ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- GroupSeminorm.map_one'
@[to_additive]
structure GroupSeminorm (G : Type*) [Group G] where
    protected toFun : G → ℝ
    protected map_one' : toFun 1 = 0
    protected mul_le' : ∀ x y, toFun (x * y) ≤ toFun x + toFun y
    protected inv' : ∀ x, toFun x⁻¹ = toFun x
Arguments: {G : Type u_7} [Group G] (self : GroupSeminorm G)
Goal: self.toFun 1 = 0
-- NonarchAddGroupSeminorm.neg'
structure NonarchAddGroupSeminorm (G : Type*) [AddGroup G] extends ZeroHom G ℝ where
    protected add_le_max' : ∀ r s, toFun (r + s) ≤ max (toFun r) (toFun s)
    protected neg' : ∀ r, toFun (-r) = toFun r
Arguments: {G : Type u_7} [AddGroup G] (self : NonarchAddGroupSeminorm G) (r : G)
Goal: self.toFun (-r) = self.toFun r
-- NonarchAddGroupSeminorm.add_le_max'
structure NonarchAddGroupSeminorm (G : Type*) [AddGroup G] extends ZeroHom G ℝ where
    protected add_le_max' : ∀ r s, toFun (r + s) ≤ max (toFun r) (toFun s)
    protected neg' : ∀ r, toFun (-r) = toFun r
Arguments: {G : Type u_7} [AddGroup G] (self : NonarchAddGroupSeminorm G) (r s : G)
Goal: self.toFun (r + s) ≤ max (self.toFun r) (self.toFun s)
-- AddGroupNorm.eq_zero_of_map_eq_zero'
structure AddGroupNorm (G : Type*) [AddGroup G] extends AddGroupSeminorm G where
    protected eq_zero_of_map_eq_zero' : ∀ x, toFun x = 0 → x = 0
Arguments: {G : Type u_7} [AddGroup G] (self : AddGroupNorm G) (x : G)
Goal: self.toFun x = 0 → x = 0
-- GroupNorm.eq_one_of_map_eq_zero'
@[to_additive]
structure GroupNorm (G : Type*) [Group G] extends GroupSeminorm G where
    protected eq_one_of_map_eq_zero' : ∀ x, toFun x = 0 → x = 1
Arguments: {G : Type u_7} [Group G] (self : GroupNorm G) (x : G)
Goal: self.toFun x = 0 → x = 1
-- NonarchAddGroupNorm.eq_zero_of_map_eq_zero'
structure NonarchAddGroupNorm (G : Type*) [AddGroup G] extends NonarchAddGroupSeminorm G where
    protected eq_zero_of_map_eq_zero' : ∀ x, toFun x = 0 → x = 0
Arguments: {G : Type u_7} [AddGroup G] (self : NonarchAddGroupNorm G) (x : G)
Goal: self.toFun x = 0 → x = 0
-- NonarchAddGroupSeminormClass.map_neg_eq_map'
class NonarchAddGroupSeminormClass (F : Type*) (α : outParam Type*) [AddGroup α] [FunLike F α ℝ]
    extends NonarchimedeanHomClass F α ℝ : Prop where
    protected map_zero (f : F) : f 0 = 0
    protected map_neg_eq_map' (f : F) (a : α) : f (-a) = f a
Arguments: {F : Type u_7} {α : outParam (Type u_8)} [AddGroup α] [FunLike F α ℝ] [self : NonarchAddGroupSeminormClass F α] (f : F) (a : α)
Goal: f (-a) = f a
-- NonarchAddGroupSeminormClass.map_zero
class NonarchAddGroupSeminormClass (F : Type*) (α : outParam Type*) [AddGroup α] [FunLike F α ℝ]
    extends NonarchimedeanHomClass F α ℝ : Prop where
    protected map_zero (f : F) : f 0 = 0
    protected map_neg_eq_map' (f : F) (a : α) : f (-a) = f a
Arguments: {F : Type u_7} {α : outParam (Type u_8)} [AddGroup α] [FunLike F α ℝ] [self : NonarchAddGroupSeminormClass F α] (f : F)
Goal: f 0 = 0
-- NonarchAddGroupNormClass.eq_zero_of_map_eq_zero
class NonarchAddGroupNormClass (F : Type*) (α : outParam Type*) [AddGroup α] [FunLike F α ℝ]
    extends NonarchAddGroupSeminormClass F α : Prop where
    protected eq_zero_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 0
Arguments: {F : Type u_7} {α : outParam (Type u_8)} [AddGroup α] [FunLike F α ℝ] [self : NonarchAddGroupNormClass F α] (f : F) {a : α}
Goal: f a = 0 → a = 0
-- map_sub_le_max
theorem map_sub_le_max : f (x - y) ≤ max (f x) (f y) 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [FunLike F E ℝ] [NonarchAddGroupSeminormClass F E] (f : F) (x y : E)
Goal: f (x - y) ≤ max (f x) (f y)
-- GroupSeminorm.toFun_eq_coe
@[to_additive (attr := simp)]
theorem toFun_eq_coe : p.toFun = p 
Arguments: {E : Type u_4} [Group E] {p : GroupSeminorm E}
Goal: p.toFun = ⇑p
-- AddGroupSeminorm.toFun_eq_coe
@[to_additive (attr := simp)]
theorem toFun_eq_coe : p.toFun = p 
Arguments: {E : Type u_4} [AddGroup E] {p : AddGroupSeminorm E}
Goal: p.toFun = ⇑p
-- GroupSeminorm.ext
@[to_additive (attr := ext)]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [Group E] {p q : GroupSeminorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- AddGroupSeminorm.ext
@[to_additive (attr := ext)]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupSeminorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- GroupSeminorm.le_def
@[to_additive]
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [Group E] {p q : GroupSeminorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- AddGroupSeminorm.le_def
@[to_additive]
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupSeminorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- GroupSeminorm.lt_def
@[to_additive]
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [Group E] {p q : GroupSeminorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- AddGroupSeminorm.lt_def
@[to_additive]
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupSeminorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- AddGroupSeminorm.coe_le_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupSeminorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- GroupSeminorm.coe_le_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [Group E] {p q : GroupSeminorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- AddGroupSeminorm.coe_lt_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupSeminorm E}
Goal: ⇑p < ⇑q ↔ p < q
-- GroupSeminorm.coe_lt_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [Group E] {p q : GroupSeminorm E}
Goal: ⇑p < ⇑q ↔ p < q
-- GroupSeminorm.coe_zero
@[to_additive (attr := simp, norm_cast)]
theorem coe_zero : ⇑(0 : GroupSeminorm E) = 0 
Arguments: {E : Type u_4} [Group E]
Goal: ⇑0 = 0
-- AddGroupSeminorm.coe_zero
@[to_additive (attr := simp, norm_cast)]
theorem coe_zero : ⇑(0 : GroupSeminorm E) = 0 
Arguments: {E : Type u_4} [AddGroup E]
Goal: ⇑0 = 0
-- GroupSeminorm.zero_apply
@[to_additive (attr := simp)]
theorem zero_apply (x : E) : (0 : GroupSeminorm E) x = 0 
Arguments: {E : Type u_4} [Group E] (x : E)
Goal: 0 x = 0
-- AddGroupSeminorm.zero_apply
@[to_additive (attr := simp)]
theorem zero_apply (x : E) : (0 : GroupSeminorm E) x = 0 
Arguments: {E : Type u_4} [AddGroup E] (x : E)
Goal: 0 x = 0
-- GroupSeminorm.coe_add
@[to_additive (attr := simp)]
theorem coe_add : ⇑(p + q) = p + q 
Arguments: {E : Type u_4} [Group E] (p q : GroupSeminorm E)
Goal: ⇑(p + q) = ⇑p + ⇑q
-- AddGroupSeminorm.coe_add
@[to_additive (attr := simp)]
theorem coe_add : ⇑(p + q) = p + q 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupSeminorm E)
Goal: ⇑(p + q) = ⇑p + ⇑q
-- AddGroupSeminorm.add_apply
@[to_additive (attr := simp)]
theorem add_apply (x : E) : (p + q) x = p x + q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupSeminorm E) (x : E)
Goal: (p + q) x = p x + q x
-- GroupSeminorm.add_apply
@[to_additive (attr := simp)]
theorem add_apply (x : E) : (p + q) x = p x + q x 
Arguments: {E : Type u_4} [Group E] (p q : GroupSeminorm E) (x : E)
Goal: (p + q) x = p x + q x
-- GroupSeminorm.coe_sup
@[to_additive (attr := simp, norm_cast)]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [Group E] (p q : GroupSeminorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- AddGroupSeminorm.coe_sup
@[to_additive (attr := simp, norm_cast)]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupSeminorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- AddGroupSeminorm.sup_apply
@[to_additive (attr := simp)]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupSeminorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- GroupSeminorm.sup_apply
@[to_additive (attr := simp)]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [Group E] (p q : GroupSeminorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- AddGroupSeminorm.coe_comp
@[to_additive (attr := simp)]
theorem coe_comp : ⇑(p.comp f) = p ∘ f 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] (p : AddGroupSeminorm E) (f : F →+ E)
Goal: ⇑(p.comp f) = ⇑p ∘ ⇑f
-- GroupSeminorm.coe_comp
@[to_additive (attr := simp)]
theorem coe_comp : ⇑(p.comp f) = p ∘ f 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] (p : GroupSeminorm E) (f : F →* E)
Goal: ⇑(p.comp f) = ⇑p ∘ ⇑f
-- GroupSeminorm.comp_apply
@[to_additive (attr := simp)]
theorem comp_apply (x : F) : (p.comp f) x = p (f x) 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] (p : GroupSeminorm E) (f : F →* E) (x : F)
Goal: (p.comp f) x = p (f x)
-- AddGroupSeminorm.comp_apply
@[to_additive (attr := simp)]
theorem comp_apply (x : F) : (p.comp f) x = p (f x) 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] (p : AddGroupSeminorm E) (f : F →+ E) (x : F)
Goal: (p.comp f) x = p (f x)
-- AddGroupSeminorm.comp_id
@[to_additive (attr := simp)]
theorem comp_id : p.comp (MonoidHom.id _) = p 
Arguments: {E : Type u_4} [AddGroup E] (p : AddGroupSeminorm E)
Goal: p.comp (AddMonoidHom.id E) = p
-- GroupSeminorm.comp_id
@[to_additive (attr := simp)]
theorem comp_id : p.comp (MonoidHom.id _) = p 
Arguments: {E : Type u_4} [Group E] (p : GroupSeminorm E)
Goal: p.comp (MonoidHom.id E) = p
-- AddGroupSeminorm.comp_zero
@[to_additive (attr := simp)]
theorem comp_zero : p.comp (1 : F →* E) = 0 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] (p : AddGroupSeminorm E)
Goal: p.comp 0 = 0
-- GroupSeminorm.comp_zero
@[to_additive (attr := simp)]
theorem comp_zero : p.comp (1 : F →* E) = 0 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] (p : GroupSeminorm E)
Goal: p.comp 1 = 0
-- GroupSeminorm.zero_comp
@[to_additive (attr := simp)]
theorem zero_comp : (0 : GroupSeminorm E).comp f = 0 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] (f : F →* E)
Goal: GroupSeminorm.comp 0 f = 0
-- AddGroupSeminorm.zero_comp
@[to_additive (attr := simp)]
theorem zero_comp : (0 : GroupSeminorm E).comp f = 0 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] (f : F →+ E)
Goal: AddGroupSeminorm.comp 0 f = 0
-- GroupSeminorm.comp_assoc
@[to_additive]
theorem comp_assoc (g : F →* E) (f : G →* F) : p.comp (g.comp f) = (p.comp g).comp f 
Arguments: {E : Type u_4} {F : Type u_5} {G : Type u_6} [Group E] [Group F] [Group G] (p : GroupSeminorm E) (g : F →* E) (f : G →* F)
Goal: p.comp (g.comp f) = (p.comp g).comp f
-- AddGroupSeminorm.comp_assoc
@[to_additive]
theorem comp_assoc (g : F →* E) (f : G →* F) : p.comp (g.comp f) = (p.comp g).comp f 
Arguments: {E : Type u_4} {F : Type u_5} {G : Type u_6} [AddGroup E] [AddGroup F] [AddGroup G] (p : AddGroupSeminorm E) (g : F →+ E) (f : G →+ F)
Goal: p.comp (g.comp f) = (p.comp g).comp f
-- GroupSeminorm.add_comp
@[to_additive]
theorem add_comp (f : F →* E) : (p + q).comp f = p.comp f + q.comp f 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] (p q : GroupSeminorm E) (f : F →* E)
Goal: (p + q).comp f = p.comp f + q.comp f
-- AddGroupSeminorm.add_comp
@[to_additive]
theorem add_comp (f : F →* E) : (p + q).comp f = p.comp f + q.comp f 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] (p q : AddGroupSeminorm E) (f : F →+ E)
Goal: (p + q).comp f = p.comp f + q.comp f
-- AddGroupSeminorm.comp_mono
@[to_additive]
theorem comp_mono (hp : p ≤ q) : p.comp f ≤ q.comp f 
Arguments: {E : Type u_4} {F : Type u_5} [AddGroup E] [AddGroup F] {p q : AddGroupSeminorm E} (f : F →+ E) (hp : p ≤ q)
Goal: p.comp f ≤ q.comp f
-- GroupSeminorm.comp_mono
@[to_additive]
theorem comp_mono (hp : p ≤ q) : p.comp f ≤ q.comp f 
Arguments: {E : Type u_4} {F : Type u_5} [Group E] [Group F] {p q : GroupSeminorm E} (f : F →* E) (hp : p ≤ q)
Goal: p.comp f ≤ q.comp f
-- AddGroupSeminorm.comp_add_le
@[to_additive]
theorem comp_mul_le (f g : F →* E) : p.comp (f * g) ≤ p.comp f + p.comp g 
Arguments: {E : Type u_4} {F : Type u_5} [AddCommGroup E] [AddCommGroup F] (p : AddGroupSeminorm E) (f g : F →+ E)
Goal: p.comp (f + g) ≤ p.comp f + p.comp g
-- GroupSeminorm.comp_mul_le
@[to_additive]
theorem comp_mul_le (f g : F →* E) : p.comp (f * g) ≤ p.comp f + p.comp g 
Arguments: {E : Type u_4} {F : Type u_5} [CommGroup E] [CommGroup F] (p : GroupSeminorm E) (f g : F →* E)
Goal: p.comp (f * g) ≤ p.comp f + p.comp g
-- GroupSeminorm.mul_bddBelow_range_add
@[to_additive]
theorem mul_bddBelow_range_add {p q : GroupSeminorm E} {x : E} :
    BddBelow (range fun y => p y + q (x / y)) 
Arguments: {E : Type u_4} [CommGroup E] {p q : GroupSeminorm E} {x : E}
Goal: BddBelow (Set.range fun y => p y + q (x / y))
-- AddGroupSeminorm.add_bddBelow_range_add
@[to_additive]
theorem mul_bddBelow_range_add {p q : GroupSeminorm E} {x : E} :
    BddBelow (range fun y => p y + q (x / y)) 
Arguments: {E : Type u_4} [AddCommGroup E] {p q : AddGroupSeminorm E} {x : E}
Goal: BddBelow (Set.range fun y => p y + q (x - y))
-- AddGroupSeminorm.inf_apply
@[to_additive (attr := simp)]
theorem inf_apply : (p ⊓ q) x = ⨅ y, p y + q (x / y) 
Arguments: {E : Type u_4} [AddCommGroup E] (p q : AddGroupSeminorm E) (x : E)
Goal: (p ⊓ q) x = ⨅ y, p y + q (x - y)
-- GroupSeminorm.inf_apply
@[to_additive (attr := simp)]
theorem inf_apply : (p ⊓ q) x = ⨅ y, p y + q (x / y) 
Arguments: {E : Type u_4} [CommGroup E] (p q : GroupSeminorm E) (x : E)
Goal: (p ⊓ q) x = ⨅ y, p y + q (x / y)
-- AddGroupSeminorm.apply_one
@[simp]
theorem apply_one [DecidableEq E] (x : E) : (1 : AddGroupSeminorm 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
-- AddGroupSeminorm.coe_smul
@[simp, norm_cast]
theorem coe_smul (r : R) (p : AddGroupSeminorm E) : ⇑(r • p) = r • ⇑p 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : AddGroupSeminorm E)
Goal: ⇑(r • p) = r • ⇑p
-- AddGroupSeminorm.smul_apply
@[simp]
theorem smul_apply (r : R) (p : AddGroupSeminorm E) (x : E) : (r • p) x = r • p x 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : AddGroupSeminorm E) (x : E)
Goal: (r • p) x = r • p x
-- AddGroupSeminorm.smul_sup
theorem smul_sup (r : R) (p q : AddGroupSeminorm E) : r • (p ⊔ q) = r • p ⊔ r • q 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : AddGroupSeminorm E)
Goal: r • (p ⊔ q) = r • p ⊔ r • q
-- NonarchAddGroupSeminorm.toZeroHom_eq_coe
@[simp]
theorem toZeroHom_eq_coe : ⇑p.toZeroHom = p 
Arguments: {E : Type u_4} [AddGroup E] {p : NonarchAddGroupSeminorm E}
Goal: ⇑p.toZeroHom = ⇑p
-- NonarchAddGroupSeminorm.ext
@[ext]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupSeminorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- NonarchAddGroupSeminorm.le_def
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupSeminorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- NonarchAddGroupSeminorm.lt_def
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupSeminorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- NonarchAddGroupSeminorm.coe_le_coe
@[simp, norm_cast]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupSeminorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- NonarchAddGroupSeminorm.coe_lt_coe
@[simp, norm_cast]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupSeminorm E}
Goal: ⇑p < ⇑q ↔ p < q
-- NonarchAddGroupSeminorm.coe_zero
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : NonarchAddGroupSeminorm E) = 0 
Arguments: {E : Type u_4} [AddGroup E]
Goal: ⇑0 = 0
-- NonarchAddGroupSeminorm.zero_apply
@[simp]
theorem zero_apply (x : E) : (0 : NonarchAddGroupSeminorm E) x = 0 
Arguments: {E : Type u_4} [AddGroup E] (x : E)
Goal: 0 x = 0
-- NonarchAddGroupSeminorm.coe_sup
@[simp, norm_cast]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [AddGroup E] (p q : NonarchAddGroupSeminorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- NonarchAddGroupSeminorm.sup_apply
@[simp]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : NonarchAddGroupSeminorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- NonarchAddGroupSeminorm.add_bddBelow_range_add
theorem add_bddBelow_range_add {p q : NonarchAddGroupSeminorm E} {x : E} :
    BddBelow (range fun y => p y + q (x - y)) 
Arguments: {E : Type u_4} [AddCommGroup E] {p q : NonarchAddGroupSeminorm E} {x : E}
Goal: BddBelow (Set.range fun y => p y + q (x - y))
-- GroupSeminorm.apply_one
@[to_additive (attr := simp) existing AddGroupSeminorm.apply_one]
theorem apply_one [DecidableEq E] (x : E) : (1 : GroupSeminorm E) x = if x = 1 then 0 else 1 
Arguments: {E : Type u_4} [Group E] [DecidableEq E] (x : E)
Goal: 1 x = if x = 1 then 0 else 1
-- GroupSeminorm.coe_smul
@[to_additive (attr := simp, norm_cast) existing AddGroupSeminorm.coe_smul]
theorem coe_smul (r : R) (p : GroupSeminorm E) : ⇑(r • p) = r • ⇑p 
Arguments: {R : Type u_2} {E : Type u_4} [Group E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : GroupSeminorm E)
Goal: ⇑(r • p) = r • ⇑p
-- GroupSeminorm.smul_apply
@[to_additive (attr := simp) existing AddGroupSeminorm.smul_apply]
theorem smul_apply (r : R) (p : GroupSeminorm E) (x : E) : (r • p) x = r • p x 
Arguments: {R : Type u_2} {E : Type u_4} [Group E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : GroupSeminorm E) (x : E)
Goal: (r • p) x = r • p x
-- GroupSeminorm.smul_sup
@[to_additive existing AddGroupSeminorm.smul_sup]
theorem smul_sup (r : R) (p q : GroupSeminorm E) : r • (p ⊔ q) = r • p ⊔ r • q 
Arguments: {R : Type u_2} {E : Type u_4} [Group E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : GroupSeminorm E)
Goal: r • (p ⊔ q) = r • p ⊔ r • q
-- NonarchAddGroupSeminorm.apply_one
@[simp]
theorem apply_one [DecidableEq E] (x : E) :
    (1 : NonarchAddGroupSeminorm 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
-- NonarchAddGroupSeminorm.coe_smul
@[simp, norm_cast]
theorem coe_smul (r : R) (p : NonarchAddGroupSeminorm E) : ⇑(r • p) = r • ⇑p 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : NonarchAddGroupSeminorm E)
Goal: ⇑(r • p) = r • ⇑p
-- NonarchAddGroupSeminorm.smul_apply
@[simp]
theorem smul_apply (r : R) (p : NonarchAddGroupSeminorm E) (x : E) : (r • p) x = r • p x 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p : NonarchAddGroupSeminorm E) (x : E)
Goal: (r • p) x = r • p x
-- NonarchAddGroupSeminorm.smul_sup
theorem smul_sup (r : R) (p q : NonarchAddGroupSeminorm E) : r • (p ⊔ q) = r • p ⊔ r • q 
Arguments: {R : Type u_2} {E : Type u_4} [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (r : R) (p q : NonarchAddGroupSeminorm E)
Goal: r • (p ⊔ q) = r • p ⊔ r • q
-- GroupNorm.toGroupSeminorm_eq_coe
@[to_additive (attr := simp)]
theorem toGroupSeminorm_eq_coe : ⇑p.toGroupSeminorm = p 
Arguments: {E : Type u_4} [Group E] {p : GroupNorm E}
Goal: ⇑p.toGroupSeminorm = ⇑p
-- AddGroupNorm.toAddGroupSeminorm_eq_coe
@[to_additive (attr := simp)]
theorem toGroupSeminorm_eq_coe : ⇑p.toGroupSeminorm = p 
Arguments: {E : Type u_4} [AddGroup E] {p : AddGroupNorm E}
Goal: ⇑p.toAddGroupSeminorm = ⇑p
-- AddGroupNorm.ext
@[to_additive (attr := ext)]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupNorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- GroupNorm.ext
@[to_additive (attr := ext)]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [Group E] {p q : GroupNorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- AddGroupNorm.le_def
@[to_additive]
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupNorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- GroupNorm.le_def
@[to_additive]
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [Group E] {p q : GroupNorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- GroupNorm.lt_def
@[to_additive]
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [Group E] {p q : GroupNorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- AddGroupNorm.lt_def
@[to_additive]
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupNorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- AddGroupNorm.coe_le_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupNorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- GroupNorm.coe_le_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [Group E] {p q : GroupNorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- AddGroupNorm.coe_lt_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : AddGroupNorm E}
Goal: ⇑p < ⇑q ↔ p < q
-- GroupNorm.coe_lt_coe
@[to_additive (attr := simp, norm_cast)]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [Group E] {p q : GroupNorm E}
Goal: ⇑p < ⇑q ↔ p < q
-- AddGroupNorm.coe_add
@[to_additive (attr := simp)]
theorem coe_add : ⇑(p + q) = p + q 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupNorm E)
Goal: ⇑(p + q) = ⇑p + ⇑q
-- GroupNorm.coe_add
@[to_additive (attr := simp)]
theorem coe_add : ⇑(p + q) = p + q 
Arguments: {E : Type u_4} [Group E] (p q : GroupNorm E)
Goal: ⇑(p + q) = ⇑p + ⇑q
-- GroupNorm.add_apply
@[to_additive (attr := simp)]
theorem add_apply (x : E) : (p + q) x = p x + q x 
Arguments: {E : Type u_4} [Group E] (p q : GroupNorm E) (x : E)
Goal: (p + q) x = p x + q x
-- AddGroupNorm.add_apply
@[to_additive (attr := simp)]
theorem add_apply (x : E) : (p + q) x = p x + q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupNorm E) (x : E)
Goal: (p + q) x = p x + q x
-- AddGroupNorm.coe_sup
@[to_additive (attr := simp, norm_cast)]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupNorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- GroupNorm.coe_sup
@[to_additive (attr := simp, norm_cast)]
theorem coe_sup : ⇑(p ⊔ q) = ⇑p ⊔ ⇑q 
Arguments: {E : Type u_4} [Group E] (p q : GroupNorm E)
Goal: ⇑(p ⊔ q) = ⇑p ⊔ ⇑q
-- AddGroupNorm.sup_apply
@[to_additive (attr := simp)]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [AddGroup E] (p q : AddGroupNorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- GroupNorm.sup_apply
@[to_additive (attr := simp)]
theorem sup_apply (x : E) : (p ⊔ q) x = p x ⊔ q x 
Arguments: {E : Type u_4} [Group E] (p q : GroupNorm E) (x : E)
Goal: (p ⊔ q) x = p x ⊔ q x
-- AddGroupNorm.apply_one
@[simp]
theorem apply_one (x : E) : (1 : AddGroupNorm 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
-- GroupNorm.apply_one
@[to_additive (attr := simp) existing AddGroupNorm.apply_one]
theorem apply_one (x : E) : (1 : GroupNorm E) x = if x = 1 then 0 else 1 
Arguments: {E : Type u_4} [Group E] [DecidableEq E] (x : E)
Goal: 1 x = if x = 1 then 0 else 1
-- NonarchAddGroupNorm.toNonarchAddGroupSeminorm_eq_coe
@[simp]
theorem toNonarchAddGroupSeminorm_eq_coe : ⇑p.toNonarchAddGroupSeminorm = p 
Arguments: {E : Type u_4} [AddGroup E] {p : NonarchAddGroupNorm E}
Goal: ⇑p.toNonarchAddGroupSeminorm = ⇑p
-- NonarchAddGroupNorm.ext
@[ext]
theorem ext : (∀ x, p x = q x) → p = q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupNorm E}
Goal: (∀ (x : E), p x = q x) → p = q
-- NonarchAddGroupNorm.le_def
theorem le_def : p ≤ q ↔ (p : E → ℝ) ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupNorm E}
Goal: p ≤ q ↔ ⇑p ≤ ⇑q
-- NonarchAddGroupNorm.lt_def
theorem lt_def : p < q ↔ (p : E → ℝ) < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupNorm E}
Goal: p < q ↔ ⇑p < ⇑q
-- NonarchAddGroupNorm.coe_le_coe
@[simp, norm_cast]
theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupNorm E}
Goal: ⇑p ≤ ⇑q ↔ p ≤ q
-- NonarchAddGroupNorm.coe_lt_coe
@[simp, norm_cast]
theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q 
Arguments: {E : Type u_4} [AddGroup E] {p q : NonarchAddGroupNorm E}
Goal: ⇑p < ⇑q ↔ p < q