Documentation

Mathlib.Algebra.Hom.Freiman

Freiman homomorphisms #

In this file, we define Freiman homomorphisms. A n-Freiman homomorphism on A is a function f : α → β→ β such that f (x₁) * ... * f (xₙ) = f (y₁) * ... * f (yₙ) for all x₁, ..., xₙ, y₁, ..., yₙ ∈ A∈ A such that x₁ * ... * xₙ = y₁ * ... * yₙ. In particular, any MulHom is a Freiman homomorphism.

They are of interest in additive combinatorics.

Main declaration #

FreimanHom: Freiman homomorphism.
AddFreimanHom: Additive Freiman homomorphism.

Notation #

A →*[n] β→*[n] β: Multiplicative n-Freiman homomorphism on A
A →+[n] β→+[n] β: Additive n-Freiman homomorphism on A

Implementation notes #

In the context of combinatorics, we are interested in Freiman homomorphisms over sets which are not necessarily closed under addition/multiplication. This means we must parametrize them with a set in an AddMonoid/Monoid instead of the AddMonoid/Monoid itself.

References #

Yufei Zhao, 18.225: Graph Theory and Additive Combinatorics

TODO #

MonoidHom.toFreimanHom could be relaxed to MulHom.toFreimanHom by proving (s.map f).prod = (t.map f).prod directly by induction instead of going through f s.prod.

Define n-Freiman isomorphisms.

Affine maps induce Freiman homs. Concretely, provide the AddFreimanHomClass (α →ₐ[𝕜] β) A β n→ₐ[𝕜] β) A β n instance.

structure AddFreimanHom {α : Type u_1} (A : Set α) (β : Type u_2) [inst : AddCommMonoid α] [inst : AddCommMonoid β] (n : ℕ) :

Type (maxu_1u_2)

The underlying function.
toFun : α → β
An additive n-Freiman homomorphism preserves sums of n elements.
map_sum_eq_map_sum' : ∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map toFun s) = Multiset.sum (Multiset.map toFun t)

An additive n-Freiman homomorphism is a map which preserves sums of n elements.

Instances For

structure FreimanHom {α : Type u_1} (A : Set α) (β : Type u_2) [inst : CommMonoid α] [inst : CommMonoid β] (n : ℕ) :

Type (maxu_1u_2)

The underlying function.
toFun : α → β
A n-Freiman homomorphism preserves products of n elements.
map_prod_eq_map_prod' : ∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.prod s = Multiset.prod t → Multiset.prod (Multiset.map toFun s) = Multiset.prod (Multiset.map toFun t)

A n-Freiman homomorphism on a set A is a map which preserves products of n elements.

Instances For

def «AddFreimanHomLocal≺» :

Lean.TrailingParserDescr

An additive n-Freiman homomorphism is a map which preserves sums of n elements.

Equations

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

def «FreimanHomLocal≺» :

Lean.TrailingParserDescr

A n-Freiman homomorphism on a set A is a map which preserves products of n elements.

Equations

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

class AddFreimanHomClass {α : Type u_1} (F : Type u_2) (A : outParam (Set α)) (β : outParam (Type u_3)) [inst : AddCommMonoid α] [inst : AddCommMonoid β] (n : ℕ) [inst : FunLike F α fun x => β] :

An additive n-Freiman homomorphism preserves sums of n elements.
map_sum_eq_map_sum' : ∀ (f : F) {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

AddFreimanHomClass F A β n states that F is a type of n-ary sums-preserving morphisms. You should extend this class when you extend AddFreimanHom.

Instances

class FreimanHomClass {α : Type u_1} (F : Type u_2) (A : outParam (Set α)) (β : outParam (Type u_3)) [inst : CommMonoid α] [inst : CommMonoid β] (n : ℕ) [inst : FunLike F α fun x => β] :

A n-Freiman homomorphism preserves products of n elements.
map_prod_eq_map_prod' : ∀ (f : F) {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.prod s = Multiset.prod t → Multiset.prod (Multiset.map (↑f) s) = Multiset.prod (Multiset.map (↑f) t)

FreimanHomClass F A β n states that F is a type of n-ary products-preserving morphisms. You should extend this class when you extend FreimanHom.

Instances

def AddFreimanHomClass.toFreimanHom.proof_1 {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} [inst : AddFreimanHomClass F A β n] (f : F) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

Equations

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

def AddFreimanHomClass.toFreimanHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} [inst : AddFreimanHomClass F A β n] (f : F) :

A →+[n] β

Turn an element of a type F satisfying AddFreimanHomClass F A β n into an actual AddFreimanHom. This is declared as the default coercion from F to AddFreimanHom A β n.

Equations

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

def FreimanHomClass.toFreimanHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} [inst : FreimanHomClass F A β n] (f : F) :

A →*[n] β

Turn an element of a type F satisfying FreimanHomClass F A β n into an actual FreimanHom. This is declared as the default coercion from F to FreimanHom A β n.

Equations

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

instance instCoeTCFreimanHom {F : Type u_1} {α : Type u_2} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} [inst : FreimanHomClass F A β n] :

CoeTC F (A →*[n] β)

Any type satisfying SMulHomClass can be cast into MulActionHom via SMulHomClass.toMulActionHom.

Equations

instCoeTCFreimanHom = { coe := FreimanHomClass.toFreimanHom }

theorem map_sum_eq_map_sum {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} [inst : AddFreimanHomClass F A β n] (f : F) {s : Multiset α} {t : Multiset α} (hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A) (hs : ↑Multiset.card s = n) (ht : ↑Multiset.card t = n) (h : Multiset.sum s = Multiset.sum t) :

Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

theorem map_prod_eq_map_prod {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} [inst : FreimanHomClass F A β n] (f : F) {s : Multiset α} {t : Multiset α} (hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A) (hs : ↑Multiset.card s = n) (ht : ↑Multiset.card t = n) (h : Multiset.prod s = Multiset.prod t) :

Multiset.prod (Multiset.map (↑f) s) = Multiset.prod (Multiset.map (↑f) t)

theorem map_add_map_eq_map_add_map {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {a : α} {b : α} {c : α} {d : α} [inst : AddFreimanHomClass F A β 2] (f : F) (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) (h : a + b = c + d) :

↑f a + ↑f b = ↑f c + ↑f d

theorem map_mul_map_eq_map_mul_map {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {a : α} {b : α} {c : α} {d : α} [inst : FreimanHomClass F A β 2] (f : F) (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) (h : a * b = c * d) :

↑f a * ↑f b = ↑f c * ↑f d

instance AddFreimanHom.funLike {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

FunLike (A →+[n] β) α fun x => β

Equations

AddFreimanHom.funLike = { coe := AddFreimanHom.toFun, coe_injective' := (_ : ∀ (f g : A →+[n] β), f.toFun = g.toFun → f = g) }

def AddFreimanHom.funLike.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) (g : A →+[n] β) (h : f.toFun = g.toFun) :

f = g

Equations

(_ : f = g) = (_ : f = g)

instance FreimanHom.funLike {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

FunLike (A →*[n] β) α fun x => β

Equations

FreimanHom.funLike = { coe := FreimanHom.toFun, coe_injective' := (_ : ∀ (f g : A →*[n] β), f.toFun = g.toFun → f = g) }

instance AddFreimanHom.addFreimanHomClass {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

AddFreimanHomClass (A →+[n] β) A β n

Equations

@AddFreimanHom.addFreimanHomClass = @AddFreimanHom.addFreimanHomClass.proof_1

def AddFreimanHom.addFreimanHomClass.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

AddFreimanHomClass (A →+[n] β) A β n

Equations

(_ : AddFreimanHomClass (A →+[n] β) A β n) = (_ : AddFreimanHomClass (A →+[n] β) A β n)

instance FreimanHom.freimanHomClass {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

FreimanHomClass (A →*[n] β) A β n

Equations

@FreimanHom.freimanHomClass = @FreimanHom.freimanHomClass.proof_1

@[simp]

theorem AddFreimanHom.toFun_eq_coe {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) :

f.toFun = ↑f

@[simp]

theorem FreimanHom.toFun_eq_coe {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : A →*[n] β) :

f.toFun = ↑f

theorem AddFreimanHom.ext {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} ⦃f : A →+[n] β⦄ ⦃g : A →+[n] β⦄ (h : ∀ (x : α), ↑f x = ↑g x) :

f = g

theorem FreimanHom.ext {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} ⦃f : A →*[n] β⦄ ⦃g : A →*[n] β⦄ (h : ∀ (x : α), ↑f x = ↑g x) :

f = g

@[simp]

theorem AddFreimanHom.coe_mk {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : α → β) (h : ∀ (s t : Multiset α), (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map f s) = Multiset.sum (Multiset.map f t)) :

↑{ toFun := f, map_sum_eq_map_sum' := fun {s t} => h s t } = f

@[simp]

theorem FreimanHom.coe_mk {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : α → β) (h : ∀ (s t : Multiset α), (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.prod s = Multiset.prod t → Multiset.prod (Multiset.map f s) = Multiset.prod (Multiset.map f t)) :

↑{ toFun := f, map_prod_eq_map_prod' := fun {s t} => h s t } = f

@[simp]

theorem AddFreimanHom.mk_coe {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) (h : ∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)) :

{ toFun := ↑f, map_sum_eq_map_sum' := h } = f

@[simp]

theorem FreimanHom.mk_coe {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : A →*[n] β) (h : ∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.prod s = Multiset.prod t → Multiset.prod (Multiset.map (↑f) s) = Multiset.prod (Multiset.map (↑f) t)) :

{ toFun := ↑f, map_prod_eq_map_prod' := h } = f

def AddFreimanHom.id {α : Type u_1} [inst : AddCommMonoid α] (A : Set α) (n : ℕ) :

A →+[n] α

The identity map from an additive commutative monoid to itself.

Equations

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

def AddFreimanHom.id.proof_1 {α : Type u_1} [inst : AddCommMonoid α] (A : Set α) (n : ℕ) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (fun x => x) s) = Multiset.sum (Multiset.map (fun x => x) t)

Equations

(_ : Multiset.sum (Multiset.map (fun x => x) s) = Multiset.sum (Multiset.map (fun x => x) t)) = (_ : Multiset.sum (Multiset.map (fun x => x) s) = Multiset.sum (Multiset.map (fun x => x) t))

@[simp]

theorem AddFreimanHom.id_apply {α : Type u_1} [inst : AddCommMonoid α] (A : Set α) (n : ℕ) (x : α) :

↑(AddFreimanHom.id A n) x = x

@[simp]

theorem FreimanHom.id_apply {α : Type u_1} [inst : CommMonoid α] (A : Set α) (n : ℕ) (x : α) :

↑(FreimanHom.id A n) x = x

def FreimanHom.id {α : Type u_1} [inst : CommMonoid α] (A : Set α) (n : ℕ) :

A →*[n] α

The identity map from a commutative monoid to itself.

Equations

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

def AddFreimanHom.comp.proof_1 {α : Type u_1} {β : Type u_3} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →+[n] γ) (g : A →+[n] β) (hAB : Set.MapsTo (↑g) A B) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f ∘ ↑g) s) = Multiset.sum (Multiset.map (↑f ∘ ↑g) t)

Equations

(_ : Multiset.sum (Multiset.map (↑f ∘ ↑g) s) = Multiset.sum (Multiset.map (↑f ∘ ↑g) t)) = (_ : Multiset.sum (Multiset.map (↑f ∘ ↑g) s) = Multiset.sum (Multiset.map (↑f ∘ ↑g) t))

def AddFreimanHom.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →+[n] γ) (g : A →+[n] β) (hAB : Set.MapsTo (↑g) A B) :

A →+[n] γ

Composition of additive Freiman homomorphisms as an additive Freiman homomorphism.

Equations

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

def FreimanHom.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →*[n] γ) (g : A →*[n] β) (hAB : Set.MapsTo (↑g) A B) :

A →*[n] γ

Composition of Freiman homomorphisms as a Freiman homomorphism.

Equations

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

@[simp]

theorem AddFreimanHom.coe_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →+[n] γ) (g : A →+[n] β) {hfg : Set.MapsTo (↑g) A B} :

↑(AddFreimanHom.comp f g hfg) = ↑f ∘ ↑g

@[simp]

theorem FreimanHom.coe_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →*[n] γ) (g : A →*[n] β) {hfg : Set.MapsTo (↑g) A B} :

↑(FreimanHom.comp f g hfg) = ↑f ∘ ↑g

theorem AddFreimanHom.comp_apply {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →+[n] γ) (g : A →+[n] β) {hfg : Set.MapsTo (↑g) A B} (x : α) :

↑(AddFreimanHom.comp f g hfg) x = ↑f (↑g x)

theorem FreimanHom.comp_apply {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : B →*[n] γ) (g : A →*[n] β) {hfg : Set.MapsTo (↑g) A B} (x : α) :

↑(FreimanHom.comp f g hfg) x = ↑f (↑g x)

theorem AddFreimanHom.comp_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] [inst : AddCommMonoid δ] {A : Set α} {B : Set β} {C : Set γ} {n : ℕ} (f : A →+[n] β) (g : B →+[n] γ) (h : C →+[n] δ) {hf : Set.MapsTo (↑f) A B} {hhg : Set.MapsTo (↑g) B C} {hgf : Set.MapsTo (↑f) A B} {hh : Set.MapsTo (↑(AddFreimanHom.comp g f hgf)) A C} :

AddFreimanHom.comp (AddFreimanHom.comp h g hhg) f hf = AddFreimanHom.comp h (AddFreimanHom.comp g f hgf) hh

theorem FreimanHom.comp_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] [inst : CommMonoid δ] {A : Set α} {B : Set β} {C : Set γ} {n : ℕ} (f : A →*[n] β) (g : B →*[n] γ) (h : C →*[n] δ) {hf : Set.MapsTo (↑f) A B} {hhg : Set.MapsTo (↑g) B C} {hgf : Set.MapsTo (↑f) A B} {hh : Set.MapsTo (↑(FreimanHom.comp g f hgf)) A C} :

FreimanHom.comp (FreimanHom.comp h g hhg) f hf = FreimanHom.comp h (FreimanHom.comp g f hgf) hh

theorem AddFreimanHom.cancel_right {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g₁ : B →+[n] γ} {g₂ : B →+[n] γ} {f : A →+[n] β} (hf : Function.Surjective ↑f) {hg₁ : Set.MapsTo (↑f) A B} {hg₂ : Set.MapsTo (↑f) A B} :

AddFreimanHom.comp g₁ f hg₁ = AddFreimanHom.comp g₂ f hg₂ ↔ g₁ = g₂

theorem FreimanHom.cancel_right {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g₁ : B →*[n] γ} {g₂ : B →*[n] γ} {f : A →*[n] β} (hf : Function.Surjective ↑f) {hg₁ : Set.MapsTo (↑f) A B} {hg₂ : Set.MapsTo (↑f) A B} :

FreimanHom.comp g₁ f hg₁ = FreimanHom.comp g₂ f hg₂ ↔ g₁ = g₂

theorem AddFreimanHom.cancel_right_on {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g₁ : B →+[n] γ} {g₂ : B →+[n] γ} {f : A →+[n] β} (hf : Set.SurjOn (↑f) A B) {hf' : Set.MapsTo (↑f) A B} :

Set.EqOn (↑(AddFreimanHom.comp g₁ f hf')) (↑(AddFreimanHom.comp g₂ f hf')) A ↔ Set.EqOn (↑g₁) (↑g₂) B

theorem FreimanHom.cancel_right_on {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g₁ : B →*[n] γ} {g₂ : B →*[n] γ} {f : A →*[n] β} (hf : Set.SurjOn (↑f) A B) {hf' : Set.MapsTo (↑f) A B} :

Set.EqOn (↑(FreimanHom.comp g₁ f hf')) (↑(FreimanHom.comp g₂ f hf')) A ↔ Set.EqOn (↑g₁) (↑g₂) B

theorem AddFreimanHom.cancel_left_on {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g : B →+[n] γ} {f₁ : A →+[n] β} {f₂ : A →+[n] β} (hg : Set.InjOn (↑g) B) {hf₁ : Set.MapsTo (↑f₁) A B} {hf₂ : Set.MapsTo (↑f₂) A B} :

Set.EqOn (↑(AddFreimanHom.comp g f₁ hf₁)) (↑(AddFreimanHom.comp g f₂ hf₂)) A ↔ Set.EqOn (↑f₁) (↑f₂) A

theorem FreimanHom.cancel_left_on {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} {g : B →*[n] γ} {f₁ : A →*[n] β} {f₂ : A →*[n] β} (hg : Set.InjOn (↑g) B) {hf₁ : Set.MapsTo (↑f₁) A B} {hf₂ : Set.MapsTo (↑f₂) A B} :

Set.EqOn (↑(FreimanHom.comp g f₁ hf₁)) (↑(FreimanHom.comp g f₂ hf₂)) A ↔ Set.EqOn (↑f₁) (↑f₂) A

@[simp]

theorem AddFreimanHom.comp_id {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) {hf : Set.MapsTo (↑(AddFreimanHom.id A n)) A A} :

AddFreimanHom.comp f (AddFreimanHom.id A n) hf = f

@[simp]

theorem FreimanHom.comp_id {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : A →*[n] β) {hf : Set.MapsTo (↑(FreimanHom.id A n)) A A} :

FreimanHom.comp f (FreimanHom.id A n) hf = f

@[simp]

theorem AddFreimanHom.id_comp {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {B : Set β} {n : ℕ} (f : A →+[n] β) {hf : Set.MapsTo (↑f) A B} :

AddFreimanHom.comp (AddFreimanHom.id B n) f hf = f

@[simp]

theorem FreimanHom.id_comp {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {B : Set β} {n : ℕ} (f : A →*[n] β) {hf : Set.MapsTo (↑f) A B} :

FreimanHom.comp (FreimanHom.id B n) f hf = f

def AddFreimanHom.const {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] (A : Set α) (n : ℕ) (b : β) :

A →+[n] β

AddFreimanHom.const An b is the Freiman homomorphism sending everything to b.

Equations

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

def AddFreimanHom.const.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] (A : Set α) (n : ℕ) (b : β) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (fun x => b) s) = Multiset.sum (Multiset.map (fun x => b) t)

Equations

(_ : Multiset.sum (Multiset.map (fun x => b) s) = Multiset.sum (Multiset.map (fun x => b) t)) = (_ : Multiset.sum (Multiset.map (fun x => b) s) = Multiset.sum (Multiset.map (fun x => b) t))

def FreimanHom.const {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] (A : Set α) (n : ℕ) (b : β) :

A →*[n] β

FreimanHom.const A n b is the Freiman homomorphism sending everything to b.

Equations

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

@[simp]

theorem AddFreimanHom.const_apply {α : Type u_2} {β : Type u_1} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} (n : ℕ) (b : β) (x : α) :

↑(AddFreimanHom.const A n b) x = b

@[simp]

theorem FreimanHom.const_apply {α : Type u_2} {β : Type u_1} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} (n : ℕ) (b : β) (x : α) :

↑(FreimanHom.const A n b) x = b

@[simp]

theorem AddFreimanHom.const_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} (n : ℕ) (c : γ) (f : A →+[n] β) {hf : Set.MapsTo (↑f) A B} :

AddFreimanHom.comp (AddFreimanHom.const B n c) f hf = AddFreimanHom.const A n c

@[simp]

theorem FreimanHom.const_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} (n : ℕ) (c : γ) (f : A →*[n] β) {hf : Set.MapsTo (↑f) A B} :

FreimanHom.comp (FreimanHom.const B n c) f hf = FreimanHom.const A n c

instance AddFreimanHom.instZeroFreimanHom {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

Zero (A →+[n] β)

0 is the Freiman homomorphism sending everything to 0.

Equations

AddFreimanHom.instZeroFreimanHom = { zero := AddFreimanHom.const A n 0 }

instance FreimanHom.instOneFreimanHom {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

One (A →*[n] β)

1 is the Freiman homomorphism sending everything to 1.

Equations

FreimanHom.instOneFreimanHom = { one := FreimanHom.const A n 1 }

@[simp]

theorem AddFreimanHom.zero_apply {α : Type u_2} {β : Type u_1} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (x : α) :

↑0 x = 0

@[simp]

theorem FreimanHom.one_apply {α : Type u_2} {β : Type u_1} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (x : α) :

↑1 x = 1

@[simp]

theorem AddFreimanHom.zero_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : A →+[n] β) {hf : Set.MapsTo (↑f) A B} :

AddFreimanHom.comp 0 f hf = 0

@[simp]

theorem FreimanHom.one_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (f : A →*[n] β) {hf : Set.MapsTo (↑f) A B} :

FreimanHom.comp 1 f hf = 1

instance AddFreimanHom.instInhabitedFreimanHom {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

Inhabited (A →+[n] β)

Equations

AddFreimanHom.instInhabitedFreimanHom = { default := 0 }

instance FreimanHom.instInhabitedFreimanHom {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

Inhabited (A →*[n] β)

Equations

FreimanHom.instInhabitedFreimanHom = { default := 1 }

def AddFreimanHom.instAddFreimanHom.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) (g : A →+[n] β) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (fun x => ↑f x + ↑g x) s) = Multiset.sum (Multiset.map (fun x => ↑f x + ↑g x) t)

Equations

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

instance AddFreimanHom.instAddFreimanHom {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

Add (A →+[n] β)

f + g is the Freiman homomorphism sending x to f x + g x.

Equations

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

instance FreimanHom.instMulFreimanHom {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

Mul (A →*[n] β)

f * g is the Freiman homomorphism sends x to f x * g x.

Equations

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

@[simp]

theorem AddFreimanHom.add_apply {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : A →+[n] β) (g : A →+[n] β) (x : α) :

↑(f + g) x = ↑f x + ↑g x

@[simp]

theorem FreimanHom.mul_apply {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : A →*[n] β) (g : A →*[n] β) (x : α) :

↑(f * g) x = ↑f x * ↑g x

theorem AddFreimanHom.add_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (g₁ : B →+[n] γ) (g₂ : B →+[n] γ) (f : A →+[n] β) {hg : Set.MapsTo (↑f) A B} {hg₁ : Set.MapsTo (↑f) A B} {hg₂ : Set.MapsTo (↑f) A B} :

AddFreimanHom.comp (g₁ + g₂) f hg = AddFreimanHom.comp g₁ f hg₁ + AddFreimanHom.comp g₂ f hg₂

theorem FreimanHom.mul_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommMonoid γ] {A : Set α} {B : Set β} {n : ℕ} (g₁ : B →*[n] γ) (g₂ : B →*[n] γ) (f : A →*[n] β) {hg : Set.MapsTo (↑f) A B} {hg₁ : Set.MapsTo (↑f) A B} {hg₂ : Set.MapsTo (↑f) A B} :

FreimanHom.comp (g₁ * g₂) f hg = FreimanHom.comp g₁ f hg₁ * FreimanHom.comp g₂ f hg₂

def AddFreimanHom.instNegFreimanHomToAddCommMonoid.proof_1 {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} (f : A →+[n] G) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (fun x => -↑f x) s) = Multiset.sum (Multiset.map (fun x => -↑f x) t)

Equations

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

instance AddFreimanHom.instNegFreimanHomToAddCommMonoid {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} :

Neg (A →+[n] G)

If f is a Freiman homomorphism to an additive commutative group, then -f is the Freiman homomorphism sending x to -f x.

Equations

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

instance FreimanHom.instInvFreimanHomToCommMonoid {α : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommGroup G] {A : Set α} {n : ℕ} :

Inv (A →*[n] G)

If f is a Freiman homomorphism to a commutative group, then f⁻¹⁻¹ is the Freiman homomorphism sending x to (f x)⁻¹⁻¹.

Equations

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

@[simp]

theorem AddFreimanHom.neg_apply {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} (f : A →+[n] G) (x : α) :

↑(-f) x = -↑f x

@[simp]

theorem FreimanHom.inv_apply {α : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommGroup G] {A : Set α} {n : ℕ} (f : A →*[n] G) (x : α) :

↑f⁻¹ x = (↑f x)⁻¹

@[simp]

theorem AddFreimanHom.neg_comp {α : Type u_3} {β : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommGroup G] {A : Set α} {B : Set β} {n : ℕ} (f : B →+[n] G) (g : A →+[n] β) {hf : Set.MapsTo (↑g) A B} {hf' : Set.MapsTo (↑g) A B} :

AddFreimanHom.comp (-f) g hf = -AddFreimanHom.comp f g hf'

@[simp]

theorem FreimanHom.inv_comp {α : Type u_3} {β : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommGroup G] {A : Set α} {B : Set β} {n : ℕ} (f : B →*[n] G) (g : A →*[n] β) {hf : Set.MapsTo (↑g) A B} {hf' : Set.MapsTo (↑g) A B} :

FreimanHom.comp f⁻¹ g hf = (FreimanHom.comp f g hf')⁻¹

def AddFreimanHom.instSubFreimanHomToAddCommMonoid.proof_1 {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} (f : A →+[n] G) (g : A →+[n] G) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (fun x => ↑f x - ↑g x) s) = Multiset.sum (Multiset.map (fun x => ↑f x - ↑g x) t)

Equations

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

instance AddFreimanHom.instSubFreimanHomToAddCommMonoid {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} :

Sub (A →+[n] G)

If f and g are additive Freiman homomorphisms to an additive commutative group, then f - g is the additive Freiman homomorphism sending x to f x - g x

Equations

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

instance FreimanHom.instDivFreimanHomToCommMonoid {α : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommGroup G] {A : Set α} {n : ℕ} :

Div (A →*[n] G)

If f and g are Freiman homomorphisms to a commutative group, then f / g is the Freiman homomorphism sending x to f x / g x.

Equations

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

@[simp]

theorem AddFreimanHom.sub_apply {α : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommGroup G] {A : Set α} {n : ℕ} (f : A →+[n] G) (g : A →+[n] G) (x : α) :

↑(f - g) x = ↑f x - ↑g x

@[simp]

theorem FreimanHom.div_apply {α : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommGroup G] {A : Set α} {n : ℕ} (f : A →*[n] G) (g : A →*[n] G) (x : α) :

↑(f / g) x = ↑f x / ↑g x

@[simp]

theorem AddFreimanHom.sub_comp {α : Type u_3} {β : Type u_1} {G : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] [inst : AddCommGroup G] {A : Set α} {B : Set β} {n : ℕ} (f₁ : B →+[n] G) (f₂ : B →+[n] G) (g : A →+[n] β) {hf : Set.MapsTo (↑g) A B} {hf₁ : Set.MapsTo (↑g) A B} {hf₂ : Set.MapsTo (↑g) A B} :

AddFreimanHom.comp (f₁ - f₂) g hf = AddFreimanHom.comp f₁ g hf₁ - AddFreimanHom.comp f₂ g hf₂

@[simp]

theorem FreimanHom.div_comp {α : Type u_3} {β : Type u_1} {G : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] [inst : CommGroup G] {A : Set α} {B : Set β} {n : ℕ} (f₁ : B →*[n] G) (f₂ : B →*[n] G) (g : A →*[n] β) {hf : Set.MapsTo (↑g) A B} {hf₁ : Set.MapsTo (↑g) A B} {hf₂ : Set.MapsTo (↑g) A B} :

FreimanHom.comp (f₁ / f₂) g hf = FreimanHom.comp f₁ g hf₁ / FreimanHom.comp f₂ g hf₂

Instances #

def AddFreimanHom.addCommMonoid.proof_3 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (a : A →+[n] β) :

a + 0 = a

Equations

(_ : a + 0 = a) = (_ : a + 0 = a)

instance AddFreimanHom.addCommMonoid {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

AddCommMonoid (A →+[n] β)

α →+[n] β→+[n] β is an AddCommMonoid.

Equations

AddFreimanHom.addCommMonoid = AddCommMonoid.mk (_ : ∀ (a b : A →+[n] β), a + b = b + a)

def AddFreimanHom.addCommMonoid.proof_5 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

∀ (n : ℕ) (x : A →+[n] β), nsmulRec (n + 1) x = nsmulRec (n + 1) x

Equations

(_ : nsmulRec (n + 1) x = nsmulRec (n + 1) x) = (_ : nsmulRec (n + 1) x = nsmulRec (n + 1) x)

def AddFreimanHom.addCommMonoid.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (a : A →+[n] β) (b : A →+[n] β) (c : A →+[n] β) :

a + b + c = a + (b + c)

Equations

(_ : a + b + c = a + (b + c)) = (_ : a + b + c = a + (b + c))

def AddFreimanHom.addCommMonoid.proof_2 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (a : A →+[n] β) :

0 + a = a

Equations

(_ : 0 + a = a) = (_ : 0 + a = a)

def AddFreimanHom.addCommMonoid.proof_6 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (a : A →+[n] β) (b : A →+[n] β) :

a + b = b + a

Equations

(_ : a + b = b + a) = (_ : a + b = b + a)

def AddFreimanHom.addCommMonoid.proof_4 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

∀ (x : A →+[n] β), nsmulRec 0 x = nsmulRec 0 x

Equations

(_ : nsmulRec 0 x = nsmulRec 0 x) = (_ : nsmulRec 0 x = nsmulRec 0 x)

instance FreimanHom.commMonoid {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

CommMonoid (A →*[n] β)

A →*[n] β→*[n] β is a CommMonoid.

Equations

FreimanHom.commMonoid = CommMonoid.mk (_ : ∀ (a b : A →*[n] β), a * b = b * a)

def AddFreimanHom.addCommGroup.proof_3 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

∀ (n : ℕ) (a : A →+[n] β), zsmulRec (Int.ofNat (Nat.succ n)) a = zsmulRec (Int.ofNat (Nat.succ n)) a

Equations

(_ : zsmulRec (Int.ofNat (Nat.succ n)) a = zsmulRec (Int.ofNat (Nat.succ n)) a) = (_ : zsmulRec (Int.ofNat (Nat.succ n)) a = zsmulRec (Int.ofNat (Nat.succ n)) a)

def AddFreimanHom.addCommGroup.proof_1 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

∀ (a b : A →+[n] β), a - b = a + -b

Equations

(_ : a - b = a + -b) = (_ : a - b = a + -b)

def AddFreimanHom.addCommGroup.proof_6 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] (a : A →+[n] β) (b : A →+[n] β) :

a + b = b + a

Equations

(_ : ∀ (a b : A →+[n] β), a + b = b + a) = (_ : ∀ (a b : A →+[n] β), a + b = b + a)

def AddFreimanHom.addCommGroup.proof_5 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

∀ (a : A →+[n] β), -a + a = 0

Equations

(_ : -a + a = 0) = (_ : -a + a = 0)

def AddFreimanHom.addCommGroup.proof_4 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

∀ (n : ℕ) (a : A →+[n] β), zsmulRec (Int.negSucc n) a = zsmulRec (Int.negSucc n) a

Equations

(_ : zsmulRec (Int.negSucc n) a = zsmulRec (Int.negSucc n) a) = (_ : zsmulRec (Int.negSucc n) a = zsmulRec (Int.negSucc n) a)

def AddFreimanHom.addCommGroup.proof_2 {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

∀ (a : A →+[n] β), zsmulRec 0 a = zsmulRec 0 a

Equations

(_ : zsmulRec 0 a = zsmulRec 0 a) = (_ : zsmulRec 0 a = zsmulRec 0 a)

instance AddFreimanHom.addCommGroup {α : Type u_1} [inst : AddCommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : AddCommGroup β] :

AddCommGroup (A →+[n] β)

If β is an additive commutative group, then A →*[n] β→*[n] β is an additive commutative group too.

Equations

AddFreimanHom.addCommGroup = let src := AddFreimanHom.addCommMonoid; AddCommGroup.mk (_ : ∀ (a b : A →+[n] β), a + b = b + a)

instance FreimanHom.commGroup {α : Type u_1} [inst : CommMonoid α] {A : Set α} {n : ℕ} {β : Type u_2} [inst : CommGroup β] :

CommGroup (A →*[n] β)

If β is a commutative group, then A →*[n] β→*[n] β is a commutative group too.

Equations

FreimanHom.commGroup = let src := FreimanHom.commMonoid; CommGroup.mk (_ : ∀ (a b : A →*[n] β), a * b = b * a)

Hom hierarchy #

instance AddMonoidHom.addFreimanHomClass {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {n : ℕ} :

AddFreimanHomClass (α →+ β) Set.univ β n

An additive monoid homomorphism is naturally an AddFreimanHom on its entire domain.

We can't leave the domain A : Set α of the AddFreimanHom a free variable, since it wouldn't be inferrable.

Equations

@AddMonoidHom.addFreimanHomClass = @AddMonoidHom.addFreimanHomClass.proof_1

def AddMonoidHom.addFreimanHomClass.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {n : ℕ} :

AddFreimanHomClass (α →+ β) Set.univ β n

Equations

(_ : AddFreimanHomClass (α →+ β) Set.univ β n) = (_ : AddFreimanHomClass (α →+ β) Set.univ β n)

instance MonoidHom.freimanHomClass {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {n : ℕ} :

FreimanHomClass (α →* β) Set.univ β n

A monoid homomorphism is naturally a FreimanHom on its entire domain.

We can't leave the domain A : Set α of the FreimanHom a free variable, since it wouldn't be inferrable.

Equations

@MonoidHom.freimanHomClass = @MonoidHom.freimanHomClass.proof_1

def AddMonoidHom.toAddFreimanHom.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] (A : Set α) (n : ℕ) (f : α →+ β) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = n → ↑Multiset.card t = n → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

Equations

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

def AddMonoidHom.toAddFreimanHom {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] (A : Set α) (n : ℕ) (f : α →+ β) :

A →+[n] β

An AddMonoidHom is naturally an AddFreimanHom

Equations

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

def MonoidHom.toFreimanHom {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] (A : Set α) (n : ℕ) (f : α →* β) :

A →*[n] β

A MonoidHom is naturally a FreimanHom.

Equations

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

@[simp]

theorem AddMonoidHom.toAddFreimanHom_coe {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} (f : α →+ β) :

↑(AddMonoidHom.toAddFreimanHom A n f) = ↑f

@[simp]

theorem MonoidHom.toFreimanHom_coe {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} (f : α →* β) :

↑(MonoidHom.toFreimanHom A n f) = ↑f

theorem AddMonoidHom.toAddFreimanHom_injective {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCommMonoid β] {A : Set α} {n : ℕ} :

Function.Injective (AddMonoidHom.toAddFreimanHom A n)

theorem MonoidHom.toFreimanHom_injective {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CommMonoid β] {A : Set α} {n : ℕ} :

Function.Injective (MonoidHom.toFreimanHom A n)

theorem map_sum_eq_map_sum_of_le {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} [inst : AddFreimanHomClass F A β n] (f : F) {s : Multiset α} {t : Multiset α} (hsA : ∀ (x : α), x ∈ s → x ∈ A) (htA : ∀ (x : α), x ∈ t → x ∈ A) (hs : ↑Multiset.card s = m) (ht : ↑Multiset.card t = m) (hst : Multiset.sum s = Multiset.sum t) (h : m ≤ n) :

Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

theorem map_prod_eq_map_prod_of_le {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} [inst : FreimanHomClass F A β n] (f : F) {s : Multiset α} {t : Multiset α} (hsA : ∀ (x : α), x ∈ s → x ∈ A) (htA : ∀ (x : α), x ∈ t → x ∈ A) (hs : ↑Multiset.card s = m) (ht : ↑Multiset.card t = m) (hst : Multiset.prod s = Multiset.prod t) (h : m ≤ n) :

Multiset.prod (Multiset.map (↑f) s) = Multiset.prod (Multiset.map (↑f) t)

def AddFreimanHom.toAddFreimanHom.proof_1 {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) (f : A →+[n] β) :

∀ {s t : Multiset α}, (∀ ⦃x : α⦄, x ∈ s → x ∈ A) → (∀ ⦃x : α⦄, x ∈ t → x ∈ A) → ↑Multiset.card s = m → ↑Multiset.card t = m → Multiset.sum s = Multiset.sum t → Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)

Equations

(_ : Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t)) = (_ : Multiset.sum (Multiset.map (↑f) s) = Multiset.sum (Multiset.map (↑f) t))

def AddFreimanHom.toAddFreimanHom {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) (f : A →+[n] β) :

A →+[m] β

α →+[n] β→+[n] β is naturally included in α →+[m] β→+[m] β for any m ≤ n≤ n

Equations

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

def FreimanHom.toFreimanHom {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) (f : A →*[n] β) :

A →*[m] β

α →*[n] β→*[n] β is naturally included in A →*[m] β→*[m] β for any m ≤ n≤ n.

Equations

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

theorem AddFreimanHom.addFreimanHomClass_of_le {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} [inst : AddFreimanHomClass F A β n] (h : m ≤ n) :

AddFreimanHomClass F A β m

An additive n-Freiman homomorphism is also an additive m-Freiman homomorphism for any m ≤ n≤ n.

theorem FreimanHom.FreimanHomClass_of_le {F : Type u_2} {α : Type u_1} {β : Type u_3} [inst : FunLike F α fun x => β] [inst : CommMonoid α] [inst : CancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} [inst : FreimanHomClass F A β n] (h : m ≤ n) :

FreimanHomClass F A β m

A n-Freiman homomorphism is also a m-Freiman homomorphism for any m ≤ n≤ n.

@[simp]

theorem AddFreimanHom.toAddFreimanHom_coe {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) (f : A →+[n] β) :

↑(AddFreimanHom.toAddFreimanHom h f) = ↑f

@[simp]

theorem FreimanHom.toFreimanHom_coe {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) (f : A →*[n] β) :

↑(FreimanHom.toFreimanHom h f) = ↑f

theorem AddFreimanHom.toAddFreimanHom_injective {α : Type u_1} {β : Type u_2} [inst : AddCommMonoid α] [inst : AddCancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) :

Function.Injective (AddFreimanHom.toAddFreimanHom h)

theorem FreimanHom.toFreimanHom_injective {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst : CancelCommMonoid β] {A : Set α} {m : ℕ} {n : ℕ} (h : m ≤ n) :

Function.Injective (FreimanHom.toFreimanHom h)