ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- NonarchAddGroupNorm.coe_sup
@[simp, norm_cast]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [AddGroup E] (p q : NonarchAddGroupNorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- NonarchAddGroupNorm.sup_apply
@[simp]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : NonarchAddGroupNorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- NonarchAddGroupNorm.apply_one
@[simp]
theorem apply_one [DecidableEq E] (x : E) :
    (1 : NonarchAddGroupNorm E) x = if x = 0 then 0 else 1 
Arguments: {E : Type u_4} [AddGroup E] [DecidableEq E] (x : E)
Goal: 1 x = if x = 0 then 0 else 1
-- NormedAddTorsor.dist_eq_norm'
class NormedAddTorsor (V : outParam Type*) (P : Type*) [SeminormedAddCommGroup V]
  [PseudoMetricSpace P] extends AddTorsor V P where
  dist_eq_norm' : ∀ x y : P, dist x y = ‖(x -ᵥ y : V)‖
Arguments: {V : outParam (Type u_1)} {P : Type u_2} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [self : NormedAddTorsor V P] (x y : P)
Goal: dist x y = ‖x -ᵥ y‖
-- dist_eq_norm_vsub
theorem dist_eq_norm_vsub (x y : P) : dist x y = ‖x -ᵥ y‖ 
Arguments: (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: dist x y = ‖x -ᵥ y‖
-- nndist_eq_nnnorm_vsub
theorem nndist_eq_nnnorm_vsub (x y : P) : nndist x y = ‖x -ᵥ y‖₊ 
Arguments: (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: nndist x y = ‖x -ᵥ y‖₊
-- dist_eq_norm_vsub'
theorem dist_eq_norm_vsub' (x y : P) : dist x y = ‖y -ᵥ x‖ 
Arguments: (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: dist x y = ‖y -ᵥ x‖
-- nndist_eq_nnnorm_vsub'
theorem nndist_eq_nnnorm_vsub' (x y : P) : nndist x y = ‖y -ᵥ x‖₊ 
Arguments: (V : Type u_2) {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y : P)
Goal: nndist x y = ‖y -ᵥ x‖₊
-- dist_vadd_cancel_left
theorem dist_vadd_cancel_left (v : V) (x y : P) : dist (v +ᵥ x) (v +ᵥ y) = dist x y 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P)
Goal: dist (v +ᵥ x) (v +ᵥ y) = dist x y
-- nndist_vadd_cancel_left
theorem nndist_vadd_cancel_left (v : V) (x y : P) : nndist (v +ᵥ x) (v +ᵥ y) = nndist x y 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x y : P)
Goal: nndist (v +ᵥ x) (v +ᵥ y) = nndist x y
-- dist_vadd_cancel_right
@[simp]
theorem dist_vadd_cancel_right (v₁ v₂ : V) (x : P) : dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P)
Goal: dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂
-- nndist_vadd_cancel_right
@[simp]
theorem nndist_vadd_cancel_right (v₁ v₂ : V) (x : P) : nndist (v₁ +ᵥ x) (v₂ +ᵥ x) = nndist v₁ v₂ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v₁ v₂ : V) (x : P)
Goal: nndist (v₁ +ᵥ x) (v₂ +ᵥ x) = nndist v₁ v₂
-- dist_vadd_left
@[simp]
theorem dist_vadd_left (v : V) (x : P) : dist (v +ᵥ x) x = ‖v‖ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P)
Goal: dist (v +ᵥ x) x = ‖v‖
-- nndist_vadd_left
@[simp]
theorem nndist_vadd_left (v : V) (x : P) : nndist (v +ᵥ x) x = ‖v‖₊ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P)
Goal: nndist (v +ᵥ x) x = ‖v‖₊
-- dist_vadd_right
@[simp]
theorem dist_vadd_right (v : V) (x : P) : dist x (v +ᵥ x) = ‖v‖ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P)
Goal: dist x (v +ᵥ x) = ‖v‖
-- nndist_vadd_right
@[simp]
theorem nndist_vadd_right (v : V) (x : P) : nndist x (v +ᵥ x) = ‖v‖₊ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v : V) (x : P)
Goal: nndist x (v +ᵥ x) = ‖v‖₊
-- IsometryEquiv.vaddConst_invFun
@[simps!]
def IsometryEquiv.vaddConst (x : P) : V ≃ᵢ P 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x p' : P)
Goal: (IsometryEquiv.vaddConst x).invFun p' = p' -ᵥ x
-- IsometryEquiv.vaddConst_symm_apply
@[simps!]
def IsometryEquiv.vaddConst (x : P) : V ≃ᵢ P 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x p' : P)
Goal: (IsometryEquiv.vaddConst x).symm p' = p' -ᵥ x
-- IsometryEquiv.vaddConst_toFun
@[simps!]
def IsometryEquiv.vaddConst (x : P) : V ≃ᵢ P 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (v : V)
Goal: (IsometryEquiv.vaddConst x) v = v +ᵥ x
-- IsometryEquiv.vaddConst_apply
@[simps!]
def IsometryEquiv.vaddConst (x : P) : V ≃ᵢ P 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P) (v : V)
Goal: (IsometryEquiv.vaddConst x) v = v +ᵥ x
-- dist_vsub_cancel_left
@[simp]
theorem dist_vsub_cancel_left (x y z : P) : dist (x -ᵥ y) (x -ᵥ z) = dist y z 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P)
Goal: dist (x -ᵥ y) (x -ᵥ z) = dist y z
-- nndist_vsub_cancel_left
@[simp]
theorem nndist_vsub_cancel_left (x y z : P) : nndist (x -ᵥ y) (x -ᵥ z) = nndist y z 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P)
Goal: nndist (x -ᵥ y) (x -ᵥ z) = nndist y z
-- IsometryEquiv.constVSub_invFun
@[simps!]
def IsometryEquiv.constVSub (x : P) : P ≃ᵢ V 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: ∀ (x_1 : V), (IsometryEquiv.constVSub x).invFun x_1 = -x_1 +ᵥ x
-- IsometryEquiv.constVSub_symm_apply
@[simps!]
def IsometryEquiv.constVSub (x : P) : P ≃ᵢ V 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: ∀ (x_1 : V), (IsometryEquiv.constVSub x).symm x_1 = -x_1 +ᵥ x
-- IsometryEquiv.constVSub_toFun
@[simps!]
def IsometryEquiv.constVSub (x : P) : P ≃ᵢ V 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: ∀ (x_1 : P), (IsometryEquiv.constVSub x) x_1 = x -ᵥ x_1
-- IsometryEquiv.constVSub_apply
@[simps!]
def IsometryEquiv.constVSub (x : P) : P ≃ᵢ V 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: ∀ (x_1 : P), (IsometryEquiv.constVSub x) x_1 = x -ᵥ x_1
-- dist_vsub_cancel_right
@[simp]
theorem dist_vsub_cancel_right (x y z : P) : dist (x -ᵥ z) (y -ᵥ z) = dist x y 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P)
Goal: dist (x -ᵥ z) (y -ᵥ z) = dist x y
-- nndist_vsub_cancel_right
@[simp]
theorem nndist_vsub_cancel_right (x y z : P) : nndist (x -ᵥ z) (y -ᵥ z) = nndist x y 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x y z : P)
Goal: nndist (x -ᵥ z) (y -ᵥ z) = nndist x y
-- dist_vadd_vadd_le
theorem dist_vadd_vadd_le (v v' : V) (p p' : P) :
    dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p' 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P)
Goal: dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'
-- nndist_vadd_vadd_le
theorem nndist_vadd_vadd_le (v v' : V) (p p' : P) :
    nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p' 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P)
Goal: nndist (v +ᵥ p) (v' +ᵥ p') ≤ nndist v v' + nndist p p'
-- dist_vsub_vsub_le
theorem dist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
    dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P)
Goal: dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄
-- nndist_vsub_vsub_le
theorem nndist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
    nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P)
Goal: nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄
-- edist_vadd_vadd_le
theorem edist_vadd_vadd_le (v v' : V) (p p' : P) :
    edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p' 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (v v' : V) (p p' : P)
Goal: edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'
-- edist_vsub_vsub_le
theorem edist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
    edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄ 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ p₄ : P)
Goal: edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄
-- LipschitzWith.vadd
theorem LipschitzWith.vadd [PseudoEMetricSpace α] {f : α → V} {g : α → P} {Kf Kg : ℝ≥0}
    (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) : LipschitzWith (Kf + Kg) (f +ᵥ g) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f : α → V} {g : α → P} {Kf Kg : ℝ≥0} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) (f +ᵥ g)
-- LipschitzWith.vsub
theorem LipschitzWith.vsub [PseudoEMetricSpace α] {f g : α → P} {Kf Kg : ℝ≥0}
    (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) : LipschitzWith (Kf + Kg) (f -ᵥ g) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [PseudoEMetricSpace α] {f g : α → P} {Kf Kg : ℝ≥0} (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g)
Goal: LipschitzWith (Kf + Kg) (f -ᵥ g)
-- uniformContinuous_vadd
theorem uniformContinuous_vadd : UniformContinuous fun x : V × P => x.1 +ᵥ x.2 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: UniformContinuous fun x => x.1 +ᵥ x.2
-- uniformContinuous_vsub
theorem uniformContinuous_vsub : UniformContinuous fun x : P × P => x.1 -ᵥ x.2 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: UniformContinuous fun x => x.1 -ᵥ x.2
-- continuous_vsub
theorem continuous_vsub : Continuous fun x : P × P => x.1 -ᵥ x.2 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: Continuous fun x => x.1 -ᵥ x.2
-- Filter.Tendsto.vsub
theorem Filter.Tendsto.vsub {l : Filter α} {f g : α → P} {x y : P} (hf : Tendsto f l (𝓝 x))
    (hg : Tendsto g l (𝓝 y)) : Tendsto (f -ᵥ g) l (𝓝 (x -ᵥ y)) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {l : Filter α} {f g : α → P} {x y : P} (hf : Filter.Tendsto f l (𝓝 x)) (hg : Filter.Tendsto g l (𝓝 y))
Goal: Filter.Tendsto (f -ᵥ g) l (𝓝 (x -ᵥ y))
-- Continuous.vsub
theorem Continuous.vsub {f g : α → P} (hf : Continuous f) (hg : Continuous g) :
    Continuous (f -ᵥ g) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} (hf : Continuous f) (hg : Continuous g)
Goal: Continuous (f -ᵥ g)
-- ContinuousAt.vsub
nonrec theorem ContinuousAt.vsub {f g : α → P} {x : α} (hf : ContinuousAt f x)
    (hg : ContinuousAt g x) :
    ContinuousAt (f -ᵥ g) x 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} (hf : ContinuousAt f x) (hg : ContinuousAt g x)
Goal: ContinuousAt (f -ᵥ g) x
-- ContinuousWithinAt.vsub
nonrec theorem ContinuousWithinAt.vsub {f g : α → P} {x : α} {s : Set α}
    (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
    ContinuousWithinAt (f -ᵥ g) s x 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {x : α} {s : Set α} (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x)
Goal: ContinuousWithinAt (f -ᵥ g) s x
-- ContinuousOn.vsub
theorem ContinuousOn.vsub {f g : α → P} {s : Set α} (hf : ContinuousOn f s)
    (hg : ContinuousOn g s) : ContinuousOn (f -ᵥ g) s 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] [TopologicalSpace α] {f g : α → P} {s : Set α} (hf : ContinuousOn f s) (hg : ContinuousOn g s)
Goal: ContinuousOn (f -ᵥ g) s
-- Filter.Tendsto.lineMap
theorem Filter.Tendsto.lineMap {l : Filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R}
    (h₁ : Tendsto f₁ l (𝓝 p₁)) (h₂ : Tendsto f₂ l (𝓝 p₂)) (hg : Tendsto g l (𝓝 c)) :
    Tendsto (fun x => AffineMap.lineMap (f₁ x) (f₂ x) (g x)) l (𝓝 <| AffineMap.lineMap p₁ p₂ c) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] {l : Filter α} {f₁ f₂ : α → P} {g : α → R} {p₁ p₂ : P} {c : R} (h₁ : Filter.Tendsto f₁ l (𝓝 p₁)) (h₂ : Filter.Tendsto f₂ l (𝓝 p₂)) (hg : Filter.Tendsto g l (𝓝 c))
Goal: Filter.Tendsto (fun x => (AffineMap.lineMap (f₁ x) (f₂ x)) (g x)) l (𝓝 ((AffineMap.lineMap p₁ p₂) c))
-- Filter.Tendsto.midpoint
theorem Filter.Tendsto.midpoint [Invertible (2 : R)] {l : Filter α} {f₁ f₂ : α → P} {p₁ p₂ : P}
    (h₁ : Tendsto f₁ l (𝓝 p₁)) (h₂ : Tendsto f₂ l (𝓝 p₂)) :
    Tendsto (fun x => midpoint R (f₁ x) (f₂ x)) l (𝓝 <| midpoint R p₁ p₂) 
Arguments: {α : Type u_1} {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {R : Type u_6} [Ring R] [TopologicalSpace R] [Module R V] [ContinuousSMul R V] [Invertible 2] {l : Filter α} {f₁ f₂ : α → P} {p₁ p₂ : P} (h₁ : Filter.Tendsto f₁ l (𝓝 p₁)) (h₂ : Filter.Tendsto f₂ l (𝓝 p₂))
Goal: Filter.Tendsto (fun x => midpoint R (f₁ x) (f₂ x)) l (𝓝 (midpoint R p₁ p₂))
-- IsClosed.vadd_right_of_isCompact
theorem IsClosed.vadd_right_of_isCompact {s : Set V} {t : Set P} (hs : IsClosed s)
    (ht : IsCompact t) : IsClosed (s +ᵥ t) 
Arguments: {V : Type u_2} {P : Type u_3} [SeminormedAddCommGroup V] [PseudoMetricSpace P] [NormedAddTorsor V P] {s : Set V} {t : Set P} (hs : IsClosed s) (ht : IsCompact t)
Goal: IsClosed (s +ᵥ t)
-- SemiNormedGrp.ext
@[ext]
lemma ext {M N : SemiNormedGrp} {f₁ f₂ : M ⟶ N} (h : ∀ (x : M), f₁ x = f₂ x) : f₁ = f₂ 
Arguments: {M N : SemiNormedGrp} {f₁ f₂ : M ⟶ N} (h : ∀ (x : ↑M), f₁ x = f₂ x)
Goal: f₁ = f₂
-- SemiNormedGrp.coe_of
@[simp]
theorem coe_of (V : Type u) [SeminormedAddCommGroup V] : (SemiNormedGrp.of V : Type u) = V 
Arguments: (V : Type u) [SeminormedAddCommGroup V]
Goal: ↑(SemiNormedGrp.of V) = V
-- SemiNormedGrp.coe_id
@[simp (high)]
theorem coe_id (V : SemiNormedGrp) : (𝟙 V : V → V) = id 
Arguments: (V : SemiNormedGrp)
Goal: ⇑(𝟙 V) = id
-- SemiNormedGrp.coe_comp
@[simp (high)]
theorem coe_comp {M N K : SemiNormedGrp} (f : M ⟶ N) (g : N ⟶ K) :
    (f ≫ g : M → K) = g ∘ f 
Arguments: {M N K : SemiNormedGrp} (f : M ⟶ N) (g : N ⟶ K)
Goal: ⇑(f ≫ g) = ⇑g ∘ ⇑f
-- SemiNormedGrp.zero_apply
@[simp]
theorem zero_apply {V W : SemiNormedGrp} (x : V) : (0 : V ⟶ W) x = 0 
Arguments: {V W : SemiNormedGrp} (x : ↑V)
Goal: 0 x = 0
-- SemiNormedGrp.isZero_of_subsingleton
theorem isZero_of_subsingleton (V : SemiNormedGrp) [Subsingleton V] : Limits.IsZero V 
Arguments: (V : SemiNormedGrp) [Subsingleton ↑V]
Goal: CategoryTheory.Limits.IsZero V
-- SemiNormedGrp.iso_isometry_of_normNoninc
theorem iso_isometry_of_normNoninc {V W : SemiNormedGrp} (i : V ≅ W) (h1 : i.hom.NormNoninc)
    (h2 : i.inv.NormNoninc) : Isometry i.hom 
Arguments: {V W : SemiNormedGrp} (i : V ≅ W) (h1 : NormedAddGroupHom.NormNoninc i.hom) (h2 : NormedAddGroupHom.NormNoninc i.inv)
Goal: Isometry ⇑i.hom
-- SemiNormedGrp₁.hom_ext
@[ext]
theorem hom_ext {M N : SemiNormedGrp₁} (f g : M ⟶ N) (w : (f : M → N) = (g : M → N)) :
    f = g 
Arguments: {M N : SemiNormedGrp₁} (f g : M ⟶ N) (w : ⇑f = ⇑g)
Goal: f = g
-- SemiNormedGrp₁.mkHom_apply
theorem mkHom_apply {M N : SemiNormedGrp} (f : M ⟶ N) (i : f.NormNoninc) (x) :
    mkHom f i x = f x 
Arguments: {M N : SemiNormedGrp} (f : M ⟶ N) (i : NormedAddGroupHom.NormNoninc f) (x : ↑(SemiNormedGrp₁.of ↑M))
Goal: (SemiNormedGrp₁.mkHom f i) x = f x
-- SemiNormedGrp₁.mkIso_hom
@[simps]
def mkIso {M N : SemiNormedGrp} (f : M ≅ N) (i : f.hom.NormNoninc) (i' : f.inv.NormNoninc) :
    SemiNormedGrp₁.of M ≅ SemiNormedGrp₁.of N 
Arguments: {M N : SemiNormedGrp} (f : M ≅ N) (i : NormedAddGroupHom.NormNoninc f.hom) (i' : NormedAddGroupHom.NormNoninc f.inv)
Goal: (SemiNormedGrp₁.mkIso f i i').hom = SemiNormedGrp₁.mkHom f.hom i
-- SemiNormedGrp₁.mkIso_inv
@[simps]
def mkIso {M N : SemiNormedGrp} (f : M ≅ N) (i : f.hom.NormNoninc) (i' : f.inv.NormNoninc) :
    SemiNormedGrp₁.of M ≅ SemiNormedGrp₁.of N 
Arguments: {M N : SemiNormedGrp} (f : M ≅ N) (i : NormedAddGroupHom.NormNoninc f.hom) (i' : NormedAddGroupHom.NormNoninc f.inv)
Goal: (SemiNormedGrp₁.mkIso f i i').inv = SemiNormedGrp₁.mkHom f.inv i'
-- SemiNormedGrp₁.coe_of
@[simp]
theorem coe_of (V : Type u) [SeminormedAddCommGroup V] : (SemiNormedGrp₁.of V : Type u) = V 
Arguments: (V : Type u) [SeminormedAddCommGroup V]
Goal: ↑(SemiNormedGrp₁.of V) = V
-- SemiNormedGrp₁.coe_id
@[simp (high)]
theorem coe_id (V : SemiNormedGrp₁) : ⇑(𝟙 V) = id 
Arguments: (V : SemiNormedGrp₁)
Goal: ⇑(𝟙 V) = id
-- SemiNormedGrp₁.coe_comp
@[simp (high)]
theorem coe_comp {M N K : SemiNormedGrp₁} (f : M ⟶ N) (g : N ⟶ K) :
    (f ≫ g : M → K) = g ∘ f 
Arguments: {M N K : SemiNormedGrp₁} (f : M ⟶ N) (g : N ⟶ K)
Goal: ⇑(f ≫ g) = ⇑g ∘ ⇑f
-- SemiNormedGrp₁.zero_apply
@[simp]
theorem zero_apply {V W : SemiNormedGrp₁} (x : V) : (0 : V ⟶ W) x = 0 
Arguments: {V W : SemiNormedGrp₁} (x : ↑V)
Goal: 0 x = 0
-- SemiNormedGrp₁.isZero_of_subsingleton
theorem isZero_of_subsingleton (V : SemiNormedGrp₁) [Subsingleton V] : Limits.IsZero V 
Arguments: (V : SemiNormedGrp₁) [Subsingleton ↑V]
Goal: CategoryTheory.Limits.IsZero V
-- SemiNormedGrp₁.iso_isometry
theorem iso_isometry {V W : SemiNormedGrp₁} (i : V ≅ W) : Isometry i.hom 
Arguments: {V W : SemiNormedGrp₁} (i : V ≅ W)
Goal: Isometry ⇑i.hom
-- SemiNormedGrp.explicitCokernelπ_surjective
theorem explicitCokernelπ_surjective {X Y : SemiNormedGrp.{u}} {f : X ⟶ Y} :
    Function.Surjective (explicitCokernelπ f) 
Arguments: {X Y : SemiNormedGrp} {f : X ⟶ Y}
Goal: Function.Surjective ⇑(SemiNormedGrp.explicitCokernelπ f)
-- SemiNormedGrp.comp_explicitCokernelπ
@[simp, reassoc]
theorem comp_explicitCokernelπ {X Y : SemiNormedGrp.{u}} (f : X ⟶ Y) :
    f ≫ explicitCokernelπ f = 0 
Arguments: {X Y : SemiNormedGrp} (f : X ⟶ Y)
Goal: f ≫ SemiNormedGrp.explicitCokernelπ f = 0
-- SemiNormedGrp.explicitCokernelπ_apply_dom_eq_zero
@[simp]
theorem explicitCokernelπ_apply_dom_eq_zero {X Y : SemiNormedGrp.{u}} {f : X ⟶ Y} (x : X) :
    (explicitCokernelπ f) (f x) = 0 
Arguments: {X Y : SemiNormedGrp} {f : X ⟶ Y} (x : ↑X)
Goal: (SemiNormedGrp.explicitCokernelπ f) (f x) = 0
-- SemiNormedGrp.explicitCokernelπ_desc
@[simp, reassoc]
theorem explicitCokernelπ_desc {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    (w : f ≫ g = 0) : explicitCokernelπ f ≫ explicitCokernelDesc w = g 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0)
Goal: SemiNormedGrp.explicitCokernelπ f ≫ SemiNormedGrp.explicitCokernelDesc w = g
-- SemiNormedGrp.explicitCokernelπ_desc_apply
@[simp]
theorem explicitCokernelπ_desc_apply {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    {cond : f ≫ g = 0} (x : Y) : explicitCokernelDesc cond (explicitCokernelπ f x) = g x 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} {cond : f ≫ g = 0} (x : ↑Y)
Goal: (SemiNormedGrp.explicitCokernelDesc cond) ((SemiNormedGrp.explicitCokernelπ f) x) = g x
-- SemiNormedGrp.explicitCokernelDesc_unique
theorem explicitCokernelDesc_unique {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    (w : f ≫ g = 0) (e : explicitCokernel f ⟶ Z) (he : explicitCokernelπ f ≫ e = g) :
    e = explicitCokernelDesc w 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0) (e : SemiNormedGrp.explicitCokernel f ⟶ Z) (he : SemiNormedGrp.explicitCokernelπ f ≫ e = g)
Goal: e = SemiNormedGrp.explicitCokernelDesc w
-- SemiNormedGrp.explicitCokernelDesc_comp_eq_desc
theorem explicitCokernelDesc_comp_eq_desc {X Y Z W : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    -- Porting note: renamed `cond` to `cond'` to avoid
    -- failed to rewrite using equation theorems for 'cond'
    {h : Z ⟶ W} {cond' : f ≫ g = 0} :
    explicitCokernelDesc cond' ≫ h =
      explicitCokernelDesc
        (show f ≫ g ≫ h = 0 by rw [← CategoryTheory.Category.assoc, cond', Limits.zero_comp]) 
Arguments: {X Y Z W : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} {h : Z ⟶ W} {cond' : f ≫ g = 0}
Goal: SemiNormedGrp.explicitCokernelDesc cond' ≫ h = SemiNormedGrp.explicitCokernelDesc ⋯
-- SemiNormedGrp.explicitCokernelDesc_zero
@[simp]
theorem explicitCokernelDesc_zero {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} :
    explicitCokernelDesc (show f ≫ (0 : Y ⟶ Z) = 0 from CategoryTheory.Limits.comp_zero) = 0 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y}
Goal: SemiNormedGrp.explicitCokernelDesc ⋯ = 0
-- SemiNormedGrp.explicitCokernel_hom_ext
@[ext]
theorem explicitCokernel_hom_ext {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y}
    (e₁ e₂ : explicitCokernel f ⟶ Z) (h : explicitCokernelπ f ≫ e₁ = explicitCokernelπ f ≫ e₂) :
    e₁ = e₂ 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} (e₁ e₂ : SemiNormedGrp.explicitCokernel f ⟶ Z) (h : SemiNormedGrp.explicitCokernelπ f ≫ e₁ = SemiNormedGrp.explicitCokernelπ f ≫ e₂)
Goal: e₁ = e₂
-- SemiNormedGrp.isQuotient_explicitCokernelπ
theorem isQuotient_explicitCokernelπ {X Y : SemiNormedGrp.{u}} (f : X ⟶ Y) :
    NormedAddGroupHom.IsQuotient (explicitCokernelπ f) 
Arguments: {X Y : SemiNormedGrp} (f : X ⟶ Y)
Goal: NormedAddGroupHom.IsQuotient (SemiNormedGrp.explicitCokernelπ f)
-- SemiNormedGrp.normNoninc_explicitCokernelπ
theorem normNoninc_explicitCokernelπ {X Y : SemiNormedGrp.{u}} (f : X ⟶ Y) :
    (explicitCokernelπ f).NormNoninc 
Arguments: {X Y : SemiNormedGrp} (f : X ⟶ Y)
Goal: NormedAddGroupHom.NormNoninc (SemiNormedGrp.explicitCokernelπ f)
-- SemiNormedGrp.explicitCokernelDesc_norm_le_of_norm_le
theorem explicitCokernelDesc_norm_le_of_norm_le {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y}
    {g : Y ⟶ Z} (w : f ≫ g = 0) (c : ℝ≥0) (h : ‖g‖ ≤ c) : ‖explicitCokernelDesc w‖ ≤ c 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0) (c : NNReal) (h : ‖g‖ ≤ ↑c)
Goal: ‖SemiNormedGrp.explicitCokernelDesc w‖ ≤ ↑c
-- SemiNormedGrp.explicitCokernelDesc_normNoninc
theorem explicitCokernelDesc_normNoninc {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    {cond : f ≫ g = 0} (hg : g.NormNoninc) : (explicitCokernelDesc cond).NormNoninc 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} {cond : f ≫ g = 0} (hg : NormedAddGroupHom.NormNoninc g)
Goal: NormedAddGroupHom.NormNoninc (SemiNormedGrp.explicitCokernelDesc cond)
-- SemiNormedGrp.explicitCokernelDesc_comp_eq_zero
theorem explicitCokernelDesc_comp_eq_zero {X Y Z W : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    {h : Z ⟶ W} (cond : f ≫ g = 0) (cond2 : g ≫ h = 0) : explicitCokernelDesc cond ≫ h = 0 
Arguments: {X Y Z W : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} {h : Z ⟶ W} (cond : f ≫ g = 0) (cond2 : g ≫ h = 0)
Goal: SemiNormedGrp.explicitCokernelDesc cond ≫ h = 0
-- SemiNormedGrp.explicitCokernelDesc_norm_le
theorem explicitCokernelDesc_norm_le {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    (w : f ≫ g = 0) : ‖explicitCokernelDesc w‖ ≤ ‖g‖ 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0)
Goal: ‖SemiNormedGrp.explicitCokernelDesc w‖ ≤ ‖g‖