x : α)
Goal: (equivShrink α) (-x) = -(equivShrink α) x
-- equivShrink_symm_inv
@[to_additive existing (attr := simp)]
lemma equivShrink_symm_inv [Inv α] [Small α] (x : Shrink α) :
    (equivShrink α).symm x⁻¹ = ((equivShrink α).symm x)⁻¹ 
Arguments: {α : Type u_1} [Inv α] [Small.{u_2, u_1} α] (x : Shrink.{u_2, u_1} α)
Goal: (equivShrink α).symm x⁻¹ = ((equivShrink α).symm x)⁻¹
-- equivShrink_inv
@[to_additive existing (attr := simp)]
lemma equivShrink_inv [Inv α] [Small α] (x : α) :
    equivShrink α x⁻¹ = (equivShrink α x)⁻¹ 
Arguments: {α : Type u_1} [Inv α] [Small.{u_2, u_1} α] (x : α)
Goal: (equivShrink α) x⁻¹ = ((equivShrink α) x)⁻¹
-- Orientation.continuousAt_oangle
theorem continuousAt_oangle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) :
    ContinuousAt (fun y : V × V => o.oangle y.1 y.2) x 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0)
Goal: ContinuousAt (fun y => o.oangle y.1 y.2) x
-- Orientation.oangle_zero_left
@[simp]
theorem oangle_zero_left (x : V) : o.oangle 0 x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: o.oangle 0 x = 0
-- Orientation.oangle_zero_right
@[simp]
theorem oangle_zero_right (x : V) : o.oangle x 0 = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: o.oangle x 0 = 0
-- Orientation.oangle_self
@[simp]
theorem oangle_self (x : V) : o.oangle x x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: o.oangle x x = 0
-- Orientation.left_ne_zero_of_oangle_ne_zero
theorem left_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y ≠ 0)
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_ne_zero
theorem right_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y ≠ 0)
Goal: y ≠ 0
-- Orientation.ne_of_oangle_ne_zero
theorem ne_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y ≠ 0)
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_eq_pi
theorem left_ne_zero_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑π)
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_eq_pi
theorem right_ne_zero_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑π)
Goal: y ≠ 0
-- Orientation.ne_of_oangle_eq_pi
theorem ne_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑π)
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_eq_pi_div_two
theorem left_ne_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_eq_pi_div_two
theorem right_ne_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: y ≠ 0
-- Orientation.ne_of_oangle_eq_pi_div_two
theorem ne_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(π / 2))
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_eq_neg_pi_div_two
theorem left_ne_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(-π / 2))
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_eq_neg_pi_div_two
theorem right_ne_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(-π / 2))
Goal: y ≠ 0
-- Orientation.ne_of_oangle_eq_neg_pi_div_two
theorem ne_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : o.oangle x y = ↑(-π / 2))
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_sign_ne_zero
theorem left_ne_zero_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign ≠ 0)
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_sign_ne_zero
theorem right_ne_zero_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign ≠ 0)
Goal: y ≠ 0
-- Orientation.ne_of_oangle_sign_ne_zero
theorem ne_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign ≠ 0)
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_sign_eq_one
theorem left_ne_zero_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = 1)
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_sign_eq_one
theorem right_ne_zero_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = 1)
Goal: y ≠ 0
-- Orientation.ne_of_oangle_sign_eq_one
theorem ne_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = 1)
Goal: x ≠ y
-- Orientation.left_ne_zero_of_oangle_sign_eq_neg_one
theorem left_ne_zero_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : x ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = -1)
Goal: x ≠ 0
-- Orientation.right_ne_zero_of_oangle_sign_eq_neg_one
theorem right_ne_zero_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : y ≠ 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = -1)
Goal: y ≠ 0
-- Orientation.ne_of_oangle_sign_eq_neg_one
theorem ne_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : x ≠ y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (h : (o.oangle x y).sign = -1)
Goal: x ≠ y
-- Orientation.oangle_rev
theorem oangle_rev (x y : V) : o.oangle y x = -o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle y x = -o.oangle x y
-- Orientation.oangle_add_oangle_rev
@[simp]
theorem oangle_add_oangle_rev (x y : V) : o.oangle x y + o.oangle y x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle x y + o.oangle y x = 0
-- Orientation.oangle_neg_left
theorem oangle_neg_left {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle (-x) y = o.oangle x y + π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: o.oangle (-x) y = o.oangle x y + ↑π
-- Orientation.oangle_neg_right
theorem oangle_neg_right {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x (-y) = o.oangle x y + π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
Goal: o.oangle x (-y) = o.oangle x y + ↑π
-- Orientation.two_zsmul_oangle_neg_left
@[simp]
theorem two_zsmul_oangle_neg_left (x y : V) :
    (2 : ℤ) • o.oangle (-x) y = (2 : ℤ) • o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: 2 • o.oangle (-x) y = 2 • o.oangle x y
-- Orientation.two_zsmul_oangle_neg_right
@[simp]
theorem two_zsmul_oangle_neg_right (x y : V) :
    (2 : ℤ) • o.oangle x (-y) = (2 : ℤ) • o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: 2 • o.oangle x (-y) = 2 • o.oangle x y
-- Orientation.oangle_neg_neg
@[simp]
theorem oangle_neg_neg (x y : V) : o.oangle (-x) (-y) = o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle (-x) (-y) = o.oangle x y
-- Orientation.oangle_neg_left_eq_neg_right
theorem oangle_neg_left_eq_neg_right (x y : V) : o.oangle (-x) y = o.oangle x (-y) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle (-x) y = o.oangle x (-y)
-- Orientation.oangle_neg_self_left
@[simp]
theorem oangle_neg_self_left {x : V} (hx : x ≠ 0) : o.oangle (-x) x = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0)
Goal: o.oangle (-x) x = ↑π
-- Orientation.oangle_neg_self_right
@[simp]
theorem oangle_neg_self_right {x : V} (hx : x ≠ 0) : o.oangle x (-x) = π 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) {x : V} (hx : x ≠ 0)
Goal: o.oangle x (-x) = ↑π
-- Orientation.two_zsmul_oangle_neg_self_left
-- @[simp] -- Porting note (#10618): simp can prove this
theorem two_zsmul_oangle_neg_self_left (x : V) : (2 : ℤ) • o.oangle (-x) x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: 2 • o.oangle (-x) x = 0
-- Orientation.two_zsmul_oangle_neg_self_right
-- @[simp] -- Porting note (#10618): simp can prove this
theorem two_zsmul_oangle_neg_self_right (x : V) : (2 : ℤ) • o.oangle x (-x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V)
Goal: 2 • o.oangle x (-x) = 0
-- Orientation.oangle_add_oangle_rev_neg_left
@[simp]
theorem oangle_add_oangle_rev_neg_left (x y : V) : o.oangle (-x) y + o.oangle (-y) x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle (-x) y + o.oangle (-y) x = 0
-- Orientation.oangle_add_oangle_rev_neg_right
@[simp]
theorem oangle_add_oangle_rev_neg_right (x y : V) : o.oangle x (-y) + o.oangle y (-x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V)
Goal: o.oangle x (-y) + o.oangle y (-x) = 0
-- Orientation.oangle_smul_left_of_pos
@[simp]
theorem oangle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
    o.oangle (r • x) y = o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : 0 < r)
Goal: o.oangle (r • x) y = o.oangle x y
-- Orientation.oangle_smul_right_of_pos
@[simp]
theorem oangle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
    o.oangle x (r • y) = o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : 0 < r)
Goal: o.oangle x (r • y) = o.oangle x y
-- Orientation.oangle_smul_left_of_neg
@[simp]
theorem oangle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
    o.oangle (r • x) y = o.oangle (-x) y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : r < 0)
Goal: o.oangle (r • x) y = o.oangle (-x) y
-- Orientation.oangle_smul_right_of_neg
@[simp]
theorem oangle_smul_right_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
    o.oangle x (r • y) = o.oangle x (-y) 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : r < 0)
Goal: o.oangle x (r • y) = o.oangle x (-y)
-- Orientation.oangle_smul_left_self_of_nonneg
@[simp]
theorem oangle_smul_left_self_of_nonneg (x : V) {r : ℝ} (hr : 0 ≤ r) : o.oangle (r • x) x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {r : ℝ} (hr : 0 ≤ r)
Goal: o.oangle (r • x) x = 0
-- Orientation.oangle_smul_right_self_of_nonneg
@[simp]
theorem oangle_smul_right_self_of_nonneg (x : V) {r : ℝ} (hr : 0 ≤ r) : o.oangle x (r • x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {r : ℝ} (hr : 0 ≤ r)
Goal: o.oangle x (r • x) = 0
-- Orientation.oangle_smul_smul_self_of_nonneg
@[simp]
theorem oangle_smul_smul_self_of_nonneg (x : V) {r₁ r₂ : ℝ} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
    o.oangle (r₁ • x) (r₂ • x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {r₁ r₂ : ℝ} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂)
Goal: o.oangle (r₁ • x) (r₂ • x) = 0
-- Orientation.two_zsmul_oangle_smul_left_of_ne_zero
@[simp]
theorem two_zsmul_oangle_smul_left_of_ne_zero (x y : V) {r : ℝ} (hr : r ≠ 0) :
    (2 : ℤ) • o.oangle (r • x) y = (2 : ℤ) • o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : r ≠ 0)
Goal: 2 • o.oangle (r • x) y = 2 • o.oangle x y
-- Orientation.two_zsmul_oangle_smul_right_of_ne_zero
@[simp]
theorem two_zsmul_oangle_smul_right_of_ne_zero (x y : V) {r : ℝ} (hr : r ≠ 0) :
    (2 : ℤ) • o.oangle x (r • y) = (2 : ℤ) • o.oangle x y 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x y : V) {r : ℝ} (hr : r ≠ 0)
Goal: 2 • o.oangle x (r • y) = 2 • o.oangle x y
-- Orientation.two_zsmul_oangle_smul_left_self
@[simp]
theorem two_zsmul_oangle_smul_left_self (x : V) {r : ℝ} : (2 : ℤ) • o.oangle (r • x) x = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {r : ℝ}
Goal: 2 • o.oangle (r • x) x = 0
-- Orientation.two_zsmul_oangle_smul_right_self
@[simp]
theorem two_zsmul_oangle_smul_right_self (x : V) {r : ℝ} : (2 : ℤ) • o.oangle x (r • x) = 0 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [Fact (FiniteDimensional.finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2)) (x : V) {r : ℝ}
Goal: 2 • o.oangle x (r • x) = 0