ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- norm_prod_le_of_le
@[to_additive]
theorem norm_prod_le_of_le (s : Finset ι) {f : ι → E} {n : ι → ℝ} (h : ∀ b ∈ s, ‖f b‖ ≤ n b) :
    ‖∏ b ∈ s, f b‖ ≤ ∑ b ∈ s, n b 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) {f : ι → E} {n : ι → ℝ} (h : ∀ b ∈ s, ‖f b‖ ≤ n b)
Goal: ‖∏ b ∈ s, f b‖ ≤ ∑ b ∈ s, n b
-- dist_sum_sum_le_of_le
@[to_additive]
theorem dist_prod_prod_le_of_le (s : Finset ι) {f a : ι → E} {d : ι → ℝ}
    (h : ∀ b ∈ s, dist (f b) (a b) ≤ d b) :
    dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, d b 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedAddCommGroup E] (s : Finset ι) {f a : ι → E} {d : ι → ℝ} (h : ∀ b ∈ s, dist (f b) (a b) ≤ d b)
Goal: dist (∑ b ∈ s, f b) (∑ b ∈ s, a b) ≤ ∑ b ∈ s, d b
-- dist_prod_prod_le_of_le
@[to_additive]
theorem dist_prod_prod_le_of_le (s : Finset ι) {f a : ι → E} {d : ι → ℝ}
    (h : ∀ b ∈ s, dist (f b) (a b) ≤ d b) :
    dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, d b 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) {f a : ι → E} {d : ι → ℝ} (h : ∀ b ∈ s, dist (f b) (a b) ≤ d b)
Goal: dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, d b
-- dist_prod_prod_le
@[to_additive]
theorem dist_prod_prod_le (s : Finset ι) (f a : ι → E) :
    dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, dist (f b) (a b) 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) (f a : ι → E)
Goal: dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, dist (f b) (a b)
-- dist_sum_sum_le
@[to_additive]
theorem dist_prod_prod_le (s : Finset ι) (f a : ι → E) :
    dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, dist (f b) (a b) 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedAddCommGroup E] (s : Finset ι) (f a : ι → E)
Goal: dist (∑ b ∈ s, f b) (∑ b ∈ s, a b) ≤ ∑ b ∈ s, dist (f b) (a b)
-- mul_mem_ball_iff_norm
@[to_additive]
theorem mul_mem_ball_iff_norm : a * b ∈ ball a r ↔ ‖b‖ < r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ}
Goal: a * b ∈ Metric.ball a r ↔ ‖b‖ < r
-- add_mem_ball_iff_norm
@[to_additive]
theorem mul_mem_ball_iff_norm : a * b ∈ ball a r ↔ ‖b‖ < r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ}
Goal: a + b ∈ Metric.ball a r ↔ ‖b‖ < r
-- add_mem_closedBall_iff_norm
@[to_additive]
theorem mul_mem_closedBall_iff_norm : a * b ∈ closedBall a r ↔ ‖b‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ}
Goal: a + b ∈ Metric.closedBall a r ↔ ‖b‖ ≤ r
-- mul_mem_closedBall_iff_norm
@[to_additive]
theorem mul_mem_closedBall_iff_norm : a * b ∈ closedBall a r ↔ ‖b‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ}
Goal: a * b ∈ Metric.closedBall a r ↔ ‖b‖ ≤ r
-- preimage_add_ball
@[to_additive (attr := simp 1001)]
-- Porting note: increase priority so that the left-hand side doesn't simplify
theorem preimage_mul_ball (a b : E) (r : ℝ) : (b * ·) ⁻¹' ball a r = ball (a / b) r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b + x) ⁻¹' Metric.ball a r = Metric.ball (a - b) r
-- preimage_mul_ball
@[to_additive (attr := simp 1001)]
-- Porting note: increase priority so that the left-hand side doesn't simplify
theorem preimage_mul_ball (a b : E) (r : ℝ) : (b * ·) ⁻¹' ball a r = ball (a / b) r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b * x) ⁻¹' Metric.ball a r = Metric.ball (a / b) r
-- preimage_add_closedBall
@[to_additive (attr := simp 1001)]
-- Porting note: increase priority so that the left-hand side doesn't simplify
theorem preimage_mul_closedBall (a b : E) (r : ℝ) :
    (b * ·) ⁻¹' closedBall a r = closedBall (a / b) r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b + x) ⁻¹' Metric.closedBall a r = Metric.closedBall (a - b) r
-- preimage_mul_closedBall
@[to_additive (attr := simp 1001)]
-- Porting note: increase priority so that the left-hand side doesn't simplify
theorem preimage_mul_closedBall (a b : E) (r : ℝ) :
    (b * ·) ⁻¹' closedBall a r = closedBall (a / b) r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b * x) ⁻¹' Metric.closedBall a r = Metric.closedBall (a / b) r
-- preimage_mul_sphere
@[to_additive (attr := simp)]
theorem preimage_mul_sphere (a b : E) (r : ℝ) : (b * ·) ⁻¹' sphere a r = sphere (a / b) r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b * x) ⁻¹' Metric.sphere a r = Metric.sphere (a / b) r
-- preimage_add_sphere
@[to_additive (attr := simp)]
theorem preimage_mul_sphere (a b : E) (r : ℝ) : (b * ·) ⁻¹' sphere a r = sphere (a / b) r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (a b : E) (r : ℝ)
Goal: (fun x => b + x) ⁻¹' Metric.sphere a r = Metric.sphere (a - b) r
-- norm_nsmul_le
@[to_additive norm_nsmul_le]
theorem norm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a ^ n‖ ≤ n * ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (n : ℕ) (a : E)
Goal: ‖n • a‖ ≤ ↑n * ‖a‖
-- norm_pow_le_mul_norm
@[to_additive norm_nsmul_le]
theorem norm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a ^ n‖ ≤ n * ‖a‖ 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (n : ℕ) (a : E)
Goal: ‖a ^ n‖ ≤ ↑n * ‖a‖
-- nnnorm_pow_le_mul_norm
@[to_additive nnnorm_nsmul_le]
theorem nnnorm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a ^ n‖₊ ≤ n * ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (n : ℕ) (a : E)
Goal: ‖a ^ n‖₊ ≤ ↑n * ‖a‖₊
-- nnnorm_nsmul_le
@[to_additive nnnorm_nsmul_le]
theorem nnnorm_pow_le_mul_norm (n : ℕ) (a : E) : ‖a ^ n‖₊ ≤ n * ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (n : ℕ) (a : E)
Goal: ‖n • a‖₊ ≤ ↑n * ‖a‖₊
-- pow_mem_closedBall
@[to_additive]
theorem pow_mem_closedBall {n : ℕ} (h : a ∈ closedBall b r) :
    a ^ n ∈ closedBall (b ^ n) (n • r) 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ} {n : ℕ} (h : a ∈ Metric.closedBall b r)
Goal: a ^ n ∈ Metric.closedBall (b ^ n) (n • r)
-- nsmul_mem_closedBall
@[to_additive]
theorem pow_mem_closedBall {n : ℕ} (h : a ∈ closedBall b r) :
    a ^ n ∈ closedBall (b ^ n) (n • r) 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ} {n : ℕ} (h : a ∈ Metric.closedBall b r)
Goal: n • a ∈ Metric.closedBall (n • b) (n • r)
-- pow_mem_ball
@[to_additive]
theorem pow_mem_ball {n : ℕ} (hn : 0 < n) (h : a ∈ ball b r) : a ^ n ∈ ball (b ^ n) (n • r) 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ} {n : ℕ} (hn : 0 < n) (h : a ∈ Metric.ball b r)
Goal: a ^ n ∈ Metric.ball (b ^ n) (n • r)
-- nsmul_mem_ball
@[to_additive]
theorem pow_mem_ball {n : ℕ} (hn : 0 < n) (h : a ∈ ball b r) : a ^ n ∈ ball (b ^ n) (n • r) 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ} {n : ℕ} (hn : 0 < n) (h : a ∈ Metric.ball b r)
Goal: n • a ∈ Metric.ball (n • b) (n • r)
-- mul_mem_closedBall_mul_iff
@[to_additive] -- Porting note (#10618): `simp` can prove this
theorem mul_mem_closedBall_mul_iff {c : E} : a * c ∈ closedBall (b * c) r ↔ a ∈ closedBall b r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ} {c : E}
Goal: a * c ∈ Metric.closedBall (b * c) r ↔ a ∈ Metric.closedBall b r
-- add_mem_closedBall_add_iff
@[to_additive] -- Porting note (#10618): `simp` can prove this
theorem mul_mem_closedBall_mul_iff {c : E} : a * c ∈ closedBall (b * c) r ↔ a ∈ closedBall b r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ} {c : E}
Goal: a + c ∈ Metric.closedBall (b + c) r ↔ a ∈ Metric.closedBall b r
-- mul_mem_ball_mul_iff
@[to_additive] -- Porting note (#10618): `simp` can prove this
theorem mul_mem_ball_mul_iff {c : E} : a * c ∈ ball (b * c) r ↔ a ∈ ball b r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ} {c : E}
Goal: a * c ∈ Metric.ball (b * c) r ↔ a ∈ Metric.ball b r
-- add_mem_ball_add_iff
@[to_additive] -- Porting note (#10618): `simp` can prove this
theorem mul_mem_ball_mul_iff {c : E} : a * c ∈ ball (b * c) r ↔ a ∈ ball b r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ} {c : E}
Goal: a + c ∈ Metric.ball (b + c) r ↔ a ∈ Metric.ball b r
-- vadd_closedBall''
@[to_additive]
theorem smul_closedBall'' : a • closedBall b r = closedBall (a • b) r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ}
Goal: a +ᵥ Metric.closedBall b r = Metric.closedBall (a +ᵥ b) r
-- smul_closedBall''
@[to_additive]
theorem smul_closedBall'' : a • closedBall b r = closedBall (a • b) r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ}
Goal: a • Metric.closedBall b r = Metric.closedBall (a • b) r
-- vadd_ball''
@[to_additive]
theorem smul_ball'' : a • ball b r = ball (a • b) r 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a b : E} {r : ℝ}
Goal: a +ᵥ Metric.ball b r = Metric.ball (a +ᵥ b) r
-- smul_ball''
@[to_additive]
theorem smul_ball'' : a • ball b r = ball (a • b) r 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a b : E} {r : ℝ}
Goal: a • Metric.ball b r = Metric.ball (a • b) r
-- controlled_sum_of_mem_closure
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i ∈ range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] {a : E} {s : AddSubgroup E} (hg : a ∈ closure ↑s) {b : ℕ → ℝ} (b_pos : ∀ (n : ℕ), 0 < b n)
Goal: ∃ v,
  Filter.Tendsto (fun n => ∑ i ∈ Finset.range (n + 1), v i) Filter.atTop (𝓝 a) ∧
    (∀ (n : ℕ), v n ∈ s) ∧ ‖v 0 - a‖ < b 0 ∧ ∀ (n : ℕ), 0 < n → ‖v n‖ < b n
-- controlled_prod_of_mem_closure
@[to_additive]
theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
    (b_pos : ∀ n, 0 < b n) :
    ∃ v : ℕ → E,
      Tendsto (fun n => ∏ i ∈ range (n + 1), v i) atTop (𝓝 a) ∧
        (∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n 
Arguments: {E : Type u_6} [SeminormedCommGroup E] {a : E} {s : Subgroup E} (hg : a ∈ closure ↑s) {b : ℕ → ℝ} (b_pos : ∀ (n : ℕ), 0 < b n)
Goal: ∃ v,
  Filter.Tendsto (fun n => ∏ i ∈ Finset.range (n + 1), v i) Filter.atTop (𝓝 a) ∧
    (∀ (n : ℕ), v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ (n : ℕ), 0 < n → ‖v n‖ < b n
-- controlled_sum_of_mem_closure_range
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i ∈ range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n 
Arguments: {E : Type u_6} {F : Type u_7} [SeminormedAddCommGroup E] [SeminormedAddCommGroup F] {j : E →+ F} {b : F} (hb : b ∈ closure ↑j.range) {f : ℕ → ℝ} (b_pos : ∀ (n : ℕ), 0 < f n)
Goal: ∃ a,
  Filter.Tendsto (fun n => ∑ i ∈ Finset.range (n + 1), j (a i)) Filter.atTop (𝓝 b) ∧
    ‖j (a 0) - b‖ < f 0 ∧ ∀ (n : ℕ), 0 < n → ‖j (a n)‖ < f n
-- controlled_prod_of_mem_closure_range
@[to_additive]
theorem controlled_prod_of_mem_closure_range {j : E →* F} {b : F}
    (hb : b ∈ closure (j.range : Set F)) {f : ℕ → ℝ} (b_pos : ∀ n, 0 < f n) :
    ∃ a : ℕ → E,
      Tendsto (fun n => ∏ i ∈ range (n + 1), j (a i)) atTop (𝓝 b) ∧
        ‖j (a 0) / b‖ < f 0 ∧ ∀ n, 0 < n → ‖j (a n)‖ < f n 
Arguments: {E : Type u_6} {F : Type u_7} [SeminormedCommGroup E] [SeminormedCommGroup F] {j : E →* F} {b : F} (hb : b ∈ closure ↑j.range) {f : ℕ → ℝ} (b_pos : ∀ (n : ℕ), 0 < f n)
Goal: ∃ a,
  Filter.Tendsto (fun n => ∏ i ∈ Finset.range (n + 1), j (a i)) Filter.atTop (𝓝 b) ∧
    ‖j (a 0) / b‖ < f 0 ∧ ∀ (n : ℕ), 0 < n → ‖j (a n)‖ < f n
-- nnnorm_multiset_sum_le
@[to_additive]
theorem nnnorm_multiset_prod_le (m : Multiset E) : ‖m.prod‖₊ ≤ (m.map fun x => ‖x‖₊).sum 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (m : Multiset E)
Goal: ‖m.sum‖₊ ≤ (Multiset.map (fun x => ‖x‖₊) m).sum
-- nnnorm_multiset_prod_le
@[to_additive]
theorem nnnorm_multiset_prod_le (m : Multiset E) : ‖m.prod‖₊ ≤ (m.map fun x => ‖x‖₊).sum 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (m : Multiset E)
Goal: ‖m.prod‖₊ ≤ (Multiset.map (fun x => ‖x‖₊) m).sum
-- nnnorm_prod_le
@[to_additive]
theorem nnnorm_prod_le (s : Finset ι) (f : ι → E) : ‖∏ a ∈ s, f a‖₊ ≤ ∑ a ∈ s, ‖f a‖₊ 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) (f : ι → E)
Goal: ‖∏ a ∈ s, f a‖₊ ≤ ∑ a ∈ s, ‖f a‖₊
-- nnnorm_sum_le
@[to_additive]
theorem nnnorm_prod_le (s : Finset ι) (f : ι → E) : ‖∏ a ∈ s, f a‖₊ ≤ ∑ a ∈ s, ‖f a‖₊ 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedAddCommGroup E] (s : Finset ι) (f : ι → E)
Goal: ‖∑ a ∈ s, f a‖₊ ≤ ∑ a ∈ s, ‖f a‖₊
-- nnnorm_prod_le_of_le
@[to_additive]
theorem nnnorm_prod_le_of_le (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b) :
    ‖∏ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b)
Goal: ‖∏ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b
-- nnnorm_sum_le_of_le
@[to_additive]
theorem nnnorm_prod_le_of_le (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b) :
    ‖∏ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedAddCommGroup E] (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b)
Goal: ‖∑ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b
-- Real.norm_eq_abs
@[simp]
theorem norm_eq_abs (r : ℝ) : ‖r‖ = |r| 
Arguments: (r : ℝ)
Goal: ‖r‖ = |r|
-- Real.norm_of_nonneg
theorem norm_of_nonneg (hr : 0 ≤ r) : ‖r‖ = r 
Arguments: {r : ℝ} (hr : 0 ≤ r)
Goal: ‖r‖ = r
-- Real.norm_of_nonpos
theorem norm_of_nonpos (hr : r ≤ 0) : ‖r‖ = -r 
Arguments: {r : ℝ} (hr : r ≤ 0)
Goal: ‖r‖ = -r
-- Real.le_norm_self
theorem le_norm_self (r : ℝ) : r ≤ ‖r‖ 
Arguments: (r : ℝ)
Goal: r ≤ ‖r‖
-- Real.norm_natCast
theorem norm_natCast (n : ℕ) : ‖(n : ℝ)‖ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖ = ↑n
-- Real.nnnorm_natCast
@[simp]
theorem nnnorm_natCast (n : ℕ) : ‖(n : ℝ)‖₊ = n 
Arguments: (n : ℕ)
Goal: ‖↑n‖₊ = ↑n
-- Real.norm_two
theorem norm_two : ‖(2 : ℝ)‖ = 2 
Goal: ‖2‖ = 2
-- Real.nnnorm_two
@[simp]
theorem nnnorm_two : ‖(2 : ℝ)‖₊ = 2 
Goal: ‖2‖₊ = 2
-- Real.nnnorm_of_nonneg
theorem nnnorm_of_nonneg (hr : 0 ≤ r) : ‖r‖₊ = ⟨r, hr⟩ 
Arguments: {r : ℝ} (hr : 0 ≤ r)
Goal: ‖r‖₊ = ⟨r, hr⟩
-- Real.nnnorm_abs
@[simp]
theorem nnnorm_abs (r : ℝ) : ‖|r|‖₊ = ‖r‖₊ 
Arguments: (r : ℝ)
Goal: ‖|r|‖₊ = ‖r‖₊
-- Real.ennnorm_eq_ofReal
theorem ennnorm_eq_ofReal (hr : 0 ≤ r) : (‖r‖₊ : ℝ≥0∞) = ENNReal.ofReal r 
Arguments: {r : ℝ} (hr : 0 ≤ r)
Goal: ↑‖r‖₊ = ENNReal.ofReal r
-- Real.ennnorm_eq_ofReal_abs
theorem ennnorm_eq_ofReal_abs (r : ℝ) : (‖r‖₊ : ℝ≥0∞) = ENNReal.ofReal |r| 
Arguments: (r : ℝ)
Goal: ↑‖r‖₊ = ENNReal.ofReal |r|
-- Real.toNNReal_eq_nnnorm_of_nonneg
theorem toNNReal_eq_nnnorm_of_nonneg (hr : 0 ≤ r) : r.toNNReal = ‖r‖₊ 
Arguments: {r : ℝ} (hr : 0 ≤ r)
Goal: r.toNNReal = ‖r‖₊
-- Real.ofReal_le_ennnorm
theorem ofReal_le_ennnorm (r : ℝ) : ENNReal.ofReal r ≤ ‖r‖₊ 
Arguments: (r : ℝ)
Goal: ENNReal.ofReal r ≤ ↑‖r‖₊
-- NNReal.nnnorm_eq_self
@[simp] lemma nnnorm_eq_self (x : ℝ≥0) : ‖x‖₊ = x 
Arguments: (x : ℝ≥0)
Goal: ‖x‖₊ = x
-- norm_eq_zero
@[to_additive (attr := simp) norm_eq_zero]
theorem norm_eq_zero'' : ‖a‖ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: ‖a‖ = 0 ↔ a = 0
-- norm_eq_zero''
@[to_additive (attr := simp) norm_eq_zero]
theorem norm_eq_zero'' : ‖a‖ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: ‖a‖ = 0 ↔ a = 1
-- norm_ne_zero_iff'
@[to_additive norm_ne_zero_iff]
theorem norm_ne_zero_iff' : ‖a‖ ≠ 0 ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: ‖a‖ ≠ 0 ↔ a ≠ 1
-- norm_ne_zero_iff
@[to_additive norm_ne_zero_iff]
theorem norm_ne_zero_iff' : ‖a‖ ≠ 0 ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: ‖a‖ ≠ 0 ↔ a ≠ 0
-- norm_pos_iff
@[to_additive (attr := simp) norm_pos_iff]
theorem norm_pos_iff'' : 0 < ‖a‖ ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: 0 < ‖a‖ ↔ a ≠ 0
-- norm_pos_iff''
@[to_additive (attr := simp) norm_pos_iff]
theorem norm_pos_iff'' : 0 < ‖a‖ ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: 0 < ‖a‖ ↔ a ≠ 1
-- norm_le_zero_iff
@[to_additive (attr := simp) norm_le_zero_iff]
theorem norm_le_zero_iff'' : ‖a‖ ≤ 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: ‖a‖ ≤ 0 ↔ a = 0
-- norm_le_zero_iff''
@[to_additive (attr := simp) norm_le_zero_iff]
theorem norm_le_zero_iff'' : ‖a‖ ≤ 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: ‖a‖ ≤ 0 ↔ a = 1
-- norm_div_eq_zero_iff
@[to_additive]
theorem norm_div_eq_zero_iff : ‖a / b‖ = 0 ↔ a = b 
Arguments: {E : Type u_6} [NormedGroup E] {a b : E}
Goal: ‖a / b‖ = 0 ↔ a = b
-- norm_sub_eq_zero_iff
@[to_additive]
theorem norm_div_eq_zero_iff : ‖a / b‖ = 0 ↔ a = b 
Arguments: {E : Type u_6} [NormedAddGroup E] {a b : E}
Goal: ‖a - b‖ = 0 ↔ a = b
-- norm_div_pos_iff
@[to_additive]
theorem norm_div_pos_iff : 0 < ‖a / b‖ ↔ a ≠ b 
Arguments: {E : Type u_6} [NormedGroup E] {a b : E}
Goal: 0 < ‖a / b‖ ↔ a ≠ b
-- norm_sub_pos_iff
@[to_additive]
theorem norm_div_pos_iff : 0 < ‖a / b‖ ↔ a ≠ b 
Arguments: {E : Type u_6} [NormedAddGroup E] {a b : E}
Goal: 0 < ‖a - b‖ ↔ a ≠ b
-- eq_of_norm_div_le_zero
@[to_additive eq_of_norm_sub_le_zero]
theorem eq_of_norm_div_le_zero (h : ‖a / b‖ ≤ 0) : a = b 
Arguments: {E : Type u_6} [NormedGroup E] {a b : E} (h : ‖a / b‖ ≤ 0)
Goal: a = b
-- eq_of_norm_sub_le_zero
@[to_additive eq_of_norm_sub_le_zero]
theorem eq_of_norm_div_le_zero (h : ‖a / b‖ ≤ 0) : a = b 
Arguments: {E : Type u_6} [NormedAddGroup E] {a b : E} (h : ‖a - b‖ ≤ 0)
Goal: a = b
-- nnnorm_eq_zero'
@[to_additive (attr := simp) nnnorm_eq_zero]
theorem nnnorm_eq_zero' : ‖a‖₊ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: ‖a‖₊ = 0 ↔ a = 1
-- nnnorm_eq_zero
@[to_additive (attr := simp) nnnorm_eq_zero]
theorem nnnorm_eq_zero' : ‖a‖₊ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: ‖a‖₊ = 0 ↔ a = 0
-- nnnorm_ne_zero_iff
@[to_additive nnnorm_ne_zero_iff]
theorem nnnorm_ne_zero_iff' : ‖a‖₊ ≠ 0 ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: ‖a‖₊ ≠ 0 ↔ a ≠ 0
-- nnnorm_ne_zero_iff'
@[to_additive nnnorm_ne_zero_iff]
theorem nnnorm_ne_zero_iff' : ‖a‖₊ ≠ 0 ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: ‖a‖₊ ≠ 0 ↔ a ≠ 1
-- nnnorm_pos
@[to_additive (attr := simp) nnnorm_pos]
lemma nnnorm_pos' : 0 < ‖a‖₊ ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedAddGroup E] {a : E}
Goal: 0 < ‖a‖₊ ↔ a ≠ 0
-- nnnorm_pos'
@[to_additive (attr := simp) nnnorm_pos]
lemma nnnorm_pos' : 0 < ‖a‖₊ ↔ a ≠ 1 
Arguments: {E : Type u_6} [NormedGroup E] {a : E}
Goal: 0 < ‖a‖₊ ↔ a ≠ 1
-- tendsto_norm_div_self_punctured_nhds
@[to_additive]
theorem tendsto_norm_div_self_punctured_nhds (a : E) :
    Tendsto (fun x => ‖x / a‖) (𝓝[≠] a) (𝓝[>] 0) 
Arguments: {E : Type u_6} [NormedGroup E] (a : E)
Goal: Filter.Tendsto (fun x => ‖x / a‖) (𝓝[≠] a) (𝓝[>] 0)
-- tendsto_norm_sub_self_punctured_nhds
@[to_additive]
theorem tendsto_norm_div_self_punctured_nhds (a : E) :
    Tendsto (fun x => ‖x / a‖) (𝓝[≠] a) (𝓝[>] 0) 
Arguments: {E : Type u_6} [NormedAddGroup E] (a : E)
Goal: Filter.Tendsto (fun x => ‖x - a‖) (𝓝[≠] a) (𝓝[>] 0)
-- tendsto_norm_nhdsWithin_zero
@[to_additive]
theorem tendsto_norm_nhdsWithin_one : Tendsto (norm : E → ℝ) (𝓝[≠] 1) (𝓝[>] 0) 
Arguments: {E : Type u_6} [NormedAddGroup E]
Goal: Filter.Tendsto norm (𝓝[≠] 0) (𝓝[>] 0)
-- tendsto_norm_nhdsWithin_one
@[to_additive]
theorem tendsto_norm_nhdsWithin_one : Tendsto (norm : E → ℝ) (𝓝[≠] 1) (𝓝[>] 0) 
Arguments: {E : Type u_6} [NormedGroup E]
Goal: Filter.Tendsto norm (𝓝[≠] 1) (𝓝[>] 0)
-- coe_normGroupNorm
@[simp]
theorem coe_normGroupNorm : ⇑(normGroupNorm E) = norm 
Arguments: (E : Type u_6) [NormedGroup E]
Goal: ⇑(normGroupNorm E) = norm
-- comap_norm_nhdsWithin_Ioi_zero'
@[to_additive comap_norm_nhdsWithin_Ioi_zero]
lemma comap_norm_nhdsWithin_Ioi_zero' : comap norm (𝓝[>] 0) = 𝓝[≠] (1 : E) 
Arguments: (E : Type u_6) [NormedGroup E]
Goal: Filter.comap norm (𝓝[>] 0) = 𝓝[≠] 1
-- comap_norm_nhdsWithin_Ioi_zero
@[to_additive comap_norm_nhdsWithin_Ioi_zero]
lemma comap_norm_nhdsWithin_Ioi_zero' : comap norm (𝓝[>] 0) = 𝓝[≠] (1 : E) 
Arguments: (E : Type u_6) [NormedAddGroup E]
Goal: Filter.comap norm (𝓝[>] 0) = 𝓝[≠] 0
-- hasCompactSupport_norm_iff
theorem hasCompactSupport_norm_iff : (HasCompactSupport fun x => ‖f x‖) ↔ HasCompactSupport f 
Arguments: {α : Type u_3} {E : Type u_6} [NormedAddGroup E] [TopologicalSpace α] {f : α → E}
Goal: (HasCompactSupport fun x => ‖f x‖) ↔ HasCompactSupport f
-- Subgroup.coe_norm
@[to_additive (attr := simp) "If `x` is an element of a subgroup `s` of a seminormed group `E`, its
norm in `s` is equal to its norm in `E`."]
theorem coe_norm (x : s) : ‖x‖ = ‖(x : E)‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] {s : Subgroup E} (x : ↥s)
Goal: ‖x‖ = ‖↑x‖
-- AddSubgroup.coe_norm
@[to_additive (attr := simp) "If `x` is an element of a subgroup `s` of a seminormed group `E`, its
norm in `s` is equal to its norm in `E`."]
theorem coe_norm (x : s) : ‖x‖ = ‖(x : E)‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {s : AddSubgroup E} (x : ↥s)
Goal: ‖x‖ = ‖↑x‖
-- Subgroup.norm_coe
@[to_additive (attr := norm_cast) "If `x` is an element of a subgroup `s` of a seminormed group `E`,
its norm in `s` is equal to its norm in `E`.

This is a reversed version of the `simp` lemma `AddSubgroup.coe_norm` for use by `norm_cast`."]
theorem norm_coe {s : Subgroup E} (x : s) : ‖(x : E)‖ = ‖x‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] {s : Subgroup E} (x : ↥s)
Goal: ‖↑x‖ = ‖x‖
-- AddSubgroup.norm_coe
@[to_additive (attr := norm_cast) "If `x` is an element of a subgroup `s` of a seminormed group `E`,
its norm in `s` is equal to its norm in `E`.

This is a reversed version of the `simp` lemma `AddSubgroup.coe_norm` for use by `norm_cast`."]
theorem norm_coe {s : Subgroup E} (x : s) : ‖(x : E)‖ = ‖x‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {s : AddSubgroup E} (x : ↥s)
Goal: ‖↑x‖ = ‖x‖
-- AddSubgroupClass.coe_norm
@[to_additive (attr := simp) "If `x` is an element of an additive subgroup `s` of a seminormed
additive group `E`, its norm in `s` is equal to its norm in `E`."]
theorem coe_norm (x : s) : ‖x‖ = ‖(x : E)‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {S : Type u_9} [SetLike S E] [AddSubgroupClass S E] (s : S) (x : ↥s)
Goal: ‖x‖ = ‖↑x‖
-- SubgroupClass.coe_norm
@[to_additive (attr := simp) "If `x` is an element of an additive subgroup `s` of a seminormed
additive group `E`, its norm in `s` is equal to its norm in `E`."]
theorem coe_norm (x : s) : ‖x‖ = ‖(x : E)‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] {S : Type u_9} [SetLike S E] [SubgroupClass S E] (s : S) (x : ↥s)
Goal: ‖x‖ = ‖↑x‖
-- controlled_closure_of_complete
theorem controlled_closure_of_complete {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C ε : ℝ}
    (hC : 0 < C) (hε : 0 < ε) (hyp : f.SurjectiveOnWith K C) :
    f.SurjectiveOnWith K.topologicalClosure (C + ε) 
Arguments: {G : Type u_1} [NormedAddCommGroup G] [CompleteSpace G] {H : Type u_2} [NormedAddCommGroup H] {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C ε : ℝ} (hC : 0 < C) (hε : 0 < ε) (hyp : f.SurjectiveOnWith K C)
Goal: f.SurjectiveOnWith K.topologicalClosure (C + ε)
-- controlled_closure_range_of_complete
theorem controlled_closure_range_of_complete {f : NormedAddGroupHom G H} {K : Type*}
    [SeminormedAddCommGroup K] {j : NormedAddGroupHom K H} (hj : ∀ x, ‖j x‖ = ‖x‖) {C ε : ℝ}
    (hC : 0 < C) (hε : 0 < ε) (hyp : ∀ k, ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖) :
    f.SurjectiveOnWith j.range.topologicalClosure (C + ε) 
Arguments: {G : Type u_1} [NormedAddCommGroup G] [CompleteSpace G] {H : Type u_2} [NormedAddCommGroup H] {f : NormedAddGroupHom G H} {K : Type u_3} [SeminormedAddCommGroup K] {j : NormedAddGroupHom K H} (hj : ∀ (x : K), ‖j x‖ = ‖x‖) {C ε : ℝ} (hC : 0 < C) (hε : 0 < ε) (hyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖)
Goal: f.SurjectiveOnWith j.range.topologicalClosure (C + ε)
-- Bornology.IsBounded.mul
@[to_additive]
theorem Bornology.IsBounded.mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) 
Arguments: {E : Type u_1} [SeminormedGroup E] {s t : Set E} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t)
Goal: Bornology.IsBounded (s * t)