Documentation

Mathlib.Algebra.Group.UniqueProds

A group G has unique products if for any two non-empty finite subsets A, B ⊆ G, there is an element g ∈ A * B that can be written uniquely as a product of an element of A and an element of B. We call the formalization this property UniqueProds. Since the condition requires no property of the group operation, we define it for a Type simply satisfying Mul. We also introduce the analogous "additive" companion, UniqueSums, and link the two so that to_additive converts UniqueProds into UniqueSums.

A common way of proving that a group satisfies the UniqueProds/Sums property is by assuming the existence of some kind of ordering on the group that is well-behaved with respect to the group operation and showing that minima/maxima are the "unique products/sums". However, the order is just a convenience and is not part of the UniqueProds/Sums setup.

Here you can see several examples of Types that have UniqueSums/Prods (inferInstance uses Covariant.to_uniqueProds_left and Covariant.to_uniqueSums_left).

import Mathlib.Data.Real.Basic
import Mathlib.Data.PNat.Basic
import Mathlib.Algebra.Group.UniqueProds

example : UniqueSums ℕ   := inferInstance
example : UniqueSums ℕ+  := inferInstance
example : UniqueSums ℤ   := inferInstance
example : UniqueSums ℚ   := inferInstance
example : UniqueSums ℝ   := inferInstance
example : UniqueProds ℕ+ := inferInstance

Use in `(Add)MonoidAlgebra`s #

UniqueProds/Sums allow to decouple certain arguments about (Add)MonoidAlgebras into an argument about the grading type and then a generic statement of the form "look at the coefficient of the 'unique product/sum'". The file Algebra/MonoidAlgebra/NoZeroDivisors contains several examples of this use.

def UniqueAdd {G : Type u_1} [Add G] (A : Finset G) (B : Finset G) (a0 : G) (b0 : G) :

Let G be a Type with addition, let A B : Finset G be finite subsets and let a0 b0 : G be two elements. UniqueAdd A B a0 b0 asserts a0 + b0 can be written in at most one way as a sum of an element from A and an element from B.

Equations

UniqueAdd A B a0 b0 = ∀ ⦃a b : G⦄, a ∈ A → b ∈ B → a + b = a0 + b0 → a = a0 ∧ b = b0

Instances For

def UniqueMul {G : Type u_1} [Mul G] (A : Finset G) (B : Finset G) (a0 : G) (b0 : G) :

Let G be a Type with multiplication, let A B : Finset G be finite subsets and let a0 b0 : G be two elements. UniqueMul A B a0 b0 asserts a0 * b0 can be written in at most one way as a product of an element of A and an element of B.

Equations

UniqueMul A B a0 b0 = ∀ ⦃a b : G⦄, a ∈ A → b ∈ B → a * b = a0 * b0 → a = a0 ∧ b = b0

Instances For

@[simp]

theorem UniqueAdd.of_subsingleton {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [Subsingleton G] :

UniqueAdd A B a0 b0

@[simp]

theorem UniqueMul.of_subsingleton {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [Subsingleton G] :

UniqueMul A B a0 b0

theorem UniqueAdd.of_card_nonpos {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :

∃ a ∈ A, ∃ b ∈ B, UniqueAdd A B a b

theorem UniqueMul.of_card_le_one {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :

∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b

theorem UniqueAdd.mt {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueAdd A B a0 b0) ⦃a : G⦄ ⦃b : G⦄ :

a ∈ A → b ∈ B → a ≠ a0 ∨ b ≠ b0 → a + b ≠ a0 + b0

theorem UniqueMul.mt {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueMul A B a0 b0) ⦃a : G⦄ ⦃b : G⦄ :

a ∈ A → b ∈ B → a ≠ a0 ∨ b ≠ b0 → a * b ≠ a0 * b0

theorem UniqueAdd.subsingleton {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueAdd A B a0 b0) :

Subsingleton { ab : G × G // ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 + ab.2 = a0 + b0 }

abbrev UniqueAdd.subsingleton.match_1 {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (motive : { ab : G × G // ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 + ab.2 = a0 + b0 } → Prop) :

∀ (x : { ab : G × G // ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 + ab.2 = a0 + b0 }), (∀ (_a' _b' : G) (ha' : (_a', _b').1 ∈ A) (hb' : (_a', _b').2 ∈ B) (ab' : (_a', _b').1 + (_a', _b').2 = a0 + b0), motive ⟨(_a', _b'), ⋯⟩) → motive x

Equations

⋯ = ⋯

Instances For

theorem UniqueMul.subsingleton {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueMul A B a0 b0) :

Subsingleton { ab : G × G // ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 }

theorem UniqueAdd.set_subsingleton {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueAdd A B a0 b0) :

{ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 + ab.2 = a0 + b0}.Subsingleton

theorem UniqueMul.set_subsingleton {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueMul A B a0 b0) :

{ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0}.Subsingleton

theorem UniqueAdd.iff_existsUnique {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (aA : a0 ∈ A) (bB : b0 ∈ B) :

UniqueAdd A B a0 b0 ↔ ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 + ab.2 = a0 + b0

theorem UniqueMul.iff_existsUnique {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (aA : a0 ∈ A) (bB : b0 ∈ B) :

UniqueMul A B a0 b0 ↔ ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0

abbrev UniqueAdd.iff_card_nonpos.match_1 {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (p2 : G × G) (motive : (p2.1 ∈ A ∧ p2.2 ∈ B) ∧ p2.1 + p2.2 = a0 + b0 → Prop) :

∀ (x : (p2.1 ∈ A ∧ p2.2 ∈ B) ∧ p2.1 + p2.2 = a0 + b0), (∀ (ha2 : p2.1 ∈ A) (hb2 : p2.2 ∈ B) (he2 : p2.1 + p2.2 = a0 + b0), motive ⋯) → motive x

Equations

⋯ = ⋯

Instances For

theorem UniqueAdd.iff_card_nonpos {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :

UniqueAdd A B a0 b0 ↔ (Finset.filter (fun (p : G × G) => p.1 + p.2 = a0 + b0) (A ×ˢ B)).card ≤ 1

theorem UniqueMul.iff_card_le_one {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :

UniqueMul A B a0 b0 ↔ (Finset.filter (fun (p : G × G) => p.1 * p.2 = a0 * b0) (A ×ˢ B)).card ≤ 1

abbrev UniqueAdd.exists_iff_exists_existsUnique.match_2 {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} (motive : (∃ (g : G), ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 + ab.2 = g) → Prop) :

∀ (x : ∃ (g : G), ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 + ab.2 = g), (∀ (g : G) (h : ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 + ab.2 = g), motive ⋯) → motive x

Equations

⋯ = ⋯

Instances For

abbrev UniqueAdd.exists_iff_exists_existsUnique.match_1 {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} (motive : (∃ (a0 : G) (b0 : G), a0 ∈ A ∧ b0 ∈ B ∧ UniqueAdd A B a0 b0) → Prop) :

∀ (x : ∃ (a0 : G) (b0 : G), a0 ∈ A ∧ b0 ∈ B ∧ UniqueAdd A B a0 b0), (∀ (a0 b0 : G) (hA : a0 ∈ A) (hB : b0 ∈ B) (h : UniqueAdd A B a0 b0), motive ⋯) → motive x

Equations

⋯ = ⋯

Instances For

theorem UniqueAdd.exists_iff_exists_existsUnique {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} :

(∃ (a0 : G) (b0 : G), a0 ∈ A ∧ b0 ∈ B ∧ UniqueAdd A B a0 b0) ↔ ∃ (g : G), ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 + ab.2 = g

theorem UniqueMul.exists_iff_exists_existsUnique {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} :

(∃ (a0 : G) (b0 : G), a0 ∈ A ∧ b0 ∈ B ∧ UniqueMul A B a0 b0) ↔ ∃ (g : G), ∃! ab : G × G, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = g

theorem UniqueAdd.addHom_preimage {G : Type u_1} {H : Type u_2} [Add G] [Add H] (f : AddHom G H) (hf : Function.Injective ⇑f) (a0 : G) (b0 : G) {A : Finset H} {B : Finset H} (u : UniqueAdd A B (f a0) (f b0)) :

UniqueAdd (A.preimage ⇑f ⋯) (B.preimage ⇑f ⋯) a0 b0

UniqueAdd is preserved by inverse images under injective, additive maps.

theorem UniqueMul.mulHom_preimage {G : Type u_1} {H : Type u_2} [Mul G] [Mul H] (f : G →ₙ* H) (hf : Function.Injective ⇑f) (a0 : G) (b0 : G) {A : Finset H} {B : Finset H} (u : UniqueMul A B (f a0) (f b0)) :

UniqueMul (A.preimage ⇑f ⋯) (B.preimage ⇑f ⋯) a0 b0

UniqueMul is preserved by inverse images under injective, multiplicative maps.

theorem UniqueAdd.of_addHom_image {G : Type u_1} {H : Type u_2} [Add G] [Add H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq H] (f : AddHom G H) (hf : ∀ ⦃a b c d : G⦄, a + b = c + d → f a = f c ∧ f b = f d → a = c ∧ b = d) (h : UniqueAdd (Finset.image (⇑f) A) (Finset.image (⇑f) B) (f a0) (f b0)) :

UniqueAdd A B a0 b0

theorem UniqueMul.of_mulHom_image {G : Type u_1} {H : Type u_2} [Mul G] [Mul H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq H] (f : G →ₙ* H) (hf : ∀ ⦃a b c d : G⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d) (h : UniqueMul (Finset.image (⇑f) A) (Finset.image (⇑f) B) (f a0) (f b0)) :

UniqueMul A B a0 b0

theorem UniqueAdd.addHom_image_iff {G : Type u_1} {H : Type u_2} [Add G] [Add H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq H] (f : AddHom G H) (hf : Function.Injective ⇑f) :

UniqueAdd (Finset.image (⇑f) A) (Finset.image (⇑f) B) (f a0) (f b0) ↔ UniqueAdd A B a0 b0

UniqueAdd is preserved under additive maps that are injective.

See UniqueAdd.addHom_map_iff for a version with swapped bundling.

theorem UniqueMul.mulHom_image_iff {G : Type u_1} {H : Type u_2} [Mul G] [Mul H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} [DecidableEq H] (f : G →ₙ* H) (hf : Function.Injective ⇑f) :

UniqueMul (Finset.image (⇑f) A) (Finset.image (⇑f) B) (f a0) (f b0) ↔ UniqueMul A B a0 b0

Unique_Mul is preserved under multiplicative maps that are injective.

See UniqueMul.mulHom_map_iff for a version with swapped bundling.

theorem UniqueAdd.addHom_map_iff {G : Type u_1} {H : Type u_2} [Add G] [Add H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (f : G ↪ H) (mul : ∀ (x y : G), f (x + y) = f x + f y) :

UniqueAdd (Finset.map f A) (Finset.map f B) (f a0) (f b0) ↔ UniqueAdd A B a0 b0

UniqueAdd is preserved under embeddings that are additive.

See UniqueAdd.addHom_image_iff for a version with swapped bundling.

theorem UniqueMul.mulHom_map_iff {G : Type u_1} {H : Type u_2} [Mul G] [Mul H] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (f : G ↪ H) (mul : ∀ (x y : G), f (x * y) = f x * f y) :

UniqueMul (Finset.map f A) (Finset.map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0

UniqueMul is preserved under embeddings that are multiplicative.

See UniqueMul.mulHom_image_iff for a version with swapped bundling.

theorem UniqueAdd.of_addOpposite {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueAdd (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } A) (AddOpposite.op b0) (AddOpposite.op a0)) :

UniqueAdd A B a0 b0

theorem UniqueMul.of_mulOpposite {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueMul (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } A) (MulOpposite.op b0) (MulOpposite.op a0)) :

UniqueMul A B a0 b0

theorem UniqueAdd.to_addOpposite {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueAdd A B a0 b0) :

UniqueAdd (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } A) (AddOpposite.op b0) (AddOpposite.op a0)

theorem UniqueMul.to_mulOpposite {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} (h : UniqueMul A B a0 b0) :

UniqueMul (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } A) (MulOpposite.op b0) (MulOpposite.op a0)

theorem UniqueAdd.iff_addOpposite {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} :

UniqueAdd (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := AddOpposite.op, inj' := ⋯ } A) (AddOpposite.op b0) (AddOpposite.op a0) ↔ UniqueAdd A B a0 b0

theorem UniqueMul.iff_mulOpposite {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} {a0 : G} {b0 : G} :

UniqueMul (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } A) (MulOpposite.op b0) (MulOpposite.op a0) ↔ UniqueMul A B a0 b0

theorem UniqueAdd.of_image_filter {G : Type u_1} {H : Type u_2} [Add G] [Add H] [DecidableEq H] (f : AddHom G H) {A : Finset G} {B : Finset G} {aG : G} {bG : G} {aH : H} {bH : H} (hae : f aG = aH) (hbe : f bG = bH) (huH : UniqueAdd (Finset.image (⇑f) A) (Finset.image (⇑f) B) aH bH) (huG : UniqueAdd (Finset.filter (fun (x : G) => f x = aH) A) (Finset.filter (fun (x : G) => f x = bH) B) aG bG) :

UniqueAdd A B aG bG

theorem UniqueMul.of_image_filter {G : Type u_1} {H : Type u_2} [Mul G] [Mul H] [DecidableEq H] (f : G →ₙ* H) {A : Finset G} {B : Finset G} {aG : G} {bG : G} {aH : H} {bH : H} (hae : f aG = aH) (hbe : f bG = bH) (huH : UniqueMul (Finset.image (⇑f) A) (Finset.image (⇑f) B) aH bH) (huG : UniqueMul (Finset.filter (fun (x : G) => f x = aH) A) (Finset.filter (fun (x : G) => f x = bH) B) aG bG) :

UniqueMul A B aG bG

class UniqueSums (G : Type u_1) [Add G] :

Let G be a Type with addition. UniqueSums G asserts that any two non-empty finite subsets of G have the UniqueAdd property, with respect to some element of their sum A + B.

uniqueAdd_of_nonempty : ∀ {A B : Finset G}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0
For A B two nonempty finite sets, there always exist a0 ∈ A, b0 ∈ B such that UniqueAdd A B a0 b0

Instances

theorem UniqueSums.uniqueAdd_of_nonempty {G : Type u_1} [Add G] [self : UniqueSums G] {A : Finset G} {B : Finset G} :

A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueAdd A B a0 b0

For A B two nonempty finite sets, there always exist a0 ∈ A, b0 ∈ B such that UniqueAdd A B a0 b0

class UniqueProds (G : Type u_1) [Mul G] :

Let G be a Type with multiplication. UniqueProds G asserts that any two non-empty finite subsets of G have the UniqueMul property, with respect to some element of their product A * B.

uniqueMul_of_nonempty : ∀ {A B : Finset G}, A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0
For A B two nonempty finite sets, there always exist a0 ∈ A, b0 ∈ B such that UniqueMul A B a0 b0

Instances

theorem UniqueProds.uniqueMul_of_nonempty {G : Type u_1} [Mul G] [self : UniqueProds G] {A : Finset G} {B : Finset G} :

A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

For A B two nonempty finite sets, there always exist a0 ∈ A, b0 ∈ B such that UniqueMul A B a0 b0

class TwoUniqueSums (G : Type u_1) [Add G] :

Let G be a Type with addition. TwoUniqueSums G asserts that any two non-empty finite subsets of G, at least one of which is not a singleton, possesses at least two pairs of elements satisfying the UniqueAdd property.

uniqueAdd_of_one_lt_card : ∀ {A B : Finset G}, 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueAdd A B p1.1 p1.2 ∧ UniqueAdd A B p2.1 p2.2
For A B two finite sets whose product has cardinality at least 2, we can find at least two unique pairs.

Instances

theorem TwoUniqueSums.uniqueAdd_of_one_lt_card {G : Type u_1} [Add G] [self : TwoUniqueSums G] {A : Finset G} {B : Finset G} :

1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueAdd A B p1.1 p1.2 ∧ UniqueAdd A B p2.1 p2.2

For A B two finite sets whose product has cardinality at least 2, we can find at least two unique pairs.

class TwoUniqueProds (G : Type u_1) [Mul G] :

Let G be a Type with multiplication. TwoUniqueProds G asserts that any two non-empty finite subsets of G, at least one of which is not a singleton, possesses at least two pairs of elements satisfying the UniqueMul property.

uniqueMul_of_one_lt_card : ∀ {A B : Finset G}, 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
For A B two finite sets whose product has cardinality at least 2, we can find at least two unique pairs.

Instances

theorem TwoUniqueProds.uniqueMul_of_one_lt_card {G : Type u_1} [Mul G] [self : TwoUniqueProds G] {A : Finset G} {B : Finset G} :

1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2

For A B two finite sets whose product has cardinality at least 2, we can find at least two unique pairs.

theorem uniqueAdd_of_twoUniqueAdd {G : Type u_1} [Add G] {A : Finset G} {B : Finset G} (h : 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueAdd A B p1.1 p1.2 ∧ UniqueAdd A B p2.1 p2.2) (hA : A.Nonempty) (hB : B.Nonempty) :

∃ a ∈ A, ∃ b ∈ B, UniqueAdd A B a b

theorem uniqueMul_of_twoUniqueMul {G : Type u_1} [Mul G] {A : Finset G} {B : Finset G} (h : 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2) (hA : A.Nonempty) (hB : B.Nonempty) :

∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b

instance TwoUniqueSums.toUniqueSums (G : Type u_1) [Add G] [TwoUniqueSums G] :

Equations

⋯ = ⋯

instance TwoUniqueProds.toUniqueProds (G : Type u_1) [Mul G] [TwoUniqueProds G] :

Equations

⋯ = ⋯

instance Multiplicative.instUniqueProdsOfUniqueSums {M : Type u_1} [Add M] [UniqueSums M] :

UniqueProds (Multiplicative M)

Equations

⋯ = ⋯

instance Multiplicative.instTwoUniqueProdsOfTwoUniqueSums {M : Type u_1} [Add M] [TwoUniqueSums M] :

TwoUniqueProds (Multiplicative M)

Equations

⋯ = ⋯

instance Additive.instUniqueSumsOfUniqueProds {M : Type u_1} [Mul M] [UniqueProds M] :

UniqueSums (Additive M)

Equations

⋯ = ⋯

instance Additive.instTwoUniqueSumsOfTwoUniqueProds {M : Type u_1} [Mul M] [TwoUniqueProds M] :

TwoUniqueSums (Additive M)

Equations

⋯ = ⋯

theorem UniqueSums.of_addHom {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : ∀ ⦃a b c d : H⦄, a + b = c + d → f a = f c ∧ f b = f d → a = c ∧ b = d) [UniqueSums G] :

theorem UniqueProds.of_mulHom {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d) [UniqueProds G] :

theorem UniqueSums.of_injective_addHom {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : Function.Injective ⇑f) :

UniqueSums G → UniqueSums H

theorem UniqueProds.of_injective_mulHom {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : Function.Injective ⇑f) :

UniqueProds G → UniqueProds H

theorem AddEquiv.uniqueSums_iff {G : Type u} {H : Type v} [Add G] [Add H] (f : G ≃+ H) :

UniqueSums G ↔ UniqueSums H

UniqueSums is preserved under additive equivalences.

theorem MulEquiv.uniqueProds_iff {G : Type u} {H : Type v} [Mul G] [Mul H] (f : G ≃* H) :

UniqueProds G ↔ UniqueProds H

UniqueProd is preserved under multiplicative equivalences.

theorem UniqueSums.of_addOpposite {G : Type u} [Add G] (h : UniqueSums Gᵃᵒᵖ) :

abbrev UniqueSums.of_addOpposite.match_1 {G : Type u_1} [Add G] :

∀ {A B : Finset G} (f : G ↪ Gᵃᵒᵖ) (motive : (∃ a0 ∈ Finset.map f B, ∃ b0 ∈ Finset.map f A, UniqueAdd (Finset.map f B) (Finset.map f A) a0 b0) → Prop) (x : ∃ a0 ∈ Finset.map f B, ∃ b0 ∈ Finset.map f A, UniqueAdd (Finset.map f B) (Finset.map f A) a0 b0), (∀ (y : Gᵃᵒᵖ) (yB : y ∈ Finset.map f B) (x : Gᵃᵒᵖ) (xA : x ∈ Finset.map f A) (hxy : UniqueAdd (Finset.map f B) (Finset.map f A) y x), motive ⋯) → motive x

Equations

⋯ = ⋯

Instances For

theorem UniqueProds.of_mulOpposite {G : Type u} [Mul G] (h : UniqueProds Gᵐᵒᵖ) :

instance UniqueSums.instAddOpposite {G : Type u} [Add G] [h : UniqueSums G] :

UniqueSums Gᵃᵒᵖ

Equations

⋯ = ⋯

instance UniqueProds.instMulOpposite {G : Type u} [Mul G] [h : UniqueProds G] :

UniqueProds Gᵐᵒᵖ

Equations

⋯ = ⋯

theorem UniqueSums.toIsAddCancel {G : Type u} [Add G] [UniqueSums G] :

theorem UniqueProds.toIsCancelMul {G : Type u} [Mul G] [UniqueProds G] :

Two theorems in [Andrzej Strojnowski, A note on u.p. groups][Strojnowski1980]

theorem UniqueSums.of_same {G : Type u_1} [AddSemigroup G] [IsCancelAdd G] (h : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueAdd A A a1 a2) :

theorem UniqueProds.of_same {G : Type u_1} [Semigroup G] [IsCancelMul G] (h : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueMul A A a1 a2) :

UniqueProds G says that for any two nonempty Finsets A and B in G, A × B contains a unique pair with the UniqueMul property. Strojnowski showed that if G is a group, then we only need to check this when A = B. Here we generalize the result to cancellative semigroups. Non-cancellative counterexample: the AddMonoid {0,1} with 1+1=1.

theorem UniqueSums.toTwoUniqueSums_of_addGroup {G : Type u_1} [AddGroup G] [UniqueSums G] :

TwoUniqueSums G

theorem UniqueProds.toTwoUniqueProds_of_group {G : Type u_1} [Group G] [UniqueProds G] :

TwoUniqueProds G

If a group has UniqueProds, then it actually has TwoUniqueProds. For an example of a semigroup G embeddable into a group that has UniqueProds but not TwoUniqueProds, see Example 10.13 in [J. Okniński, Semigroup Algebras][Okninski1991].

instance UniqueSums.instForall {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → Add (G i)] [∀ (i : ι), UniqueSums (G i)] :

UniqueSums ((i : ι) → G i)

Equations

⋯ = ⋯

instance UniqueProds.instForall {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → Mul (G i)] [∀ (i : ι), UniqueProds (G i)] :

UniqueProds ((i : ι) → G i)

Equations

⋯ = ⋯

instance UniqueSums.instSum {G : Type u} {H : Type v} [Add G] [Add H] [UniqueSums G] [UniqueSums H] :

UniqueSums (G × H)

Equations

⋯ = ⋯

instance UniqueProds.instProd {G : Type u} {H : Type v} [Mul G] [Mul H] [UniqueProds G] [UniqueProds H] :

UniqueProds (G × H)

Equations

⋯ = ⋯

instance instUniqueSumsDFinsupp {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → AddZeroClass (G i)] [∀ (i : ι), UniqueSums (G i)] :

UniqueSums (Π₀ (i : ι), G i)

Equations

⋯ = ⋯

instance instUniqueSumsFinsupp {ι : Type u_1} {G : Type u_2} [AddZeroClass G] [UniqueSums G] :

UniqueSums (ι →₀ G)

Equations

⋯ = ⋯

theorem TwoUniqueSums.of_addHom {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : ∀ ⦃a b c d : H⦄, a + b = c + d → f a = f c ∧ f b = f d → a = c ∧ b = d) [TwoUniqueSums G] :

TwoUniqueSums H

theorem TwoUniqueProds.of_mulHom {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d) [TwoUniqueProds G] :

TwoUniqueProds H

theorem TwoUniqueSums.of_injective_addHom {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : Function.Injective ⇑f) :

TwoUniqueSums G → TwoUniqueSums H

theorem TwoUniqueProds.of_injective_mulHom {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : Function.Injective ⇑f) :

TwoUniqueProds G → TwoUniqueProds H

theorem AddEquiv.twoUniqueSums_iff {G : Type u} {H : Type v} [Add G] [Add H] (f : G ≃+ H) :

TwoUniqueSums G ↔ TwoUniqueSums H

TwoUniqueSums is preserved under additive equivalences.

theorem MulEquiv.twoUniqueProds_iff {G : Type u} {H : Type v} [Mul G] [Mul H] (f : G ≃* H) :

TwoUniqueProds G ↔ TwoUniqueProds H

TwoUniqueProd is preserved under multiplicative equivalences.

instance TwoUniqueSums.instForall {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → Add (G i)] [∀ (i : ι), TwoUniqueSums (G i)] :

TwoUniqueSums ((i : ι) → G i)

Equations

⋯ = ⋯

instance TwoUniqueProds.instForall {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → Mul (G i)] [∀ (i : ι), TwoUniqueProds (G i)] :

TwoUniqueProds ((i : ι) → G i)

Equations

⋯ = ⋯

instance TwoUniqueSums.instSum {G : Type u} {H : Type v} [Add G] [Add H] [TwoUniqueSums G] [TwoUniqueSums H] :

TwoUniqueSums (G × H)

Equations

⋯ = ⋯

instance TwoUniqueProds.instProd {G : Type u} {H : Type v} [Mul G] [Mul H] [TwoUniqueProds G] [TwoUniqueProds H] :

TwoUniqueProds (G × H)

Equations

⋯ = ⋯

theorem TwoUniqueSums.of_addOpposite {G : Type u} [Add G] (h : TwoUniqueSums Gᵃᵒᵖ) :

TwoUniqueSums G

theorem TwoUniqueProds.of_mulOpposite {G : Type u} [Mul G] (h : TwoUniqueProds Gᵐᵒᵖ) :

TwoUniqueProds G

instance TwoUniqueSums.instAddOpposite {G : Type u} [Add G] [h : TwoUniqueSums G] :

TwoUniqueSums Gᵃᵒᵖ

Equations

⋯ = ⋯

instance TwoUniqueProds.instMulOpposite {G : Type u} [Mul G] [h : TwoUniqueProds G] :

TwoUniqueProds Gᵐᵒᵖ

Equations

⋯ = ⋯

@[instance 100]

instance TwoUniqueSums.of_covariant_right {G : Type u} [Add G] [IsRightCancelAdd G] [LinearOrder G] [CovariantClass G G (fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x < x_1] :

TwoUniqueSums G

This instance asserts that if G has a right-cancellative addition, a linear order, and addition is strictly monotone w.r.t. the second argument, then G has TwoUniqueSums.

Equations

⋯ = ⋯

@[instance 100]

instance TwoUniqueProds.of_covariant_right {G : Type u} [Mul G] [IsRightCancelMul G] [LinearOrder G] [CovariantClass G G (fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x < x_1] :

TwoUniqueProds G

This instance asserts that if G has a right-cancellative multiplication, a linear order, and multiplication is strictly monotone w.r.t. the second argument, then G has TwoUniqueProds.

Equations

⋯ = ⋯

@[instance 100]

instance TwoUniqueSums.of_covariant_left {G : Type u} [Add G] [IsLeftCancelAdd G] [LinearOrder G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x + x_1) fun (x x_1 : G) => x < x_1] :

TwoUniqueSums G

This instance asserts that if G has a left-cancellative addition, a linear order, and addition is strictly monotone w.r.t. the first argument, then G has TwoUniqueSums.

Equations

⋯ = ⋯

@[instance 100]

instance TwoUniqueProds.of_covariant_left {G : Type u} [Mul G] [IsLeftCancelMul G] [LinearOrder G] [CovariantClass G G (Function.swap fun (x x_1 : G) => x * x_1) fun (x x_1 : G) => x < x_1] :

TwoUniqueProds G

This instance asserts that if G has a left-cancellative multiplication, a linear order, and multiplication is strictly monotone w.r.t. the first argument, then G has TwoUniqueProds.

Equations

⋯ = ⋯

@[deprecated UniqueProds.of_injective_mulHom]

theorem UniqueProds.mulHom_image_of_injective {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : Function.Injective ⇑f) :

UniqueProds G → UniqueProds H

Alias of UniqueProds.of_injective_mulHom.

@[deprecated UniqueSums.of_injective_addHom]

theorem UniqueSums.addHom_image_of_injective {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : Function.Injective ⇑f) :

UniqueSums G → UniqueSums H

Alias of UniqueSums.of_injective_addHom.

@[deprecated MulEquiv.uniqueProds_iff]

theorem UniqueProds.mulHom_image_iff {G : Type u} {H : Type v} [Mul G] [Mul H] (f : G ≃* H) :

UniqueProds G ↔ UniqueProds H

Alias of MulEquiv.uniqueProds_iff.

UniqueProd is preserved under multiplicative equivalences.

@[deprecated AddEquiv.uniqueSums_iff]

theorem UniqueSums.addHom_image_iff {G : Type u} {H : Type v} [Add G] [Add H] (f : G ≃+ H) :

UniqueSums G ↔ UniqueSums H

Alias of AddEquiv.uniqueSums_iff.

UniqueSums is preserved under additive equivalences.

@[deprecated TwoUniqueProds.of_injective_mulHom]

theorem TwoUniqueProds.mulHom_image_of_injective {G : Type u} {H : Type v} [Mul G] [Mul H] (f : H →ₙ* G) (hf : Function.Injective ⇑f) :

TwoUniqueProds G → TwoUniqueProds H

Alias of TwoUniqueProds.of_injective_mulHom.

@[deprecated TwoUniqueSums.of_injective_addHom]

theorem TwoUniqueSums.addHom_image_of_injective {G : Type u} {H : Type v} [Add G] [Add H] (f : AddHom H G) (hf : Function.Injective ⇑f) :

TwoUniqueSums G → TwoUniqueSums H

Alias of TwoUniqueSums.of_injective_addHom.

@[deprecated MulEquiv.twoUniqueProds_iff]

theorem TwoUniqueProds.mulHom_image_iff {G : Type u} {H : Type v} [Mul G] [Mul H] (f : G ≃* H) :

TwoUniqueProds G ↔ TwoUniqueProds H

Alias of MulEquiv.twoUniqueProds_iff.

TwoUniqueProd is preserved under multiplicative equivalences.

@[deprecated AddEquiv.twoUniqueSums_iff]

theorem TwoUniqueSums.addHom_image_iff {G : Type u} {H : Type v} [Add G] [Add H] (f : G ≃+ H) :

TwoUniqueSums G ↔ TwoUniqueSums H

Alias of AddEquiv.twoUniqueSums_iff.

TwoUniqueSums is preserved under additive equivalences.

instance instTwoUniqueSumsDFinsupp {ι : Type u_2} (G : ι → Type u_1) [(i : ι) → AddZeroClass (G i)] [∀ (i : ι), TwoUniqueSums (G i)] :

TwoUniqueSums (Π₀ (i : ι), G i)

Equations

⋯ = ⋯

instance instTwoUniqueSumsFinsupp {ι : Type u_1} {G : Type u_2} [AddZeroClass G] [TwoUniqueSums G] :

TwoUniqueSums (ι →₀ G)

Equations

⋯ = ⋯

instance instTwoUniqueSumsOfModuleRat {G : Type u} [AddCommGroup G] [Module ℚ G] :

TwoUniqueSums G

Any ℚ-vector space has TwoUniqueSums, because it is isomorphic to some (Basis.ofVectorSpaceIndex ℚ G) →₀ ℚ by choosing a basis, and ℚ already has TwoUniqueSums because it's ordered.

Equations

⋯ = ⋯

instance FreeMonoid.instTwoUniqueProds {κ : Type u_1} :

TwoUniqueProds (FreeMonoid κ)

Any FreeMonoid has the TwoUniqueProds property.

Equations

⋯ = ⋯

instance FreeAddMonoid.instTwoUniqueSums {κ : Type u_1} :

TwoUniqueSums (FreeAddMonoid κ)

Any FreeAddMonoid has the TwoUniqueSums property.

Equations

⋯ = ⋯