ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- norm_div_le_norm_div_add_norm_div
@[to_additive]
lemma norm_div_le_norm_div_add_norm_div (a b c : E) : ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b c : E)
Goal: ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖
-- norm_nonneg'
@[to_additive (attr := simp) norm_nonneg]
theorem norm_nonneg' (a : E) : 0 ≤ ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: 0 ≤ ‖a‖
-- norm_nonneg
@[to_additive (attr := simp) norm_nonneg]
theorem norm_nonneg' (a : E) : 0 ≤ ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: 0 ≤ ‖a‖
-- abs_norm'
@[to_additive (attr := simp) abs_norm]
theorem abs_norm' (z : E) : |‖z‖| = ‖z‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (z : E)
Goal: |‖z‖| = ‖z‖
-- abs_norm
@[to_additive (attr := simp) abs_norm]
theorem abs_norm' (z : E) : |‖z‖| = ‖z‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (z : E)
Goal: |‖z‖| = ‖z‖
-- norm_one'
@[to_additive (attr := simp) norm_zero]
theorem norm_one' : ‖(1 : E)‖ = 0 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: ‖1‖ = 0
-- norm_zero
@[to_additive (attr := simp) norm_zero]
theorem norm_one' : ‖(1 : E)‖ = 0 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: ‖0‖ = 0
-- ne_zero_of_norm_ne_zero
@[to_additive]
theorem ne_one_of_norm_ne_zero : ‖a‖ ≠ 0 → a ≠ 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E}
Goal: ‖a‖ ≠ 0 → a ≠ 0
-- ne_one_of_norm_ne_zero
@[to_additive]
theorem ne_one_of_norm_ne_zero : ‖a‖ ≠ 0 → a ≠ 1 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E}
Goal: ‖a‖ ≠ 0 → a ≠ 1
-- norm_of_subsingleton'
@[to_additive (attr := nontriviality) norm_of_subsingleton]
theorem norm_of_subsingleton' [Subsingleton E] (a : E) : ‖a‖ = 0 
Arguments: {E : Type u_6} [SeminormedGroup E] [Subsingleton E] (a : E)
Goal: ‖a‖ = 0
-- norm_of_subsingleton
@[to_additive (attr := nontriviality) norm_of_subsingleton]
theorem norm_of_subsingleton' [Subsingleton E] (a : E) : ‖a‖ = 0 
Arguments: {E : Type u_6} [SeminormedAddGroup E] [Subsingleton E] (a : E)
Goal: ‖a‖ = 0
-- zero_lt_one_add_norm_sq'
@[to_additive zero_lt_one_add_norm_sq]
theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 
Arguments: {E : Type u_6} [SeminormedGroup E] (x : E)
Goal: 0 < 1 + ‖x‖ ^ 2
-- zero_lt_one_add_norm_sq
@[to_additive zero_lt_one_add_norm_sq]
theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (x : E)
Goal: 0 < 1 + ‖x‖ ^ 2
-- norm_sub_le
@[to_additive]
theorem norm_div_le (a b : E) : ‖a / b‖ ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a - b‖ ≤ ‖a‖ + ‖b‖
-- norm_div_le
@[to_additive]
theorem norm_div_le (a b : E) : ‖a / b‖ ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a / b‖ ≤ ‖a‖ + ‖b‖
-- norm_div_le_of_le
@[to_additive]
theorem norm_div_le_of_le {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂) : ‖a₁ / a₂‖ ≤ r₁ + r₂ 
Arguments: {E : Type u_6} [SeminormedGroup E] {a₁ a₂ : E} {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂)
Goal: ‖a₁ / a₂‖ ≤ r₁ + r₂
-- norm_sub_le_of_le
@[to_additive]
theorem norm_div_le_of_le {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂) : ‖a₁ / a₂‖ ≤ r₁ + r₂ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a₁ a₂ : E} {r₁ r₂ : ℝ} (H₁ : ‖a₁‖ ≤ r₁) (H₂ : ‖a₂‖ ≤ r₂)
Goal: ‖a₁ - a₂‖ ≤ r₁ + r₂
-- dist_le_norm_add_norm'
@[to_additive dist_le_norm_add_norm]
theorem dist_le_norm_add_norm' (a b : E) : dist a b ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: dist a b ≤ ‖a‖ + ‖b‖
-- dist_le_norm_add_norm
@[to_additive dist_le_norm_add_norm]
theorem dist_le_norm_add_norm' (a b : E) : dist a b ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: dist a b ≤ ‖a‖ + ‖b‖
-- abs_norm_sub_norm_le
@[to_additive abs_norm_sub_norm_le]
theorem abs_norm_sub_norm_le' (a b : E) : |‖a‖ - ‖b‖| ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: |‖a‖ - ‖b‖| ≤ ‖a - b‖
-- abs_norm_sub_norm_le'
@[to_additive abs_norm_sub_norm_le]
theorem abs_norm_sub_norm_le' (a b : E) : |‖a‖ - ‖b‖| ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: |‖a‖ - ‖b‖| ≤ ‖a / b‖
-- norm_sub_norm_le
@[to_additive norm_sub_norm_le]
theorem norm_sub_norm_le' (a b : E) : ‖a‖ - ‖b‖ ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a‖ - ‖b‖ ≤ ‖a - b‖
-- norm_sub_norm_le'
@[to_additive norm_sub_norm_le]
theorem norm_sub_norm_le' (a b : E) : ‖a‖ - ‖b‖ ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a‖ - ‖b‖ ≤ ‖a / b‖
-- dist_norm_norm_le'
@[to_additive dist_norm_norm_le]
theorem dist_norm_norm_le' (a b : E) : dist ‖a‖ ‖b‖ ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: dist ‖a‖ ‖b‖ ≤ ‖a / b‖
-- dist_norm_norm_le
@[to_additive dist_norm_norm_le]
theorem dist_norm_norm_le' (a b : E) : dist ‖a‖ ‖b‖ ≤ ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: dist ‖a‖ ‖b‖ ≤ ‖a - b‖
-- norm_le_norm_add_norm_div'
@[to_additive]
theorem norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (u v : E)
Goal: ‖u‖ ≤ ‖v‖ + ‖u / v‖
-- norm_le_norm_add_norm_sub'
@[to_additive]
theorem norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (u v : E)
Goal: ‖u‖ ≤ ‖v‖ + ‖u - v‖
-- norm_le_norm_add_norm_sub
@[to_additive]
theorem norm_le_norm_add_norm_div (u v : E) : ‖v‖ ≤ ‖u‖ + ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (u v : E)
Goal: ‖v‖ ≤ ‖u‖ + ‖u - v‖
-- norm_le_norm_add_norm_div
@[to_additive]
theorem norm_le_norm_add_norm_div (u v : E) : ‖v‖ ≤ ‖u‖ + ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (u v : E)
Goal: ‖v‖ ≤ ‖u‖ + ‖u / v‖
-- norm_le_mul_norm_add
@[to_additive]
theorem norm_le_mul_norm_add (u v : E) : ‖u‖ ≤ ‖u * v‖ + ‖v‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (u v : E)
Goal: ‖u‖ ≤ ‖u * v‖ + ‖v‖
-- norm_le_add_norm_add
@[to_additive]
theorem norm_le_mul_norm_add (u v : E) : ‖u‖ ≤ ‖u * v‖ + ‖v‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (u v : E)
Goal: ‖u‖ ≤ ‖u + v‖ + ‖v‖
-- ball_eq
@[to_additive ball_eq]
theorem ball_eq' (y : E) (ε : ℝ) : ball y ε = { x | ‖x / y‖ < ε } 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (y : E) (ε : ℝ)
Goal: Metric.ball y ε = {x | ‖x - y‖ < ε}
-- ball_eq'
@[to_additive ball_eq]
theorem ball_eq' (y : E) (ε : ℝ) : ball y ε = { x | ‖x / y‖ < ε } 
Arguments: {E : Type u_6} [SeminormedGroup E] (y : E) (ε : ℝ)
Goal: Metric.ball y ε = {x | ‖x / y‖ < ε}
-- ball_one_eq
@[to_additive]
theorem ball_one_eq (r : ℝ) : ball (1 : E) r = { x | ‖x‖ < r } 
Arguments: {E : Type u_6} [SeminormedGroup E] (r : ℝ)
Goal: Metric.ball 1 r = {x | ‖x‖ < r}
-- ball_zero_eq
@[to_additive]
theorem ball_one_eq (r : ℝ) : ball (1 : E) r = { x | ‖x‖ < r } 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (r : ℝ)
Goal: Metric.ball 0 r = {x | ‖x‖ < r}
-- mem_ball_iff_norm
@[to_additive mem_ball_iff_norm]
theorem mem_ball_iff_norm'' : b ∈ ball a r ↔ ‖b / a‖ < r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.ball a r ↔ ‖b - a‖ < r
-- mem_ball_iff_norm''
@[to_additive mem_ball_iff_norm]
theorem mem_ball_iff_norm'' : b ∈ ball a r ↔ ‖b / a‖ < r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.ball a r ↔ ‖b / a‖ < r
-- mem_ball_iff_norm'
@[to_additive mem_ball_iff_norm']
theorem mem_ball_iff_norm''' : b ∈ ball a r ↔ ‖a / b‖ < r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.ball a r ↔ ‖a - b‖ < r
-- mem_ball_iff_norm'''
@[to_additive mem_ball_iff_norm']
theorem mem_ball_iff_norm''' : b ∈ ball a r ↔ ‖a / b‖ < r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.ball a r ↔ ‖a / b‖ < r
-- mem_ball_one_iff
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.ball 1 r ↔ ‖a‖ < r
-- mem_ball_zero_iff
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.ball 0 r ↔ ‖a‖ < r
-- mem_closedBall_iff_norm''
@[to_additive mem_closedBall_iff_norm]
theorem mem_closedBall_iff_norm'' : b ∈ closedBall a r ↔ ‖b / a‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.closedBall a r ↔ ‖b / a‖ ≤ r
-- mem_closedBall_iff_norm
@[to_additive mem_closedBall_iff_norm]
theorem mem_closedBall_iff_norm'' : b ∈ closedBall a r ↔ ‖b / a‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.closedBall a r ↔ ‖b - a‖ ≤ r
-- mem_closedBall_zero_iff
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_closedBall_one_iff : a ∈ closedBall (1 : E) r ↔ ‖a‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.closedBall 0 r ↔ ‖a‖ ≤ r
-- mem_closedBall_one_iff
@[to_additive] -- Porting note (#10618): `simp` can prove it
theorem mem_closedBall_one_iff : a ∈ closedBall (1 : E) r ↔ ‖a‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.closedBall 1 r ↔ ‖a‖ ≤ r
-- mem_closedBall_iff_norm'
@[to_additive mem_closedBall_iff_norm']
theorem mem_closedBall_iff_norm''' : b ∈ closedBall a r ↔ ‖a / b‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.closedBall a r ↔ ‖a - b‖ ≤ r
-- mem_closedBall_iff_norm'''
@[to_additive mem_closedBall_iff_norm']
theorem mem_closedBall_iff_norm''' : b ∈ closedBall a r ↔ ‖a / b‖ ≤ r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.closedBall a r ↔ ‖a / b‖ ≤ r
-- norm_le_of_mem_closedBall'
@[to_additive norm_le_of_mem_closedBall]
theorem norm_le_of_mem_closedBall' (h : b ∈ closedBall a r) : ‖b‖ ≤ ‖a‖ + r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ} (h : b ∈ Metric.closedBall a r)
Goal: ‖b‖ ≤ ‖a‖ + r
-- norm_le_of_mem_closedBall
@[to_additive norm_le_of_mem_closedBall]
theorem norm_le_of_mem_closedBall' (h : b ∈ closedBall a r) : ‖b‖ ≤ ‖a‖ + r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ} (h : b ∈ Metric.closedBall a r)
Goal: ‖b‖ ≤ ‖a‖ + r
-- norm_le_norm_add_const_of_dist_le
@[to_additive norm_le_norm_add_const_of_dist_le]
theorem norm_le_norm_add_const_of_dist_le' : dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r
-- norm_le_norm_add_const_of_dist_le'
@[to_additive norm_le_norm_add_const_of_dist_le]
theorem norm_le_norm_add_const_of_dist_le' : dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: dist a b ≤ r → ‖a‖ ≤ ‖b‖ + r
-- norm_lt_of_mem_ball'
@[to_additive norm_lt_of_mem_ball]
theorem norm_lt_of_mem_ball' (h : b ∈ ball a r) : ‖b‖ < ‖a‖ + r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ} (h : b ∈ Metric.ball a r)
Goal: ‖b‖ < ‖a‖ + r
-- norm_lt_of_mem_ball
@[to_additive norm_lt_of_mem_ball]
theorem norm_lt_of_mem_ball' (h : b ∈ ball a r) : ‖b‖ < ‖a‖ + r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ} (h : b ∈ Metric.ball a r)
Goal: ‖b‖ < ‖a‖ + r
-- norm_div_sub_norm_div_le_norm_div
@[to_additive]
theorem norm_div_sub_norm_div_le_norm_div (u v w : E) : ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (u v w : E)
Goal: ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖
-- norm_sub_sub_norm_sub_le_norm_sub
@[to_additive]
theorem norm_div_sub_norm_div_le_norm_div (u v w : E) : ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (u v w : E)
Goal: ‖u - w‖ - ‖v - w‖ ≤ ‖u - v‖
-- mem_sphere_iff_norm
@[to_additive (attr := simp 1001) mem_sphere_iff_norm]
-- Porting note: increase priority so the left-hand side doesn't reduce
theorem mem_sphere_iff_norm' : b ∈ sphere a r ↔ ‖b / a‖ = r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.sphere a r ↔ ‖b - a‖ = r
-- mem_sphere_iff_norm'
@[to_additive (attr := simp 1001) mem_sphere_iff_norm]
-- Porting note: increase priority so the left-hand side doesn't reduce
theorem mem_sphere_iff_norm' : b ∈ sphere a r ↔ ‖b / a‖ = r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a b : E} {r : ℝ}
Goal: b ∈ Metric.sphere a r ↔ ‖b / a‖ = r
-- mem_sphere_one_iff_norm
@[to_additive] -- `simp` can prove this
theorem mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.sphere 1 r ↔ ‖a‖ = r
-- mem_sphere_zero_iff_norm
@[to_additive] -- `simp` can prove this
theorem mem_sphere_one_iff_norm : a ∈ sphere (1 : E) r ↔ ‖a‖ = r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E} {r : ℝ}
Goal: a ∈ Metric.sphere 0 r ↔ ‖a‖ = r
-- norm_eq_of_mem_sphere
@[to_additive (attr := simp) norm_eq_of_mem_sphere]
theorem norm_eq_of_mem_sphere' (x : sphere (1 : E) r) : ‖(x : E)‖ = r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {r : ℝ} (x : ↑(Metric.sphere 0 r))
Goal: ‖↑x‖ = r
-- norm_eq_of_mem_sphere'
@[to_additive (attr := simp) norm_eq_of_mem_sphere]
theorem norm_eq_of_mem_sphere' (x : sphere (1 : E) r) : ‖(x : E)‖ = r 
Arguments: {E : Type u_6} [SeminormedGroup E] {r : ℝ} (x : ↑(Metric.sphere 1 r))
Goal: ‖↑x‖ = r
-- ne_zero_of_mem_sphere
@[to_additive]
theorem ne_one_of_mem_sphere (hr : r ≠ 0) (x : sphere (1 : E) r) : (x : E) ≠ 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {r : ℝ} (hr : r ≠ 0) (x : ↑(Metric.sphere 0 r))
Goal: ↑x ≠ 0
-- ne_one_of_mem_sphere
@[to_additive]
theorem ne_one_of_mem_sphere (hr : r ≠ 0) (x : sphere (1 : E) r) : (x : E) ≠ 1 
Arguments: {E : Type u_6} [SeminormedGroup E] {r : ℝ} (hr : r ≠ 0) (x : ↑(Metric.sphere 1 r))
Goal: ↑x ≠ 1
-- ne_zero_of_mem_unit_sphere
@[to_additive ne_zero_of_mem_unit_sphere]
theorem ne_one_of_mem_unit_sphere (x : sphere (1 : E) 1) : (x : E) ≠ 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (x : ↑(Metric.sphere 0 1))
Goal: ↑x ≠ 0
-- ne_one_of_mem_unit_sphere
@[to_additive ne_zero_of_mem_unit_sphere]
theorem ne_one_of_mem_unit_sphere (x : sphere (1 : E) 1) : (x : E) ≠ 1 
Arguments: {E : Type u_6} [SeminormedGroup E] (x : ↑(Metric.sphere 1 1))
Goal: ↑x ≠ 1
-- coe_normAddGroupSeminorm
@[to_additive (attr := simp)]
theorem coe_normGroupSeminorm : ⇑(normGroupSeminorm E) = norm 
Arguments: (E : Type u_6) [SeminormedAddGroup E]
Goal: ⇑(normAddGroupSeminorm E) = norm
-- coe_normGroupSeminorm
@[to_additive (attr := simp)]
theorem coe_normGroupSeminorm : ⇑(normGroupSeminorm E) = norm 
Arguments: (E : Type u_6) [SeminormedGroup E]
Goal: ⇑(normGroupSeminorm E) = norm
-- NormedCommGroup.tendsto_nhds_one
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_one {f : α → E} {l : Filter α} :
    Tendsto f l (𝓝 1) ↔ ∀ ε > 0, ∀ᶠ x in l, ‖f x‖ < ε 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {f : α → E} {l : Filter α}
Goal: Filter.Tendsto f l (𝓝 1) ↔ ∀ ε > 0, ∀ᶠ (x : α) in l, ‖f x‖ < ε
-- NormedAddCommGroup.tendsto_nhds_zero
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_one {f : α → E} {l : Filter α} :
    Tendsto f l (𝓝 1) ↔ ∀ ε > 0, ∀ᶠ x in l, ‖f x‖ < ε 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {f : α → E} {l : Filter α}
Goal: Filter.Tendsto f l (𝓝 0) ↔ ∀ ε > 0, ∀ᶠ (x : α) in l, ‖f x‖ < ε
-- NormedAddCommGroup.tendsto_nhds_nhds
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_nhds {f : E → F} {x : E} {y : F} :
    Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ x', ‖x' / x‖ < δ → ‖f x' / y‖ < ε 
Arguments: {E : Type u_6} {F : Type u_7} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {x : E} {y : F}
Goal: Filter.Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' - x‖ < δ → ‖f x' - y‖ < ε
-- NormedCommGroup.tendsto_nhds_nhds
@[to_additive]
theorem NormedCommGroup.tendsto_nhds_nhds {f : E → F} {x : E} {y : F} :
    Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ x', ‖x' / x‖ < δ → ‖f x' / y‖ < ε 
Arguments: {E : Type u_6} {F : Type u_7} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {x : E} {y : F}
Goal: Filter.Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε
-- NormedAddCommGroup.nhds_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.nhds_basis_norm_lt (x : E) :
    (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y / x‖ < ε } 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (x : E)
Goal: (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y - x‖ < ε}
-- NormedCommGroup.nhds_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.nhds_basis_norm_lt (x : E) :
    (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y / x‖ < ε } 
Arguments: {E : Type u_6} [SeminormedGroup E] (x : E)
Goal: (𝓝 x).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y / x‖ < ε}
-- NormedAddCommGroup.nhds_zero_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.nhds_one_basis_norm_lt :
    (𝓝 (1 : E)).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y‖ < ε } 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: (𝓝 0).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y‖ < ε}
-- NormedCommGroup.nhds_one_basis_norm_lt
@[to_additive]
theorem NormedCommGroup.nhds_one_basis_norm_lt :
    (𝓝 (1 : E)).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y‖ < ε } 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: (𝓝 1).HasBasis (fun ε => 0 < ε) fun ε => {y | ‖y‖ < ε}
-- NormedCommGroup.uniformity_basis_dist
@[to_additive]
theorem NormedCommGroup.uniformity_basis_dist :
    (𝓤 E).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : E × E | ‖p.fst / p.snd‖ < ε } 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: (𝓤 E).HasBasis (fun ε => 0 < ε) fun ε => {p | ‖p.1 / p.2‖ < ε}
-- NormedAddCommGroup.uniformity_basis_dist
@[to_additive]
theorem NormedCommGroup.uniformity_basis_dist :
    (𝓤 E).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : E × E | ‖p.fst / p.snd‖ < ε } 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: (𝓤 E).HasBasis (fun ε => 0 < ε) fun ε => {p | ‖p.1 - p.2‖ < ε}
-- coe_nnnorm
@[to_additive (attr := simp, norm_cast) coe_nnnorm]
theorem coe_nnnorm' (a : E) : (‖a‖₊ : ℝ) = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: ↑‖a‖₊ = ‖a‖
-- coe_nnnorm'
@[to_additive (attr := simp, norm_cast) coe_nnnorm]
theorem coe_nnnorm' (a : E) : (‖a‖₊ : ℝ) = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: ↑‖a‖₊ = ‖a‖
-- coe_comp_nnnorm'
@[to_additive (attr := simp) coe_comp_nnnorm]
theorem coe_comp_nnnorm' : (toReal : ℝ≥0 → ℝ) ∘ (nnnorm : E → ℝ≥0) = norm 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: NNReal.toReal ∘ nnnorm = norm
-- coe_comp_nnnorm
@[to_additive (attr := simp) coe_comp_nnnorm]
theorem coe_comp_nnnorm' : (toReal : ℝ≥0 → ℝ) ∘ (nnnorm : E → ℝ≥0) = norm 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: NNReal.toReal ∘ nnnorm = norm
-- norm_toNNReal'
@[to_additive norm_toNNReal]
theorem norm_toNNReal' : ‖a‖.toNNReal = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E}
Goal: ‖a‖.toNNReal = ‖a‖₊
-- norm_toNNReal
@[to_additive norm_toNNReal]
theorem norm_toNNReal' : ‖a‖.toNNReal = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E}
Goal: ‖a‖.toNNReal = ‖a‖₊
-- nndist_eq_nnnorm_div
@[to_additive]
theorem nndist_eq_nnnorm_div (a b : E) : nndist a b = ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: nndist a b = ‖a / b‖₊
-- nndist_eq_nnnorm_sub
@[to_additive]
theorem nndist_eq_nnnorm_div (a b : E) : nndist a b = ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: nndist a b = ‖a - b‖₊
-- nnnorm_one'
@[to_additive (attr := simp) nnnorm_zero]
theorem nnnorm_one' : ‖(1 : E)‖₊ = 0 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: ‖1‖₊ = 0
-- nnnorm_zero
@[to_additive (attr := simp) nnnorm_zero]
theorem nnnorm_one' : ‖(1 : E)‖₊ = 0 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: ‖0‖₊ = 0
-- ne_zero_of_nnnorm_ne_zero
@[to_additive]
theorem ne_one_of_nnnorm_ne_zero {a : E} : ‖a‖₊ ≠ 0 → a ≠ 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E}
Goal: ‖a‖₊ ≠ 0 → a ≠ 0
-- ne_one_of_nnnorm_ne_zero
@[to_additive]
theorem ne_one_of_nnnorm_ne_zero {a : E} : ‖a‖₊ ≠ 0 → a ≠ 1 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E}
Goal: ‖a‖₊ ≠ 0 → a ≠ 1
-- nnnorm_add_le
@[to_additive nnnorm_add_le]
theorem nnnorm_mul_le' (a b : E) : ‖a * b‖₊ ≤ ‖a‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a + b‖₊ ≤ ‖a‖₊ + ‖b‖₊
-- nnnorm_mul_le'
@[to_additive nnnorm_add_le]
theorem nnnorm_mul_le' (a b : E) : ‖a * b‖₊ ≤ ‖a‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a * b‖₊ ≤ ‖a‖₊ + ‖b‖₊
-- nnnorm_neg
@[to_additive (attr := simp) nnnorm_neg]
theorem nnnorm_inv' (a : E) : ‖a⁻¹‖₊ = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: ‖-a‖₊ = ‖a‖₊
-- nnnorm_inv'
@[to_additive (attr := simp) nnnorm_neg]
theorem nnnorm_inv' (a : E) : ‖a⁻¹‖₊ = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: ‖a⁻¹‖₊ = ‖a‖₊
-- nndist_mulIndicator
@[to_additive]
theorem nndist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    nndist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖₊ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] (s t : Set α) (f : α → E) (x : α)
Goal: nndist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(symmDiff s t).mulIndicator f x‖₊
-- nndist_indicator
@[to_additive]
theorem nndist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    nndist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖₊ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] (s t : Set α) (f : α → E) (x : α)
Goal: nndist (s.indicator f x) (t.indicator f x) = ‖(symmDiff s t).indicator f x‖₊
-- nnnorm_sub_le
@[to_additive]
theorem nnnorm_div_le (a b : E) : ‖a / b‖₊ ≤ ‖a‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a - b‖₊ ≤ ‖a‖₊ + ‖b‖₊
-- nnnorm_div_le
@[to_additive]
theorem nnnorm_div_le (a b : E) : ‖a / b‖₊ ≤ ‖a‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a / b‖₊ ≤ ‖a‖₊ + ‖b‖₊
-- nndist_nnnorm_nnnorm_le
@[to_additive nndist_nnnorm_nnnorm_le]
theorem nndist_nnnorm_nnnorm_le' (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: nndist ‖a‖₊ ‖b‖₊ ≤ ‖a - b‖₊
-- nndist_nnnorm_nnnorm_le'
@[to_additive nndist_nnnorm_nnnorm_le]
theorem nndist_nnnorm_nnnorm_le' (a b : E) : nndist ‖a‖₊ ‖b‖₊ ≤ ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: nndist ‖a‖₊ ‖b‖₊ ≤ ‖a / b‖₊
-- nnnorm_le_nnnorm_add_nnnorm_sub
@[to_additive]
theorem nnnorm_le_nnnorm_add_nnnorm_div (a b : E) : ‖b‖₊ ≤ ‖a‖₊ + ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖b‖₊ ≤ ‖a‖₊ + ‖a - b‖₊
-- nnnorm_le_nnnorm_add_nnnorm_div
@[to_additive]
theorem nnnorm_le_nnnorm_add_nnnorm_div (a b : E) : ‖b‖₊ ≤ ‖a‖₊ + ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖b‖₊ ≤ ‖a‖₊ + ‖a / b‖₊
-- nnnorm_le_nnnorm_add_nnnorm_div'
@[to_additive]
theorem nnnorm_le_nnnorm_add_nnnorm_div' (a b : E) : ‖a‖₊ ≤ ‖b‖₊ + ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a‖₊ ≤ ‖b‖₊ + ‖a / b‖₊
-- nnnorm_le_nnnorm_add_nnnorm_sub'
@[to_additive]
theorem nnnorm_le_nnnorm_add_nnnorm_div' (a b : E) : ‖a‖₊ ≤ ‖b‖₊ + ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a‖₊ ≤ ‖b‖₊ + ‖a - b‖₊
-- nnnorm_le_mul_nnnorm_add
@[to_additive]
theorem nnnorm_le_mul_nnnorm_add (a b : E) : ‖a‖₊ ≤ ‖a * b‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a‖₊ ≤ ‖a * b‖₊ + ‖b‖₊
-- nnnorm_le_add_nnnorm_add
@[to_additive]
theorem nnnorm_le_mul_nnnorm_add (a b : E) : ‖a‖₊ ≤ ‖a * b‖₊ + ‖b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a‖₊ ≤ ‖a + b‖₊ + ‖b‖₊
-- ofReal_norm_eq_coe_nnnorm'
@[to_additive ofReal_norm_eq_coe_nnnorm]
theorem ofReal_norm_eq_coe_nnnorm' (a : E) : ENNReal.ofReal ‖a‖ = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: ENNReal.ofReal ‖a‖ = ↑‖a‖₊
-- ofReal_norm_eq_coe_nnnorm
@[to_additive ofReal_norm_eq_coe_nnnorm]
theorem ofReal_norm_eq_coe_nnnorm' (a : E) : ENNReal.ofReal ‖a‖ = ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: ENNReal.ofReal ‖a‖ = ↑‖a‖₊