Documentation

Mathlib.Algebra.Order.Hom.Basic

Algebraic order homomorphism classes #

This file defines hom classes for common properties at the intersection of order theory and algebra.

Typeclasses #

Basic typeclasses

NonnegHomClass: Homs are nonnegative: ∀ f a, 0 ≤ f a
SubadditiveHomClass: Homs are subadditive: ∀ f a b, f (a + b) ≤ f a + f b
SubmultiplicativeHomClass: Homs are submultiplicative: ∀ f a b, f (a * b) ≤ f a * f b
MulLEAddHomClass: ∀ f a b, f (a * b) ≤ f a + f b
NonarchimedeanHomClass: ∀ a b, f (a + b) ≤ max (f a) (f b)

Group norms

AddGroupSeminormClass: Homs are nonnegative, subadditive, even and preserve zero.
GroupSeminormClass: Homs are nonnegative, respect f (a * b) ≤ f a + f b, f a⁻¹ = f a and preserve zero.
AddGroupNormClass: Homs are seminorms such that f x = 0 → x = 0 for all x.
GroupNormClass: Homs are seminorms such that f x = 0 → x = 1 for all x.

Ring norms

RingSeminormClass: Homs are submultiplicative group norms.
RingNormClass: Homs are ring seminorms that are also additive group norms.
MulRingSeminormClass: Homs are ring seminorms that are multiplicative.
MulRingNormClass: Homs are ring norms that are multiplicative.

Notes #

Typeclasses for seminorms are defined here while types of seminorms are defined in Analysis.Normed.Group.Seminorm and Analysis.Normed.Ring.Seminorm because absolute values are multiplicative ring norms but outside of this use we only consider real-valued seminorms.

TODO #

Finitary versions of the current lemmas.

Basics #

class NonnegHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Zero β] [LE β] [FunLike F α β] :

NonnegHomClass F α β states that F is a type of nonnegative morphisms.

apply_nonneg : ∀ (f : F) (a : α), 0 ≤ f a
the image of any element is non negative.

Instances

@[simp]

theorem NonnegHomClass.apply_nonneg {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Zero β} {inst_1 : LE β} {inst_2 : FunLike F α β} [self : NonnegHomClass F α β] (f : F) (a : α), 0 ≤ f a

the image of any element is non negative.

class SubadditiveHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Add α] [Add β] [LE β] [FunLike F α β] :

SubadditiveHomClass F α β states that F is a type of subadditive morphisms.

map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
the image of a sum is less or equal than the sum of the images.

Instances

theorem SubadditiveHomClass.map_add_le_add {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Add α} {inst_1 : Add β} {inst_2 : LE β} {inst_3 : FunLike F α β} [self : SubadditiveHomClass F α β] (f : F) (a b : α), f (a + b) ≤ f a + f b

the image of a sum is less or equal than the sum of the images.

class SubmultiplicativeHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Mul α] [Mul β] [LE β] [FunLike F α β] :

SubmultiplicativeHomClass F α β states that F is a type of submultiplicative morphisms.

map_mul_le_mul : ∀ (f : F) (a b : α), f (a * b) ≤ f a * f b
the image of a product is less or equal than the product of the images.

Instances

theorem SubmultiplicativeHomClass.map_mul_le_mul {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Mul α} {inst_1 : Mul β} {inst_2 : LE β} {inst_3 : FunLike F α β} [self : SubmultiplicativeHomClass F α β] (f : F) (a b : α), f (a * b) ≤ f a * f b

the image of a product is less or equal than the product of the images.

class MulLEAddHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Mul α] [Add β] [LE β] [FunLike F α β] :

MulLEAddHomClass F α β states that F is a type of subadditive morphisms.

map_mul_le_add : ∀ (f : F) (a b : α), f (a * b) ≤ f a + f b
the image of a product is less or equal than the sum of the images.

Instances

theorem MulLEAddHomClass.map_mul_le_add {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Mul α} {inst_1 : Add β} {inst_2 : LE β} {inst_3 : FunLike F α β} [self : MulLEAddHomClass F α β] (f : F) (a b : α), f (a * b) ≤ f a + f b

the image of a product is less or equal than the sum of the images.

class NonarchimedeanHomClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Add α] [LinearOrder β] [FunLike F α β] :

NonarchimedeanHomClass F α β states that F is a type of non-archimedean morphisms.

map_add_le_max : ∀ (f : F) (a b : α), f (a + b) ≤ f a ⊔ f b
the image of a sum is less or equal than the maximum of the images.

Instances

theorem NonarchimedeanHomClass.map_add_le_max {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Add α} {inst_1 : LinearOrder β} {inst_2 : FunLike F α β} [self : NonarchimedeanHomClass F α β] (f : F) (a b : α), f (a + b) ≤ f a ⊔ f b

the image of a sum is less or equal than the maximum of the images.

theorem le_map_mul_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [CommSemigroup β] [LE β] [SubmultiplicativeHomClass F α β] (f : F) (a : α) (b : α) :

f a ≤ f b * f (a / b)

theorem le_map_add_map_sub {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [AddCommSemigroup β] [LE β] [SubadditiveHomClass F α β] (f : F) (a : α) (b : α) :

f a ≤ f b + f (a - b)

theorem le_map_add_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [AddCommSemigroup β] [LE β] [MulLEAddHomClass F α β] (f : F) (a : α) (b : α) :

f a ≤ f b + f (a / b)

theorem le_map_div_mul_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [CommSemigroup β] [LE β] [SubmultiplicativeHomClass F α β] (f : F) (a : α) (b : α) (c : α) :

f (a / c) ≤ f (a / b) * f (b / c)

theorem le_map_sub_add_map_sub {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [AddCommSemigroup β] [LE β] [SubadditiveHomClass F α β] (f : F) (a : α) (b : α) (c : α) :

f (a - c) ≤ f (a - b) + f (b - c)

theorem le_map_div_add_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [AddCommSemigroup β] [LE β] [MulLEAddHomClass F α β] (f : F) (a : α) (b : α) (c : α) :

f (a / c) ≤ f (a / b) + f (b / c)

def Mathlib.Meta.Positivity.evalMap :

Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: nonnegative maps take nonnegative values.

Equations

One or more equations did not get rendered due to their size.

Instances For

Group (semi)norms #

class AddGroupSeminormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β] extends SubadditiveHomClass :

AddGroupSeminormClass F α states that F is a type of β-valued seminorms on the additive group α.

You should extend this class when you extend AddGroupSeminorm.

map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
map_zero : ∀ (f : F), f 0 = 0
The image of zero is zero.
map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
The map is invariant under negation of its argument.

Instances

theorem AddGroupSeminormClass.map_zero {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : AddGroup α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : AddGroupSeminormClass F α β] (f : F), f 0 = 0

The image of zero is zero.

@[simp]

theorem AddGroupSeminormClass.map_neg_eq_map {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : AddGroup α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : AddGroupSeminormClass F α β] (f : F) (a : α), f (-a) = f a

The map is invariant under negation of its argument.

class GroupSeminormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Group α] [OrderedAddCommMonoid β] [FunLike F α β] extends MulLEAddHomClass :

GroupSeminormClass F α states that F is a type of β-valued seminorms on the group α.

You should extend this class when you extend GroupSeminorm.

map_mul_le_add : ∀ (f : F) (a b : α), f (a * b) ≤ f a + f b
map_one_eq_zero : ∀ (f : F), f 1 = 0
The image of one is zero.
map_inv_eq_map : ∀ (f : F) (a : α), f a⁻¹ = f a
The map is invariant under inversion of its argument.

Instances

@[simp]

theorem GroupSeminormClass.map_one_eq_zero {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Group α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : GroupSeminormClass F α β] (f : F), f 1 = 0

The image of one is zero.

@[simp]

theorem GroupSeminormClass.map_inv_eq_map {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Group α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : GroupSeminormClass F α β] (f : F) (a : α), f a⁻¹ = f a

The map is invariant under inversion of its argument.

class AddGroupNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β] extends AddGroupSeminormClass , SubadditiveHomClass :

AddGroupNormClass F α states that F is a type of β-valued norms on the additive group α.

You should extend this class when you extend AddGroupNorm.

Instances

theorem AddGroupNormClass.eq_zero_of_map_eq_zero {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : AddGroup α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : AddGroupNormClass F α β] (f : F) {a : α}, f a = 0 → a = 0

The argument is zero if its image under the map is zero.

class GroupNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [Group α] [OrderedAddCommMonoid β] [FunLike F α β] extends GroupSeminormClass , MulLEAddHomClass :

GroupNormClass F α states that F is a type of β-valued norms on the group α.

You should extend this class when you extend GroupNorm.

Instances

theorem GroupNormClass.eq_one_of_map_eq_zero {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : Group α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : GroupNormClass F α β] (f : F) {a : α}, f a = 0 → a = 1

The argument is one if its image under the map is zero.

theorem AddGroupSeminormClass.toAddLEAddHomClass {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} :

∀ {inst : AddGroup α} {inst_1 : OrderedAddCommMonoid β} {inst_2 : FunLike F α β} [self : AddGroupSeminormClass F α β], SubadditiveHomClass F α β

@[instance 100]

instance AddGroupSeminormClass.toZeroHomClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupSeminormClass F α β] :

ZeroHomClass F α β

Equations

⋯ = ⋯

theorem map_div_le_add {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [OrderedAddCommMonoid β] [GroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f (x / y) ≤ f x + f y

theorem map_sub_le_add {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f (x - y) ≤ f x + f y

theorem map_div_rev {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [OrderedAddCommMonoid β] [GroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f (x / y) = f (y / x)

theorem map_sub_rev {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f (x - y) = f (y - x)

theorem le_map_add_map_div' {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [OrderedAddCommMonoid β] [GroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f x ≤ f y + f (y / x)

theorem le_map_add_map_sub' {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupSeminormClass F α β] (f : F) (x : α) (y : α) :

f x ≤ f y + f (y - x)

theorem abs_sub_map_le_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [LinearOrderedAddCommGroup β] [GroupSeminormClass F α β] (f : F) (x : α) (y : α) :

|f x - f y| ≤ f (x / y)

theorem abs_sub_map_le_sub {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [LinearOrderedAddCommGroup β] [AddGroupSeminormClass F α β] (f : F) (x : α) (y : α) :

|f x - f y| ≤ f (x - y)

@[instance 100]

instance GroupSeminormClass.toNonnegHomClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [LinearOrderedAddCommMonoid β] [GroupSeminormClass F α β] :

NonnegHomClass F α β

Equations

⋯ = ⋯

@[instance 100]

instance AddGroupSeminormClass.toNonnegHomClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [LinearOrderedAddCommMonoid β] [AddGroupSeminormClass F α β] :

NonnegHomClass F α β

Equations

⋯ = ⋯

@[simp]

theorem map_eq_zero_iff_eq_one {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [OrderedAddCommMonoid β] [GroupNormClass F α β] (f : F) {x : α} :

f x = 0 ↔ x = 1

@[simp]

theorem map_eq_zero_iff_eq_zero {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupNormClass F α β] (f : F) {x : α} :

f x = 0 ↔ x = 0

theorem map_ne_zero_iff_ne_one {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [OrderedAddCommMonoid β] [GroupNormClass F α β] (f : F) {x : α} :

f x ≠ 0 ↔ x ≠ 1

theorem map_ne_zero_iff_ne_zero {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [OrderedAddCommMonoid β] [AddGroupNormClass F α β] (f : F) {x : α} :

f x ≠ 0 ↔ x ≠ 0

theorem map_pos_of_ne_one {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [Group α] [LinearOrderedAddCommMonoid β] [GroupNormClass F α β] (f : F) {x : α} (hx : x ≠ 1) :

0 < f x

theorem map_pos_of_ne_zero {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [AddGroup α] [LinearOrderedAddCommMonoid β] [AddGroupNormClass F α β] (f : F) {x : α} (hx : x ≠ 0) :

0 < f x

Ring (semi)norms #

class RingSeminormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonUnitalNonAssocRing α] [OrderedSemiring β] [FunLike F α β] extends AddGroupSeminormClass , SubmultiplicativeHomClass , SubadditiveHomClass :

RingSeminormClass F α states that F is a type of β-valued seminorms on the ring α.

You should extend this class when you extend RingSeminorm.

Instances

class RingNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonUnitalNonAssocRing α] [OrderedSemiring β] [FunLike F α β] extends RingSeminormClass , AddGroupNormClass , AddGroupSeminormClass , SubmultiplicativeHomClass , SubadditiveHomClass :

RingNormClass F α states that F is a type of β-valued norms on the ring α.

You should extend this class when you extend RingNorm.

Instances

class MulRingSeminormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonAssocRing α] [OrderedSemiring β] [FunLike F α β] extends AddGroupSeminormClass , MonoidWithZeroHomClass , SubadditiveHomClass , MonoidHomClass , ZeroHomClass , MulHomClass , OneHomClass :

MulRingSeminormClass F α states that F is a type of β-valued multiplicative seminorms on the ring α.

You should extend this class when you extend MulRingSeminorm.

Instances

class MulRingNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonAssocRing α] [OrderedSemiring β] [FunLike F α β] extends MulRingSeminormClass , AddGroupNormClass , AddGroupSeminormClass , MonoidWithZeroHomClass , SubadditiveHomClass , MonoidHomClass , ZeroHomClass , MulHomClass , OneHomClass :

MulRingNormClass F α states that F is a type of β-valued multiplicative norms on the ring α.

You should extend this class when you extend MulRingNorm.

Instances

@[instance 100]

instance RingSeminormClass.toNonnegHomClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [NonUnitalNonAssocRing α] [LinearOrderedSemiring β] [RingSeminormClass F α β] :

NonnegHomClass F α β

Equations

⋯ = ⋯

@[instance 100]

instance MulRingSeminormClass.toRingSeminormClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [NonAssocRing α] [OrderedSemiring β] [MulRingSeminormClass F α β] :

RingSeminormClass F α β

Equations

⋯ = ⋯

@[instance 100]

instance MulRingNormClass.toRingNormClass {F : Type u_2} {α : Type u_3} {β : Type u_4} [FunLike F α β] [NonAssocRing α] [OrderedSemiring β] [MulRingNormClass F α β] :

RingNormClass F α β

Equations

⋯ = ⋯