ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- summable_of_ratio_test_tendsto_lt_one
theorem summable_of_ratio_test_tendsto_lt_one {α : Type*} [NormedAddCommGroup α] [CompleteSpace α]
    {f : ℕ → α} {l : ℝ} (hl₁ : l < 1) (hf : ∀ᶠ n in atTop, f n ≠ 0)
    (h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)) : Summable f 
Arguments: {α : Type u_4} [NormedAddCommGroup α] [CompleteSpace α] {f : ℕ → α} {l : ℝ} (hl₁ : l < 1) (hf : ∀ᶠ (n : ℕ) in Filter.atTop, f n ≠ 0) (h : Filter.Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) Filter.atTop (𝓝 l))
Goal: Summable f
-- not_summable_of_ratio_norm_eventually_ge
theorem not_summable_of_ratio_norm_eventually_ge {α : Type*} [SeminormedAddCommGroup α] {f : ℕ → α}
    {r : ℝ} (hr : 1 < r) (hf : ∃ᶠ n in atTop, ‖f n‖ ≠ 0)
    (h : ∀ᶠ n in atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖) : ¬Summable f 
Arguments: {α : Type u_4} [SeminormedAddCommGroup α] {f : ℕ → α} {r : ℝ} (hr : 1 < r) (hf : ∃ᶠ (n : ℕ) in Filter.atTop, ‖f n‖ ≠ 0) (h : ∀ᶠ (n : ℕ) in Filter.atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖)
Goal: ¬Summable f
-- not_summable_of_ratio_test_tendsto_gt_one
theorem not_summable_of_ratio_test_tendsto_gt_one {α : Type*} [SeminormedAddCommGroup α]
    {f : ℕ → α} {l : ℝ} (hl : 1 < l) (h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)) :
    ¬Summable f 
Arguments: {α : Type u_4} [SeminormedAddCommGroup α] {f : ℕ → α} {l : ℝ} (hl : 1 < l) (h : Filter.Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) Filter.atTop (𝓝 l))
Goal: ¬Summable f
-- summable_powerSeries_of_norm_lt
theorem summable_powerSeries_of_norm_lt {w z : α}
    (h : CauchySeq fun n ↦ ∑ i ∈ range n, f i * w ^ i) (hz : ‖z‖ < ‖w‖) :
    Summable fun n ↦ f n * z ^ n 
Arguments: {α : Type u_1} [NormedDivisionRing α] [CompleteSpace α] {f : ℕ → α} {w z : α} (h : CauchySeq fun n => ∑ i ∈ Finset.range n, f i * w ^ i) (hz : ‖z‖ < ‖w‖)
Goal: Summable fun n => f n * z ^ n
-- summable_powerSeries_of_norm_lt_one
theorem summable_powerSeries_of_norm_lt_one {z : α}
    (h : CauchySeq fun n ↦ ∑ i ∈ range n, f i) (hz : ‖z‖ < 1) :
    Summable fun n ↦ f n * z ^ n 
Arguments: {α : Type u_1} [NormedDivisionRing α] [CompleteSpace α] {f : ℕ → α} {z : α} (h : CauchySeq fun n => ∑ i ∈ Finset.range n, f i) (hz : ‖z‖ < 1)
Goal: Summable fun n => f n * z ^ n
-- Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded
theorem Monotone.cauchySeq_series_mul_of_tendsto_zero_of_bounded (hfa : Monotone f)
    (hf0 : Tendsto f atTop (𝓝 0)) (hgb : ∀ n, ‖∑ i ∈ range n, z i‖ ≤ b) :
    CauchySeq fun n ↦ ∑ i ∈ range n, f i • z i 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0)) (hgb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b)
Goal: CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i
-- Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded
theorem Antitone.cauchySeq_series_mul_of_tendsto_zero_of_bounded (hfa : Antitone f)
    (hf0 : Tendsto f atTop (𝓝 0)) (hzb : ∀ n, ‖∑ i ∈ range n, z i‖ ≤ b) :
    CauchySeq fun n ↦ ∑ i ∈ range n, f i • z i 
Arguments: {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℕ → ℝ} {z : ℕ → E} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0)) (hzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b)
Goal: CauchySeq fun n => ∑ i ∈ Finset.range n, f i • z i
-- norm_sum_neg_one_pow_le
theorem norm_sum_neg_one_pow_le (n : ℕ) : ‖∑ i ∈ range n, (-1 : ℝ) ^ i‖ ≤ 1 
Arguments: (n : ℕ)
Goal: ‖∑ i ∈ Finset.range n, (-1) ^ i‖ ≤ 1
-- Monotone.cauchySeq_alternating_series_of_tendsto_zero
theorem Monotone.cauchySeq_alternating_series_of_tendsto_zero (hfa : Monotone f)
    (hf0 : Tendsto f atTop (𝓝 0)) : CauchySeq fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i 
Arguments: {f : ℕ → ℝ} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0))
Goal: CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i
-- Monotone.tendsto_alternating_series_of_tendsto_zero
theorem Monotone.tendsto_alternating_series_of_tendsto_zero (hfa : Monotone f)
    (hf0 : Tendsto f atTop (𝓝 0)) :
    ∃ l, Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l) 
Arguments: {f : ℕ → ℝ} (hfa : Monotone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0))
Goal: ∃ l, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)
-- Antitone.cauchySeq_alternating_series_of_tendsto_zero
theorem Antitone.cauchySeq_alternating_series_of_tendsto_zero (hfa : Antitone f)
    (hf0 : Tendsto f atTop (𝓝 0)) : CauchySeq fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i 
Arguments: {f : ℕ → ℝ} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0))
Goal: CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i
-- Antitone.tendsto_alternating_series_of_tendsto_zero
theorem Antitone.tendsto_alternating_series_of_tendsto_zero (hfa : Antitone f)
    (hf0 : Tendsto f atTop (𝓝 0)) :
    ∃ l, Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l) 
Arguments: {f : ℕ → ℝ} (hfa : Antitone f) (hf0 : Filter.Tendsto f Filter.atTop (𝓝 0))
Goal: ∃ l, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)
-- Monotone.tendsto_le_alternating_series
theorem Monotone.tendsto_le_alternating_series
    (hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
    (hfm : Monotone f) (k : ℕ) : l ≤ ∑ i ∈ range (2 * k), (-1) ^ i * f i 
Arguments: {E : Type u_4} [OrderedRing E] [TopologicalSpace E] [OrderClosedTopology E] {l : E} {f : ℕ → E} (hfl : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)) (hfm : Monotone f) (k : ℕ)
Goal: l ≤ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i
-- Monotone.alternating_series_le_tendsto
theorem Monotone.alternating_series_le_tendsto
    (hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
    (hfm : Monotone f) (k : ℕ) : ∑ i ∈ range (2 * k + 1), (-1) ^ i * f i ≤ l 
Arguments: {E : Type u_4} [OrderedRing E] [TopologicalSpace E] [OrderClosedTopology E] {l : E} {f : ℕ → E} (hfl : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)) (hfm : Monotone f) (k : ℕ)
Goal: ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i ≤ l
-- Antitone.alternating_series_le_tendsto
theorem Antitone.alternating_series_le_tendsto
    (hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
    (hfa : Antitone f) (k : ℕ) : ∑ i ∈ range (2 * k), (-1) ^ i * f i ≤ l 
Arguments: {E : Type u_4} [OrderedRing E] [TopologicalSpace E] [OrderClosedTopology E] {l : E} {f : ℕ → E} (hfl : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)) (hfa : Antitone f) (k : ℕ)
Goal: ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i ≤ l
-- Antitone.tendsto_le_alternating_series
theorem Antitone.tendsto_le_alternating_series
    (hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
    (hfa : Antitone f) (k : ℕ) : l ≤ ∑ i ∈ range (2 * k + 1), (-1) ^ i * f i 
Arguments: {E : Type u_4} [OrderedRing E] [TopologicalSpace E] [OrderClosedTopology E] {l : E} {f : ℕ → E} (hfl : Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) Filter.atTop (𝓝 l)) (hfa : Antitone f) (k : ℕ)
Goal: l ≤ ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i
-- Real.summable_pow_div_factorial
theorem Real.summable_pow_div_factorial (x : ℝ) : Summable (fun n ↦ x ^ n / n ! : ℕ → ℝ) 
Arguments: (x : ℝ)
Goal: Summable fun n => x ^ n / ↑n !
-- Real.tendsto_pow_div_factorial_atTop
theorem Real.tendsto_pow_div_factorial_atTop (x : ℝ) :
    Tendsto (fun n ↦ x ^ n / n ! : ℕ → ℝ) atTop (𝓝 0) 
Arguments: (x : ℝ)
Goal: Filter.Tendsto (fun n => x ^ n / ↑n !) Filter.atTop (𝓝 0)
-- NormedStarGroup.norm_star
class NormedStarGroup (E : Type*) [SeminormedAddCommGroup E] [StarAddMonoid E] : Prop where
  norm_star : ∀ x : E, ‖x⋆‖ = ‖x‖
Arguments: {E : Type u_1} [SeminormedAddCommGroup E] [StarAddMonoid E] [self : NormedStarGroup E] (x : E)
Goal: ‖x⋆‖ = ‖x‖
-- nnnorm_star
@[simp]
theorem nnnorm_star (x : E) : ‖star x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] (x : E)
Goal: ‖x⋆‖₊ = ‖x‖₊
-- star_isometry
theorem star_isometry : Isometry (star : E → E) 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E]
Goal: Isometry star
-- CstarRing.norm_mul_self_le
class CstarRing (E : Type*) [NonUnitalNormedRing E] [StarRing E] : Prop where
  norm_mul_self_le : ∀ x : E, ‖x‖ * ‖x‖ ≤ ‖x⋆ * x‖
Arguments: {E : Type u_4} [NonUnitalNormedRing E] [StarRing E] [self : CstarRing E] (x : E)
Goal: ‖x‖ * ‖x‖ ≤ ‖x⋆ * x‖
-- CstarRing.norm_star_mul_self
theorem norm_star_mul_self {x : E} : ‖x⋆ * x‖ = ‖x‖ * ‖x‖ 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E}
Goal: ‖x⋆ * x‖ = ‖x‖ * ‖x‖
-- CstarRing.norm_self_mul_star
theorem norm_self_mul_star {x : E} : ‖x * x⋆‖ = ‖x‖ * ‖x‖ 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E}
Goal: ‖x * x⋆‖ = ‖x‖ * ‖x‖
-- CstarRing.norm_star_mul_self'
theorem norm_star_mul_self' {x : E} : ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖ 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E}
Goal: ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖
-- CstarRing.nnnorm_self_mul_star
theorem nnnorm_self_mul_star {x : E} : ‖x * x⋆‖₊ = ‖x‖₊ * ‖x‖₊ 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E}
Goal: ‖x * x⋆‖₊ = ‖x‖₊ * ‖x‖₊
-- CstarRing.nnnorm_star_mul_self
theorem nnnorm_star_mul_self {x : E} : ‖x⋆ * x‖₊ = ‖x‖₊ * ‖x‖₊ 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] {x : E}
Goal: ‖x⋆ * x‖₊ = ‖x‖₊ * ‖x‖₊
-- CstarRing.star_mul_self_eq_zero_iff
@[simp]
theorem star_mul_self_eq_zero_iff (x : E) : x⋆ * x = 0 ↔ x = 0 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E)
Goal: x⋆ * x = 0 ↔ x = 0
-- CstarRing.star_mul_self_ne_zero_iff
theorem star_mul_self_ne_zero_iff (x : E) : x⋆ * x ≠ 0 ↔ x ≠ 0 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E)
Goal: x⋆ * x ≠ 0 ↔ x ≠ 0
-- CstarRing.mul_star_self_eq_zero_iff
@[simp]
theorem mul_star_self_eq_zero_iff (x : E) : x * x⋆ = 0 ↔ x = 0 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E)
Goal: x * x⋆ = 0 ↔ x = 0
-- CstarRing.mul_star_self_ne_zero_iff
theorem mul_star_self_ne_zero_iff (x : E) : x * x⋆ ≠ 0 ↔ x ≠ 0 
Arguments: {E : Type u_2} [NonUnitalNormedRing E] [StarRing E] [CstarRing E] (x : E)
Goal: x * x⋆ ≠ 0 ↔ x ≠ 0
-- CstarRing.norm_one
@[simp, nolint simpNF] -- Porting note (#10959): simp cannot prove this
theorem norm_one [Nontrivial E] : ‖(1 : E)‖ = 1 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E]
Goal: ‖1‖ = 1
-- CstarRing.norm_coe_unitary
theorem norm_coe_unitary [Nontrivial E] (U : unitary E) : ‖(U : E)‖ = 1 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] (U : ↥(unitary E))
Goal: ‖↑U‖ = 1
-- CstarRing.norm_of_mem_unitary
@[simp]
theorem norm_of_mem_unitary [Nontrivial E] {U : E} (hU : U ∈ unitary E) : ‖U‖ = 1 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] [Nontrivial E] {U : E} (hU : U ∈ unitary E)
Goal: ‖U‖ = 1
-- CstarRing.norm_coe_unitary_mul
@[simp]
theorem norm_coe_unitary_mul (U : unitary E) (A : E) : ‖(U : E) * A‖ = ‖A‖ 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : ↥(unitary E)) (A : E)
Goal: ‖↑U * A‖ = ‖A‖
-- CstarRing.norm_unitary_smul
@[simp]
theorem norm_unitary_smul (U : unitary E) (A : E) : ‖U • A‖ = ‖A‖ 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (U : ↥(unitary E)) (A : E)
Goal: ‖U • A‖ = ‖A‖
-- CstarRing.norm_mem_unitary_mul
theorem norm_mem_unitary_mul {U : E} (A : E) (hU : U ∈ unitary E) : ‖U * A‖ = ‖A‖ 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {U : E} (A : E) (hU : U ∈ unitary E)
Goal: ‖U * A‖ = ‖A‖
-- CstarRing.norm_mul_coe_unitary
@[simp]
theorem norm_mul_coe_unitary (A : E) (U : unitary E) : ‖A * U‖ = ‖A‖ 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (A : E) (U : ↥(unitary E))
Goal: ‖A * ↑U‖ = ‖A‖
-- CstarRing.norm_mul_mem_unitary
theorem norm_mul_mem_unitary (A : E) {U : E} (hU : U ∈ unitary E) : ‖A * U‖ = ‖A‖ 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (A : E) {U : E} (hU : U ∈ unitary E)
Goal: ‖A * U‖ = ‖A‖
-- IsSelfAdjoint.nnnorm_pow_two_pow
theorem IsSelfAdjoint.nnnorm_pow_two_pow [NormedRing E] [StarRing E] [CstarRing E] {x : E}
    (hx : IsSelfAdjoint x) (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] {x : E} (hx : IsSelfAdjoint x) (n : ℕ)
Goal: ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n
-- selfAdjoint.nnnorm_pow_two_pow
theorem selfAdjoint.nnnorm_pow_two_pow [NormedRing E] [StarRing E] [CstarRing E] (x : selfAdjoint E)
    (n : ℕ) : ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n 
Arguments: {E : Type u_2} [NormedRing E] [StarRing E] [CstarRing E] (x : ↥(selfAdjoint E)) (n : ℕ)
Goal: ‖x ^ 2 ^ n‖₊ = ‖x‖₊ ^ 2 ^ n
-- coe_starₗᵢ
@[simp]
theorem coe_starₗᵢ : (starₗᵢ 𝕜 : E → E) = star 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E]
Goal: ⇑(starₗᵢ 𝕜) = star
-- starₗᵢ_apply
theorem starₗᵢ_apply {x : E} : starₗᵢ 𝕜 x = star x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E] {x : E}
Goal: (starₗᵢ 𝕜) x = x⋆
-- starₗᵢ_toContinuousLinearEquiv
@[simp]
theorem starₗᵢ_toContinuousLinearEquiv :
    (starₗᵢ 𝕜 : E ≃ₗᵢ⋆[𝕜] E).toContinuousLinearEquiv = (starL 𝕜 : E ≃L⋆[𝕜] E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [CommSemiring 𝕜] [StarRing 𝕜] [SeminormedAddCommGroup E] [StarAddMonoid E] [NormedStarGroup E] [Module 𝕜 E] [StarModule 𝕜 E]
Goal: (starₗᵢ 𝕜).toContinuousLinearEquiv = starL 𝕜
-- FiniteDimensional.of_fact_finrank_eq_two
lemma FiniteDimensional.of_fact_finrank_eq_two {K V : Type*} [DivisionRing K]
    [AddCommGroup V] [Module K V] [Fact (finrank K V = 2)] : FiniteDimensional K V 
Arguments: {K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] [Fact (FiniteDimensional.finrank K V = 2)]
Goal: FiniteDimensional K V
-- Orientation.areaForm_def
irreducible_def areaForm : E →ₗ[ℝ] E →ₗ[ℝ] ℝ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: o.areaForm =
  let z := AlternatingMap.constLinearEquivOfIsEmpty.symm;
  let y := (LinearMap.llcomp ℝ E (E [⋀^Fin 0]→ₗ[ℝ] ℝ) ℝ) ↑z ∘ₗ AlternatingMap.curryLeftLinearMap;
  y ∘ₗ AlternatingMap.curryLeftLinearMap o.volumeForm
-- Orientation.areaForm_to_volumeForm
theorem areaForm_to_volumeForm (x y : E) : ω x y = o.volumeForm ![x, y] 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm x) y = o.volumeForm ![x, y]
-- Orientation.areaForm_apply_self
@[simp]
theorem areaForm_apply_self (x : E) : ω x x = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: (o.areaForm x) x = 0
-- Orientation.areaForm_swap
theorem areaForm_swap (x y : E) : ω x y = -ω y x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm x) y = -(o.areaForm y) x
-- Orientation.areaForm_neg_orientation
@[simp]
theorem areaForm_neg_orientation : (-o).areaForm = -o.areaForm 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: (-o).areaForm = -o.areaForm
-- Orientation.areaForm'_apply
@[simp]
theorem areaForm'_apply (x : E) :
    o.areaForm' x = LinearMap.toContinuousLinearMap (o.areaForm x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: o.areaForm' x = LinearMap.toContinuousLinearMap (o.areaForm x)
-- Orientation.abs_areaForm_le
theorem abs_areaForm_le (x y : E) : |ω x y| ≤ ‖x‖ * ‖y‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖
-- Orientation.areaForm_le
theorem areaForm_le (x y : E) : ω x y ≤ ‖x‖ * ‖y‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm x) y ≤ ‖x‖ * ‖y‖
-- Orientation.abs_areaForm_of_orthogonal
theorem abs_areaForm_of_orthogonal {x y : E} (h : ⟪x, y⟫ = 0) : |ω x y| = ‖x‖ * ‖y‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {x y : E} (h : ⟪x, y⟫_ℝ = 0)
Goal: |(o.areaForm x) y| = ‖x‖ * ‖y‖
-- Orientation.areaForm_map
theorem areaForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
    [hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
    (Orientation.map (Fin 2) φ.toLinearEquiv o).areaForm x y =
    o.areaForm (φ.symm x) (φ.symm y) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [hF : Fact (FiniteDimensional.finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F)
Goal: (((Orientation.map (Fin 2) φ.toLinearEquiv) o).areaForm x) y = (o.areaForm (φ.symm x)) (φ.symm y)
-- Orientation.areaForm_comp_linearIsometryEquiv
theorem areaForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
    (hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
    o.areaForm (φ x) (φ y) = o.areaForm x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (φ : E ≃ₗᵢ[ℝ] E) (hφ : 0 < LinearMap.det ↑φ.toLinearEquiv) (x y : E)
Goal: (o.areaForm (φ x)) (φ y) = (o.areaForm x) y
-- Orientation.rightAngleRotationAux₁_def
irreducible_def rightAngleRotationAux₁ : E →ₗ[ℝ] E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: o.rightAngleRotationAux₁ =
  let to_dual := (InnerProductSpace.toDual ℝ E).toLinearEquiv ≪≫ₗ LinearMap.toContinuousLinearMap.symm;
  ↑to_dual.symm ∘ₗ o.areaForm
-- Orientation.inner_rightAngleRotationAux₁_left
@[simp]
theorem inner_rightAngleRotationAux₁_left (x y : E) : ⟪o.rightAngleRotationAux₁ x, y⟫ = ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪o.rightAngleRotationAux₁ x, y⟫_ℝ = (o.areaForm x) y
-- Orientation.inner_rightAngleRotationAux₁_right
@[simp]
theorem inner_rightAngleRotationAux₁_right (x y : E) :
    ⟪x, o.rightAngleRotationAux₁ y⟫ = -ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪x, o.rightAngleRotationAux₁ y⟫_ℝ = -(o.areaForm x) y
-- Orientation.rightAngleRotationAux₁_rightAngleRotationAux₁
@[simp]
theorem rightAngleRotationAux₁_rightAngleRotationAux₁ (x : E) :
    o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x) = -x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x) = -x
-- Orientation.rightAngleRotation_def
irreducible_def rightAngleRotation : E ≃ₗᵢ[ℝ] E 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: o.rightAngleRotation = LinearIsometryEquiv.ofLinearIsometry o.rightAngleRotationAux₂ (-o.rightAngleRotationAux₁) ⋯ ⋯
-- Orientation.inner_rightAngleRotation_left
@[simp]
theorem inner_rightAngleRotation_left (x y : E) : ⟪J x, y⟫ = ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪o.rightAngleRotation x, y⟫_ℝ = (o.areaForm x) y
-- Orientation.inner_rightAngleRotation_right
@[simp]
theorem inner_rightAngleRotation_right (x y : E) : ⟪x, J y⟫ = -ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪x, o.rightAngleRotation y⟫_ℝ = -(o.areaForm x) y
-- Orientation.rightAngleRotation_rightAngleRotation
@[simp]
theorem rightAngleRotation_rightAngleRotation (x : E) : J (J x) = -x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: o.rightAngleRotation (o.rightAngleRotation x) = -x
-- Orientation.rightAngleRotation_symm
@[simp]
theorem rightAngleRotation_symm :
    LinearIsometryEquiv.symm J = LinearIsometryEquiv.trans J (LinearIsometryEquiv.neg ℝ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: o.rightAngleRotation.symm = o.rightAngleRotation.trans (LinearIsometryEquiv.neg ℝ)
-- Orientation.inner_rightAngleRotation_self
theorem inner_rightAngleRotation_self (x : E) : ⟪J x, x⟫ = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x : E)
Goal: ⟪o.rightAngleRotation x, x⟫_ℝ = 0
-- Orientation.inner_rightAngleRotation_swap
theorem inner_rightAngleRotation_swap (x y : E) : ⟪x, J y⟫ = -⟪J x, y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪x, o.rightAngleRotation y⟫_ℝ = -⟪o.rightAngleRotation x, y⟫_ℝ
-- Orientation.inner_rightAngleRotation_swap'
theorem inner_rightAngleRotation_swap' (x y : E) : ⟪J x, y⟫ = -⟪x, J y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪o.rightAngleRotation x, y⟫_ℝ = -⟪x, o.rightAngleRotation y⟫_ℝ
-- Orientation.inner_comp_rightAngleRotation
theorem inner_comp_rightAngleRotation (x y : E) : ⟪J x, J y⟫ = ⟪x, y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: ⟪o.rightAngleRotation x, o.rightAngleRotation y⟫_ℝ = ⟪x, y⟫_ℝ
-- Orientation.areaForm_rightAngleRotation_left
@[simp]
theorem areaForm_rightAngleRotation_left (x y : E) : ω (J x) y = -⟪x, y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm (o.rightAngleRotation x)) y = -⟪x, y⟫_ℝ
-- Orientation.areaForm_rightAngleRotation_right
@[simp]
theorem areaForm_rightAngleRotation_right (x y : E) : ω x (J y) = ⟪x, y⟫ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm x) (o.rightAngleRotation y) = ⟪x, y⟫_ℝ
-- Orientation.areaForm_comp_rightAngleRotation
theorem areaForm_comp_rightAngleRotation (x y : E) : ω (J x) (J y) = ω x y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2)) (x y : E)
Goal: (o.areaForm (o.rightAngleRotation x)) (o.rightAngleRotation y) = (o.areaForm x) y
-- Orientation.rightAngleRotation_trans_rightAngleRotation
@[simp]
theorem rightAngleRotation_trans_rightAngleRotation :
    LinearIsometryEquiv.trans J J = LinearIsometryEquiv.neg ℝ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fact (FiniteDimensional.finrank ℝ E = 2)] (o : Orientation ℝ E (Fin 2))
Goal: o.rightAngleRotation.trans o.rightAngleRotation = LinearIsometryEquiv.neg ℝ