ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- lipschitzOnWith_inv_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} [SeminormedCommGroup E] [PseudoEMetricSpace α] {K : ℝ≥0} {f : α → E} {s : Set α}
Goal: LipschitzOnWith K f⁻¹ s ↔ LipschitzOnWith K f s
-- locallyLipschitz_neg_iff
@[to_additive (attr := simp)]
lemma locallyLipschitz_inv_iff : LocallyLipschitz f⁻¹ ↔ LocallyLipschitz f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {f : α → E}
Goal: LocallyLipschitz (-f) ↔ LocallyLipschitz f
-- locallyLipschitz_inv_iff
@[to_additive (attr := simp)]
lemma locallyLipschitz_inv_iff : LocallyLipschitz f⁻¹ ↔ LocallyLipschitz f 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {f : α → E}
Goal: LocallyLipschitz f⁻¹ ↔ LocallyLipschitz f
-- LipschitzWith.add
@[to_additive add]
theorem mul' (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (Kf + Kg) fun x => f x * g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) fun x => f x + g x
-- LipschitzWith.mul'
@[to_additive add]
theorem mul' (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (Kf + Kg) fun x => f x * g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) fun x => f x * g x
-- LipschitzWith.div
@[to_additive]
theorem div (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (Kf + Kg) fun x => f x / g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) fun x => f x / g x
-- LipschitzWith.sub
@[to_additive]
theorem div (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) :
    LipschitzWith (Kf + Kg) fun x => f x / g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) fun x => f x - g x
-- AntilipschitzWith.add_lipschitzWith
@[to_additive]
theorem mul_lipschitzWith (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹) :
    AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹)
Goal: AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x + g x
-- AntilipschitzWith.mul_lipschitzWith
@[to_additive]
theorem mul_lipschitzWith (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹) :
    AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg g) (hK : Kg < Kf⁻¹)
Goal: AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x
-- AntilipschitzWith.add_sub_lipschitzWith
@[to_additive]
theorem mul_div_lipschitzWith (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g / f))
    (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedAddCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g - f)) (hK : Kg < Kf⁻¹)
Goal: AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g
-- AntilipschitzWith.mul_div_lipschitzWith
@[to_additive]
theorem mul_div_lipschitzWith (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g / f))
    (hK : Kg < Kf⁻¹) : AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g 
Arguments: {α : Type u_5} {E : Type u_6} [SeminormedCommGroup E] [PseudoEMetricSpace α] {Kf Kg : ℝ≥0} {f g : α → E} (hf : AntilipschitzWith Kf f) (hg : LipschitzWith Kg (g / f)) (hK : Kg < Kf⁻¹)
Goal: AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ g
-- AntilipschitzWith.le_mul_norm_sub
@[to_additive le_mul_norm_sub]
theorem le_mul_norm_div {f : E → F} (hf : AntilipschitzWith K f) (x y : E) :
    ‖x / y‖ ≤ K * ‖f x / f y‖ 
Arguments: {F : Type u_4} [SeminormedAddCommGroup F] {E : Type u_6} [SeminormedAddCommGroup E] {K : ℝ≥0} {f : E → F} (hf : AntilipschitzWith K f) (x y : E)
Goal: ‖x - y‖ ≤ ↑K * ‖f x - f y‖
-- AntilipschitzWith.le_mul_norm_div
@[to_additive le_mul_norm_sub]
theorem le_mul_norm_div {f : E → F} (hf : AntilipschitzWith K f) (x y : E) :
    ‖x / y‖ ≤ K * ‖f x / f y‖ 
Arguments: {F : Type u_4} [SeminormedCommGroup F] {E : Type u_6} [SeminormedCommGroup E] {K : ℝ≥0} {f : E → F} (hf : AntilipschitzWith K f) (x y : E)
Goal: ‖x / y‖ ≤ ↑K * ‖f x / f y‖
-- cauchySeq_sum_of_eventually_eq
@[to_additive]
theorem cauchySeq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n)
    (hv : CauchySeq fun n => ∏ k ∈ range (n + 1), v k) :
    CauchySeq fun n => ∏ k ∈ range (n + 1), u k 
Arguments: {E : Type u_3} [SeminormedAddCommGroup E] {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n) (hv : CauchySeq fun n => ∑ k ∈ Finset.range (n + 1), v k)
Goal: CauchySeq fun n => ∑ k ∈ Finset.range (n + 1), u k
-- cauchySeq_prod_of_eventually_eq
@[to_additive]
theorem cauchySeq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n)
    (hv : CauchySeq fun n => ∏ k ∈ range (n + 1), v k) :
    CauchySeq fun n => ∏ k ∈ range (n + 1), u k 
Arguments: {E : Type u_3} [SeminormedCommGroup E] {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n) (hv : CauchySeq fun n => ∏ k ∈ Finset.range (n + 1), v k)
Goal: CauchySeq fun n => ∏ k ∈ Finset.range (n + 1), u k
-- UniformSpace.Completion.norm_coe
@[simp]
theorem norm_coe {E} [SeminormedAddCommGroup E] (x : E) : ‖(x : Completion E)‖ = ‖x‖ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] (x : E)
Goal: ‖↑E x‖ = ‖x‖
-- UniformSpace.Completion.nnnorm_coe
@[simp]
theorem nnnorm_coe {E} [SeminormedAddCommGroup E] (x : E) : ‖(x : Completion E)‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [SeminormedAddCommGroup E] (x : E)
Goal: ‖↑E x‖₊ = ‖x‖₊
-- eVariationOn.nonempty_monotone_mem
theorem nonempty_monotone_mem {s : Set α} (hs : s.Nonempty) :
    Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} (hs : s.Nonempty)
Goal: Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
-- eVariationOn.eq_of_edist_zero_on
theorem eq_of_edist_zero_on {f f' : α → E} {s : Set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) :
    eVariationOn f s = eVariationOn f' s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f f' : α → E} {s : Set α} (h : ∀ ⦃x : α⦄, x ∈ s → edist (f x) (f' x) = 0)
Goal: eVariationOn f s = eVariationOn f' s
-- eVariationOn.eq_of_eqOn
theorem eq_of_eqOn {f f' : α → E} {s : Set α} (h : EqOn f f' s) :
    eVariationOn f s = eVariationOn f' s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f f' : α → E} {s : Set α} (h : Set.EqOn f f' s)
Goal: eVariationOn f s = eVariationOn f' s
-- eVariationOn.sum_le
theorem sum_le (f : α → E) {s : Set α} (n : ℕ) {u : ℕ → α} (hu : Monotone u) (us : ∀ i, u i ∈ s) :
    (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} (n : ℕ) {u : ℕ → α} (hu : Monotone u) (us : ∀ (i : ℕ), u i ∈ s)
Goal: ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
-- eVariationOn.sum_le_of_monotoneOn_Icc
theorem sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α}
    (hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) :
    (∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α} (hu : MonotoneOn u (Set.Icc m n)) (us : ∀ i ∈ Set.Icc m n, u i ∈ s)
Goal: ∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
-- eVariationOn.sum_le_of_monotoneOn_Iic
theorem sum_le_of_monotoneOn_Iic (f : α → E) {s : Set α} {n : ℕ} {u : ℕ → α}
    (hu : MonotoneOn u (Iic n)) (us : ∀ i ≤ n, u i ∈ s) :
    (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {n : ℕ} {u : ℕ → α} (hu : MonotoneOn u (Set.Iic n)) (us : ∀ i ≤ n, u i ∈ s)
Goal: ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s
-- eVariationOn.mono
theorem mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s t : Set α} (hst : t ⊆ s)
Goal: eVariationOn f t ≤ eVariationOn f s
-- BoundedVariationOn.mono
theorem _root_.BoundedVariationOn.mono {f : α → E} {s : Set α} (h : BoundedVariationOn f s)
    {t : Set α} (ht : t ⊆ s) : BoundedVariationOn f t 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (h : BoundedVariationOn f s) {t : Set α} (ht : t ⊆ s)
Goal: BoundedVariationOn f t
-- BoundedVariationOn.locallyBoundedVariationOn
theorem _root_.BoundedVariationOn.locallyBoundedVariationOn {f : α → E} {s : Set α}
    (h : BoundedVariationOn f s) : LocallyBoundedVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (h : BoundedVariationOn f s)
Goal: LocallyBoundedVariationOn f s
-- eVariationOn.edist_le
theorem edist_le (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
    edist (f x) (f y) ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: edist (f x) (f y) ≤ eVariationOn f s
-- eVariationOn.eq_zero_iff
theorem eq_zero_iff (f : α → E) {s : Set α} :
    eVariationOn f s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α}
Goal: eVariationOn f s = 0 ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0
-- eVariationOn.constant_on
theorem constant_on {f : α → E} {s : Set α} (hf : (f '' s).Subsingleton) :
    eVariationOn f s = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : (f '' s).Subsingleton)
Goal: eVariationOn f s = 0
-- eVariationOn.subsingleton
@[simp]
protected theorem subsingleton (f : α → E) {s : Set α} (hs : s.Subsingleton) :
    eVariationOn f s = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} (hs : s.Subsingleton)
Goal: eVariationOn f s = 0
-- eVariationOn.lowerSemicontinuous_aux
theorem lowerSemicontinuous_aux {ι : Type*} {F : ι → α → E} {p : Filter ι} {f : α → E} {s : Set α}
    (Ffs : ∀ x ∈ s, Tendsto (fun i => F i x) p (𝓝 (f x))) {v : ℝ≥0∞} (hv : v < eVariationOn f s) :
    ∀ᶠ n : ι in p, v < eVariationOn (F n) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {ι : Type u_3} {F : ι → α → E} {p : Filter ι} {f : α → E} {s : Set α} (Ffs : ∀ x ∈ s, Filter.Tendsto (fun i => F i x) p (𝓝 (f x))) {v : ℝ≥0∞} (hv : v < eVariationOn f s)
Goal: ∀ᶠ (n : ι) in p, v < eVariationOn (F n) s
-- eVariationOn.lowerSemicontinuous
protected theorem lowerSemicontinuous (s : Set α) :
    LowerSemicontinuous fun f : α →ᵤ[s.image singleton] E => eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (s : Set α)
Goal: LowerSemicontinuous fun f => eVariationOn f s
-- eVariationOn.lowerSemicontinuous_uniformOn
theorem lowerSemicontinuous_uniformOn (s : Set α) :
    LowerSemicontinuous fun f : α →ᵤ[{s}] E => eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (s : Set α)
Goal: LowerSemicontinuous fun f => eVariationOn f s
-- BoundedVariationOn.dist_le
theorem _root_.BoundedVariationOn.dist_le {E : Type*} [PseudoMetricSpace E] {f : α → E}
    {s : Set α} (h : BoundedVariationOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
    dist (f x) (f y) ≤ (eVariationOn f s).toReal 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_3} [PseudoMetricSpace E] {f : α → E} {s : Set α} (h : BoundedVariationOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: dist (f x) (f y) ≤ (eVariationOn f s).toReal
-- BoundedVariationOn.sub_le
theorem _root_.BoundedVariationOn.sub_le {f : α → ℝ} {s : Set α} (h : BoundedVariationOn f s)
    {x y : α} (hx : x ∈ s) (hy : y ∈ s) : f x - f y ≤ (eVariationOn f s).toReal 
Arguments: {α : Type u_1} [LinearOrder α] {f : α → ℝ} {s : Set α} (h : BoundedVariationOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: f x - f y ≤ (eVariationOn f s).toReal
-- eVariationOn.add_point
theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
    (us : ∀ i, u i ∈ s) (n : ℕ) :
    ∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
      (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
        ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j)) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u) (us : ∀ (i : ℕ), u i ∈ s) (n : ℕ)
Goal: ∃ v m,
  Monotone v ∧
    (∀ (i : ℕ), v i ∈ s) ∧
      x ∈ v '' Set.Iio m ∧
        ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ ∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
-- eVariationOn.add_le_union
theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
    eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y)
Goal: eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t)
-- eVariationOn.union
theorem union (f : α → E) {s t : Set α} {x : α} (hs : IsGreatest s x) (ht : IsLeast t x) :
    eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s t : Set α} {x : α} (hs : IsGreatest s x) (ht : IsLeast t x)
Goal: eVariationOn f (s ∪ t) = eVariationOn f s + eVariationOn f t
-- eVariationOn.Icc_add_Icc
theorem Icc_add_Icc (f : α → E) {s : Set α} {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) (hb : b ∈ s) :
    eVariationOn f (s ∩ Icc a b) + eVariationOn f (s ∩ Icc b c) = eVariationOn f (s ∩ Icc a c) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) {s : Set α} {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) (hb : b ∈ s)
Goal: eVariationOn f (s ∩ Set.Icc a b) + eVariationOn f (s ∩ Set.Icc b c) = eVariationOn f (s ∩ Set.Icc a c)
-- eVariationOn.comp_le_of_monotoneOn
theorem comp_le_of_monotoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
    (φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) (φst : Set.MapsTo φ t s)
Goal: eVariationOn (f ∘ φ) t ≤ eVariationOn f s
-- eVariationOn.comp_le_of_antitoneOn
theorem comp_le_of_antitoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
    (φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) (φst : Set.MapsTo φ t s)
Goal: eVariationOn (f ∘ φ) t ≤ eVariationOn f s
-- eVariationOn.comp_eq_of_monotoneOn
theorem comp_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) :
    eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
Goal: eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)
-- eVariationOn.comp_inter_Icc_eq_of_monotoneOn
theorem comp_inter_Icc_eq_of_monotoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t)
    {x y : β} (hx : x ∈ t) (hy : y ∈ t) :
    eVariationOn (f ∘ φ) (t ∩ Icc x y) = eVariationOn f (φ '' t ∩ Icc (φ x) (φ y)) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {t : Set β} (φ : β → α) (hφ : MonotoneOn φ t) {x y : β} (hx : x ∈ t) (hy : y ∈ t)
Goal: eVariationOn (f ∘ φ) (t ∩ Set.Icc x y) = eVariationOn f (φ '' t ∩ Set.Icc (φ x) (φ y))
-- eVariationOn.comp_eq_of_antitoneOn
theorem comp_eq_of_antitoneOn (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t) :
    eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t) 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {β : Type u_3} [LinearOrder β] (f : α → E) {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
Goal: eVariationOn (f ∘ φ) t = eVariationOn f (φ '' t)
-- eVariationOn.comp_ofDual
theorem comp_ofDual (f : α → E) (s : Set α) :
    eVariationOn (f ∘ ofDual) (ofDual ⁻¹' s) = eVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α)
Goal: eVariationOn (f ∘ ⇑OrderDual.ofDual) (⇑OrderDual.ofDual ⁻¹' s) = eVariationOn f s
-- MonotoneOn.eVariationOn_le
theorem MonotoneOn.eVariationOn_le {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) {a b : α}
    (as : a ∈ s) (bs : b ∈ s) : eVariationOn f (s ∩ Icc a b) ≤ ENNReal.ofReal (f b - f a) 
Arguments: {α : Type u_1} [LinearOrder α] {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) {a b : α} (as : a ∈ s) (bs : b ∈ s)
Goal: eVariationOn f (s ∩ Set.Icc a b) ≤ ENNReal.ofReal (f b - f a)
-- MonotoneOn.locallyBoundedVariationOn
theorem MonotoneOn.locallyBoundedVariationOn {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) :
    LocallyBoundedVariationOn f s 
Arguments: {α : Type u_1} [LinearOrder α] {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s)
Goal: LocallyBoundedVariationOn f s
-- variationOnFromTo.self
protected theorem self (a : α) : variationOnFromTo f s a a = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) (a : α)
Goal: variationOnFromTo f s a a = 0
-- variationOnFromTo.nonneg_of_le
protected theorem nonneg_of_le {a b : α} (h : a ≤ b) : 0 ≤ variationOnFromTo f s a b 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) {a b : α} (h : a ≤ b)
Goal: 0 ≤ variationOnFromTo f s a b
-- variationOnFromTo.eq_neg_swap
protected theorem eq_neg_swap (a b : α) :
    variationOnFromTo f s a b = -variationOnFromTo f s b a 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) (a b : α)
Goal: variationOnFromTo f s a b = -variationOnFromTo f s b a
-- variationOnFromTo.nonpos_of_ge
protected theorem nonpos_of_ge {a b : α} (h : b ≤ a) : variationOnFromTo f s a b ≤ 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) {a b : α} (h : b ≤ a)
Goal: variationOnFromTo f s a b ≤ 0
-- variationOnFromTo.eq_of_le
protected theorem eq_of_le {a b : α} (h : a ≤ b) :
    variationOnFromTo f s a b = (eVariationOn f (s ∩ Icc a b)).toReal 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) {a b : α} (h : a ≤ b)
Goal: variationOnFromTo f s a b = (eVariationOn f (s ∩ Set.Icc a b)).toReal
-- variationOnFromTo.eq_of_ge
protected theorem eq_of_ge {a b : α} (h : b ≤ a) :
    variationOnFromTo f s a b = -(eVariationOn f (s ∩ Icc b a)).toReal 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] (f : α → E) (s : Set α) {a b : α} (h : b ≤ a)
Goal: variationOnFromTo f s a b = -(eVariationOn f (s ∩ Set.Icc b a)).toReal
-- variationOnFromTo.add
protected theorem add {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b c : α}
    (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
    variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s)
Goal: variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c
-- variationOnFromTo.edist_zero_of_eq_zero
protected theorem edist_zero_of_eq_zero {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : variationOnFromTo f s a b = 0) :
    edist (f a) (f b) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : variationOnFromTo f s a b = 0)
Goal: edist (f a) (f b) = 0
-- variationOnFromTo.eq_left_iff
protected theorem eq_left_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s) :
    variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hc : c ∈ s)
Goal: variationOnFromTo f s a b = variationOnFromTo f s a c ↔ variationOnFromTo f s b c = 0
-- variationOnFromTo.eq_zero_iff_of_le
protected theorem eq_zero_iff_of_le {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ab : a ≤ b) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc a b) ⦃y⦄ (_hy : y ∈ s ∩ Icc a b), edist (f x) (f y) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ab : a ≤ b)
Goal: variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Set.Icc a b → ∀ ⦃y : α⦄, y ∈ s ∩ Set.Icc a b → edist (f x) (f y) = 0
-- variationOnFromTo.eq_zero_iff_of_ge
protected theorem eq_zero_iff_of_ge {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s)
    {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ba : b ≤ a) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ Icc b a) ⦃y⦄ (_hy : y ∈ s ∩ Icc b a), edist (f x) (f y) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (ba : b ≤ a)
Goal: variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Set.Icc b a → ∀ ⦃y : α⦄, y ∈ s ∩ Set.Icc b a → edist (f x) (f y) = 0
-- variationOnFromTo.eq_zero_iff
protected theorem eq_zero_iff {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α}
    (ha : a ∈ s) (hb : b ∈ s) :
    variationOnFromTo f s a b = 0 ↔
      ∀ ⦃x⦄ (_hx : x ∈ s ∩ uIcc a b) ⦃y⦄ (_hy : y ∈ s ∩ uIcc a b), edist (f x) (f y) = 0 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: variationOnFromTo f s a b = 0 ↔
  ∀ ⦃x : α⦄, x ∈ s ∩ Set.uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ Set.uIcc a b → edist (f x) (f y) = 0
-- variationOnFromTo.monotoneOn
protected theorem monotoneOn (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s) :
    MonotoneOn (variationOnFromTo f s a) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {a : α} (as : a ∈ s)
Goal: MonotoneOn (variationOnFromTo f s a) s
-- variationOnFromTo.antitoneOn
protected theorem antitoneOn (hf : LocallyBoundedVariationOn f s) {b : α} (bs : b ∈ s) :
    AntitoneOn (fun a => variationOnFromTo f s a b) s 
Arguments: {α : Type u_1} [LinearOrder α] {E : Type u_2} [PseudoEMetricSpace E] {f : α → E} {s : Set α} (hf : LocallyBoundedVariationOn f s) {b : α} (bs : b ∈ s)
Goal: AntitoneOn (fun a => variationOnFromTo f s a b) s