ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- toReal_coe_nnnorm
@[to_additive toReal_coe_nnnorm "The non negative norm seen as an `ENNReal` and
then as a `Real` is equal to the norm."]
theorem toReal_coe_nnnorm' (a : E) : (‖a‖₊ : ℝ≥0∞).toReal = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: (↑‖a‖₊).toReal = ‖a‖
-- toReal_coe_nnnorm'
@[to_additive toReal_coe_nnnorm "The non negative norm seen as an `ENNReal` and
then as a `Real` is equal to the norm."]
theorem toReal_coe_nnnorm' (a : E) : (‖a‖₊ : ℝ≥0∞).toReal = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: (↑‖a‖₊).toReal = ‖a‖
-- edist_eq_coe_nnnorm_sub
@[to_additive]
theorem edist_eq_coe_nnnorm_div (a b : E) : edist a b = ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: edist a b = ↑‖a - b‖₊
-- edist_eq_coe_nnnorm_div
@[to_additive]
theorem edist_eq_coe_nnnorm_div (a b : E) : edist a b = ‖a / b‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: edist a b = ↑‖a / b‖₊
-- edist_eq_coe_nnnorm
@[to_additive edist_eq_coe_nnnorm]
theorem edist_eq_coe_nnnorm' (x : E) : edist x 1 = (‖x‖₊ : ℝ≥0∞) 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (x : E)
Goal: edist x 0 = ↑‖x‖₊
-- edist_eq_coe_nnnorm'
@[to_additive edist_eq_coe_nnnorm]
theorem edist_eq_coe_nnnorm' (x : E) : edist x 1 = (‖x‖₊ : ℝ≥0∞) 
Arguments: {E : Type u_6} [SeminormedGroup E] (x : E)
Goal: edist x 1 = ↑‖x‖₊
-- edist_indicator
@[to_additive]
theorem edist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    edist (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: edist (s.indicator f x) (t.indicator f x) = ↑‖(symmDiff s t).indicator f x‖₊
-- edist_mulIndicator
@[to_additive]
theorem edist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    edist (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: edist (s.mulIndicator f x) (t.mulIndicator f x) = ↑‖(symmDiff s t).mulIndicator f x‖₊
-- mem_emetric_ball_one_iff
@[to_additive]
theorem mem_emetric_ball_one_iff {r : ℝ≥0∞} : a ∈ EMetric.ball (1 : E) r ↔ ↑‖a‖₊ < r 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E} {r : ℝ≥0∞}
Goal: a ∈ EMetric.ball 1 r ↔ ↑‖a‖₊ < r
-- mem_emetric_ball_zero_iff
@[to_additive]
theorem mem_emetric_ball_one_iff {r : ℝ≥0∞} : a ∈ EMetric.ball (1 : E) r ↔ ↑‖a‖₊ < r 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E} {r : ℝ≥0∞}
Goal: a ∈ EMetric.ball 0 r ↔ ↑‖a‖₊ < r
-- tendsto_iff_norm_div_tendsto_zero
@[to_additive]
theorem tendsto_iff_norm_div_tendsto_zero {f : α → E} {a : Filter α} {b : E} :
    Tendsto f a (𝓝 b) ↔ Tendsto (fun e => ‖f e / b‖) a (𝓝 0) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {f : α → E} {a : Filter α} {b : E}
Goal: Filter.Tendsto f a (𝓝 b) ↔ Filter.Tendsto (fun e => ‖f e / b‖) a (𝓝 0)
-- tendsto_iff_norm_sub_tendsto_zero
@[to_additive]
theorem tendsto_iff_norm_div_tendsto_zero {f : α → E} {a : Filter α} {b : E} :
    Tendsto f a (𝓝 b) ↔ Tendsto (fun e => ‖f e / b‖) a (𝓝 0) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {f : α → E} {a : Filter α} {b : E}
Goal: Filter.Tendsto f a (𝓝 b) ↔ Filter.Tendsto (fun e => ‖f e - b‖) a (𝓝 0)
-- tendsto_zero_iff_norm_tendsto_zero
@[to_additive]
theorem tendsto_one_iff_norm_tendsto_zero {f : α → E} {a : Filter α} :
    Tendsto f a (𝓝 1) ↔ Tendsto (‖f ·‖) a (𝓝 0) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {f : α → E} {a : Filter α}
Goal: Filter.Tendsto f a (𝓝 0) ↔ Filter.Tendsto (fun x => ‖f x‖) a (𝓝 0)
-- tendsto_one_iff_norm_tendsto_zero
@[to_additive]
theorem tendsto_one_iff_norm_tendsto_zero {f : α → E} {a : Filter α} :
    Tendsto f a (𝓝 1) ↔ Tendsto (‖f ·‖) a (𝓝 0) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {f : α → E} {a : Filter α}
Goal: Filter.Tendsto f a (𝓝 1) ↔ Filter.Tendsto (fun x => ‖f x‖) a (𝓝 0)
-- comap_norm_nhds_zero
@[to_additive]
theorem comap_norm_nhds_one : comap norm (𝓝 0) = 𝓝 (1 : E) 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: Filter.comap norm (𝓝 0) = 𝓝 0
-- comap_norm_nhds_one
@[to_additive]
theorem comap_norm_nhds_one : comap norm (𝓝 0) = 𝓝 (1 : E) 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: Filter.comap norm (𝓝 0) = 𝓝 1
-- squeeze_zero_norm'
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is eventually bounded by a
real function `a` which tends to `0`, then `f` tends to `0`. In this pair of lemmas
(`squeeze_zero_norm'` and `squeeze_zero_norm`), following a convention of similar lemmas in
`Topology.MetricSpace.Pseudo.Defs` and `Topology.Algebra.Order`, the `'` version is phrased using
\"eventually\" and the non-`'` version is phrased absolutely."]
theorem squeeze_one_norm' {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ᶠ n in t₀, ‖f n‖ ≤ a n)
    (h' : Tendsto a t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 1) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ᶠ (n : α) in t₀, ‖f n‖ ≤ a n) (h' : Filter.Tendsto a t₀ (𝓝 0))
Goal: Filter.Tendsto f t₀ (𝓝 0)
-- squeeze_one_norm'
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is eventually bounded by a
real function `a` which tends to `0`, then `f` tends to `0`. In this pair of lemmas
(`squeeze_zero_norm'` and `squeeze_zero_norm`), following a convention of similar lemmas in
`Topology.MetricSpace.Pseudo.Defs` and `Topology.Algebra.Order`, the `'` version is phrased using
\"eventually\" and the non-`'` version is phrased absolutely."]
theorem squeeze_one_norm' {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ᶠ n in t₀, ‖f n‖ ≤ a n)
    (h' : Tendsto a t₀ (𝓝 0)) : Tendsto f t₀ (𝓝 1) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ᶠ (n : α) in t₀, ‖f n‖ ≤ a n) (h' : Filter.Tendsto a t₀ (𝓝 0))
Goal: Filter.Tendsto f t₀ (𝓝 1)
-- squeeze_one_norm
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is bounded by a real
function `a` which tends to `0`, then `f` tends to `0`."]
theorem squeeze_one_norm {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ n, ‖f n‖ ≤ a n) :
    Tendsto a t₀ (𝓝 0) → Tendsto f t₀ (𝓝 1) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ (n : α), ‖f n‖ ≤ a n)
Goal: Filter.Tendsto a t₀ (𝓝 0) → Filter.Tendsto f t₀ (𝓝 1)
-- squeeze_zero_norm
@[to_additive "Special case of the sandwich theorem: if the norm of `f` is bounded by a real
function `a` which tends to `0`, then `f` tends to `0`."]
theorem squeeze_one_norm {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ n, ‖f n‖ ≤ a n) :
    Tendsto a t₀ (𝓝 0) → Tendsto f t₀ (𝓝 1) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {f : α → E} {a : α → ℝ} {t₀ : Filter α} (h : ∀ (n : α), ‖f n‖ ≤ a n)
Goal: Filter.Tendsto a t₀ (𝓝 0) → Filter.Tendsto f t₀ (𝓝 0)
-- tendsto_norm_div_self
@[to_additive]
theorem tendsto_norm_div_self (x : E) : Tendsto (fun a => ‖a / x‖) (𝓝 x) (𝓝 0) 
Arguments: {E : Type u_6} [SeminormedGroup E] (x : E)
Goal: Filter.Tendsto (fun a => ‖a / x‖) (𝓝 x) (𝓝 0)
-- tendsto_norm_sub_self
@[to_additive]
theorem tendsto_norm_div_self (x : E) : Tendsto (fun a => ‖a / x‖) (𝓝 x) (𝓝 0) 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (x : E)
Goal: Filter.Tendsto (fun a => ‖a - x‖) (𝓝 x) (𝓝 0)
-- tendsto_norm'
@[to_additive tendsto_norm]
theorem tendsto_norm' {x : E} : Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖) 
Arguments: {E : Type u_6} [SeminormedGroup E] {x : E}
Goal: Filter.Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖)
-- tendsto_norm
@[to_additive tendsto_norm]
theorem tendsto_norm' {x : E} : Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖) 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {x : E}
Goal: Filter.Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖)
-- tendsto_norm_one
@[to_additive]
theorem tendsto_norm_one : Tendsto (fun a : E => ‖a‖) (𝓝 1) (𝓝 0) 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: Filter.Tendsto (fun a => ‖a‖) (𝓝 1) (𝓝 0)
-- tendsto_norm_zero
@[to_additive]
theorem tendsto_norm_one : Tendsto (fun a : E => ‖a‖) (𝓝 1) (𝓝 0) 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: Filter.Tendsto (fun a => ‖a‖) (𝓝 0) (𝓝 0)
-- continuous_norm'
@[to_additive (attr := continuity) continuous_norm]
theorem continuous_norm' : Continuous fun a : E => ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: Continuous fun a => ‖a‖
-- continuous_norm
@[to_additive (attr := continuity) continuous_norm]
theorem continuous_norm' : Continuous fun a : E => ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: Continuous fun a => ‖a‖
-- continuous_nnnorm
@[to_additive (attr := continuity) continuous_nnnorm]
theorem continuous_nnnorm' : Continuous fun a : E => ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: Continuous fun a => ‖a‖₊
-- continuous_nnnorm'
@[to_additive (attr := continuity) continuous_nnnorm]
theorem continuous_nnnorm' : Continuous fun a : E => ‖a‖₊ 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: Continuous fun a => ‖a‖₊
-- mem_closure_zero_iff_norm
@[to_additive]
theorem mem_closure_one_iff_norm {x : E} : x ∈ closure ({1} : Set E) ↔ ‖x‖ = 0 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {x : E}
Goal: x ∈ closure {0} ↔ ‖x‖ = 0
-- mem_closure_one_iff_norm
@[to_additive]
theorem mem_closure_one_iff_norm {x : E} : x ∈ closure ({1} : Set E) ↔ ‖x‖ = 0 
Arguments: {E : Type u_6} [SeminormedGroup E] {x : E}
Goal: x ∈ closure {1} ↔ ‖x‖ = 0
-- closure_one_eq
@[to_additive]
theorem closure_one_eq : closure ({1} : Set E) = { x | ‖x‖ = 0 } 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: closure {1} = {x | ‖x‖ = 0}
-- closure_zero_eq
@[to_additive]
theorem closure_one_eq : closure ({1} : Set E) = { x | ‖x‖ = 0 } 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: closure {0} = {x | ‖x‖ = 0}
-- Filter.Tendsto.norm
@[to_additive Filter.Tendsto.norm]
theorem Filter.Tendsto.norm' (h : Tendsto f l (𝓝 a)) : Tendsto (fun x => ‖f x‖) l (𝓝 ‖a‖) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {a : E} {l : Filter α} {f : α → E} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun x => ‖f x‖) l (𝓝 ‖a‖)
-- Filter.Tendsto.norm'
@[to_additive Filter.Tendsto.norm]
theorem Filter.Tendsto.norm' (h : Tendsto f l (𝓝 a)) : Tendsto (fun x => ‖f x‖) l (𝓝 ‖a‖) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {a : E} {l : Filter α} {f : α → E} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun x => ‖f x‖) l (𝓝 ‖a‖)
-- Filter.Tendsto.nnnorm'
@[to_additive Filter.Tendsto.nnnorm]
theorem Filter.Tendsto.nnnorm' (h : Tendsto f l (𝓝 a)) : Tendsto (fun x => ‖f x‖₊) l (𝓝 ‖a‖₊) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {a : E} {l : Filter α} {f : α → E} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun x => ‖f x‖₊) l (𝓝 ‖a‖₊)
-- Filter.Tendsto.nnnorm
@[to_additive Filter.Tendsto.nnnorm]
theorem Filter.Tendsto.nnnorm' (h : Tendsto f l (𝓝 a)) : Tendsto (fun x => ‖f x‖₊) l (𝓝 ‖a‖₊) 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {a : E} {l : Filter α} {f : α → E} (h : Filter.Tendsto f l (𝓝 a))
Goal: Filter.Tendsto (fun x => ‖f x‖₊) l (𝓝 ‖a‖₊)
-- Continuous.norm'
@[to_additive (attr := fun_prop) Continuous.norm]
theorem Continuous.norm' : Continuous f → Continuous fun x => ‖f x‖ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E}
Goal: Continuous f → Continuous fun x => ‖f x‖
-- Continuous.norm
@[to_additive (attr := fun_prop) Continuous.norm]
theorem Continuous.norm' : Continuous f → Continuous fun x => ‖f x‖ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E}
Goal: Continuous f → Continuous fun x => ‖f x‖
-- Continuous.nnnorm
@[to_additive (attr := fun_prop) Continuous.nnnorm]
theorem Continuous.nnnorm' : Continuous f → Continuous fun x => ‖f x‖₊ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E}
Goal: Continuous f → Continuous fun x => ‖f x‖₊
-- Continuous.nnnorm'
@[to_additive (attr := fun_prop) Continuous.nnnorm]
theorem Continuous.nnnorm' : Continuous f → Continuous fun x => ‖f x‖₊ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E}
Goal: Continuous f → Continuous fun x => ‖f x‖₊
-- ContinuousAt.norm
@[to_additive (attr := fun_prop) ContinuousAt.norm]
theorem ContinuousAt.norm' {a : α} (h : ContinuousAt f a) : ContinuousAt (fun x => ‖f x‖) a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {a : α} (h : ContinuousAt f a)
Goal: ContinuousAt (fun x => ‖f x‖) a
-- ContinuousAt.norm'
@[to_additive (attr := fun_prop) ContinuousAt.norm]
theorem ContinuousAt.norm' {a : α} (h : ContinuousAt f a) : ContinuousAt (fun x => ‖f x‖) a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {a : α} (h : ContinuousAt f a)
Goal: ContinuousAt (fun x => ‖f x‖) a
-- ContinuousAt.nnnorm
@[to_additive (attr := fun_prop) ContinuousAt.nnnorm]
theorem ContinuousAt.nnnorm' {a : α} (h : ContinuousAt f a) : ContinuousAt (fun x => ‖f x‖₊) a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {a : α} (h : ContinuousAt f a)
Goal: ContinuousAt (fun x => ‖f x‖₊) a
-- ContinuousAt.nnnorm'
@[to_additive (attr := fun_prop) ContinuousAt.nnnorm]
theorem ContinuousAt.nnnorm' {a : α} (h : ContinuousAt f a) : ContinuousAt (fun x => ‖f x‖₊) a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {a : α} (h : ContinuousAt f a)
Goal: ContinuousAt (fun x => ‖f x‖₊) a
-- ContinuousWithinAt.norm
@[to_additive ContinuousWithinAt.norm]
theorem ContinuousWithinAt.norm' {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => ‖f x‖) s a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => ‖f x‖) s a
-- ContinuousWithinAt.norm'
@[to_additive ContinuousWithinAt.norm]
theorem ContinuousWithinAt.norm' {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => ‖f x‖) s a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => ‖f x‖) s a
-- ContinuousWithinAt.nnnorm'
@[to_additive ContinuousWithinAt.nnnorm]
theorem ContinuousWithinAt.nnnorm' {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => ‖f x‖₊) s a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => ‖f x‖₊) s a
-- ContinuousWithinAt.nnnorm
@[to_additive ContinuousWithinAt.nnnorm]
theorem ContinuousWithinAt.nnnorm' {s : Set α} {a : α} (h : ContinuousWithinAt f s a) :
    ContinuousWithinAt (fun x => ‖f x‖₊) s a 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} {a : α} (h : ContinuousWithinAt f s a)
Goal: ContinuousWithinAt (fun x => ‖f x‖₊) s a
-- ContinuousOn.norm
@[to_additive (attr := fun_prop) ContinuousOn.norm]
theorem ContinuousOn.norm' {s : Set α} (h : ContinuousOn f s) : ContinuousOn (fun x => ‖f x‖) s 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => ‖f x‖) s
-- ContinuousOn.norm'
@[to_additive (attr := fun_prop) ContinuousOn.norm]
theorem ContinuousOn.norm' {s : Set α} (h : ContinuousOn f s) : ContinuousOn (fun x => ‖f x‖) s 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => ‖f x‖) s
-- ContinuousOn.nnnorm
@[to_additive (attr := fun_prop) ContinuousOn.nnnorm]
theorem ContinuousOn.nnnorm' {s : Set α} (h : ContinuousOn f s) :
    ContinuousOn (fun x => ‖f x‖₊) s 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => ‖f x‖₊) s
-- ContinuousOn.nnnorm'
@[to_additive (attr := fun_prop) ContinuousOn.nnnorm]
theorem ContinuousOn.nnnorm' {s : Set α} (h : ContinuousOn f s) :
    ContinuousOn (fun x => ‖f x‖₊) s 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => ‖f x‖₊) s
-- eventually_ne_of_tendsto_norm_atTop'
@[to_additive eventually_ne_of_tendsto_norm_atTop "If `‖y‖→∞`, then we can assume `y≠x` for any
fixed `x`"]
theorem eventually_ne_of_tendsto_norm_atTop' {l : Filter α} {f : α → E}
    (h : Tendsto (fun y => ‖f y‖) l atTop) (x : E) : ∀ᶠ y in l, f y ≠ x 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] {l : Filter α} {f : α → E} (h : Filter.Tendsto (fun y => ‖f y‖) l Filter.atTop) (x : E)
Goal: ∀ᶠ (y : α) in l, f y ≠ x
-- eventually_ne_of_tendsto_norm_atTop
@[to_additive eventually_ne_of_tendsto_norm_atTop "If `‖y‖→∞`, then we can assume `y≠x` for any
fixed `x`"]
theorem eventually_ne_of_tendsto_norm_atTop' {l : Filter α} {f : α → E}
    (h : Tendsto (fun y => ‖f y‖) l atTop) (x : E) : ∀ᶠ y in l, f y ≠ x 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] {l : Filter α} {f : α → E} (h : Filter.Tendsto (fun y => ‖f y‖) l Filter.atTop) (x : E)
Goal: ∀ᶠ (y : α) in l, f y ≠ x
-- SeminormedCommGroup.mem_closure_iff
@[to_additive]
theorem SeminormedCommGroup.mem_closure_iff :
    a ∈ closure s ↔ ∀ ε, 0 < ε → ∃ b ∈ s, ‖a / b‖ < ε 
Arguments: {E : Type u_6} [SeminormedGroup E] {s : Set E} {a : E}
Goal: a ∈ closure s ↔ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ s, ‖a / b‖ < ε
-- SeminormedAddCommGroup.mem_closure_iff
@[to_additive]
theorem SeminormedCommGroup.mem_closure_iff :
    a ∈ closure s ↔ ∀ ε, 0 < ε → ∃ b ∈ s, ‖a / b‖ < ε 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {s : Set E} {a : E}
Goal: a ∈ closure s ↔ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ s, ‖a - b‖ < ε
-- norm_le_zero_iff'
@[to_additive norm_le_zero_iff']
theorem norm_le_zero_iff''' [T0Space E] {a : E} : ‖a‖ ≤ 0 ↔ a = 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] [T0Space E] {a : E}
Goal: ‖a‖ ≤ 0 ↔ a = 0
-- norm_le_zero_iff'''
@[to_additive norm_le_zero_iff']
theorem norm_le_zero_iff''' [T0Space E] {a : E} : ‖a‖ ≤ 0 ↔ a = 1 
Arguments: {E : Type u_6} [SeminormedGroup E] [T0Space E] {a : E}
Goal: ‖a‖ ≤ 0 ↔ a = 1
-- norm_eq_zero'''
@[to_additive norm_eq_zero']
theorem norm_eq_zero''' [T0Space E] {a : E} : ‖a‖ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [SeminormedGroup E] [T0Space E] {a : E}
Goal: ‖a‖ = 0 ↔ a = 1
-- norm_eq_zero'
@[to_additive norm_eq_zero']
theorem norm_eq_zero''' [T0Space E] {a : E} : ‖a‖ = 0 ↔ a = 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] [T0Space E] {a : E}
Goal: ‖a‖ = 0 ↔ a = 0
-- norm_pos_iff'
@[to_additive norm_pos_iff']
theorem norm_pos_iff''' [T0Space E] {a : E} : 0 < ‖a‖ ↔ a ≠ 1 
Arguments: {E : Type u_6} [SeminormedAddGroup E] [T0Space E] {a : E}
Goal: 0 < ‖a‖ ↔ a ≠ 0
-- norm_pos_iff'''
@[to_additive norm_pos_iff']
theorem norm_pos_iff''' [T0Space E] {a : E} : 0 < ‖a‖ ↔ a ≠ 1 
Arguments: {E : Type u_6} [SeminormedGroup E] [T0Space E] {a : E}
Goal: 0 < ‖a‖ ↔ a ≠ 1
-- SeminormedAddGroup.tendstoUniformlyOn_zero
@[to_additive]
theorem SeminormedGroup.tendstoUniformlyOn_one {f : ι → κ → G} {s : Set κ} {l : Filter ι} :
    TendstoUniformlyOn f 1 l s ↔ ∀ ε > 0, ∀ᶠ i in l, ∀ x ∈ s, ‖f i x‖ < ε 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedAddGroup G] {f : ι → κ → G} {s : Set κ} {l : Filter ι}
Goal: TendstoUniformlyOn f 0 l s ↔ ∀ ε > 0, ∀ᶠ (i : ι) in l, ∀ x ∈ s, ‖f i x‖ < ε
-- SeminormedGroup.tendstoUniformlyOn_one
@[to_additive]
theorem SeminormedGroup.tendstoUniformlyOn_one {f : ι → κ → G} {s : Set κ} {l : Filter ι} :
    TendstoUniformlyOn f 1 l s ↔ ∀ ε > 0, ∀ᶠ i in l, ∀ x ∈ s, ‖f i x‖ < ε 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedGroup G] {f : ι → κ → G} {s : Set κ} {l : Filter ι}
Goal: TendstoUniformlyOn f 1 l s ↔ ∀ ε > 0, ∀ᶠ (i : ι) in l, ∀ x ∈ s, ‖f i x‖ < ε
-- SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one
@[to_additive]
theorem SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one {f : ι → κ → G}
    {l : Filter ι} {l' : Filter κ} :
    UniformCauchySeqOnFilter f l l' ↔
      TendstoUniformlyOnFilter (fun n : ι × ι => fun z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l' 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedGroup G] {f : ι → κ → G} {l : Filter ι} {l' : Filter κ}
Goal: UniformCauchySeqOnFilter f l l' ↔ TendstoUniformlyOnFilter (fun n z => f n.1 z / f n.2 z) 1 (l ×ˢ l) l'
-- SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero
@[to_additive]
theorem SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one {f : ι → κ → G}
    {l : Filter ι} {l' : Filter κ} :
    UniformCauchySeqOnFilter f l l' ↔
      TendstoUniformlyOnFilter (fun n : ι × ι => fun z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l' 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedAddGroup G] {f : ι → κ → G} {l : Filter ι} {l' : Filter κ}
Goal: UniformCauchySeqOnFilter f l l' ↔ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) l'
-- SeminormedAddGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_zero
@[to_additive]
theorem SeminormedGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_one {f : ι → κ → G} {s : Set κ}
    {l : Filter ι} :
    UniformCauchySeqOn f l s ↔
      TendstoUniformlyOn (fun n : ι × ι => fun z => f n.fst z / f n.snd z) 1 (l ×ˢ l) s 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedAddGroup G] {f : ι → κ → G} {s : Set κ} {l : Filter ι}
Goal: UniformCauchySeqOn f l s ↔ TendstoUniformlyOn (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) s
-- SeminormedGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_one
@[to_additive]
theorem SeminormedGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_one {f : ι → κ → G} {s : Set κ}
    {l : Filter ι} :
    UniformCauchySeqOn f l s ↔
      TendstoUniformlyOn (fun n : ι × ι => fun z => f n.fst z / f n.snd z) 1 (l ×ˢ l) s 
Arguments: {ι : Type u_4} {κ : Type u_5} {G : Type u_8} [SeminormedGroup G] {f : ι → κ → G} {s : Set κ} {l : Filter ι}
Goal: UniformCauchySeqOn f l s ↔ TendstoUniformlyOn (fun n z => f n.1 z / f n.2 z) 1 (l ×ˢ l) s
-- dist_neg
@[to_additive]
theorem dist_inv (x y : E) : dist x⁻¹ y = dist x y⁻¹ 
Arguments: {E : Type u_6} [SeminormedAddCommGroup E] (x y : E)
Goal: dist (-x) y = dist x (-y)
-- dist_inv
@[to_additive]
theorem dist_inv (x y : E) : dist x⁻¹ y = dist x y⁻¹ 
Arguments: {E : Type u_6} [SeminormedCommGroup E] (x y : E)
Goal: dist x⁻¹ y = dist x y⁻¹
-- norm_multiset_sum_le
theorem norm_multiset_sum_le {E} [SeminormedAddCommGroup E] (m : Multiset E) :
    ‖m.sum‖ ≤ (m.map fun x => ‖x‖).sum 
Arguments: {E : Type u_9} [SeminormedAddCommGroup E] (m : Multiset E)
Goal: ‖m.sum‖ ≤ (Multiset.map (fun x => ‖x‖) m).sum
-- norm_multiset_prod_le
@[to_additive existing]
theorem norm_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
-- norm_sum_le
theorem norm_sum_le {ι E} [SeminormedAddCommGroup E] (s : Finset ι) (f : ι → E) :
    ‖∑ i ∈ s, f i‖ ≤ ∑ i ∈ s, ‖f i‖ 
Arguments: {ι : Type u_9} {E : Type u_10} [SeminormedAddCommGroup E] (s : Finset ι) (f : ι → E)
Goal: ‖∑ i ∈ s, f i‖ ≤ ∑ i ∈ s, ‖f i‖
-- norm_prod_le
@[to_additive existing]
theorem norm_prod_le (s : Finset ι) (f : ι → E) : ‖∏ i ∈ s, f i‖ ≤ ∑ i ∈ s, ‖f i‖ 
Arguments: {ι : Type u_4} {E : Type u_6} [SeminormedCommGroup E] (s : Finset ι) (f : ι → E)
Goal: ‖∏ i ∈ s, f i‖ ≤ ∑ i ∈ s, ‖f i‖
-- norm_sum_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} [SeminormedAddCommGroup 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