ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- dist_add_self_right
@[to_additive (attr := simp)]
theorem dist_mul_self_right (a b : E) : dist b (a * b) = ‖a‖ 
Arguments: {E : Type u_3} [SeminormedAddGroup E] (a b : E)
Goal: dist b (a + b) = ‖a‖
-- dist_mul_self_right
@[to_additive (attr := simp)]
theorem dist_mul_self_right (a b : E) : dist b (a * b) = ‖a‖ 
Arguments: {E : Type u_3} [SeminormedGroup E] (a b : E)
Goal: dist b (a * b) = ‖a‖
-- dist_add_self_left
@[to_additive (attr := simp)]
theorem dist_mul_self_left (a b : E) : dist (a * b) b = ‖a‖ 
Arguments: {E : Type u_3} [SeminormedAddGroup E] (a b : E)
Goal: dist (a + b) b = ‖a‖
-- dist_mul_self_left
@[to_additive (attr := simp)]
theorem dist_mul_self_left (a b : E) : dist (a * b) b = ‖a‖ 
Arguments: {E : Type u_3} [SeminormedGroup E] (a b : E)
Goal: dist (a * b) b = ‖a‖
-- dist_div_eq_dist_mul_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_left (a b c : E) : dist (a / b) c = dist a (c * b) 
Arguments: {E : Type u_3} [SeminormedGroup E] (a b c : E)
Goal: dist (a / b) c = dist a (c * b)
-- dist_sub_eq_dist_add_left
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_left (a b c : E) : dist (a / b) c = dist a (c * b) 
Arguments: {E : Type u_3} [SeminormedAddGroup E] (a b c : E)
Goal: dist (a - b) c = dist a (c + b)
-- dist_div_eq_dist_mul_right
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_right (a b c : E) : dist a (b / c) = dist (a * c) b 
Arguments: {E : Type u_3} [SeminormedGroup E] (a b c : E)
Goal: dist a (b / c) = dist (a * c) b
-- dist_sub_eq_dist_add_right
@[to_additive (attr := simp)]
theorem dist_div_eq_dist_mul_right (a b c : E) : dist a (b / c) = dist (a * c) b 
Arguments: {E : Type u_3} [SeminormedAddGroup E] (a b c : E)
Goal: dist a (b - c) = dist (a + c) b
-- AddMonoidHomClass.lipschitz_of_bound
@[to_additive "A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant
`C` such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `Mathlib/Analysis/NormedSpace/OperatorNorm.lean`."]
theorem MonoidHomClass.lipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : LipschitzWith (Real.toNNReal C) f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: LipschitzWith C.toNNReal ⇑f
-- MonoidHomClass.lipschitz_of_bound
@[to_additive "A homomorphism `f` of seminormed groups is Lipschitz, if there exists a constant
`C` such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`. The analogous condition for a linear map of
(semi)normed spaces is in `Mathlib/Analysis/NormedSpace/OperatorNorm.lean`."]
theorem MonoidHomClass.lipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : LipschitzWith (Real.toNNReal C) f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: LipschitzWith C.toNNReal ⇑f
-- lipschitzOnWith_iff_norm_sub_le
@[to_additive]
theorem lipschitzOnWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
    LipschitzOnWith C f s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ‖f x / f y‖ ≤ C * ‖x / y‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {s : Set E} {f : E → F} {C : ℝ≥0}
Goal: LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x - f y‖ ≤ ↑C * ‖x - y‖
-- lipschitzOnWith_iff_norm_div_le
@[to_additive]
theorem lipschitzOnWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
    LipschitzOnWith C f s ↔ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → ‖f x / f y‖ ≤ C * ‖x / y‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {s : Set E} {f : E → F} {C : ℝ≥0}
Goal: LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x / f y‖ ≤ ↑C * ‖x / y‖
-- LipschitzOnWith.norm_sub_le_of_le
@[to_additive]
theorem LipschitzOnWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzOnWith C f s)
    (ha : a ∈ s) (hb : b ∈ s) (hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {s : Set E} {a b : E} {r : ℝ} {f : E → F} {C : ℝ≥0} (h : LipschitzOnWith C f s) (ha : a ∈ s) (hb : b ∈ s) (hr : ‖a - b‖ ≤ r)
Goal: ‖f a - f b‖ ≤ ↑C * r
-- LipschitzOnWith.norm_div_le_of_le
@[to_additive]
theorem LipschitzOnWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzOnWith C f s)
    (ha : a ∈ s) (hb : b ∈ s) (hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {s : Set E} {a b : E} {r : ℝ} {f : E → F} {C : ℝ≥0} (h : LipschitzOnWith C f s) (ha : a ∈ s) (hb : b ∈ s) (hr : ‖a / b‖ ≤ r)
Goal: ‖f a / f b‖ ≤ ↑C * r
-- lipschitzWith_iff_norm_sub_le
@[to_additive]
theorem lipschitzWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
    LipschitzWith C f ↔ ∀ x y, ‖f x / f y‖ ≤ C * ‖x / y‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {C : ℝ≥0}
Goal: LipschitzWith C f ↔ ∀ (x y : E), ‖f x - f y‖ ≤ ↑C * ‖x - y‖
-- lipschitzWith_iff_norm_div_le
@[to_additive]
theorem lipschitzWith_iff_norm_div_le {f : E → F} {C : ℝ≥0} :
    LipschitzWith C f ↔ ∀ x y, ‖f x / f y‖ ≤ C * ‖x / y‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {C : ℝ≥0}
Goal: LipschitzWith C f ↔ ∀ (x y : E), ‖f x / f y‖ ≤ ↑C * ‖x / y‖
-- LipschitzWith.norm_div_le_of_le
@[to_additive]
theorem LipschitzWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzWith C f)
    (hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {a b : E} {r : ℝ} {f : E → F} {C : ℝ≥0} (h : LipschitzWith C f) (hr : ‖a / b‖ ≤ r)
Goal: ‖f a / f b‖ ≤ ↑C * r
-- LipschitzWith.norm_sub_le_of_le
@[to_additive]
theorem LipschitzWith.norm_div_le_of_le {f : E → F} {C : ℝ≥0} (h : LipschitzWith C f)
    (hr : ‖a / b‖ ≤ r) : ‖f a / f b‖ ≤ C * r 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {a b : E} {r : ℝ} {f : E → F} {C : ℝ≥0} (h : LipschitzWith C f) (hr : ‖a - b‖ ≤ r)
Goal: ‖f a - f b‖ ≤ ↑C * r
-- AddMonoidHomClass.continuous_of_bound
@[to_additive "A homomorphism `f` of seminormed groups is continuous, if there exists a constant `C`
such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`"]
theorem MonoidHomClass.continuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : Continuous f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: Continuous ⇑f
-- MonoidHomClass.continuous_of_bound
@[to_additive "A homomorphism `f` of seminormed groups is continuous, if there exists a constant `C`
such that for all `x`, one has `‖f x‖ ≤ C * ‖x‖`"]
theorem MonoidHomClass.continuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : Continuous f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: Continuous ⇑f
-- MonoidHomClass.uniformContinuous_of_bound
@[to_additive]
theorem MonoidHomClass.uniformContinuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : UniformContinuous f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: UniformContinuous ⇑f
-- AddMonoidHomClass.uniformContinuous_of_bound
@[to_additive]
theorem MonoidHomClass.uniformContinuous_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ)
    (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) : UniformContinuous f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ) (h : ∀ (x : E), ‖f x‖ ≤ C * ‖x‖)
Goal: UniformContinuous ⇑f
-- MonoidHomClass.isometry_iff_norm
@[to_additive]
theorem MonoidHomClass.isometry_iff_norm [MonoidHomClass 𝓕 E F] (f : 𝓕) :
    Isometry f ↔ ∀ x, ‖f x‖ = ‖x‖ 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕)
Goal: Isometry ⇑f ↔ ∀ (x : E), ‖f x‖ = ‖x‖
-- AddMonoidHomClass.isometry_iff_norm
@[to_additive]
theorem MonoidHomClass.isometry_iff_norm [MonoidHomClass 𝓕 E F] (f : 𝓕) :
    Isometry f ↔ ∀ x, ‖f x‖ = ‖x‖ 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕)
Goal: Isometry ⇑f ↔ ∀ (x : E), ‖f x‖ = ‖x‖
-- AddMonoidHomClass.lipschitz_of_bound_nnnorm
@[to_additive]
theorem MonoidHomClass.lipschitz_of_bound_nnnorm [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ≥0)
    (h : ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊) : LipschitzWith C f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ≥0) (h : ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊)
Goal: LipschitzWith C ⇑f
-- MonoidHomClass.lipschitz_of_bound_nnnorm
@[to_additive]
theorem MonoidHomClass.lipschitz_of_bound_nnnorm [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ≥0)
    (h : ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊) : LipschitzWith C f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) (C : ℝ≥0) (h : ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊)
Goal: LipschitzWith C ⇑f
-- AddMonoidHomClass.antilipschitz_of_bound
@[to_additive]
theorem MonoidHomClass.antilipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0}
    (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) : AntilipschitzWith K f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [AddMonoidHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖)
Goal: AntilipschitzWith K ⇑f
-- MonoidHomClass.antilipschitz_of_bound
@[to_additive]
theorem MonoidHomClass.antilipschitz_of_bound [MonoidHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0}
    (h : ∀ x, ‖x‖ ≤ K * ‖f x‖) : AntilipschitzWith K f 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [MonoidHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0} (h : ∀ (x : E), ‖x‖ ≤ ↑K * ‖f x‖)
Goal: AntilipschitzWith K ⇑f
-- LipschitzWith.norm_le_mul
@[to_additive LipschitzWith.norm_le_mul]
theorem LipschitzWith.norm_le_mul' {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1)
    (x) : ‖f x‖ ≤ K * ‖x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 0 = 0) (x : E)
Goal: ‖f x‖ ≤ ↑K * ‖x‖
-- LipschitzWith.norm_le_mul'
@[to_additive LipschitzWith.norm_le_mul]
theorem LipschitzWith.norm_le_mul' {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1)
    (x) : ‖f x‖ ≤ K * ‖x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1) (x : E)
Goal: ‖f x‖ ≤ ↑K * ‖x‖
-- LipschitzWith.nnorm_le_mul
@[to_additive LipschitzWith.nnorm_le_mul]
theorem LipschitzWith.nnorm_le_mul' {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1)
    (x) : ‖f x‖₊ ≤ K * ‖x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 0 = 0) (x : E)
Goal: ‖f x‖₊ ≤ K * ‖x‖₊
-- LipschitzWith.nnorm_le_mul'
@[to_additive LipschitzWith.nnorm_le_mul]
theorem LipschitzWith.nnorm_le_mul' {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1)
    (x) : ‖f x‖₊ ≤ K * ‖x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : ℝ≥0} (h : LipschitzWith K f) (hf : f 1 = 1) (x : E)
Goal: ‖f x‖₊ ≤ K * ‖x‖₊
-- AntilipschitzWith.le_mul_norm'
@[to_additive AntilipschitzWith.le_mul_norm]
theorem AntilipschitzWith.le_mul_norm' {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f)
    (hf : f 1 = 1) (x) : ‖x‖ ≤ K * ‖f x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f) (hf : f 1 = 1) (x : E)
Goal: ‖x‖ ≤ ↑K * ‖f x‖
-- AntilipschitzWith.le_mul_norm
@[to_additive AntilipschitzWith.le_mul_norm]
theorem AntilipschitzWith.le_mul_norm' {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f)
    (hf : f 1 = 1) (x) : ‖x‖ ≤ K * ‖f x‖ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f) (hf : f 0 = 0) (x : E)
Goal: ‖x‖ ≤ ↑K * ‖f x‖
-- AntilipschitzWith.le_mul_nnnorm
@[to_additive AntilipschitzWith.le_mul_nnnorm]
theorem AntilipschitzWith.le_mul_nnnorm' {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f)
    (hf : f 1 = 1) (x) : ‖x‖₊ ≤ K * ‖f x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f) (hf : f 0 = 0) (x : E)
Goal: ‖x‖₊ ≤ K * ‖f x‖₊
-- AntilipschitzWith.le_mul_nnnorm'
@[to_additive AntilipschitzWith.le_mul_nnnorm]
theorem AntilipschitzWith.le_mul_nnnorm' {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f)
    (hf : f 1 = 1) (x) : ‖x‖₊ ≤ K * ‖f x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} {K : ℝ≥0} (h : AntilipschitzWith K f) (hf : f 1 = 1) (x : E)
Goal: ‖x‖₊ ≤ K * ‖f x‖₊
-- ZeroHomClass.bound_of_antilipschitz
@[to_additive]
theorem OneHomClass.bound_of_antilipschitz [OneHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0}
    (h : AntilipschitzWith K f) (x) : ‖x‖ ≤ K * ‖f x‖ 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] [FunLike 𝓕 E F] [ZeroHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0} (h : AntilipschitzWith K ⇑f) (x : E)
Goal: ‖x‖ ≤ ↑K * ‖f x‖
-- OneHomClass.bound_of_antilipschitz
@[to_additive]
theorem OneHomClass.bound_of_antilipschitz [OneHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0}
    (h : AntilipschitzWith K f) (x) : ‖x‖ ≤ K * ‖f x‖ 
Arguments: {𝓕 : Type u_1} {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] [FunLike 𝓕 E F] [OneHomClass 𝓕 E F] (f : 𝓕) {K : ℝ≥0} (h : AntilipschitzWith K ⇑f) (x : E)
Goal: ‖x‖ ≤ ↑K * ‖f x‖
-- Isometry.nnnorm_map_of_map_zero
@[to_additive]
theorem Isometry.nnnorm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
    ‖f x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedAddGroup E] [SeminormedAddGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 0 = 0) (x : E)
Goal: ‖f x‖₊ = ‖x‖₊
-- Isometry.nnnorm_map_of_map_one
@[to_additive]
theorem Isometry.nnnorm_map_of_map_one {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E) :
    ‖f x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_3} {F : Type u_4} [SeminormedGroup E] [SeminormedGroup F] {f : E → F} (hi : Isometry f) (h₁ : f 1 = 1) (x : E)
Goal: ‖f x‖₊ = ‖x‖₊
-- lipschitzWith_one_norm
@[to_additive lipschitzWith_one_norm]
theorem lipschitzWith_one_norm' : LipschitzWith 1 (norm : E → ℝ) 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: LipschitzWith 1 norm
-- lipschitzWith_one_norm'
@[to_additive lipschitzWith_one_norm]
theorem lipschitzWith_one_norm' : LipschitzWith 1 (norm : E → ℝ) 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: LipschitzWith 1 norm
-- lipschitzWith_one_nnnorm
@[to_additive lipschitzWith_one_nnnorm]
theorem lipschitzWith_one_nnnorm' : LipschitzWith 1 (NNNorm.nnnorm : E → ℝ≥0) 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: LipschitzWith 1 nnnorm
-- lipschitzWith_one_nnnorm'
@[to_additive lipschitzWith_one_nnnorm]
theorem lipschitzWith_one_nnnorm' : LipschitzWith 1 (NNNorm.nnnorm : E → ℝ≥0) 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: LipschitzWith 1 nnnorm
-- uniformContinuous_norm
@[to_additive uniformContinuous_norm]
theorem uniformContinuous_norm' : UniformContinuous (norm : E → ℝ) 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: UniformContinuous norm
-- uniformContinuous_norm'
@[to_additive uniformContinuous_norm]
theorem uniformContinuous_norm' : UniformContinuous (norm : E → ℝ) 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: UniformContinuous norm
-- uniformContinuous_nnnorm'
@[to_additive uniformContinuous_nnnorm]
theorem uniformContinuous_nnnorm' : UniformContinuous fun a : E => ‖a‖₊ 
Arguments: {E : Type u_3} [SeminormedGroup E]
Goal: UniformContinuous fun a => ‖a‖₊
-- uniformContinuous_nnnorm
@[to_additive uniformContinuous_nnnorm]
theorem uniformContinuous_nnnorm' : UniformContinuous fun a : E => ‖a‖₊ 
Arguments: {E : Type u_3} [SeminormedAddGroup E]
Goal: UniformContinuous fun a => ‖a‖₊
-- dist_self_mul_right
@[to_additive (attr := simp)]
theorem dist_self_mul_right (a b : E) : dist a (a * b) = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a b : E)
Goal: dist a (a * b) = ‖b‖
-- dist_self_add_right
@[to_additive (attr := simp)]
theorem dist_self_mul_right (a b : E) : dist a (a * b) = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a b : E)
Goal: dist a (a + b) = ‖b‖
-- dist_self_mul_left
@[to_additive (attr := simp)]
theorem dist_self_mul_left (a b : E) : dist (a * b) a = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a b : E)
Goal: dist (a * b) a = ‖b‖
-- dist_self_add_left
@[to_additive (attr := simp)]
theorem dist_self_mul_left (a b : E) : dist (a * b) a = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a b : E)
Goal: dist (a + b) a = ‖b‖
-- dist_self_div_right
@[to_additive (attr := simp 1001)]
-- porting note (#10618): increase priority because `simp` can prove this
theorem dist_self_div_right (a b : E) : dist a (a / b) = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a b : E)
Goal: dist a (a / b) = ‖b‖
-- dist_self_sub_right
@[to_additive (attr := simp 1001)]
-- porting note (#10618): increase priority because `simp` can prove this
theorem dist_self_div_right (a b : E) : dist a (a / b) = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a b : E)
Goal: dist a (a - b) = ‖b‖
-- dist_self_div_left
@[to_additive (attr := simp 1001)]
-- porting note (#10618): increase priority because `simp` can prove this
theorem dist_self_div_left (a b : E) : dist (a / b) a = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a b : E)
Goal: dist (a / b) a = ‖b‖
-- dist_self_sub_left
@[to_additive (attr := simp 1001)]
-- porting note (#10618): increase priority because `simp` can prove this
theorem dist_self_div_left (a b : E) : dist (a / b) a = ‖b‖ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a b : E)
Goal: dist (a - b) a = ‖b‖
-- dist_add_add_le
@[to_additive]
theorem dist_mul_mul_le (a₁ a₂ b₁ b₂ : E) : dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: dist (a₁ + a₂) (b₁ + b₂) ≤ dist a₁ b₁ + dist a₂ b₂
-- dist_mul_mul_le
@[to_additive]
theorem dist_mul_mul_le (a₁ a₂ b₁ b₂ : E) : dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: dist (a₁ * a₂) (b₁ * b₂) ≤ dist a₁ b₁ + dist a₂ b₂
-- dist_add_add_le_of_le
@[to_additive]
theorem dist_mul_mul_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
    dist (a₁ * a₂) (b₁ * b₂) ≤ r₁ + r₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] {a₁ a₂ b₁ b₂ : E} {r₁ r₂ : ℝ} (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂)
Goal: dist (a₁ + a₂) (b₁ + b₂) ≤ r₁ + r₂
-- dist_mul_mul_le_of_le
@[to_additive]
theorem dist_mul_mul_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
    dist (a₁ * a₂) (b₁ * b₂) ≤ r₁ + r₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] {a₁ a₂ b₁ b₂ : E} {r₁ r₂ : ℝ} (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂)
Goal: dist (a₁ * a₂) (b₁ * b₂) ≤ r₁ + r₂
-- dist_sub_sub_le
@[to_additive]
theorem dist_div_div_le (a₁ a₂ b₁ b₂ : E) : dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: dist (a₁ - a₂) (b₁ - b₂) ≤ dist a₁ b₁ + dist a₂ b₂
-- dist_div_div_le
@[to_additive]
theorem dist_div_div_le (a₁ a₂ b₁ b₂ : E) : dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: dist (a₁ / a₂) (b₁ / b₂) ≤ dist a₁ b₁ + dist a₂ b₂
-- dist_div_div_le_of_le
@[to_additive]
theorem dist_div_div_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
    dist (a₁ / a₂) (b₁ / b₂) ≤ r₁ + r₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] {a₁ a₂ b₁ b₂ : E} {r₁ r₂ : ℝ} (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂)
Goal: dist (a₁ / a₂) (b₁ / b₂) ≤ r₁ + r₂
-- dist_sub_sub_le_of_le
@[to_additive]
theorem dist_div_div_le_of_le (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂) :
    dist (a₁ / a₂) (b₁ / b₂) ≤ r₁ + r₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] {a₁ a₂ b₁ b₂ : E} {r₁ r₂ : ℝ} (h₁ : dist a₁ b₁ ≤ r₁) (h₂ : dist a₂ b₂ ≤ r₂)
Goal: dist (a₁ - a₂) (b₁ - b₂) ≤ r₁ + r₂
-- abs_dist_sub_le_dist_add_add
@[to_additive]
theorem abs_dist_sub_le_dist_mul_mul (a₁ a₂ b₁ b₂ : E) :
    |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂) 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ + a₂) (b₁ + b₂)
-- abs_dist_sub_le_dist_mul_mul
@[to_additive]
theorem abs_dist_sub_le_dist_mul_mul (a₁ a₂ b₁ b₂ : E) :
    |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂) 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂)
-- nndist_add_add_le
@[to_additive]
theorem nndist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
    nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: nndist (a₁ + a₂) (b₁ + b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
-- nndist_mul_mul_le
@[to_additive]
theorem nndist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
    nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: nndist (a₁ * a₂) (b₁ * b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂
-- edist_mul_mul_le
@[to_additive]
theorem edist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
    edist (a₁ * a₂) (b₁ * b₂) ≤ edist a₁ b₁ + edist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: edist (a₁ * a₂) (b₁ * b₂) ≤ edist a₁ b₁ + edist a₂ b₂
-- edist_add_add_le
@[to_additive]
theorem edist_mul_mul_le (a₁ a₂ b₁ b₂ : E) :
    edist (a₁ * a₂) (b₁ * b₂) ≤ edist a₁ b₁ + edist a₂ b₂ 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] (a₁ a₂ b₁ b₂ : E)
Goal: edist (a₁ + a₂) (b₁ + b₂) ≤ edist a₁ b₁ + edist a₂ b₂
-- lipschitzWith_inv_iff
@[to_additive (attr := simp)]
lemma lipschitzWith_inv_iff : LipschitzWith K f⁻¹ ↔ LipschitzWith K f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E}
Goal: LipschitzWith K f⁻¹ ↔ LipschitzWith K f
-- lipschitzWith_neg_iff
@[to_additive (attr := simp)]
lemma lipschitzWith_inv_iff : LipschitzWith K f⁻¹ ↔ LipschitzWith K f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E}
Goal: LipschitzWith K (-f) ↔ LipschitzWith K f
-- antilipschitzWith_inv_iff
@[to_additive (attr := simp)]
lemma antilipschitzWith_inv_iff : AntilipschitzWith K f⁻¹ ↔ AntilipschitzWith K f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E}
Goal: AntilipschitzWith K f⁻¹ ↔ AntilipschitzWith K f
-- antilipschitzWith_neg_iff
@[to_additive (attr := simp)]
lemma antilipschitzWith_inv_iff : AntilipschitzWith K f⁻¹ ↔ AntilipschitzWith K f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E}
Goal: AntilipschitzWith K (-f) ↔ AntilipschitzWith K f
-- lipschitzOnWith_neg_iff
@[to_additive (attr := simp)]
lemma lipschitzOnWith_inv_iff : LipschitzOnWith K f⁻¹ s ↔ LipschitzOnWith K f s 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E} {s : Set α}
Goal: LipschitzOnWith K (-f) s ↔ LipschitzOnWith K f s