ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- dist_anti_right_pi
lemma dist_anti_right_pi : AntitoneOn (dist x) (Iic x) 
Arguments: {ι : Type u_2} [Fintype ι] {x : ι → ℝ}
Goal: AntitoneOn (dist x) (Set.Iic x)
-- dist_le_dist_of_le_pi
lemma dist_le_dist_of_le_pi (ha : a₂ ≤ a₁) (h₁ : a₁ ≤ b₁) (hb : b₁ ≤ b₂) :
    dist a₁ b₁ ≤ dist a₂ b₂ 
Arguments: {ι : Type u_2} [Fintype ι] {a₁ a₂ b₁ b₂ : ι → ℝ} (ha : a₂ ≤ a₁) (h₁ : a₁ ≤ b₁) (hb : b₁ ≤ b₂)
Goal: dist a₁ b₁ ≤ dist a₂ b₂
-- IsUpperSet.exists_subset_ball
theorem IsUpperSet.exists_subset_ball (hs : IsUpperSet s) (hx : x ∈ closure s) (hδ : 0 < δ) :
    ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s 
Arguments: {ι : Type u_2} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ} {δ : ℝ} (hs : IsUpperSet s) (hx : x ∈ closure s) (hδ : 0 < δ)
Goal: ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior s
-- IsLowerSet.exists_subset_ball
theorem IsLowerSet.exists_subset_ball (hs : IsLowerSet s) (hx : x ∈ closure s) (hδ : 0 < δ) :
    ∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s 
Arguments: {ι : Type u_2} [Fintype ι] {s : Set (ι → ℝ)} {x : ι → ℝ} {δ : ℝ} (hs : IsLowerSet s) (hx : x ∈ closure s) (hδ : 0 < δ)
Goal: ∃ y, Metric.closedBall y (δ / 4) ⊆ Metric.closedBall x δ ∧ Metric.closedBall y (δ / 4) ⊆ interior s
-- IsClosed.upperClosure_pi
protected lemma IsClosed.upperClosure_pi (hs : IsClosed s) (hs' : BddBelow s) :
    IsClosed (upperClosure s : Set (ι → ℝ)) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : IsClosed s) (hs' : BddBelow s)
Goal: IsClosed ↑(upperClosure s)
-- IsClosed.lowerClosure_pi
protected lemma IsClosed.lowerClosure_pi (hs : IsClosed s) (hs' : BddAbove s) :
    IsClosed (lowerClosure s : Set (ι → ℝ)) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : IsClosed s) (hs' : BddAbove s)
Goal: IsClosed ↑(lowerClosure s)
-- IsClopen.upperClosure_pi
protected lemma IsClopen.upperClosure_pi (hs : IsClopen s) (hs' : BddBelow s) :
    IsClopen (upperClosure s : Set (ι → ℝ)) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : IsClopen s) (hs' : BddBelow s)
Goal: IsClopen ↑(upperClosure s)
-- IsClopen.lowerClosure_pi
protected lemma IsClopen.lowerClosure_pi (hs : IsClopen s) (hs' : BddAbove s) :
    IsClopen (lowerClosure s : Set (ι → ℝ)) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : IsClopen s) (hs' : BddAbove s)
Goal: IsClopen ↑(lowerClosure s)
-- closure_upperClosure_comm_pi
lemma closure_upperClosure_comm_pi (hs : BddBelow s) :
    closure (upperClosure s : Set (ι → ℝ)) = upperClosure (closure s) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : BddBelow s)
Goal: closure ↑(upperClosure s) = ↑(upperClosure (closure s))
-- closure_lowerClosure_comm_pi
lemma closure_lowerClosure_comm_pi (hs : BddAbove s) :
    closure (lowerClosure s : Set (ι → ℝ)) = lowerClosure (closure s) 
Arguments: {ι : Type u_2} [Finite ι] {s : Set (ι → ℝ)} (hs : BddAbove s)
Goal: closure ↑(lowerClosure s) = ↑(lowerClosure (closure s))
-- Real.pow_arith_mean_le_arith_mean_pow
theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ)
Goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n
-- Real.pow_arith_mean_le_arith_mean_pow_of_even
theorem pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) {n : ℕ} (hn : Even n) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) {n : ℕ} (hn : Even n)
Goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n
-- Real.pow_sum_div_card_le_sum_pow
theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) 
Arguments: {ι : Type u} (s : Finset ι) {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a)
Goal: (∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ∑ x ∈ s, f x ^ (n + 1)
-- Real.zpow_arith_mean_le_arith_mean_zpow
theorem zpow_arith_mean_le_arith_mean_zpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) :
    (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ)
Goal: (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m
-- Real.rpow_arith_mean_le_arith_mean_rpow
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
    (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p
-- Real.arith_mean_le_rpow_mean
theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
    (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
    ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p)
Goal: ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)
-- NNReal.pow_arith_mean_le_arith_mean_pow
theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) (n : ℕ) :
    (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) (n : ℕ)
Goal: (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n
-- NNReal.pow_sum_div_card_le_sum_pow
theorem pow_sum_div_card_le_sum_pow (f : ι → ℝ≥0) (n : ℕ) :
    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) 
Arguments: {ι : Type u} (s : Finset ι) (f : ι → ℝ≥0) (n : ℕ)
Goal: ↑(∑ x ∈ s, f x) ^ (n + 1) / ↑s.card ^ n ≤ ↑(∑ x ∈ s, f x ^ (n + 1))
-- NNReal.rpow_arith_mean_le_arith_mean_rpow
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p
-- NNReal.rpow_arith_mean_le_arith_mean2_rpow
theorem rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0) (hw' : w₁ + w₂ = 1) {p : ℝ}
    (hp : 1 ≤ p) : (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p 
Arguments: (w₁ w₂ z₁ z₂ : ℝ≥0) (hw' : w₁ + w₂ = 1) {p : ℝ} (hp : 1 ≤ p)
Goal: (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
-- NNReal.rpow_add_le_mul_rpow_add_rpow
theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
    (z₁ + z₂) ^ p ≤ (2 : ℝ≥0) ^ (p - 1) * (z₁ ^ p + z₂ ^ p) 
Arguments: (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 ≤ p)
Goal: (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)
-- NNReal.arith_mean_le_rpow_mean
theorem arith_mean_le_rpow_mean (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ} (hp : 1 ≤ p) :
    ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ} (hp : 1 ≤ p)
Goal: ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)
-- _private.Mathlib.Analysis.MeanInequalitiesPow.0.NNReal.add_rpow_le_one_of_add_le_one
private theorem add_rpow_le_one_of_add_le_one {p : ℝ} (a b : ℝ≥0) (hab : a + b ≤ 1) (hp1 : 1 ≤ p) :
    a ^ p + b ^ p ≤ 1 
Arguments: {p : ℝ} (a b : ℝ≥0) (hab : a + b ≤ 1) (hp1 : 1 ≤ p)
Goal: a ^ p + b ^ p ≤ 1
-- NNReal.add_rpow_le_rpow_add
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p 
Arguments: {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p)
Goal: a ^ p + b ^ p ≤ (a + b) ^ p
-- NNReal.rpow_add_rpow_le_add
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b 
Arguments: {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p)
Goal: (a ^ p + b ^ p) ^ (1 / p) ≤ a + b
-- NNReal.rpow_add_rpow_le
theorem rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0) (hp_pos : 0 < p) (hpq : p ≤ q) :
    (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p) 
Arguments: {p q : ℝ} (a b : ℝ≥0) (hp_pos : 0 < p) (hpq : p ≤ q)
Goal: (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)
-- NNReal.rpow_add_le_add_rpow
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p 
Arguments: {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1)
Goal: (a + b) ^ p ≤ a ^ p + b ^ p
-- ENNReal.rpow_arith_mean_le_arith_mean_rpow
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ≥0∞) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ} (hp : 1 ≤ p)
Goal: (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p
-- ENNReal.rpow_arith_mean_le_arith_mean2_rpow
theorem rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0∞) (hw' : w₁ + w₂ = 1) {p : ℝ}
    (hp : 1 ≤ p) : (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p 
Arguments: (w₁ w₂ z₁ z₂ : ℝ≥0∞) (hw' : w₁ + w₂ = 1) {p : ℝ} (hp : 1 ≤ p)
Goal: (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
-- ENNReal.rpow_add_le_mul_rpow_add_rpow
theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p) :
    (z₁ + z₂) ^ p ≤ (2 : ℝ≥0∞) ^ (p - 1) * (z₁ ^ p + z₂ ^ p) 
Arguments: (z₁ z₂ : ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p)
Goal: (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)
-- ENNReal.add_rpow_le_rpow_add
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p 
Arguments: {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p)
Goal: a ^ p + b ^ p ≤ (a + b) ^ p
-- ENNReal.rpow_add_rpow_le_add
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b 
Arguments: {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p)
Goal: (a ^ p + b ^ p) ^ (1 / p) ≤ a + b
-- ENNReal.rpow_add_rpow_le
theorem rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0∞) (hp_pos : 0 < p) (hpq : p ≤ q) :
    (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p) 
Arguments: {p q : ℝ} (a b : ℝ≥0∞) (hp_pos : 0 < p) (hpq : p ≤ q)
Goal: (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)
-- ENNReal.rpow_add_le_add_rpow
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p 
Arguments: {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1)
Goal: (a + b) ^ p ≤ a ^ p + b ^ p
-- banach_steinhaus
theorem banach_steinhaus {ι : Type*} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F}
    (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' 
Arguments: {E : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {𝕜₂ : Type u_4} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type u_5} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F} (h : ∀ (x : E), ∃ C, ∀ (i : ι), ‖(g i) x‖ ≤ C)
Goal: ∃ C', ∀ (i : ι), ‖g i‖ ≤ C'
-- banach_steinhaus_iSup_nnnorm
theorem banach_steinhaus_iSup_nnnorm {ι : Type*} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F}
    (h : ∀ x, (⨆ i, ↑‖g i x‖₊) < ∞) : (⨆ i, ↑‖g i‖₊) < ∞ 
Arguments: {E : Type u_1} {F : Type u_2} {𝕜 : Type u_3} {𝕜₂ : Type u_4} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] {ι : Type u_5} [CompleteSpace E] {g : ι → E →SL[σ₁₂] F} (h : ∀ (x : E), ⨆ i, ↑‖(g i) x‖₊ < ⊤)
Goal: ⨆ i, ↑‖g i‖₊ < ⊤
-- NormedSpace.Dual.coe_toWeakDual
@[simp]
theorem coe_toWeakDual (x' : Dual 𝕜 E) : toWeakDual x' = x' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : NormedSpace.Dual 𝕜 E)
Goal: NormedSpace.Dual.toWeakDual x' = x'
-- NormedSpace.Dual.toWeakDual_eq_iff
@[simp]
theorem toWeakDual_eq_iff (x' y' : Dual 𝕜 E) : toWeakDual x' = toWeakDual y' ↔ x' = y' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' y' : NormedSpace.Dual 𝕜 E)
Goal: NormedSpace.Dual.toWeakDual x' = NormedSpace.Dual.toWeakDual y' ↔ x' = y'
-- NormedSpace.Dual.toWeakDual_continuous
theorem toWeakDual_continuous : Continuous fun x' : Dual 𝕜 E => toWeakDual x' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: Continuous fun x' => NormedSpace.Dual.toWeakDual x'
-- NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology
theorem dual_norm_topology_le_weak_dual_topology :
    (UniformSpace.toTopologicalSpace : TopologicalSpace (Dual 𝕜 E)) ≤
      (WeakDual.instTopologicalSpace : TopologicalSpace (WeakDual 𝕜 E)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: UniformSpace.toTopologicalSpace ≤ WeakDual.instTopologicalSpace
-- WeakDual.toNormedDual_apply
theorem toNormedDual_apply (x : WeakDual 𝕜 E) (y : E) : (toNormedDual x) y = x y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : WeakDual 𝕜 E) (y : E)
Goal: (WeakDual.toNormedDual x) y = x y
-- WeakDual.coe_toNormedDual
@[simp]
theorem coe_toNormedDual (x' : WeakDual 𝕜 E) : toNormedDual x' = x' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : WeakDual 𝕜 E)
Goal: WeakDual.toNormedDual x' = x'
-- WeakDual.toNormedDual_eq_iff
@[simp]
theorem toNormedDual_eq_iff (x' y' : WeakDual 𝕜 E) : toNormedDual x' = toNormedDual y' ↔ x' = y' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' y' : WeakDual 𝕜 E)
Goal: WeakDual.toNormedDual x' = WeakDual.toNormedDual y' ↔ x' = y'
-- WeakDual.isClosed_closedBall
theorem isClosed_closedBall (x' : Dual 𝕜 E) (r : ℝ) : IsClosed (toNormedDual ⁻¹' closedBall x' r) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : NormedSpace.Dual 𝕜 E) (r : ℝ)
Goal: IsClosed (⇑WeakDual.toNormedDual ⁻¹' Metric.closedBall x' r)
-- WeakDual.polar_def
theorem polar_def (s : Set E) : polar 𝕜 s = { f : WeakDual 𝕜 E | ∀ x ∈ s, ‖f x‖ ≤ 1 } 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E)
Goal: WeakDual.polar 𝕜 s = {f | ∀ x ∈ s, ‖f x‖ ≤ 1}
-- WeakDual.isClosed_polar
theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E)
Goal: IsClosed (WeakDual.polar 𝕜 s)
-- WeakDual.isClosed_image_coe_of_bounded_of_closed
theorem isClosed_image_coe_of_bounded_of_closed {s : Set (WeakDual 𝕜 E)}
    (hb : IsBounded (Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s) :
    IsClosed (((↑) : WeakDual 𝕜 E → E → 𝕜) '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set (WeakDual 𝕜 E)} (hb : Bornology.IsBounded (⇑NormedSpace.Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s)
Goal: IsClosed (DFunLike.coe '' s)
-- WeakDual.isCompact_of_bounded_of_closed
theorem isCompact_of_bounded_of_closed [ProperSpace 𝕜] {s : Set (WeakDual 𝕜 E)}
    (hb : IsBounded (Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s) : IsCompact s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] {s : Set (WeakDual 𝕜 E)} (hb : Bornology.IsBounded (⇑NormedSpace.Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s)
Goal: IsCompact s
-- WeakDual.isClosed_image_polar_of_mem_nhds
theorem isClosed_image_polar_of_mem_nhds {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsClosed (((↑) : WeakDual 𝕜 E → E → 𝕜) '' polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (s_nhd : s ∈ 𝓝 0)
Goal: IsClosed (DFunLike.coe '' WeakDual.polar 𝕜 s)
-- NormedSpace.Dual.isClosed_image_polar_of_mem_nhds
theorem _root_.NormedSpace.Dual.isClosed_image_polar_of_mem_nhds {s : Set E}
    (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsClosed (((↑) : Dual 𝕜 E → E → 𝕜) '' NormedSpace.polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (s_nhd : s ∈ 𝓝 0)
Goal: IsClosed (DFunLike.coe '' NormedSpace.polar 𝕜 s)
-- WeakDual.isCompact_polar
theorem isCompact_polar [ProperSpace 𝕜] {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsCompact (polar 𝕜 s) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] {s : Set E} (s_nhd : s ∈ 𝓝 0)
Goal: IsCompact (WeakDual.polar 𝕜 s)
-- WeakDual.isCompact_closedBall
theorem isCompact_closedBall [ProperSpace 𝕜] (x' : Dual 𝕜 E) (r : ℝ) :
    IsCompact (toNormedDual ⁻¹' closedBall x' r) 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] (x' : NormedSpace.Dual 𝕜 E) (r : ℝ)
Goal: IsCompact (⇑WeakDual.toNormedDual ⁻¹' Metric.closedBall x' r)
-- ContinuousLinearMap.ker_closedComplemented_of_finiteDimensional_range
theorem ker_closedComplemented_of_finiteDimensional_range (f : E →L[𝕜] F)
    [FiniteDimensional 𝕜 (range f)] : (ker f).ClosedComplemented 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace 𝕜] (f : E →L[𝕜] F) [FiniteDimensional 𝕜 ↥(LinearMap.range f)]
Goal: (LinearMap.ker f).ClosedComplemented
-- ContinuousLinearMap.coe_equivProdOfSurjectiveOfIsCompl
@[simp]
theorem coe_equivProdOfSurjectiveOfIsCompl {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : range f = ⊤)
    (hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) :
    (equivProdOfSurjectiveOfIsCompl f g hf hg hfg : E →ₗ[𝕜] F × G) = f.prod g 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace E] [CompleteSpace (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : LinearMap.range f = ⊤) (hg : LinearMap.range g = ⊤) (hfg : IsCompl (LinearMap.ker f) (LinearMap.ker g))
Goal: ↑↑(f.equivProdOfSurjectiveOfIsCompl g hf hg hfg) = ↑(f.prod g)
-- ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl_toLinearEquiv
@[simp]
theorem equivProdOfSurjectiveOfIsCompl_toLinearEquiv {f : E →L[𝕜] F} {g : E →L[𝕜] G}
    (hf : range f = ⊤) (hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) :
    (equivProdOfSurjectiveOfIsCompl f g hf hg hfg).toLinearEquiv =
      LinearMap.equivProdOfSurjectiveOfIsCompl f g hf hg hfg 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace E] [CompleteSpace (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : LinearMap.range f = ⊤) (hg : LinearMap.range g = ⊤) (hfg : IsCompl (LinearMap.ker f) (LinearMap.ker g))
Goal: (f.equivProdOfSurjectiveOfIsCompl g hf hg hfg).toLinearEquiv = (↑f).equivProdOfSurjectiveOfIsCompl (↑g) hf hg hfg
-- ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl_apply
@[simp]
theorem equivProdOfSurjectiveOfIsCompl_apply {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : range f = ⊤)
    (hg : range g = ⊤) (hfg : IsCompl (ker f) (ker g)) (x : E) :
    equivProdOfSurjectiveOfIsCompl f g hf hg hfg x = (f x, g x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace E] [CompleteSpace (F × G)] {f : E →L[𝕜] F} {g : E →L[𝕜] G} (hf : LinearMap.range f = ⊤) (hg : LinearMap.range g = ⊤) (hfg : IsCompl (LinearMap.ker f) (LinearMap.ker g)) (x : E)
Goal: (f.equivProdOfSurjectiveOfIsCompl g hf hg hfg) x = (f x, g x)
-- Submodule.coe_prodEquivOfClosedCompl
@[simp]
theorem coe_prodEquivOfClosedCompl (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) :
    ⇑(p.prodEquivOfClosedCompl q h hp hq) = p.prodEquivOfIsCompl q h 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p q : Subspace 𝕜 E} (h : IsCompl p q) (hp : IsClosed ↑p) (hq : IsClosed ↑q)
Goal: ⇑(Submodule.prodEquivOfClosedCompl p q h hp hq) = ⇑(Submodule.prodEquivOfIsCompl p q h)
-- Submodule.coe_prodEquivOfClosedCompl_symm
@[simp]
theorem coe_prodEquivOfClosedCompl_symm (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) :
    ⇑(p.prodEquivOfClosedCompl q h hp hq).symm = (p.prodEquivOfIsCompl q h).symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p q : Subspace 𝕜 E} (h : IsCompl p q) (hp : IsClosed ↑p) (hq : IsClosed ↑q)
Goal: ⇑(Submodule.prodEquivOfClosedCompl p q h hp hq).symm = ⇑(Submodule.prodEquivOfIsCompl p q h).symm
-- Submodule.coe_continuous_linearProjOfClosedCompl
@[simp]
theorem coe_continuous_linearProjOfClosedCompl (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) :
    (p.linearProjOfClosedCompl q h hp hq : E →ₗ[𝕜] p) = p.linearProjOfIsCompl q h 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p q : Subspace 𝕜 E} (h : IsCompl p q) (hp : IsClosed ↑p) (hq : IsClosed ↑q)
Goal: ↑(Submodule.linearProjOfClosedCompl p q h hp hq) = Submodule.linearProjOfIsCompl p q h
-- Submodule.coe_continuous_linearProjOfClosedCompl'
@[simp]
theorem coe_continuous_linearProjOfClosedCompl' (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) :
    ⇑(p.linearProjOfClosedCompl q h hp hq) = p.linearProjOfIsCompl q h 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p q : Subspace 𝕜 E} (h : IsCompl p q) (hp : IsClosed ↑p) (hq : IsClosed ↑q)
Goal: ⇑(Submodule.linearProjOfClosedCompl p q h hp hq) = ⇑(Submodule.linearProjOfIsCompl p q h)
-- Submodule.ClosedComplemented.of_isCompl_isClosed
theorem ClosedComplemented.of_isCompl_isClosed (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) : p.ClosedComplemented 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p q : Subspace 𝕜 E} (h : IsCompl p q) (hp : IsClosed ↑p) (hq : IsClosed ↑q)
Goal: Submodule.ClosedComplemented p
-- Submodule.closedComplemented_iff_isClosed_exists_isClosed_isCompl
theorem closedComplemented_iff_isClosed_exists_isClosed_isCompl :
    p.ClosedComplemented ↔
      IsClosed (p : Set E) ∧ ∃ q : Submodule 𝕜 E, IsClosed (q : Set E) ∧ IsCompl p q 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p : Subspace 𝕜 E}
Goal: Submodule.ClosedComplemented p ↔ IsClosed ↑p ∧ ∃ q, IsClosed ↑q ∧ IsCompl p q
-- Submodule.ClosedComplemented.of_quotient_finiteDimensional
theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
    [FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {p : Subspace 𝕜 E} [CompleteSpace 𝕜] [FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed ↑p)
Goal: Submodule.ClosedComplemented p
-- NormedSpace.dual_def
@[simp]
theorem dual_def (x : E) (f : Dual 𝕜 E) : inclusionInDoubleDual 𝕜 E x f = f x 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (f : NormedSpace.Dual 𝕜 E)
Goal: ((NormedSpace.inclusionInDoubleDual 𝕜 E) x) f = f x