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] extends FunLike : Type (max (max u_7 u_8) u_9)

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_nonneg : ∀ (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] extends FunLike : Type (max (max u_7 u_8) u_9)

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

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

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_max : ∀ (f : F) (a b : α), f (a + b) ≤ max (f a) (f b)

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

theorem le_map_add_map_sub {F : Type u_2} {α : Type u_3} {β : Type u_4} [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_mul_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [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_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [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_sub_add_map_sub {F : Type u_2} {α : Type u_3} {β : Type u_4} [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_mul_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [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_div_add_map_div {F : Type u_2} {α : Type u_3} {β : Type u_4} [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.

Group (semi)norms

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

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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.

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

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
• map_zero : ∀ (f : F), f 0 = 0
• map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
• eq_zero_of_map_eq_zero : ∀ (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] extends GroupSeminormClass : Type (max (max u_7 u_8) u_9)

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• 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
• map_inv_eq_map : ∀ (f : F) (a : α), f a⁻¹ = f a
• eq_one_of_map_eq_zero : ∀ (f : F) {a : α}, f a = 0 → a = 1

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

instance AddGroupSeminormClass.toAddLEAddHomClass {F : Type u_7} {α : outParam (Type u_8)} {β : outParam (Type u_9)} [AddGroup α] [OrderedAddCommMonoid β] [self : AddGroupSeminormClass F α β] : SubadditiveHomClass F α β

Equations

• AddGroupSeminormClass.toAddLEAddHomClass = self.1

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

Equations

• AddGroupSeminormClass.toZeroHomClass = let src := inst; ZeroHomClass.mk (_ : ∀ (f : F), f 0 = 0)

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

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

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

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

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

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

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

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

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

Equations

• AddGroupSeminormClass.toNonnegHomClass = let src := inst; NonnegHomClass.mk (_ : ∀ (f : F) (a : α), 0 ≤ f a)

theorem AddGroupSeminormClass.toNonnegHomClass.proof_1 {F : Type u_3} {α : Type u_2} {β : Type u_1} [AddGroup α] [LinearOrderedAddCommMonoid β] [AddGroupSeminormClass F α β] (f : F) (a : α) : 0 ≤ f a

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

Equations

• GroupSeminormClass.toNonnegHomClass = let src := inst; NonnegHomClass.mk (_ : ∀ (f : F) (a : α), 0 ≤ f a)

@[simp]

theorem map_eq_zero_iff_eq_zero {F : Type u_2} {α : Type u_3} {β : Type u_4} [AddGroup α] [OrderedAddCommMonoid β] [AddGroupNormClass F α β] (f : F) {x : α} : f x = 0 ↔ x = 0

@[simp]

theorem map_eq_zero_iff_eq_one {F : Type u_2} {α : Type u_3} {β : Type u_4} [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} [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} [Group α] [OrderedAddCommMonoid β] [GroupNormClass F α β] (f : F) {x : α} : f x ≠ 0 ↔ x ≠ 1

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

theorem map_pos_of_ne_one {F : Type u_2} {α : Type u_3} {β : Type u_4} [Group α] [LinearOrderedAddCommMonoid β] [GroupNormClass F α β] (f : F) {x : α} (hx : x ≠ 1) : 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] extends AddGroupSeminormClass : Type (max (max u_7 u_8) u_9)

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
• map_zero : ∀ (f : F), f 0 = 0
• map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
• 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.

class RingNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonUnitalNonAssocRing α] [OrderedSemiring β] extends RingSeminormClass : Type (max (max u_7 u_8) u_9)

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
• map_zero : ∀ (f : F), f 0 = 0
• map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
• map_mul_le_mul : ∀ (f : F) (a b : α), f (a * b) ≤ f a * f b
• eq_zero_of_map_eq_zero : ∀ (f : F) {a : α}, f a = 0 → a = 0

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

class MulRingSeminormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonAssocRing α] [OrderedSemiring β] extends AddGroupSeminormClass : Type (max (max u_7 u_8) u_9)

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
• map_zero : ∀ (f : F), f 0 = 0
• map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
• map_mul : ∀ (f : F) (x y : α), f (x * y) = f x * f y

  The proposition that the function preserves multiplication

• map_one : ∀ (f : F), f 1 = 1

  The proposition that the function preserves 1

class MulRingNormClass (F : Type u_7) (α : outParam (Type u_8)) (β : outParam (Type u_9)) [NonAssocRing α] [OrderedSemiring β] extends MulRingSeminormClass : Type (max (max u_7 u_8) u_9)

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.

• coe : F → α → β
• coe_injective' : Function.Injective FunLike.coe
• map_add_le_add : ∀ (f : F) (a b : α), f (a + b) ≤ f a + f b
• map_zero : ∀ (f : F), f 0 = 0
• map_neg_eq_map : ∀ (f : F) (a : α), f (-a) = f a
• map_mul : ∀ (f : F) (x y : α), f (x * y) = f x * f y
• map_one : ∀ (f : F), f 1 = 1
• eq_zero_of_map_eq_zero : ∀ (f : F) {a : α}, f a = 0 → a = 0

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

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

Equations

• RingSeminormClass.toNonnegHomClass = AddGroupSeminormClass.toNonnegHomClass

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

Equations

• MulRingSeminormClass.toRingSeminormClass = let src := inst; RingSeminormClass.mk (_ : ∀ (f : F) (a b : α), f (a * b) ≤ f a * f b)

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

Equations

• MulRingNormClass.toRingNormClass = let src := inst; let src_1 := MulRingSeminormClass.toRingSeminormClass; RingNormClass.mk (_ : ∀ (f : F) {a : α}, f a = 0 → a = 0)