ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SemiNormedGrp.explicitCokernelIso_hom_π
@[simp]
theorem explicitCokernelIso_hom_π {X Y : SemiNormedGrp.{u}} (f : X ⟶ Y) :
    explicitCokernelπ f ≫ (explicitCokernelIso f).hom = cokernel.π _ 
Arguments: {X Y : SemiNormedGrp} (f : X ⟶ Y)
Goal: SemiNormedGrp.explicitCokernelπ f ≫ (SemiNormedGrp.explicitCokernelIso f).hom = CategoryTheory.Limits.cokernel.π f
-- SemiNormedGrp.explicitCokernelIso_inv_π
@[simp]
theorem explicitCokernelIso_inv_π {X Y : SemiNormedGrp.{u}} (f : X ⟶ Y) :
    cokernel.π f ≫ (explicitCokernelIso f).inv = explicitCokernelπ f 
Arguments: {X Y : SemiNormedGrp} (f : X ⟶ Y)
Goal: CategoryTheory.Limits.cokernel.π f ≫ (SemiNormedGrp.explicitCokernelIso f).inv = SemiNormedGrp.explicitCokernelπ f
-- SemiNormedGrp.explicitCokernelIso_hom_desc
@[simp]
theorem explicitCokernelIso_hom_desc {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
    (w : f ≫ g = 0) :
    (explicitCokernelIso f).hom ≫ cokernel.desc f g w = explicitCokernelDesc w 
Arguments: {X Y Z : SemiNormedGrp} {f : X ⟶ Y} {g : Y ⟶ Z} (w : f ≫ g = 0)
Goal: (SemiNormedGrp.explicitCokernelIso f).hom ≫ CategoryTheory.Limits.cokernel.desc f g w =
  SemiNormedGrp.explicitCokernelDesc w
-- SemiNormedGrp.ExplicitCoker.map_desc
theorem ExplicitCoker.map_desc {A B C D B' D' : SemiNormedGrp.{u}}
    {fab : A ⟶ B} {fbd : B ⟶ D} {fac : A ⟶ C} {fcd : C ⟶ D} {h : fab ≫ fbd = fac ≫ fcd}
    {fbb' : B ⟶ B'} {fdd' : D ⟶ D'} {condb : fab ≫ fbb' = 0} {condd : fcd ≫ fdd' = 0} {g : B' ⟶ D'}
    (h' : fbb' ≫ g = fbd ≫ fdd') :
    explicitCokernelDesc condb ≫ g = explicitCokernel.map h ≫ explicitCokernelDesc condd 
Arguments: {A B C D B' D' : SemiNormedGrp} {fab : A ⟶ B} {fbd : B ⟶ D} {fac : A ⟶ C} {fcd : C ⟶ D} {h : fab ≫ fbd = fac ≫ fcd} {fbb' : B ⟶ B'} {fdd' : D ⟶ D'} {condb : fab ≫ fbb' = 0} {condd : fcd ≫ fdd' = 0} {g : B' ⟶ D'} (h' : fbb' ≫ g = fbd ≫ fdd')
Goal: SemiNormedGrp.explicitCokernelDesc condb ≫ g =
  SemiNormedGrp.explicitCokernel.map h ≫ SemiNormedGrp.explicitCokernelDesc condd
-- PUnit.norm_eq_zero
@[simp] lemma norm_eq_zero (x : PUnit) : ‖x‖ = 0 
Arguments: (x : PUnit.{u_5 + 1})
Goal: ‖x‖ = 0
-- ULift.norm_def
lemma norm_def (x : ULift E) : ‖x‖ = ‖x.down‖ 
Arguments: {E : Type u_2} [Norm E] (x : ULift.{u_5, u_2} E)
Goal: ‖x‖ = ‖x.down‖
-- ULift.norm_up
@[simp] lemma norm_up (x : E) : ‖ULift.up x‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : E)
Goal: ‖{ down := x }‖ = ‖x‖
-- ULift.norm_down
@[simp] lemma norm_down (x : ULift E) : ‖x.down‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : ULift.{u_5, u_2} E)
Goal: ‖x.down‖ = ‖x‖
-- ULift.nnnorm_def
lemma nnnorm_def (x : ULift E) : ‖x‖₊ = ‖x.down‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : ULift.{u_5, u_2} E)
Goal: ‖x‖₊ = ‖x.down‖₊
-- ULift.nnnorm_up
@[simp] lemma nnnorm_up (x : E) : ‖ULift.up x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : E)
Goal: ‖{ down := x }‖₊ = ‖x‖₊
-- ULift.nnnorm_down
@[simp] lemma nnnorm_down (x : ULift E) : ‖x.down‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : ULift.{u_5, u_2} E)
Goal: ‖x.down‖₊ = ‖x‖₊
-- norm_toMul
@[simp] lemma norm_toMul (x) : ‖(toMul x : E)‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : Additive E)
Goal: ‖Additive.toMul x‖ = ‖x‖
-- norm_ofMul
@[simp] lemma norm_ofMul (x : E) : ‖ofMul x‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : E)
Goal: ‖Additive.ofMul x‖ = ‖x‖
-- norm_toAdd
@[simp] lemma norm_toAdd (x) : ‖(toAdd x : E)‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : Multiplicative E)
Goal: ‖Multiplicative.toAdd x‖ = ‖x‖
-- norm_ofAdd
@[simp] lemma norm_ofAdd (x : E) : ‖ofAdd x‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : E)
Goal: ‖Multiplicative.ofAdd x‖ = ‖x‖
-- nnnorm_toMul
@[simp] lemma nnnorm_toMul (x) : ‖(toMul x : E)‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : Additive E)
Goal: ‖Additive.toMul x‖₊ = ‖x‖₊
-- nnnorm_ofMul
@[simp] lemma nnnorm_ofMul (x : E) : ‖ofMul x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : E)
Goal: ‖Additive.ofMul x‖₊ = ‖x‖₊
-- nnnorm_toAdd
@[simp] lemma nnnorm_toAdd (x) : ‖(toAdd x : E)‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : Multiplicative E)
Goal: ‖Multiplicative.toAdd x‖₊ = ‖x‖₊
-- nnnorm_ofAdd
@[simp] lemma nnnorm_ofAdd (x : E) : ‖ofAdd x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : E)
Goal: ‖Multiplicative.ofAdd x‖₊ = ‖x‖₊
-- norm_toDual
@[simp] lemma norm_toDual (x : E) : ‖toDual x‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : E)
Goal: ‖OrderDual.toDual x‖ = ‖x‖
-- norm_ofDual
@[simp] lemma norm_ofDual (x : Eᵒᵈ) : ‖ofDual x‖ = ‖x‖ 
Arguments: {E : Type u_2} [Norm E] (x : Eᵒᵈ)
Goal: ‖OrderDual.ofDual x‖ = ‖x‖
-- nnnorm_toDual
@[simp] lemma nnnorm_toDual (x : E) : ‖toDual x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : E)
Goal: ‖OrderDual.toDual x‖₊ = ‖x‖₊
-- nnnorm_ofDual
@[simp] lemma nnnorm_ofDual (x : Eᵒᵈ) : ‖ofDual x‖₊ = ‖x‖₊ 
Arguments: {E : Type u_2} [NNNorm E] (x : Eᵒᵈ)
Goal: ‖OrderDual.ofDual x‖₊ = ‖x‖₊
-- Prod.norm_def
lemma Prod.norm_def (x : E × F) : ‖x‖ = max ‖x.1‖ ‖x.2‖ 
Arguments: {E : Type u_2} {F : Type u_3} [Norm E] [Norm F] (x : E × F)
Goal: ‖x‖ = max ‖x.1‖ ‖x.2‖
-- norm_fst_le
lemma norm_fst_le (x : E × F) : ‖x.1‖ ≤ ‖x‖ 
Arguments: {E : Type u_2} {F : Type u_3} [Norm E] [Norm F] (x : E × F)
Goal: ‖x.1‖ ≤ ‖x‖
-- norm_snd_le
lemma norm_snd_le (x : E × F) : ‖x.2‖ ≤ ‖x‖ 
Arguments: {E : Type u_2} {F : Type u_3} [Norm E] [Norm F] (x : E × F)
Goal: ‖x.2‖ ≤ ‖x‖
-- norm_prod_le_iff
lemma norm_prod_le_iff : ‖x‖ ≤ r ↔ ‖x.1‖ ≤ r ∧ ‖x.2‖ ≤ r 
Arguments: {E : Type u_2} {F : Type u_3} [Norm E] [Norm F] {x : E × F} {r : ℝ}
Goal: ‖x‖ ≤ r ↔ ‖x.1‖ ≤ r ∧ ‖x.2‖ ≤ r
-- Prod.nnnorm_def'
@[to_additive Prod.nnnorm_def']
lemma Prod.nnorm_def (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ 
Arguments: {E : Type u_2} {F : Type u_3} [SeminormedAddGroup E] [SeminormedAddGroup F] (x : E × F)
Goal: ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊
-- Prod.nnorm_def
@[to_additive Prod.nnnorm_def']
lemma Prod.nnorm_def (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ 
Arguments: {E : Type u_2} {F : Type u_3} [SeminormedGroup E] [SeminormedGroup F] (x : E × F)
Goal: ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊
-- Pi.norm_def
@[to_additive Pi.norm_def]
lemma Pi.norm_def' : ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊) 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i)
Goal: ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊)
-- Pi.norm_def'
@[to_additive Pi.norm_def]
lemma Pi.norm_def' : ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊) 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i)
Goal: ‖f‖ = ↑(Finset.univ.sup fun b => ‖f b‖₊)
-- Pi.nnnorm_def
@[to_additive Pi.nnnorm_def]
lemma Pi.nnnorm_def' : ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i)
Goal: ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊
-- Pi.nnnorm_def'
@[to_additive Pi.nnnorm_def]
lemma Pi.nnnorm_def' : ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i)
Goal: ‖f‖₊ = Finset.univ.sup fun b => ‖f b‖₊
-- pi_norm_le_iff_of_nonneg
@[to_additive pi_norm_le_iff_of_nonneg "The seminorm of an element in a product space is `≤ r` if
and only if the norm of each component is."]
lemma pi_norm_le_iff_of_nonneg' (hr : 0 ≤ r) : ‖x‖ ≤ r ↔ ∀ i, ‖x i‖ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 ≤ r)
Goal: ‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r
-- pi_norm_le_iff_of_nonneg'
@[to_additive pi_norm_le_iff_of_nonneg "The seminorm of an element in a product space is `≤ r` if
and only if the norm of each component is."]
lemma pi_norm_le_iff_of_nonneg' (hr : 0 ≤ r) : ‖x‖ ≤ r ↔ ∀ i, ‖x i‖ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 ≤ r)
Goal: ‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r
-- pi_nnnorm_le_iff
@[to_additive pi_nnnorm_le_iff]
lemma pi_nnnorm_le_iff' {r : ℝ≥0} : ‖x‖₊ ≤ r ↔ ∀ i, ‖x i‖₊ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ≥0}
Goal: ‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r
-- pi_nnnorm_le_iff'
@[to_additive pi_nnnorm_le_iff]
lemma pi_nnnorm_le_iff' {r : ℝ≥0} : ‖x‖₊ ≤ r ↔ ∀ i, ‖x i‖₊ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ≥0}
Goal: ‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r
-- pi_norm_le_iff_of_nonempty'
@[to_additive pi_norm_le_iff_of_nonempty]
lemma pi_norm_le_iff_of_nonempty' [Nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) {r : ℝ} [Nonempty ι]
Goal: ‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r
-- pi_norm_le_iff_of_nonempty
@[to_additive pi_norm_le_iff_of_nonempty]
lemma pi_norm_le_iff_of_nonempty' [Nonempty ι] : ‖f‖ ≤ r ↔ ∀ b, ‖f b‖ ≤ r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) {r : ℝ} [Nonempty ι]
Goal: ‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r
-- pi_norm_lt_iff
@[to_additive pi_norm_lt_iff "The seminorm of an element in a product space is `< r` if and only
if the norm of each component is."]
lemma pi_norm_lt_iff' (hr : 0 < r) : ‖x‖ < r ↔ ∀ i, ‖x i‖ < r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 < r)
Goal: ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r
-- pi_norm_lt_iff'
@[to_additive pi_norm_lt_iff "The seminorm of an element in a product space is `< r` if and only
if the norm of each component is."]
lemma pi_norm_lt_iff' (hr : 0 < r) : ‖x‖ < r ↔ ∀ i, ‖x i‖ < r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 < r)
Goal: ‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r
-- pi_nnnorm_lt_iff'
@[to_additive pi_nnnorm_lt_iff]
lemma pi_nnnorm_lt_iff' {r : ℝ≥0} (hr : 0 < r) : ‖x‖₊ < r ↔ ∀ i, ‖x i‖₊ < r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ≥0} (hr : 0 < r)
Goal: ‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r
-- pi_nnnorm_lt_iff
@[to_additive pi_nnnorm_lt_iff]
lemma pi_nnnorm_lt_iff' {r : ℝ≥0} (hr : 0 < r) : ‖x‖₊ < r ↔ ∀ i, ‖x i‖₊ < r 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ≥0} (hr : 0 < r)
Goal: ‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r
-- norm_le_pi_norm
@[to_additive norm_le_pi_norm]
lemma norm_le_pi_norm' (i : ι) : ‖f i‖ ≤ ‖f‖ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) (i : ι)
Goal: ‖f i‖ ≤ ‖f‖
-- norm_le_pi_norm'
@[to_additive norm_le_pi_norm]
lemma norm_le_pi_norm' (i : ι) : ‖f i‖ ≤ ‖f‖ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) (i : ι)
Goal: ‖f i‖ ≤ ‖f‖
-- nnnorm_le_pi_nnnorm
@[to_additive nnnorm_le_pi_nnnorm]
lemma nnnorm_le_pi_nnnorm' (i : ι) : ‖f i‖₊ ≤ ‖f‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) (i : ι)
Goal: ‖f i‖₊ ≤ ‖f‖₊
-- nnnorm_le_pi_nnnorm'
@[to_additive nnnorm_le_pi_nnnorm]
lemma nnnorm_le_pi_nnnorm' (i : ι) : ‖f i‖₊ ≤ ‖f‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) (i : ι)
Goal: ‖f i‖₊ ≤ ‖f‖₊
-- pi_norm_const_le
@[to_additive pi_norm_const_le]
lemma pi_norm_const_le' (a : E) : ‖fun _ : ι => a‖ ≤ ‖a‖ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] (a : E)
Goal: ‖fun x => a‖ ≤ ‖a‖
-- pi_norm_const_le'
@[to_additive pi_norm_const_le]
lemma pi_norm_const_le' (a : E) : ‖fun _ : ι => a‖ ≤ ‖a‖ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] (a : E)
Goal: ‖fun x => a‖ ≤ ‖a‖
-- pi_nnnorm_const_le
@[to_additive pi_nnnorm_const_le]
lemma pi_nnnorm_const_le' (a : E) : ‖fun _ : ι => a‖₊ ≤ ‖a‖₊ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] (a : E)
Goal: ‖fun x => a‖₊ ≤ ‖a‖₊
-- pi_nnnorm_const_le'
@[to_additive pi_nnnorm_const_le]
lemma pi_nnnorm_const_le' (a : E) : ‖fun _ : ι => a‖₊ ≤ ‖a‖₊ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] (a : E)
Goal: ‖fun x => a‖₊ ≤ ‖a‖₊
-- pi_norm_const'
@[to_additive (attr := simp) pi_norm_const]
lemma pi_norm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖ = ‖a‖ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] [Nonempty ι] (a : E)
Goal: ‖fun _i => a‖ = ‖a‖
-- pi_norm_const
@[to_additive (attr := simp) pi_norm_const]
lemma pi_norm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖ = ‖a‖ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] [Nonempty ι] (a : E)
Goal: ‖fun _i => a‖ = ‖a‖
-- pi_nnnorm_const'
@[to_additive (attr := simp) pi_nnnorm_const]
lemma pi_nnnorm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖₊ = ‖a‖₊ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] [Nonempty ι] (a : E)
Goal: ‖fun _i => a‖₊ = ‖a‖₊
-- pi_nnnorm_const
@[to_additive (attr := simp) pi_nnnorm_const]
lemma pi_nnnorm_const' [Nonempty ι] (a : E) : ‖fun _i : ι => a‖₊ = ‖a‖₊ 
Arguments: {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] [Nonempty ι] (a : E)
Goal: ‖fun _i => a‖₊ = ‖a‖₊
-- Pi.sum_norm_apply_le_norm'
@[to_additive Pi.sum_norm_apply_le_norm "The $L^1$ norm is less than the $L^\\infty$ norm scaled by
the cardinality."]
lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f‖ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i)
Goal: ∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖
-- Pi.sum_norm_apply_le_norm
@[to_additive Pi.sum_norm_apply_le_norm "The $L^1$ norm is less than the $L^\\infty$ norm scaled by
the cardinality."]
lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f‖ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i)
Goal: ∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖
-- Pi.sum_nnnorm_apply_le_nnnorm
@[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
scaled by the cardinality."]
lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i)
Goal: ∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊
-- Pi.sum_nnnorm_apply_le_nnnorm'
@[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
scaled by the cardinality."]
lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i)
Goal: ∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊
-- Pi.nnnorm_single
theorem Pi.nnnorm_single [DecidableEq ι] [∀ i, NormedAddCommGroup (π i)] {i : ι} (y : π i) :
    ‖Pi.single i y‖₊ = ‖y‖₊ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [DecidableEq ι] [(i : ι) → NormedAddCommGroup (π i)] {i : ι} (y : π i)
Goal: ‖Pi.single i y‖₊ = ‖y‖₊
-- Pi.norm_single
theorem Pi.norm_single [DecidableEq ι] [∀ i, NormedAddCommGroup (π i)] {i : ι} (y : π i) :
    ‖Pi.single i y‖ = ‖y‖ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [DecidableEq ι] [(i : ι) → NormedAddCommGroup (π i)] {i : ι} (y : π i)
Goal: ‖Pi.single i y‖ = ‖y‖
-- MulOpposite.norm_op
lemma norm_op [SeminormedAddGroup E] (a : E) : ‖MulOpposite.op a‖ = ‖a‖ 
Arguments: {E : Type u_2} [SeminormedAddGroup E] (a : E)
Goal: ‖MulOpposite.op a‖ = ‖a‖
-- MulOpposite.norm_unop
lemma norm_unop [SeminormedAddGroup E] (a : Eᵐᵒᵖ) : ‖MulOpposite.unop a‖ = ‖a‖ 
Arguments: {E : Type u_2} [SeminormedAddGroup E] (a : Eᵐᵒᵖ)
Goal: ‖MulOpposite.unop a‖ = ‖a‖
-- MulOpposite.nnnorm_op
lemma nnnorm_op [SeminormedAddGroup E] (a : E) : ‖MulOpposite.op a‖₊ = ‖a‖₊ 
Arguments: {E : Type u_2} [SeminormedAddGroup E] (a : E)
Goal: ‖MulOpposite.op a‖₊ = ‖a‖₊
-- MulOpposite.nnnorm_unop
lemma nnnorm_unop [SeminormedAddGroup E] (a : Eᵐᵒᵖ) : ‖MulOpposite.unop a‖₊ = ‖a‖₊ 
Arguments: {E : Type u_2} [SeminormedAddGroup E] (a : Eᵐᵒᵖ)
Goal: ‖MulOpposite.unop a‖₊ = ‖a‖₊
-- WeakDual.CharacterSpace.norm_le_norm_one
theorem norm_le_norm_one (φ : characterSpace 𝕜 A) : ‖toNormedDual (φ : WeakDual 𝕜 A)‖ ≤ ‖(1 : A)‖ 
Arguments: {𝕜 : Type u_1} {A : Type u_2} [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A] [CompleteSpace A] (φ : ↑(WeakDual.characterSpace 𝕜 A))
Goal: ‖WeakDual.toNormedDual ↑φ‖ ≤ ‖1‖
-- SeminormedAddGroup.dist_eq
class SeminormedAddGroup (E : Type*) extends Norm E, AddGroup E, PseudoMetricSpace E where
  dist := fun x y => ‖x - y‖
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {E : Type u_9} [self : SeminormedAddGroup E] (x y : E)
Goal: dist x y = ‖x - y‖
-- SeminormedGroup.dist_eq
@[to_additive]
class SeminormedGroup (E : Type*) extends Norm E, Group E, PseudoMetricSpace E where
  dist := fun x y => ‖x / y‖
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {E : Type u_9} [self : SeminormedGroup E] (x y : E)
Goal: dist x y = ‖x / y‖
-- NormedAddGroup.dist_eq
class NormedAddGroup (E : Type*) extends Norm E, AddGroup E, MetricSpace E where
  dist := fun x y => ‖x - y‖
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {E : Type u_9} [self : NormedAddGroup E] (x y : E)
Goal: dist x y = ‖x - y‖
-- NormedGroup.dist_eq
@[to_additive]
class NormedGroup (E : Type*) extends Norm E, Group E, MetricSpace E where
  dist := fun x y => ‖x / y‖
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {E : Type u_9} [self : NormedGroup E] (x y : E)
Goal: dist x y = ‖x / y‖
-- SeminormedAddCommGroup.dist_eq
class SeminormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E,
  PseudoMetricSpace E where
  dist := fun x y => ‖x - y‖
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {E : Type u_9} [self : SeminormedAddCommGroup E] (x y : E)
Goal: dist x y = ‖x - y‖
-- SeminormedCommGroup.dist_eq
@[to_additive]
class SeminormedCommGroup (E : Type*) extends Norm E, CommGroup E, PseudoMetricSpace E where
  dist := fun x y => ‖x / y‖
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {E : Type u_9} [self : SeminormedCommGroup E] (x y : E)
Goal: dist x y = ‖x / y‖
-- NormedAddCommGroup.dist_eq
class NormedAddCommGroup (E : Type*) extends Norm E, AddCommGroup E, MetricSpace E where
  dist := fun x y => ‖x - y‖
    dist_eq : ∀ x y, dist x y = ‖x - y‖ := by aesop
Arguments: {E : Type u_9} [self : NormedAddCommGroup E] (x y : E)
Goal: dist x y = ‖x - y‖
-- NormedCommGroup.dist_eq
@[to_additive]
class NormedCommGroup (E : Type*) extends Norm E, CommGroup E, MetricSpace E where
  dist := fun x y => ‖x / y‖
    dist_eq : ∀ x y, dist x y = ‖x / y‖ := by aesop
Arguments: {E : Type u_9} [self : NormedCommGroup E] (x y : E)
Goal: dist x y = ‖x / y‖
-- dist_eq_norm_sub
@[to_additive]
theorem dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: dist a b = ‖a - b‖
-- dist_eq_norm_div
@[to_additive]
theorem dist_eq_norm_div (a b : E) : dist a b = ‖a / b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: dist a b = ‖a / b‖
-- dist_eq_norm_div'
@[to_additive]
theorem dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: dist a b = ‖b / a‖
-- dist_eq_norm_sub'
@[to_additive]
theorem dist_eq_norm_div' (a b : E) : dist a b = ‖b / a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: dist a b = ‖b - a‖
-- DiscreteTopology.of_forall_le_norm
@[to_additive of_forall_le_norm]
lemma DiscreteTopology.of_forall_le_norm' (hpos : 0 < r) (hr : ∀ x : E, x ≠ 1 → r ≤ ‖x‖) :
    DiscreteTopology E 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {r : ℝ} (hpos : 0 < r) (hr : ∀ (x : E), x ≠ 0 → r ≤ ‖x‖)
Goal: DiscreteTopology E
-- DiscreteTopology.of_forall_le_norm'
@[to_additive of_forall_le_norm]
lemma DiscreteTopology.of_forall_le_norm' (hpos : 0 < r) (hr : ∀ x : E, x ≠ 1 → r ≤ ‖x‖) :
    DiscreteTopology E 
Arguments: {E : Type u_6} [SeminormedGroup E] {r : ℝ} (hpos : 0 < r) (hr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖)
Goal: DiscreteTopology E
-- dist_zero_right
@[to_additive (attr := simp)]
theorem dist_one_right (a : E) : dist a 1 = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: dist a 0 = ‖a‖
-- dist_one_right
@[to_additive (attr := simp)]
theorem dist_one_right (a : E) : dist a 1 = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: dist a 1 = ‖a‖
-- inseparable_zero_iff_norm
@[to_additive]
theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a : E}
Goal: Inseparable a 0 ↔ ‖a‖ = 0
-- inseparable_one_iff_norm
@[to_additive]
theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 
Arguments: {E : Type u_6} [SeminormedGroup E] {a : E}
Goal: Inseparable a 1 ↔ ‖a‖ = 0
-- dist_one_left
@[to_additive (attr := simp)]
theorem dist_one_left : dist (1 : E) = norm 
Arguments: {E : Type u_6} [SeminormedGroup E]
Goal: dist 1 = norm
-- dist_zero_left
@[to_additive (attr := simp)]
theorem dist_one_left : dist (1 : E) = norm 
Arguments: {E : Type u_6} [SeminormedAddGroup E]
Goal: dist 0 = norm
-- norm_sub_rev
@[to_additive]
theorem norm_div_rev (a b : E) : ‖a / b‖ = ‖b / a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a - b‖ = ‖b - a‖
-- norm_div_rev
@[to_additive]
theorem norm_div_rev (a b : E) : ‖a / b‖ = ‖b / a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a / b‖ = ‖b / a‖
-- norm_neg
@[to_additive (attr := simp) norm_neg]
theorem norm_inv' (a : E) : ‖a⁻¹‖ = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a : E)
Goal: ‖-a‖ = ‖a‖
-- norm_inv'
@[to_additive (attr := simp) norm_neg]
theorem norm_inv' (a : E) : ‖a⁻¹‖ = ‖a‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a : E)
Goal: ‖a⁻¹‖ = ‖a‖
-- dist_mulIndicator
@[to_additive]
theorem dist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedGroup E] (s t : Set α) (f : α → E) (x : α)
Goal: dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(symmDiff s t).mulIndicator f x‖
-- dist_indicator
@[to_additive]
theorem dist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
    dist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖ 
Arguments: {α : Type u_3} {E : Type u_6} [SeminormedAddGroup E] (s t : Set α) (f : α → E) (x : α)
Goal: dist (s.indicator f x) (t.indicator f x) = ‖(symmDiff s t).indicator f x‖
-- norm_add_le
@[to_additive norm_add_le "**Triangle inequality** for the norm."]
theorem norm_mul_le' (a b : E) : ‖a * b‖ ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b : E)
Goal: ‖a + b‖ ≤ ‖a‖ + ‖b‖
-- norm_mul_le'
@[to_additive norm_add_le "**Triangle inequality** for the norm."]
theorem norm_mul_le' (a b : E) : ‖a * b‖ ≤ ‖a‖ + ‖b‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b : E)
Goal: ‖a * b‖ ≤ ‖a‖ + ‖b‖
-- norm_add_le_of_le
@[to_additive]
theorem norm_mul_le_of_le (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂) : ‖a₁ * a₂‖ ≤ r₁ + r₂ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] {a₁ a₂ : E} {r₁ r₂ : ℝ} (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂)
Goal: ‖a₁ + a₂‖ ≤ r₁ + r₂
-- norm_mul_le_of_le
@[to_additive]
theorem norm_mul_le_of_le (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂) : ‖a₁ * a₂‖ ≤ r₁ + r₂ 
Arguments: {E : Type u_6} [SeminormedGroup E] {a₁ a₂ : E} {r₁ r₂ : ℝ} (h₁ : ‖a₁‖ ≤ r₁) (h₂ : ‖a₂‖ ≤ r₂)
Goal: ‖a₁ * a₂‖ ≤ r₁ + r₂
-- norm_mul₃_le
@[to_additive norm_add₃_le]
theorem norm_mul₃_le (a b c : E) : ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ 
Arguments: {E : Type u_6} [SeminormedGroup E] (a b c : E)
Goal: ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖
-- norm_add₃_le
@[to_additive norm_add₃_le]
theorem norm_mul₃_le (a b c : E) : ‖a * b * c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b c : E)
Goal: ‖a + b + c‖ ≤ ‖a‖ + ‖b‖ + ‖c‖
-- norm_sub_le_norm_sub_add_norm_sub
@[to_additive]
lemma norm_div_le_norm_div_add_norm_div (a b c : E) : ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖ 
Arguments: {E : Type u_6} [SeminormedAddGroup E] (a b c : E)
Goal: ‖a - c‖ ≤ ‖a - b‖ + ‖b - c‖