Monoid algebras

When the domain of a Finsupp has a multiplicative or additive structure, we can define a convolution product. To mathematicians this structure is known as the "monoid algebra", i.e. the finite formal linear combinations over a given semiring of elements of the monoid. The "group ring" ℤ[G] or the "group algebra" k[G] are typical uses.

In fact the construction of the "monoid algebra" makes sense when G is not even a monoid, but merely a magma, i.e., when G carries a multiplication which is not required to satisfy any conditions at all. In this case the construction yields a not-necessarily-unital, not-necessarily-associative algebra but it is still adjoint to the forgetful functor from such algebras to magmas, and we prove this as MonoidAlgebra.liftMagma.

In this file we define MonoidAlgebra k G := G →₀ k, and AddMonoidAlgebra k G in the same way, and then define the convolution product on these.

When the domain is additive, this is used to define polynomials:

Polynomial R := AddMonoidAlgebra R ℕ
MvPolynomial σ α := AddMonoidAlgebra R (σ →₀ ℕ)

When the domain is multiplicative, e.g. a group, this will be used to define the group ring.

Notation

We introduce the notation R[A] for AddMonoidAlgebra R A.

Implementation note

Unfortunately because additive and multiplicative structures both appear in both cases, it doesn't appear to be possible to make much use of to_additive, and we just settle for saying everything twice.

Similarly, I attempted to just define k[G] := MonoidAlgebra k (Multiplicative G), but the definitional equality Multiplicative G = G leaks through everywhere, and seems impossible to use.

Multiplicative monoids

def MonoidAlgebra (k : Type u₁) (G : Type u₂) [Semiring k] : Type (max u₁ u₂)

The monoid algebra over a semiring k generated by the monoid G. It is the type of finite formal k-linear combinations of terms of G, endowed with the convolution product.

instance MonoidAlgebra.inhabited (k : Type u₁) (G : Type u₂) [Semiring k] : Inhabited (MonoidAlgebra k G)

instance MonoidAlgebra.addCommMonoid (k : Type u₁) (G : Type u₂) [Semiring k] : AddCommMonoid (MonoidAlgebra k G)

instance MonoidAlgebra.instIsCancelAdd (k : Type u₁) (G : Type u₂) [Semiring k] [IsCancelAdd k] : IsCancelAdd (MonoidAlgebra k G)

instance MonoidAlgebra.coeFun (k : Type u₁) (G : Type u₂) [Semiring k] : CoeFun (MonoidAlgebra k G) fun (x : MonoidAlgebra k G) => G → k

@[reducible, inline]

abbrev MonoidAlgebra.single {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b : k) : MonoidAlgebra k G

MonoidAlgebra.single a r for a : M, r : R is the element ar : R[M].

theorem MonoidAlgebra.single_zero {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) : single a 0 = 0

theorem MonoidAlgebra.single_add {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂

theorem MonoidAlgebra.sum_single_index {k : Type u₁} {G : Type u₂} [Semiring k] {N : Type u_5} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N} (h_zero : h a 0 = 0) : Finsupp.sum (single a b) h = h a b

@[simp]

theorem MonoidAlgebra.sum_single {k : Type u₁} {G : Type u₂} [Semiring k] (f : MonoidAlgebra k G) : Finsupp.sum f single = f

theorem MonoidAlgebra.single_apply {k : Type u₁} {G : Type u₂} [Semiring k] {a a' : G} {b : k} [Decidable (a = a')] : (single a b) a' = if a = a' then b else 0

@[simp]

theorem MonoidAlgebra.single_eq_zero {k : Type u₁} {G : Type u₂} [Semiring k] {a : G} {b : k} : single a b = 0 ↔ b = 0

theorem MonoidAlgebra.single_ne_zero {k : Type u₁} {G : Type u₂} [Semiring k] {a : G} {b : k} : single a b ≠ 0 ↔ b ≠ 0

theorem MonoidAlgebra.induction_linear {k : Type u₁} {G : Type u₂} [Semiring k] {p : MonoidAlgebra k G → Prop} (f : MonoidAlgebra k G) (zero : p 0) (add : ∀ (f g : MonoidAlgebra k G), p f → p g → p (f + g)) (single : ∀ (a : G) (b : k), p (single a b)) : p f

@[irreducible]

def MonoidAlgebra.mul' {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (f g : MonoidAlgebra k G) : MonoidAlgebra k G

The multiplication in a monoid algebra. We make it irreducible so that Lean doesn't unfold it trying to unify two things that are different.

instance MonoidAlgebra.instMul {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] : Mul (MonoidAlgebra k G)

The product of f g : MonoidAlgebra k G is the finitely supported function whose value at a is the sum of f x * g y over all pairs x, y such that x * y = a. (Think of the group ring of a group.)

theorem MonoidAlgebra.mul_def {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] {f g : MonoidAlgebra k G} : f * g = Finsupp.sum f fun (a₁ : G) (b₁ : k) => Finsupp.sum g fun (a₂ : G) (b₂ : k) => single (a₁ * a₂) (b₁ * b₂)

instance MonoidAlgebra.nonUnitalNonAssocSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] : NonUnitalNonAssocSemiring (MonoidAlgebra k G)

instance MonoidAlgebra.nonUnitalSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [Semigroup G] : NonUnitalSemiring (MonoidAlgebra k G)

instance MonoidAlgebra.one {k : Type u₁} {G : Type u₂} [Semiring k] [One G] : One (MonoidAlgebra k G)

The unit of the multiplication is single 1 1, i.e. the function that is 1 at 1 and zero elsewhere.

theorem MonoidAlgebra.one_def {k : Type u₁} {G : Type u₂} [Semiring k] [One G] : 1 = single 1 1

instance MonoidAlgebra.nonAssocSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] : NonAssocSemiring (MonoidAlgebra k G)

theorem MonoidAlgebra.natCast_def {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (n : ℕ) : ↑n = single 1 ↑n

Basic scalar multiplication instances

This section collects instances needed for the algebraic structure of Polynomial, which is defined in terms of MonoidAlgebra. Further results on scalar multiplication can be found in Mathlib/Algebra/MonoidAlgebra/Module.lean.

instance MonoidAlgebra.smulZeroClass {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] : SMulZeroClass R (MonoidAlgebra k G)

instance MonoidAlgebra.distribSMul {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [DistribSMul R k] : DistribSMul R (MonoidAlgebra k G)

instance MonoidAlgebra.isScalarTower {k : Type u₁} {G : Type u₂} {R : Type u_2} {S : Type u_5} [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S] [IsScalarTower R S k] : IsScalarTower R S (MonoidAlgebra k G)

instance MonoidAlgebra.smulCommClass {k : Type u₁} {G : Type u₂} {R : Type u_2} {S : Type u_5} [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMulCommClass R S k] : SMulCommClass R S (MonoidAlgebra k G)

instance MonoidAlgebra.isCentralScalar {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k] [IsCentralScalar R k] : IsCentralScalar R (MonoidAlgebra k G)

Semiring structure

instance MonoidAlgebra.semiring {k : Type u₁} {G : Type u₂} [Semiring k] [Monoid G] : Semiring (MonoidAlgebra k G)

@[simp]

theorem MonoidAlgebra.coe_add {G : Type u₂} {R : Type u_2} [Monoid G] [Semiring R] (f g : MonoidAlgebra R G) : ⇑(f + g) = ⇑f + ⇑g

instance MonoidAlgebra.nonUnitalCommSemiring {k : Type u₁} {G : Type u₂} [CommSemiring k] [CommSemigroup G] : NonUnitalCommSemiring (MonoidAlgebra k G)

instance MonoidAlgebra.nontrivial {k : Type u₁} {G : Type u₂} [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial (MonoidAlgebra k G)

Derived instances

instance MonoidAlgebra.commSemiring {k : Type u₁} {G : Type u₂} [CommSemiring k] [CommMonoid G] : CommSemiring (MonoidAlgebra k G)

instance MonoidAlgebra.unique {k : Type u₁} {G : Type u₂} [Semiring k] [Subsingleton k] : Unique (MonoidAlgebra k G)

instance MonoidAlgebra.addCommGroup {k : Type u₁} {G : Type u₂} [Ring k] : AddCommGroup (MonoidAlgebra k G)

instance MonoidAlgebra.nonUnitalNonAssocRing {k : Type u₁} {G : Type u₂} [Ring k] [Mul G] : NonUnitalNonAssocRing (MonoidAlgebra k G)

instance MonoidAlgebra.nonUnitalRing {k : Type u₁} {G : Type u₂} [Ring k] [Semigroup G] : NonUnitalRing (MonoidAlgebra k G)

instance MonoidAlgebra.nonAssocRing {k : Type u₁} {G : Type u₂} [Ring k] [MulOneClass G] : NonAssocRing (MonoidAlgebra k G)

theorem MonoidAlgebra.intCast_def {k : Type u₁} {G : Type u₂} [Ring k] [MulOneClass G] (z : ℤ) : ↑z = single 1 ↑z

instance MonoidAlgebra.ring {G : Type u₂} {R : Type u_2} [Ring R] [Monoid G] : Ring (MonoidAlgebra R G)

theorem MonoidAlgebra.single_neg {R : Type u_2} {M : Type u_4} [Ring R] (a : M) (b : R) : single a (-b) = -single a b

@[simp]

theorem MonoidAlgebra.neg_apply {R : Type u_2} {M : Type u_4} [Ring R] (m : M) (x : MonoidAlgebra R M) : (-x) m = -x m

instance MonoidAlgebra.nonUnitalCommRing {k : Type u₁} {G : Type u₂} [CommRing k] [CommSemigroup G] : NonUnitalCommRing (MonoidAlgebra k G)

instance MonoidAlgebra.commRing {k : Type u₁} {G : Type u₂} [CommRing k] [CommMonoid G] : CommRing (MonoidAlgebra k G)

@[simp]

theorem MonoidAlgebra.smul_apply {R : Type u_2} {S : Type u_3} {M : Type u_4} [Semiring S] [SMulZeroClass R S] (r : R) (m : M) (x : MonoidAlgebra S M) : (r • x) m = r • x m

@[simp]

theorem MonoidAlgebra.smul_single {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] (a : G) (c : R) (b : k) : c • single a b = single a (c • b)

Copies of ext lemmas and bundled singles from Finsupp

As MonoidAlgebra is a type synonym, ext will not unfold it to find ext lemmas. We need bundled version of Finsupp.single with the right types to state these lemmas. It is good practice to have those, regardless of the ext issue.

theorem MonoidAlgebra.ext {k : Type u₁} {G : Type u₂} [Semiring k] ⦃f g : MonoidAlgebra k G⦄ (H : ∀ (x : G), f x = g x) : f = g

A copy of Finsupp.ext for MonoidAlgebra.

theorem MonoidAlgebra.ext_iff {k : Type u₁} {G : Type u₂} [Semiring k] {f g : MonoidAlgebra k G} : f = g ↔ ∀ (x : G), f x = g x

@[reducible, inline]

abbrev MonoidAlgebra.singleAddHom {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) : k →+ MonoidAlgebra k G

A copy of Finsupp.singleAddHom for MonoidAlgebra.

@[simp]

theorem MonoidAlgebra.singleAddHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b : k) : (singleAddHom a) b = single a b

theorem MonoidAlgebra.addHom_ext' {k : Type u₁} {G : Type u₂} {N : Type u_5} [Semiring k] [AddZeroClass N] ⦃f g : MonoidAlgebra k G →+ N⦄ (H : ∀ (x : G), f.comp (singleAddHom x) = g.comp (singleAddHom x)) : f = g

A copy of Finsupp.addHom_ext' for MonoidAlgebra.

theorem MonoidAlgebra.addHom_ext'_iff {k : Type u₁} {G : Type u₂} {N : Type u_5} [Semiring k] [AddZeroClass N] {f g : MonoidAlgebra k G →+ N} : f = g ↔ ∀ (x : G), f.comp (singleAddHom x) = g.comp (singleAddHom x)

theorem MonoidAlgebra.mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Mul G] (f g : MonoidAlgebra k G) (x : G) : (f * g) x = Finsupp.sum f fun (a₁ : G) (b₁ : k) => Finsupp.sum g fun (a₂ : G) (b₂ : k) => if a₁ * a₂ = x then b₁ * b₂ else 0

theorem MonoidAlgebra.mul_apply_antidiagonal {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (f g : MonoidAlgebra k G) (x : G) (s : Finset (G × G)) (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2

@[simp]

theorem MonoidAlgebra.single_mul_single {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] {a₁ a₂ : G} {b₁ b₂ : k} : single a₁ b₁ * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂)

theorem MonoidAlgebra.single_commute_single {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] {a₁ a₂ : G} {b₁ b₂ : k} (ha : Commute a₁ a₂) (hb : Commute b₁ b₂) : Commute (single a₁ b₁) (single a₂ b₂)

theorem MonoidAlgebra.single_commute {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] {a : G} {b : k} (ha : ∀ (a' : G), Commute a a') (hb : ∀ (b' : k), Commute b b') (f : MonoidAlgebra k G) : Commute (single a b) f

@[simp]

theorem MonoidAlgebra.single_pow {k : Type u₁} {G : Type u₂} [Semiring k] [Monoid G] {a : G} {b : k} (n : ℕ) : single a b ^ n = single (a ^ n) (b ^ n)

def MonoidAlgebra.ofMagma (k : Type u₁) (G : Type u₂) [Semiring k] [Mul G] : G →ₙ* MonoidAlgebra k G

The embedding of a magma into its magma algebra.

@[simp]

theorem MonoidAlgebra.ofMagma_apply (k : Type u₁) (G : Type u₂) [Semiring k] [Mul G] (a : G) : (ofMagma k G) a = single a 1

def MonoidAlgebra.of (k : Type u₁) (G : Type u₂) [Semiring k] [MulOneClass G] : G →* MonoidAlgebra k G

The embedding of a unital magma into its magma algebra.

@[simp]

theorem MonoidAlgebra.of_apply (k : Type u₁) (G : Type u₂) [Semiring k] [MulOneClass G] (a : G) : (of k G) a = single a 1

theorem MonoidAlgebra.of_injective {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] [Nontrivial k] : Function.Injective ⇑(of k G)

theorem MonoidAlgebra.of_commute {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] {a : G} (h : ∀ (a' : G), Commute a a') (f : MonoidAlgebra k G) : Commute ((of k G) a) f

def MonoidAlgebra.singleHom {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] : k × G →* MonoidAlgebra k G

Finsupp.single as a MonoidHom from the product type into the monoid algebra.

Note the order of the elements of the product are reversed compared to the arguments of Finsupp.single.

@[simp]

theorem MonoidAlgebra.singleHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (a : k × G) : singleHom a = single a.2 a.1

theorem MonoidAlgebra.mul_single_apply_aux {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G} (H : ∀ a ∈ f.support, a * x = z ↔ a = y) : (f * single x r) z = f y * r

theorem MonoidAlgebra.mul_single_one_apply {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) : (f * single 1 r) x = f x * r

theorem MonoidAlgebra.mul_single_apply_of_not_exists_mul {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G) (h : ¬∃ (d : G), g' = d * g) : (x * single g r) g' = 0

theorem MonoidAlgebra.single_mul_apply_aux {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G} (H : ∀ a ∈ f.support, x * a = y ↔ a = z) : (single x r * f) y = r * f z

theorem MonoidAlgebra.single_one_mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) : (single 1 r * f) x = r * f x

theorem MonoidAlgebra.single_mul_apply_of_not_exists_mul {k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G) (h : ¬∃ (d : G), g' = g * d) : (single g r * x) g' = 0

theorem MonoidAlgebra.single_one_comm {k : Type u₁} {G : Type u₂} [CommSemiring k] [MulOneClass G] (r : k) (f : MonoidAlgebra k G) : single 1 r * f = f * single 1 r

def MonoidAlgebra.singleOneRingHom {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] : k →+* MonoidAlgebra k G

Finsupp.single 1 as a RingHom

@[simp]

theorem MonoidAlgebra.singleOneRingHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] [MulOneClass G] (b : k) : singleOneRingHom b = single 1 b

theorem MonoidAlgebra.ringHom_ext {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R} (h₁ : ∀ (b : k), f (single 1 b) = g (single 1 b)) (h_of : ∀ (a : G), f (single a 1) = g (single a 1)) : f = g

If two ring homomorphisms from MonoidAlgebra k G are equal on all single a 1 and single 1 b, then they are equal.

theorem MonoidAlgebra.ringHom_ext' {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R} (h₁ : f.comp singleOneRingHom = g.comp singleOneRingHom) (h_of : (↑f).comp (of k G) = (↑g).comp (of k G)) : f = g

If two ring homomorphisms from MonoidAlgebra k G are equal on all single a 1 and single 1 b, then they are equal.

See note [partially-applied ext lemmas].

theorem MonoidAlgebra.ringHom_ext'_iff {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R} : f = g ↔ f.comp singleOneRingHom = g.comp singleOneRingHom ∧ (↑f).comp (of k G) = (↑g).comp (of k G)

theorem MonoidAlgebra.induction_on {k : Type u₁} {G : Type u₂} [Semiring k] [Monoid G] {p : MonoidAlgebra k G → Prop} (f : MonoidAlgebra k G) (hM : ∀ (g : G), p ((of k G) g)) (hadd : ∀ (f g : MonoidAlgebra k G), p f → p g → p (f + g)) (hsmul : ∀ (r : k) (f : MonoidAlgebra k G), p f → p (r • f)) : p f

theorem MonoidAlgebra.prod_single {k : Type u₁} {G : Type u₂} {ι : Type ui} [CommSemiring k] [CommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} : ∏ i ∈ s, single (a i) (b i) = single (∏ i ∈ s, a i) (∏ i ∈ s, b i)

@[simp]

theorem MonoidAlgebra.mul_single_apply {k : Type u₁} {G : Type u₂} [Semiring k] [Group G] (f : MonoidAlgebra k G) (r : k) (x y : G) : (f * single x r) y = f (y * x⁻¹) * r

@[simp]

theorem MonoidAlgebra.single_mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [Group G] (r : k) (x : G) (f : MonoidAlgebra k G) (y : G) : (single x r * f) y = r * f (x⁻¹ * y)

theorem MonoidAlgebra.mul_apply_left {k : Type u₁} {G : Type u₂} [Semiring k] [Group G] (f g : MonoidAlgebra k G) (x : G) : (f * g) x = Finsupp.sum f fun (a : G) (b : k) => b * g (a⁻¹ * x)

theorem MonoidAlgebra.mul_apply_right {k : Type u₁} {G : Type u₂} [Semiring k] [Group G] (f g : MonoidAlgebra k G) (x : G) : (f * g) x = Finsupp.sum g fun (a : G) (b : k) => f (x * a⁻¹) * b

instance MonoidAlgebra.isLocalHom_singleOneRingHom {k : Type u₁} {G : Type u₂} [Semiring k] [Monoid G] : IsLocalHom singleOneRingHom

Additive monoids

def AddMonoidAlgebra (k : Type u₁) (G : Type u₂) [Semiring k] : Type (max u₂ u₁)

The monoid algebra over a semiring k generated by the additive monoid G, denoted by k[G]. It is the type of finite formal k-linear combinations of terms of G, endowed with the convolution product.

def AddMonoidAlgebra.«term__[_]» : Lean.TrailingParserDescr

The monoid algebra over a semiring k generated by the additive monoid G, denoted by k[G]. It is the type of finite formal k-linear combinations of terms of G, endowed with the convolution product.

def AddMonoidAlgebra.unexpander : Lean.PrettyPrinter.Unexpander

Unexpander for AddMonoidAlgebra.

instance AddMonoidAlgebra.inhabited (k : Type u₁) (G : Type u₂) [Semiring k] : Inhabited (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.addCommMonoid (k : Type u₁) (G : Type u₂) [Semiring k] : AddCommMonoid (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.instIsCancelAdd (k : Type u₁) (G : Type u₂) [Semiring k] [IsCancelAdd k] : IsCancelAdd (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.coeFun (k : Type u₁) (G : Type u₂) [Semiring k] : CoeFun (AddMonoidAlgebra k G) fun (x : AddMonoidAlgebra k G) => G → k

@[reducible, inline]

abbrev AddMonoidAlgebra.single {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b : k) : AddMonoidAlgebra k G

MonoidAlgebra.single a r for a : M, r : R is the element ar : R[M].

theorem AddMonoidAlgebra.single_zero {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) : single a 0 = 0

theorem AddMonoidAlgebra.single_add {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂

theorem AddMonoidAlgebra.sum_single_index {k : Type u₁} {G : Type u₂} [Semiring k] {N : Type u_5} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N} (h_zero : h a 0 = 0) : Finsupp.sum (single a b) h = h a b

@[simp]

theorem AddMonoidAlgebra.sum_single {k : Type u₁} {G : Type u₂} [Semiring k] (f : AddMonoidAlgebra k G) : Finsupp.sum f single = f

theorem AddMonoidAlgebra.single_apply {k : Type u₁} {G : Type u₂} [Semiring k] {a a' : G} {b : k} [Decidable (a = a')] : (single a b) a' = if a = a' then b else 0

@[simp]

theorem AddMonoidAlgebra.single_eq_zero {k : Type u₁} {G : Type u₂} [Semiring k] {a : G} {b : k} : single a b = 0 ↔ b = 0

theorem AddMonoidAlgebra.single_ne_zero {k : Type u₁} {G : Type u₂} [Semiring k] {a : G} {b : k} : single a b ≠ 0 ↔ b ≠ 0

instance AddMonoidAlgebra.hasMul {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] : Mul (AddMonoidAlgebra k G)

The product of f g : k[G] is the finitely supported function whose value at a is the sum of f x * g y over all pairs x, y such that x + y = a. (Think of the product of multivariate polynomials where α is the additive monoid of monomial exponents.)

theorem AddMonoidAlgebra.mul_def {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] {f g : AddMonoidAlgebra k G} : f * g = Finsupp.sum f fun (a₁ : G) (b₁ : k) => Finsupp.sum g fun (a₂ : G) (b₂ : k) => single (a₁ + a₂) (b₁ * b₂)

instance AddMonoidAlgebra.nonUnitalNonAssocSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] : NonUnitalNonAssocSemiring (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.one {k : Type u₁} {G : Type u₂} [Semiring k] [Zero G] : One (AddMonoidAlgebra k G)

The unit of the multiplication is single 0 1, i.e. the function that is 1 at 0 and zero elsewhere.

theorem AddMonoidAlgebra.one_def {k : Type u₁} {G : Type u₂} [Semiring k] [Zero G] : 1 = single 0 1

instance AddMonoidAlgebra.nonUnitalSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [AddSemigroup G] : NonUnitalSemiring (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.nonAssocSemiring {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] : NonAssocSemiring (AddMonoidAlgebra k G)

theorem AddMonoidAlgebra.natCast_def {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (n : ℕ) : ↑n = single 0 ↑n

Basic scalar multiplication instances

The SMul section for MonoidAlgebra collects instances needed for the algebraic structure of Polynomial, which is defined in terms of MonoidAlgebra. This section mirrors the MonoidAlgebra section.

Further results on scalar multiplication can be found in Mathlib/Algebra/MonoidAlgebra/Module.lean.

instance AddMonoidAlgebra.smulZeroClass {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] : SMulZeroClass R (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.distribSMul {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [DistribSMul R k] : DistribSMul R (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.isScalarTower {k : Type u₁} {G : Type u₂} {R : Type u_2} {S : Type u_5} [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S] [IsScalarTower R S k] : IsScalarTower R S (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.isCentralScalar {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k] [IsCentralScalar R k] : IsCentralScalar R (AddMonoidAlgebra k G)

Semiring structure

instance AddMonoidAlgebra.semiring {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] : Semiring (AddMonoidAlgebra k G)

@[simp]

theorem AddMonoidAlgebra.coe_add {G : Type u₂} {R : Type u_2} [AddMonoid G] [Semiring R] (f g : AddMonoidAlgebra R G) : ⇑(f + g) = ⇑f + ⇑g

instance AddMonoidAlgebra.nonUnitalCommSemiring {k : Type u₁} {G : Type u₂} [CommSemiring k] [AddCommSemigroup G] : NonUnitalCommSemiring (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.nontrivial {k : Type u₁} {G : Type u₂} [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial (AddMonoidAlgebra k G)

Derived instances

instance AddMonoidAlgebra.commSemiring {k : Type u₁} {G : Type u₂} [CommSemiring k] [AddCommMonoid G] : CommSemiring (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.unique {k : Type u₁} {G : Type u₂} [Semiring k] [Subsingleton k] : Unique (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.addCommGroup {k : Type u₁} {G : Type u₂} [Ring k] : AddCommGroup (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.nonUnitalNonAssocRing {k : Type u₁} {G : Type u₂} [Ring k] [Add G] : NonUnitalNonAssocRing (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.nonUnitalRing {k : Type u₁} {G : Type u₂} [Ring k] [AddSemigroup G] : NonUnitalRing (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.nonAssocRing {k : Type u₁} {G : Type u₂} [Ring k] [AddZeroClass G] : NonAssocRing (AddMonoidAlgebra k G)

theorem AddMonoidAlgebra.intCast_def {k : Type u₁} {G : Type u₂} [Ring k] [AddZeroClass G] (z : ℤ) : ↑z = single 0 ↑z

instance AddMonoidAlgebra.ring {G : Type u₂} {R : Type u_2} [Ring R] [AddMonoid G] : Ring (AddMonoidAlgebra R G)

theorem AddMonoidAlgebra.single_neg {R : Type u_2} {M : Type u_4} [Ring R] (a : M) (b : R) : single a (-b) = -single a b

@[simp]

theorem AddMonoidAlgebra.neg_apply {R : Type u_2} {M : Type u_4} [Ring R] (m : M) (x : MonoidAlgebra R M) : (-x) m = -x m

instance AddMonoidAlgebra.nonUnitalCommRing {k : Type u₁} {G : Type u₂} [CommRing k] [AddCommSemigroup G] : NonUnitalCommRing (AddMonoidAlgebra k G)

instance AddMonoidAlgebra.commRing {k : Type u₁} {G : Type u₂} [CommRing k] [AddCommMonoid G] : CommRing (AddMonoidAlgebra k G)

@[simp]

theorem AddMonoidAlgebra.smul_apply {R : Type u_2} {S : Type u_3} {M : Type u_4} [Semiring S] [SMulZeroClass R S] (r : R) (m : M) (x : MonoidAlgebra S M) : (r • x) m = r • x m

@[simp]

theorem AddMonoidAlgebra.smul_single {k : Type u₁} {G : Type u₂} {R : Type u_2} [Semiring k] [SMulZeroClass R k] (a : G) (c : R) (b : k) : c • single a b = single a (c • b)

Copies of ext lemmas and bundled singles from Finsupp

As AddMonoidAlgebra is a type synonym, ext will not unfold it to find ext lemmas. We need bundled version of Finsupp.single with the right types to state these lemmas. It is good practice to have those, regardless of the ext issue.

theorem AddMonoidAlgebra.ext {k : Type u₁} {G : Type u₂} [Semiring k] ⦃f g : AddMonoidAlgebra k G⦄ (H : ∀ (x : G), f x = g x) : f = g

A copy of Finsupp.ext for AddMonoidAlgebra.

theorem AddMonoidAlgebra.ext_iff {k : Type u₁} {G : Type u₂} [Semiring k] {f g : AddMonoidAlgebra k G} : f = g ↔ ∀ (x : G), f x = g x

@[reducible, inline]

abbrev AddMonoidAlgebra.singleAddHom {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) : k →+ AddMonoidAlgebra k G

A copy of Finsupp.singleAddHom for AddMonoidAlgebra.

@[simp]

theorem AddMonoidAlgebra.singleAddHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) (b : k) : (singleAddHom a) b = single a b

theorem AddMonoidAlgebra.addHom_ext' {k : Type u₁} {G : Type u₂} {N : Type u_5} [Semiring k] [AddZeroClass N] ⦃f g : AddMonoidAlgebra k G →+ N⦄ (H : ∀ (x : G), f.comp (singleAddHom x) = g.comp (singleAddHom x)) : f = g

A copy of Finsupp.addHom_ext' for AddMonoidAlgebra.

theorem AddMonoidAlgebra.addHom_ext'_iff {k : Type u₁} {G : Type u₂} {N : Type u_5} [Semiring k] [AddZeroClass N] {f g : AddMonoidAlgebra k G →+ N} : f = g ↔ ∀ (x : G), f.comp (singleAddHom x) = g.comp (singleAddHom x)

theorem AddMonoidAlgebra.mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [DecidableEq G] [Add G] (f g : AddMonoidAlgebra k G) (x : G) : (f * g) x = Finsupp.sum f fun (a₁ : G) (b₁ : k) => Finsupp.sum g fun (a₂ : G) (b₂ : k) => if a₁ + a₂ = x then b₁ * b₂ else 0

theorem AddMonoidAlgebra.mul_apply_antidiagonal {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] (f g : AddMonoidAlgebra k G) (x : G) (s : Finset (G × G)) (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 + p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2

theorem AddMonoidAlgebra.single_mul_single {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] {a₁ a₂ : G} {b₁ b₂ : k} : single a₁ b₁ * single a₂ b₂ = single (a₁ + a₂) (b₁ * b₂)

theorem AddMonoidAlgebra.single_commute_single {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] {a₁ a₂ : G} {b₁ b₂ : k} (ha : AddCommute a₁ a₂) (hb : Commute b₁ b₂) : Commute (single a₁ b₁) (single a₂ b₂)

theorem AddMonoidAlgebra.single_pow {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] {a : G} {b : k} (n : ℕ) : single a b ^ n = single (n • a) (b ^ n)

def AddMonoidAlgebra.ofMagma (k : Type u₁) (G : Type u₂) [Semiring k] [Add G] : Multiplicative G →ₙ* AddMonoidAlgebra k G

The embedding of an additive magma into its additive magma algebra.

@[simp]

theorem AddMonoidAlgebra.ofMagma_apply (k : Type u₁) (G : Type u₂) [Semiring k] [Add G] (a : Multiplicative G) : (ofMagma k G) a = single a 1

def AddMonoidAlgebra.of (k : Type u₁) (G : Type u₂) [Semiring k] [AddZeroClass G] : Multiplicative G →* AddMonoidAlgebra k G

Embedding of a magma with zero into its magma algebra.

def AddMonoidAlgebra.of' (k : Type u₁) (G : Type u₂) [Semiring k] : G → AddMonoidAlgebra k G

Embedding of a magma with zero G, into its magma algebra, having G as source.

@[simp]

theorem AddMonoidAlgebra.of_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (a : Multiplicative G) : (of k G) a = single (Multiplicative.toAdd a) 1

@[simp]

theorem AddMonoidAlgebra.of'_apply {k : Type u₁} {G : Type u₂} [Semiring k] (a : G) : of' k G a = single a 1

theorem AddMonoidAlgebra.of'_eq_of {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (a : G) : of' k G a = (of k G) (Multiplicative.ofAdd a)

theorem AddMonoidAlgebra.of_injective {k : Type u₁} {G : Type u₂} [Semiring k] [Nontrivial k] [AddZeroClass G] : Function.Injective ⇑(of k G)

theorem AddMonoidAlgebra.of'_commute {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] {a : G} (h : ∀ (a' : G), AddCommute a a') (f : AddMonoidAlgebra k G) : Commute (of' k G a) f

def AddMonoidAlgebra.singleHom {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] : k × Multiplicative G →* AddMonoidAlgebra k G

Finsupp.single as a MonoidHom from the product type into the additive monoid algebra.

Note the order of the elements of the product are reversed compared to the arguments of Finsupp.single.

@[simp]

theorem AddMonoidAlgebra.singleHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (a : k × Multiplicative G) : singleHom a = single (Multiplicative.toAdd a.2) a.1

theorem AddMonoidAlgebra.smul_single' {k : Type u₁} {G : Type u₂} [Semiring k] (c : k) (a : G) (b : k) : c • single a b = single a (c * b)

Copy of Finsupp.smul_single' that avoids the AddMonoidAlgebra = Finsupp defeq abuse.

theorem AddMonoidAlgebra.mul_single_apply_aux {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] (f : AddMonoidAlgebra k G) (r : k) (x y z : G) (H : ∀ a ∈ f.support, a + x = z ↔ a = y) : (f * single x r) z = f y * r

theorem AddMonoidAlgebra.mul_single_zero_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (f : AddMonoidAlgebra k G) (r : k) (x : G) : (f * single 0 r) x = f x * r

theorem AddMonoidAlgebra.mul_single_apply_of_not_exists_add {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] (r : k) {g g' : G} (x : AddMonoidAlgebra k G) (h : ¬∃ (d : G), g' = d + g) : (x * single g r) g' = 0

theorem AddMonoidAlgebra.single_mul_apply_aux {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] (f : AddMonoidAlgebra k G) (r : k) (x y z : G) (H : ∀ a ∈ f.support, x + a = y ↔ a = z) : (single x r * f) y = r * f z

theorem AddMonoidAlgebra.single_zero_mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddZeroClass G] (f : AddMonoidAlgebra k G) (r : k) (x : G) : (single 0 r * f) x = r * f x

theorem AddMonoidAlgebra.single_mul_apply_of_not_exists_add {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] (r : k) {g g' : G} (x : AddMonoidAlgebra k G) (h : ¬∃ (d : G), g' = g + d) : (single g r * x) g' = 0

theorem AddMonoidAlgebra.mul_single_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddGroup G] (f : AddMonoidAlgebra k G) (r : k) (x y : G) : (f * single x r) y = f (y - x) * r

theorem AddMonoidAlgebra.single_mul_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddGroup G] (r : k) (x : G) (f : AddMonoidAlgebra k G) (y : G) : (single x r * f) y = r * f (-x + y)

theorem AddMonoidAlgebra.induction_on {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] {p : AddMonoidAlgebra k G → Prop} (f : AddMonoidAlgebra k G) (hM : ∀ (g : G), p ((of k G) (Multiplicative.ofAdd g))) (hadd : ∀ (f g : AddMonoidAlgebra k G), p f → p g → p (f + g)) (hsmul : ∀ (r : k) (f : AddMonoidAlgebra k G), p f → p (r • f)) : p f

Algebra structure

def AddMonoidAlgebra.singleZeroRingHom {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] : k →+* AddMonoidAlgebra k G

Finsupp.single 0 as a RingHom

@[simp]

theorem AddMonoidAlgebra.singleZeroRingHom_apply {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] (b : k) : singleZeroRingHom b = single 0 b

theorem AddMonoidAlgebra.ringHom_ext {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [AddMonoid G] [Semiring R] {f g : AddMonoidAlgebra k G →+* R} (h₀ : ∀ (b : k), f (single 0 b) = g (single 0 b)) (h_of : ∀ (a : G), f (single a 1) = g (single a 1)) : f = g

If two ring homomorphisms from k[G] are equal on all single a 1 and single 0 b, then they are equal.

theorem AddMonoidAlgebra.ringHom_ext' {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [AddMonoid G] [Semiring R] {f g : AddMonoidAlgebra k G →+* R} (h₁ : f.comp singleZeroRingHom = g.comp singleZeroRingHom) (h_of : (↑f).comp (of k G) = (↑g).comp (of k G)) : f = g

If two ring homomorphisms from k[G] are equal on all single a 1 and single 0 b, then they are equal.

See note [partially-applied ext lemmas].

theorem AddMonoidAlgebra.ringHom_ext'_iff {k : Type u₁} {G : Type u₂} {R : Type u_5} [Semiring k] [AddMonoid G] [Semiring R] {f g : AddMonoidAlgebra k G →+* R} : f = g ↔ f.comp singleZeroRingHom = g.comp singleZeroRingHom ∧ (↑f).comp (of k G) = (↑g).comp (of k G)

theorem AddMonoidAlgebra.prod_single {k : Type u₁} {G : Type u₂} {ι : Type ui} [CommSemiring k] [AddCommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} : ∏ i ∈ s, single (a i) (b i) = single (∑ i ∈ s, a i) (∏ i ∈ s, b i)

instance AddMonoidAlgebra.isLocalHom_singleZeroRingHom {k : Type u₁} {G : Type u₂} [Semiring k] [AddMonoid G] : IsLocalHom singleZeroRingHom