ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Bornology.IsBounded.exists_pos_norm_lt'
@[to_additive Bornology.IsBounded.exists_pos_norm_lt]
lemma Bornology.IsBounded.exists_pos_norm_lt' (hs : IsBounded s) : ∃ R > 0, ∀ x ∈ s, ‖x‖ < R 
Arguments: {E : Type u_3} [SeminormedGroup E] {s : Set E} (hs : Bornology.IsBounded s)
Goal: ∃ R > 0, ∀ x ∈ s, ‖x‖ < R
-- NormedCommGroup.cauchySeq_iff
@[to_additive]
lemma NormedCommGroup.cauchySeq_iff [Nonempty α] [SemilatticeSup α] {u : α → E} :
    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → ‖u m / u n‖ < ε 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedGroup E] [Nonempty α] [SemilatticeSup α] {u : α → E}
Goal: CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ (m : α), N ≤ m → ∀ (n : α), N ≤ n → ‖u m / u n‖ < ε
-- NormedAddCommGroup.cauchySeq_iff
@[to_additive]
lemma NormedCommGroup.cauchySeq_iff [Nonempty α] [SemilatticeSup α] {u : α → E} :
    CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → ‖u m / u n‖ < ε 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddGroup E] [Nonempty α] [SemilatticeSup α] {u : α → E}
Goal: CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ (m : α), N ≤ m → ∀ (n : α), N ≤ n → ‖u m - u n‖ < ε
-- IsCompact.exists_bound_of_continuousOn
@[to_additive IsCompact.exists_bound_of_continuousOn]
lemma IsCompact.exists_bound_of_continuousOn' [TopologicalSpace α] {s : Set α} (hs : IsCompact s)
    {f : α → E} (hf : ContinuousOn f s) : ∃ C, ∀ x ∈ s, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddGroup E] [TopologicalSpace α] {s : Set α} (hs : IsCompact s) {f : α → E} (hf : ContinuousOn f s)
Goal: ∃ C, ∀ x ∈ s, ‖f x‖ ≤ C
-- IsCompact.exists_bound_of_continuousOn'
@[to_additive IsCompact.exists_bound_of_continuousOn]
lemma IsCompact.exists_bound_of_continuousOn' [TopologicalSpace α] {s : Set α} (hs : IsCompact s)
    {f : α → E} (hf : ContinuousOn f s) : ∃ C, ∀ x ∈ s, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedGroup E] [TopologicalSpace α] {s : Set α} (hs : IsCompact s) {f : α → E} (hf : ContinuousOn f s)
Goal: ∃ C, ∀ x ∈ s, ‖f x‖ ≤ C
-- HasCompactMulSupport.exists_bound_of_continuous
@[to_additive]
lemma HasCompactMulSupport.exists_bound_of_continuous [TopologicalSpace α]
    {f : α → E} (hf : HasCompactMulSupport f) (h'f : Continuous f) : ∃ C, ∀ x, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedGroup E] [TopologicalSpace α] {f : α → E} (hf : HasCompactMulSupport f) (h'f : Continuous f)
Goal: ∃ C, ∀ (x : α), ‖f x‖ ≤ C
-- HasCompactSupport.exists_bound_of_continuous
@[to_additive]
lemma HasCompactMulSupport.exists_bound_of_continuous [TopologicalSpace α]
    {f : α → E} (hf : HasCompactMulSupport f) (h'f : Continuous f) : ∃ C, ∀ x, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [SeminormedAddGroup E] [TopologicalSpace α] {f : α → E} (hf : HasCompactSupport f) (h'f : Continuous f)
Goal: ∃ C, ∀ (x : α), ‖f x‖ ≤ C
-- Filter.Tendsto.op_zero_isBoundedUnder_le'
@[to_additive "A helper lemma used to prove that the (scalar or usual) product of a function that
tends to zero and a bounded function tends to zero. This lemma is formulated for any binary
operation `op : E → F → G` with an estimate `‖op x y‖ ≤ A * ‖x‖ * ‖y‖` for some constant A instead
of multiplication so that it can be applied to `(*)`, `flip (*)`, `(•)`, and `flip (•)`."]
lemma Filter.Tendsto.op_one_isBoundedUnder_le' {f : α → E} {g : α → F} {l : Filter α}
    (hf : Tendsto f l (𝓝 1)) (hg : IsBoundedUnder (· ≤ ·) l (Norm.norm ∘ g)) (op : E → F → G)
    (h_op : ∃ A, ∀ x y, ‖op x y‖ ≤ A * ‖x‖ * ‖y‖) : Tendsto (fun x => op (f x) (g x)) l (𝓝 1) 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [SeminormedAddGroup E] [SeminormedAddGroup F] [SeminormedAddGroup G] {f : α → E} {g : α → F} {l : Filter α} (hf : Filter.Tendsto f l (𝓝 0)) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g)) (op : E → F → G) (h_op : ∃ A, ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖)
Goal: Filter.Tendsto (fun x => op (f x) (g x)) l (𝓝 0)
-- Filter.Tendsto.op_one_isBoundedUnder_le'
@[to_additive "A helper lemma used to prove that the (scalar or usual) product of a function that
tends to zero and a bounded function tends to zero. This lemma is formulated for any binary
operation `op : E → F → G` with an estimate `‖op x y‖ ≤ A * ‖x‖ * ‖y‖` for some constant A instead
of multiplication so that it can be applied to `(*)`, `flip (*)`, `(•)`, and `flip (•)`."]
lemma Filter.Tendsto.op_one_isBoundedUnder_le' {f : α → E} {g : α → F} {l : Filter α}
    (hf : Tendsto f l (𝓝 1)) (hg : IsBoundedUnder (· ≤ ·) l (Norm.norm ∘ g)) (op : E → F → G)
    (h_op : ∃ A, ∀ x y, ‖op x y‖ ≤ A * ‖x‖ * ‖y‖) : Tendsto (fun x => op (f x) (g x)) l (𝓝 1) 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [SeminormedGroup E] [SeminormedGroup F] [SeminormedGroup G] {f : α → E} {g : α → F} {l : Filter α} (hf : Filter.Tendsto f l (𝓝 1)) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g)) (op : E → F → G) (h_op : ∃ A, ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖)
Goal: Filter.Tendsto (fun x => op (f x) (g x)) l (𝓝 1)
-- Filter.Tendsto.op_one_isBoundedUnder_le
@[to_additive "A helper lemma used to prove that the (scalar or usual) product of a function that
tends to zero and a bounded function tends to zero. This lemma is formulated for any binary
operation `op : E → F → G` with an estimate `‖op x y‖ ≤ ‖x‖ * ‖y‖` instead of multiplication so
that it can be applied to `(*)`, `flip (*)`, `(•)`, and `flip (•)`."]
theorem Filter.Tendsto.op_one_isBoundedUnder_le {f : α → E} {g : α → F} {l : Filter α}
    (hf : Tendsto f l (𝓝 1)) (hg : IsBoundedUnder (· ≤ ·) l (Norm.norm ∘ g)) (op : E → F → G)
    (h_op : ∀ x y, ‖op x y‖ ≤ ‖x‖ * ‖y‖) : Tendsto (fun x => op (f x) (g x)) l (𝓝 1) 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [SeminormedGroup E] [SeminormedGroup F] [SeminormedGroup G] {f : α → E} {g : α → F} {l : Filter α} (hf : Filter.Tendsto f l (𝓝 1)) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g)) (op : E → F → G) (h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ ‖x‖ * ‖y‖)
Goal: Filter.Tendsto (fun x => op (f x) (g x)) l (𝓝 1)
-- Filter.Tendsto.op_zero_isBoundedUnder_le
@[to_additive "A helper lemma used to prove that the (scalar or usual) product of a function that
tends to zero and a bounded function tends to zero. This lemma is formulated for any binary
operation `op : E → F → G` with an estimate `‖op x y‖ ≤ ‖x‖ * ‖y‖` instead of multiplication so
that it can be applied to `(*)`, `flip (*)`, `(•)`, and `flip (•)`."]
theorem Filter.Tendsto.op_one_isBoundedUnder_le {f : α → E} {g : α → F} {l : Filter α}
    (hf : Tendsto f l (𝓝 1)) (hg : IsBoundedUnder (· ≤ ·) l (Norm.norm ∘ g)) (op : E → F → G)
    (h_op : ∀ x y, ‖op x y‖ ≤ ‖x‖ * ‖y‖) : Tendsto (fun x => op (f x) (g x)) l (𝓝 1) 
Arguments: {α : Type u_1} {E : Type u_3} {F : Type u_4} {G : Type u_5} [SeminormedAddGroup E] [SeminormedAddGroup F] [SeminormedAddGroup G] {f : α → E} {g : α → F} {l : Filter α} (hf : Filter.Tendsto f l (𝓝 0)) (hg : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l (norm ∘ g)) (op : E → F → G) (h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ ‖x‖ * ‖y‖)
Goal: Filter.Tendsto (fun x => op (f x) (g x)) l (𝓝 0)
-- Continuous.bounded_above_of_compact_support
lemma Continuous.bounded_above_of_compact_support (hf : Continuous f) (h : HasCompactSupport f) :
    ∃ C, ∀ x, ‖f x‖ ≤ C 
Arguments: {α : Type u_1} {E : Type u_3} [NormedAddGroup E] [TopologicalSpace α] {f : α → E} (hf : Continuous f) (h : HasCompactSupport f)
Goal: ∃ C, ∀ (x : α), ‖f x‖ ≤ C
-- HasCompactMulSupport.exists_pos_le_norm
@[to_additive]
lemma HasCompactMulSupport.exists_pos_le_norm [One E] (hf : HasCompactMulSupport f) :
    ∃ R : ℝ, 0 < R ∧ ∀ x : α, R ≤ ‖x‖ → f x = 1 
Arguments: {α : Type u_1} {E : Type u_3} [NormedAddGroup α] {f : α → E} [One E] (hf : HasCompactMulSupport f)
Goal: ∃ R, 0 < R ∧ ∀ (x : α), R ≤ ‖x‖ → f x = 1
-- HasCompactSupport.exists_pos_le_norm
@[to_additive]
lemma HasCompactMulSupport.exists_pos_le_norm [One E] (hf : HasCompactMulSupport f) :
    ∃ R : ℝ, 0 < R ∧ ∀ x : α, R ≤ ‖x‖ → f x = 1 
Arguments: {α : Type u_1} {E : Type u_3} [NormedAddGroup α] {f : α → E} [Zero E] (hf : HasCompactSupport f)
Goal: ∃ R, 0 < R ∧ ∀ (x : α), R ≤ ‖x‖ → f x = 0
-- WithLp.zero_fst
@[simp]
theorem zero_fst : (0 : WithLp p (α × β)).fst = 0 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β]
Goal: 0.1 = 0
-- WithLp.zero_snd
@[simp]
theorem zero_snd : (0 : WithLp p (α × β)).snd = 0 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β]
Goal: 0.2 = 0
-- WithLp.add_fst
@[simp]
theorem add_fst : (x + y).fst = x.fst + y.fst 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x y : WithLp p (α × β))
Goal: (x + y).1 = x.1 + y.1
-- WithLp.add_snd
@[simp]
theorem add_snd : (x + y).snd = x.snd + y.snd 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x y : WithLp p (α × β))
Goal: (x + y).2 = x.2 + y.2
-- WithLp.sub_fst
@[simp]
theorem sub_fst : (x - y).fst = x.fst - y.fst 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x y : WithLp p (α × β))
Goal: (x - y).1 = x.1 - y.1
-- WithLp.sub_snd
@[simp]
theorem sub_snd : (x - y).snd = x.snd - y.snd 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x y : WithLp p (α × β))
Goal: (x - y).2 = x.2 - y.2
-- WithLp.neg_fst
@[simp]
theorem neg_fst : (-x).fst = -x.fst 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x : WithLp p (α × β))
Goal: (-x).1 = -x.1
-- WithLp.neg_snd
@[simp]
theorem neg_snd : (-x).snd = -x.snd 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [AddCommGroup α] [AddCommGroup β] (x : WithLp p (α × β))
Goal: (-x).2 = -x.2
-- WithLp.smul_fst
@[simp]
theorem smul_fst : (c • x).fst = c • x.fst 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {α : Type u_2} {β : Type u_3} [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β] (x : WithLp p (α × β)) (c : 𝕜) [Module 𝕜 α] [Module 𝕜 β]
Goal: (c • x).1 = c • x.1
-- WithLp.smul_snd
@[simp]
theorem smul_snd : (c • x).snd = c • x.snd 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {α : Type u_2} {β : Type u_3} [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β] (x : WithLp p (α × β)) (c : 𝕜) [Module 𝕜 α] [Module 𝕜 β]
Goal: (c • x).2 = c • x.2
-- WithLp.equiv_fst
@[simp]
theorem equiv_fst (x : WithLp p (α × β)) : (WithLp.equiv p (α × β) x).fst = x.fst 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} (x : WithLp p (α × β))
Goal: ((WithLp.equiv p (α × β)) x).1 = x.1
-- WithLp.equiv_snd
@[simp]
theorem equiv_snd (x : WithLp p (α × β)) : (WithLp.equiv p (α × β) x).snd = x.snd 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} (x : WithLp p (α × β))
Goal: ((WithLp.equiv p (α × β)) x).2 = x.2
-- WithLp.equiv_symm_fst
@[simp]
theorem equiv_symm_fst (x : α × β) : ((WithLp.equiv p (α × β)).symm x).fst = x.fst 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} (x : α × β)
Goal: ((WithLp.equiv p (α × β)).symm x).1 = x.1
-- WithLp.equiv_symm_snd
@[simp]
theorem equiv_symm_snd (x : α × β) : ((WithLp.equiv p (α × β)).symm x).snd = x.snd 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} (x : α × β)
Goal: ((WithLp.equiv p (α × β)).symm x).2 = x.2
-- WithLp.prod_edist_eq_card
@[simp]
theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
    edist f g =
      (if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) 
Arguments: {α : Type u_2} {β : Type u_3} [EDist α] [EDist β] (f g : WithLp 0 (α × β))
Goal: edist f g = (if edist f.1 g.1 = 0 then 0 else 1) + if edist f.2 g.2 = 0 then 0 else 1
-- WithLp.prod_edist_eq_add
theorem prod_edist_eq_add (hp : 0 < p.toReal) (f g : WithLp p (α × β)) :
    edist f g = (edist f.fst g.fst ^ p.toReal + edist f.snd g.snd ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [EDist α] [EDist β] (hp : 0 < p.toReal) (f g : WithLp p (α × β))
Goal: edist f g = (edist f.1 g.1 ^ p.toReal + edist f.2 g.2 ^ p.toReal) ^ (1 / p.toReal)
-- WithLp.prod_edist_eq_sup
theorem prod_edist_eq_sup (f g : WithLp ∞ (α × β)) :
    edist f g = edist f.fst g.fst ⊔ edist f.snd g.snd 
Arguments: {α : Type u_2} {β : Type u_3} [EDist α] [EDist β] (f g : WithLp ⊤ (α × β))
Goal: edist f g = edist f.1 g.1 ⊔ edist f.2 g.2
-- WithLp.prod_edist_self
theorem prod_edist_self (f : WithLp p (α × β)) : edist f f = 0 
Arguments: (p : ℝ≥0∞) {α : Type u_2} {β : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : WithLp p (α × β))
Goal: edist f f = 0
-- WithLp.prod_edist_comm
theorem prod_edist_comm (f g : WithLp p (α × β)) : edist f g = edist g f 
Arguments: (p : ℝ≥0∞) {α : Type u_2} {β : Type u_3} [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f g : WithLp p (α × β))
Goal: edist f g = edist g f
-- WithLp.prod_dist_eq_card
theorem prod_dist_eq_card (f g : WithLp 0 (α × β)) : dist f g =
    (if dist f.fst g.fst = 0 then 0 else 1) + (if dist f.snd g.snd = 0 then 0 else 1) 
Arguments: {α : Type u_2} {β : Type u_3} [Dist α] [Dist β] (f g : WithLp 0 (α × β))
Goal: dist f g = (if dist f.1 g.1 = 0 then 0 else 1) + if dist f.2 g.2 = 0 then 0 else 1
-- WithLp.prod_dist_eq_add
theorem prod_dist_eq_add (hp : 0 < p.toReal) (f g : WithLp p (α × β)) :
    dist f g = (dist f.fst g.fst ^ p.toReal + dist f.snd g.snd ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [Dist α] [Dist β] (hp : 0 < p.toReal) (f g : WithLp p (α × β))
Goal: dist f g = (dist f.1 g.1 ^ p.toReal + dist f.2 g.2 ^ p.toReal) ^ (1 / p.toReal)
-- WithLp.prod_dist_eq_sup
theorem prod_dist_eq_sup (f g : WithLp ∞ (α × β)) :
    dist f g = dist f.fst g.fst ⊔ dist f.snd g.snd 
Arguments: {α : Type u_2} {β : Type u_3} [Dist α] [Dist β] (f g : WithLp ⊤ (α × β))
Goal: dist f g = dist f.1 g.1 ⊔ dist f.2 g.2
-- WithLp.prod_norm_eq_card
@[simp]
theorem prod_norm_eq_card (f : WithLp 0 (α × β)) :
    ‖f‖ = (if ‖f.fst‖ = 0 then 0 else 1) + (if ‖f.snd‖ = 0 then 0 else 1) 
Arguments: {α : Type u_2} {β : Type u_3} [Norm α] [Norm β] (f : WithLp 0 (α × β))
Goal: ‖f‖ = (if ‖f.1‖ = 0 then 0 else 1) + if ‖f.2‖ = 0 then 0 else 1
-- WithLp.prod_norm_eq_sup
theorem prod_norm_eq_sup (f : WithLp ∞ (α × β)) : ‖f‖ = ‖f.fst‖ ⊔ ‖f.snd‖ 
Arguments: {α : Type u_2} {β : Type u_3} [Norm α] [Norm β] (f : WithLp ⊤ (α × β))
Goal: ‖f‖ = ‖f.1‖ ⊔ ‖f.2‖
-- WithLp.prod_norm_eq_add
theorem prod_norm_eq_add (hp : 0 < p.toReal) (f : WithLp p (α × β)) :
    ‖f‖ = (‖f.fst‖ ^ p.toReal + ‖f.snd‖ ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [Norm α] [Norm β] (hp : 0 < p.toReal) (f : WithLp p (α × β))
Goal: ‖f‖ = (‖f.1‖ ^ p.toReal + ‖f.2‖ ^ p.toReal) ^ (1 / p.toReal)
-- WithLp.prod_sup_edist_ne_top_aux
theorem prod_sup_edist_ne_top_aux [PseudoMetricSpace α] [PseudoMetricSpace β]
    (f g : WithLp ∞ (α × β)) :
    edist f.fst g.fst ⊔ edist f.snd g.snd ≠ ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] (f g : WithLp ⊤ (α × β))
Goal: edist f.1 g.1 ⊔ edist f.2 g.2 ≠ ⊤
-- WithLp.prod_lipschitzWith_equiv_aux
theorem prod_lipschitzWith_equiv_aux [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    LipschitzWith 1 (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: LipschitzWith 1 ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_antilipschitzWith_equiv_aux
theorem prod_antilipschitzWith_equiv_aux [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    AntilipschitzWith ((2 : ℝ≥0) ^ (1 / p).toReal) (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: AntilipschitzWith (2 ^ (1 / p).toReal) ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_aux_uniformity_eq
theorem prod_aux_uniformity_eq [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    𝓤 (WithLp p (α × β)) = 𝓤[instUniformSpaceProd] 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: 𝓤 (WithLp p (α × β)) = 𝓤 (α × β)
-- WithLp.prod_aux_cobounded_eq
theorem prod_aux_cobounded_eq [PseudoMetricSpace α] [PseudoMetricSpace β] :
    cobounded (WithLp p (α × β)) = @cobounded _ Prod.instBornology 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoMetricSpace α] [PseudoMetricSpace β]
Goal: Bornology.cobounded (WithLp p (α × β)) = Bornology.cobounded (α × β)
-- WithLp.prod_continuous_equiv
@[continuity]
theorem prod_continuous_equiv : Continuous (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β]
Goal: Continuous ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_continuous_equiv_symm
@[continuity]
theorem prod_continuous_equiv_symm : Continuous (WithLp.equiv p (α × β)).symm 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β]
Goal: Continuous ⇑(WithLp.equiv p (α × β)).symm
-- WithLp.prod_uniformContinuous_equiv
theorem prod_uniformContinuous_equiv : UniformContinuous (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [UniformSpace α] [UniformSpace β]
Goal: UniformContinuous ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_uniformContinuous_equiv_symm
theorem prod_uniformContinuous_equiv_symm : UniformContinuous (WithLp.equiv p (α × β)).symm 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [UniformSpace α] [UniformSpace β]
Goal: UniformContinuous ⇑(WithLp.equiv p (α × β)).symm
-- WithLp.prodContinuousLinearEquiv_symm_apply
@[simps! (config := .asFn) apply symm_apply]
protected def prodContinuousLinearEquiv : WithLp p (α × β) ≃L[𝕜] α × β 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β] [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β] [Module 𝕜 α] [Module 𝕜 β]
Goal: ⇑(WithLp.prodContinuousLinearEquiv p 𝕜 α β).symm = ⇑(WithLp.equiv p (α × β)).symm
-- WithLp.prodContinuousLinearEquiv_apply
@[simps! (config := .asFn) apply symm_apply]
protected def prodContinuousLinearEquiv : WithLp p (α × β) ≃L[𝕜] α × β 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (α : Type u_2) (β : Type u_3) [TopologicalSpace α] [TopologicalSpace β] [Semiring 𝕜] [AddCommGroup α] [AddCommGroup β] [Module 𝕜 α] [Module 𝕜 β]
Goal: ⇑(WithLp.prodContinuousLinearEquiv p 𝕜 α β) = ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_nndist_eq_add
theorem prod_nndist_eq_add [PseudoMetricSpace α] [PseudoMetricSpace β]
    (hp : p ≠ ∞) (x y : WithLp p (α × β)) :
    nndist x y = (nndist x.fst y.fst ^ p.toReal + nndist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [PseudoMetricSpace α] [PseudoMetricSpace β]
Goal: p ≠ ⊤ →
  ∀ (x y : WithLp p (α × β)), nndist x y = (nndist x.1 y.1 ^ p.toReal + nndist x.2 y.2 ^ p.toReal) ^ (1 / p.toReal)
-- WithLp.prod_nndist_eq_sup
theorem prod_nndist_eq_sup [PseudoMetricSpace α] [PseudoMetricSpace β] (x y : WithLp ∞ (α × β)) :
    nndist x y = nndist x.fst y.fst ⊔ nndist x.snd y.snd 
Arguments: {α : Type u_2} {β : Type u_3} [PseudoMetricSpace α] [PseudoMetricSpace β] (x y : WithLp ⊤ (α × β))
Goal: nndist x y = nndist x.1 y.1 ⊔ nndist x.2 y.2
-- WithLp.prod_lipschitzWith_equiv
theorem prod_lipschitzWith_equiv [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    LipschitzWith 1 (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: LipschitzWith 1 ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_antilipschitzWith_equiv
theorem prod_antilipschitzWith_equiv [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    AntilipschitzWith ((2 : ℝ≥0) ^ (1 / p).toReal) (WithLp.equiv p (α × β)) 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: AntilipschitzWith (2 ^ (1 / p).toReal) ⇑(WithLp.equiv p (α × β))
-- WithLp.prod_infty_equiv_isometry
theorem prod_infty_equiv_isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
    Isometry (WithLp.equiv ∞ (α × β)) 
Arguments: (α : Type u_2) (β : Type u_3) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
Goal: Isometry ⇑(WithLp.equiv ⊤ (α × β))
-- WithLp.prod_norm_eq_of_nat
theorem prod_norm_eq_of_nat [Norm α] [Norm β] (n : ℕ) (h : p = n) (f : WithLp p (α × β)) :
    ‖f‖ = (‖f.fst‖ ^ n + ‖f.snd‖ ^ n) ^ (1 / (n : ℝ)) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [Norm α] [Norm β] (n : ℕ) (h : p = ↑n) (f : WithLp p (α × β))
Goal: ‖f‖ = (‖f.1‖ ^ n + ‖f.2‖ ^ n) ^ (1 / ↑n)
-- WithLp.prod_nnnorm_eq_add
theorem prod_nnnorm_eq_add (hp : p ≠ ∞) (f : WithLp p (α × β)) :
    ‖f‖₊ = (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal) ^ (1 / p.toReal) 
Arguments: {p : ℝ≥0∞} {α : Type u_2} {β : Type u_3} [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β]
Goal: p ≠ ⊤ → ∀ (f : WithLp p (α × β)), ‖f‖₊ = (‖f.1‖₊ ^ p.toReal + ‖f.2‖₊ ^ p.toReal) ^ (1 / p.toReal)
-- WithLp.prod_nnnorm_eq_sup
theorem prod_nnnorm_eq_sup (f : WithLp ∞ (α × β)) : ‖f‖₊ = ‖f.fst‖₊ ⊔  ‖f.snd‖₊ 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (f : WithLp ⊤ (α × β))
Goal: ‖f‖₊ = ‖f.1‖₊ ⊔ ‖f.2‖₊
-- WithLp.prod_nnnorm_equiv
@[simp] theorem prod_nnnorm_equiv (f : WithLp ∞ (α × β)) : ‖WithLp.equiv ⊤ _ f‖₊ = ‖f‖₊ 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (f : WithLp ⊤ (α × β))
Goal: ‖(WithLp.equiv ⊤ (α × β)) f‖₊ = ‖f‖₊
-- WithLp.prod_nnnorm_equiv_symm
@[simp] theorem prod_nnnorm_equiv_symm (f : α × β) : ‖(WithLp.equiv ⊤ _).symm f‖₊ = ‖f‖₊ 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (f : α × β)
Goal: ‖(WithLp.equiv ⊤ (α × β)).symm f‖₊ = ‖f‖₊
-- WithLp.prod_norm_equiv
@[simp] theorem prod_norm_equiv (f : WithLp ∞ (α × β)) : ‖WithLp.equiv ⊤ _ f‖ = ‖f‖ 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (f : WithLp ⊤ (α × β))
Goal: ‖(WithLp.equiv ⊤ (α × β)) f‖ = ‖f‖
-- WithLp.prod_norm_equiv_symm
@[simp] theorem prod_norm_equiv_symm (f : α × β) : ‖(WithLp.equiv ⊤ _).symm f‖ = ‖f‖ 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (f : α × β)
Goal: ‖(WithLp.equiv ⊤ (α × β)).symm f‖ = ‖f‖
-- WithLp.prod_norm_eq_of_L2
theorem prod_norm_eq_of_L2 (x : WithLp 2 (α × β)) :
    ‖x‖ = √(‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2) 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x : WithLp 2 (α × β))
Goal: ‖x‖ = √(‖x.1‖ ^ 2 + ‖x.2‖ ^ 2)
-- WithLp.prod_nnnorm_eq_of_L2
theorem prod_nnnorm_eq_of_L2 (x : WithLp 2 (α × β)) :
    ‖x‖₊ = NNReal.sqrt (‖x.fst‖₊ ^ 2 + ‖x.snd‖₊ ^ 2) 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x : WithLp 2 (α × β))
Goal: ‖x‖₊ = NNReal.sqrt (‖x.1‖₊ ^ 2 + ‖x.2‖₊ ^ 2)
-- WithLp.prod_norm_sq_eq_of_L2
theorem prod_norm_sq_eq_of_L2 (x : WithLp 2 (α × β)) : ‖x‖ ^ 2 = ‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x : WithLp 2 (α × β))
Goal: ‖x‖ ^ 2 = ‖x.1‖ ^ 2 + ‖x.2‖ ^ 2
-- WithLp.prod_dist_eq_of_L2
theorem prod_dist_eq_of_L2 (x y : WithLp 2 (α × β)) :
    dist x y = √(dist x.fst y.fst ^ 2 + dist x.snd y.snd ^ 2) 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x y : WithLp 2 (α × β))
Goal: dist x y = √(dist x.1 y.1 ^ 2 + dist x.2 y.2 ^ 2)
-- WithLp.prod_nndist_eq_of_L2
theorem prod_nndist_eq_of_L2 (x y : WithLp 2 (α × β)) :
    nndist x y = NNReal.sqrt (nndist x.fst y.fst ^ 2 + nndist x.snd y.snd ^ 2) 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x y : WithLp 2 (α × β))
Goal: nndist x y = NNReal.sqrt (nndist x.1 y.1 ^ 2 + nndist x.2 y.2 ^ 2)
-- WithLp.prod_edist_eq_of_L2
theorem prod_edist_eq_of_L2 (x y : WithLp 2 (α × β)) :
    edist x y = (edist x.fst y.fst ^ 2 + edist x.snd y.snd ^ 2) ^ (1 / 2 : ℝ) 
Arguments: {α : Type u_2} {β : Type u_3} [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x y : WithLp 2 (α × β))
Goal: edist x y = (edist x.1 y.1 ^ 2 + edist x.2 y.2 ^ 2) ^ (1 / 2)
-- WithLp.nnnorm_equiv_symm_fst
@[simp]
theorem nnnorm_equiv_symm_fst (x : α) :
    ‖(WithLp.equiv p (α × β)).symm (x, 0)‖₊ = ‖x‖₊ 
Arguments: (p : ℝ≥0∞) (α : Type u_2) (β : Type u_3) [hp : Fact (1 ≤ p)] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (x : α)
Goal: ‖(WithLp.equiv p (α × β)).symm (x, 0)‖₊ = ‖x‖₊