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Result

Found 357 declarations mentioning AddMonoidAlgebra.coeff. Of these, only the first 200 are shown.

• AddMonoidAlgebra.coeff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_8} {M : Type u_9} [Semiring R] (self : AddMonoidAlgebra R M) : M →₀ R
• AddMonoidAlgebra.coeff_injective 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] : Function.Injective AddMonoidAlgebra.coeff
• AddMonoidAlgebra.ofCoeff_coeff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (x : AddMonoidAlgebra R M) : AddMonoidAlgebra.ofCoeff x.coeff = x
• AddMonoidAlgebra.coeff_ofCoeff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (x : M →₀ R) : (AddMonoidAlgebra.ofCoeff x).coeff = x
• AddMonoidAlgebra.coeff_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (m : M) (r : R) : (AddMonoidAlgebra.single m r).coeff = fun₀ | m => r
• AddMonoidAlgebra.ext 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x y : AddMonoidAlgebra R M} : x.coeff = y.coeff → x = y
• AddMonoidAlgebra.coeff_inj 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x y : AddMonoidAlgebra R M} : x.coeff = y.coeff ↔ x = y
• AddMonoidAlgebra.ext_iff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x y : AddMonoidAlgebra R M} : x = y ↔ x.coeff = y.coeff
• AddMonoidAlgebra.sum_coeff_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : AddMonoidAlgebra R M) : f.coeff.sum AddMonoidAlgebra.single = f
• AddMonoidAlgebra.sum_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f : AddMonoidAlgebra R M) : f.coeff.sum AddMonoidAlgebra.single = f
• AddMonoidAlgebra.coeff_erase 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (m : M) (x : AddMonoidAlgebra R M) : (AddMonoidAlgebra.erase m x).coeff = Finsupp.erase m x.coeff
• AddMonoidAlgebra.coeff_update 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (m : M) (r : R) (x : AddMonoidAlgebra R M) : (AddMonoidAlgebra.update m r x).coeff = x.coeff.update m r
• AddMonoidAlgebra.coeff_single_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {a a' : M} {b : R} [Decidable (a = a')] : (AddMonoidAlgebra.single a b).coeff a' = if a = a' then b else 0
• AddMonoidAlgebra.coeff_sum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {ι : Type u_7} [Semiring R] (s : Finset ι) (f : ι → AddMonoidAlgebra R M) : (∑ i ∈ s, f i).coeff = ∑ i ∈ s, (f i).coeff
• AddMonoidAlgebra.sum_single_index 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} [Semiring R] [AddCommMonoid N] {m : M} {r : R} {h : M → R → N} (h_zero : h m 0 = 0) : (AddMonoidAlgebra.single m r).coeff.sum h = h m r
• AddMonoidAlgebra.coeff_one_zero 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Zero M] : (AddMonoidAlgebra.coeff 1) 0 = 1
• AddMonoidAlgebra.coeff_zero_zero 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Zero M] : (AddMonoidAlgebra.coeff 1) 0 = 1
• AddMonoidAlgebra.coeff_zero 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] : AddMonoidAlgebra.coeff 0 = 0
• AddMonoidAlgebra.coeff_natCast 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddZeroClass M] (n : ℕ) : (↑n).coeff = fun₀ | 0 => ↑n
• AddMonoidAlgebra.coeff_eq_zero 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x : AddMonoidAlgebra R M} : x.coeff = 0 ↔ x = 0
• AddMonoidAlgebra.erase_add_single 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (m : M) (x : AddMonoidAlgebra R M) : AddMonoidAlgebra.erase m x + AddMonoidAlgebra.single m (x.coeff m) = x
• AddMonoidAlgebra.single_add_erase 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (m : M) (x : AddMonoidAlgebra R M) : AddMonoidAlgebra.single m (x.coeff m) + AddMonoidAlgebra.erase m x = x
• AddMonoidAlgebra.coeff_ofNat 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddZeroClass M] (n : ℕ) [n.AtLeastTwo] : (OfNat.ofNat n).coeff = fun₀ | 0 => OfNat.ofNat n
• AddMonoidAlgebra.coeff_mul_single_of_forall_add_ne 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {m m' : M} [Add M] (r : R) (x : AddMonoidAlgebra R M) (h : ∀ (d : M), d + m ≠ m') : (x * AddMonoidAlgebra.single m r).coeff m' = 0
• AddMonoidAlgebra.coeff_single_mul_of_forall_add_ne 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {m m' : M} [Add M] (r : R) (x : AddMonoidAlgebra R M) (h : ∀ (d : M), m + d ≠ m') : (AddMonoidAlgebra.single m r * x).coeff m' = 0
• AddMonoidAlgebra.add_single_apply_of_not_exists_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (r : R) {g g' : M} (x : AddMonoidAlgebra R M) (h : ¬∃ d, g' = d + g) : (x * AddMonoidAlgebra.single g r).coeff g' = 0
• AddMonoidAlgebra.coeffEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (self : AddMonoidAlgebra R M) : AddMonoidAlgebra.coeffEquiv self = self.coeff
• AddMonoidAlgebra.single_add_apply_of_not_exists_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (r : R) {g g' : M} (x : AddMonoidAlgebra R M) (h : ¬∃ d, g' = g + d) : (AddMonoidAlgebra.single g r * x).coeff g' = 0
• AddMonoidAlgebra.coeff_finsuppSum 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} {N : Type u_5} {ι : Type u_7} [Semiring R] [AddCommMonoid N] (f : ι →₀ N) (g : ι → N → AddMonoidAlgebra R M) : (f.sum g).coeff = f.sum fun i n => (g i n).coeff
• AddMonoidAlgebra.coeff_neg 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Ring R] (x : AddMonoidAlgebra R M) : (-x).coeff = -x.coeff
• AddMonoidAlgebra.mul_def 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (x y : AddMonoidAlgebra R M) : x * y = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => AddMonoidAlgebra.single (m₁ + m₂) (r₁ * r₂)
• AddMonoidAlgebra.coeff_mul_single_zero 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddZeroClass M] (x : AddMonoidAlgebra R M) (r : R) (m : M) : (x * AddMonoidAlgebra.single 0 r).coeff m = x.coeff m * r
• AddMonoidAlgebra.coeff_single_zero_mul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddZeroClass M] (x : AddMonoidAlgebra R M) (r : R) (m : M) : (AddMonoidAlgebra.single 0 r * x).coeff m = r * x.coeff m
• AddMonoidAlgebra.coeff_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (x y : AddMonoidAlgebra R M) : (x + y).coeff = x.coeff + y.coeff
• AddMonoidAlgebra.coeff_mul_single_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddMonoid M] [IsCancelAdd M] (x : AddMonoidAlgebra R M) (r : R) (m m' : M) : (x * AddMonoidAlgebra.single m r).coeff (m' + m) = x.coeff m' * r
• AddMonoidAlgebra.coeff_single_mul_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddMonoid M] [IsCancelAdd M] (x : AddMonoidAlgebra R M) (r : R) (m m' : M) : (AddMonoidAlgebra.single m r * x).coeff (m + m') = r * x.coeff m'
• AddMonoidAlgebra.induction 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {motive : AddMonoidAlgebra R M → Prop} (x : AddMonoidAlgebra R M) (zero : motive 0) (single_add : ∀ (m : M) (r : R) (x : AddMonoidAlgebra R M), m ∉ x.coeff.support → r ≠ 0 → motive x → motive (AddMonoidAlgebra.single m r + x)) : motive x
• AddMonoidAlgebra.coeff_smul_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {A : Type u_8} [SMulZeroClass A R] (a : A) (x : AddMonoidAlgebra R M) (m : M) : (a • x).coeff m = a • x.coeff m
• AddMonoidAlgebra.coeff_smul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {A : Type u_8} [SMulZeroClass A R] (a : A) (x : AddMonoidAlgebra R M) : (a • x).coeff = a • x.coeff
• AddMonoidAlgebra.coeff_mul_single_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x : AddMonoidAlgebra R G) (r : R) (g h : G) : (x * AddMonoidAlgebra.single g r).coeff h = x.coeff (h + -g) * r
• AddMonoidAlgebra.coeff_single_mul_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x : AddMonoidAlgebra R G) (r : R) (g h : G) : (AddMonoidAlgebra.single g r * x).coeff h = r * x.coeff (-g + h)
• AddMonoidAlgebra.coe_add 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (f g : AddMonoidAlgebra R M) : ⇑(f + g).coeff = ⇑f.coeff + ⇑g.coeff
• AddMonoidAlgebra.coeff_mul_single_eq_coeff_mul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x : AddMonoidAlgebra R M} {r : R} {m m₁ : M} [Add M] (m₂ : M) (H : ∀ m' ∈ x.coeff.support, m' + m = m₁ ↔ m' = m₂) : (x * AddMonoidAlgebra.single m r).coeff m₁ = x.coeff m₂ * r
• AddMonoidAlgebra.coeff_single_mul_eq_mul_coeff 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] {x : AddMonoidAlgebra R M} {r : R} {m m₁ : M} [Add M] (m₂ : M) (H : ∀ m' ∈ x.coeff.support, m + m' = m₁ ↔ m' = m₂) : (AddMonoidAlgebra.single m r * x).coeff m₁ = r * x.coeff m₂
• AddMonoidAlgebra.coeff_sub 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Ring R] (x y : AddMonoidAlgebra R M) : (x - y).coeff = x.coeff - y.coeff
• AddMonoidAlgebra.coeff_mul 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] [DecidableEq M] (x y : AddMonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ + m₂ = m then r₁ * r₂ else 0
• AddMonoidAlgebra.mul_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] [DecidableEq M] (x y : AddMonoidAlgebra R M) (m : M) : (x * y).coeff m = x.coeff.sum fun m₁ r₁ => y.coeff.sum fun m₂ r₂ => if m₁ + m₂ = m then r₁ * r₂ else 0
• AddMonoidAlgebra.coeff_mul_apply_left 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x y : AddMonoidAlgebra R G) (g : G) : (x * y).coeff g = x.coeff.sum fun h r => r * y.coeff (-h + g)
• AddMonoidAlgebra.coeff_mul_apply_right 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {G : Type u_3} [Semiring R] [AddGroup G] (x y : AddMonoidAlgebra R G) (g : G) : (x * y).coeff g = y.coeff.sum fun h r => x.coeff (g + -h) * r
• AddMonoidAlgebra.coeffAddEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] (self : AddMonoidAlgebra R M) : AddMonoidAlgebra.coeffAddEquiv self = self.coeff
• AddMonoidAlgebra.coeff_mul_antidiag 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (x y : AddMonoidAlgebra R M) (m : M) (s : Finset (M × M)) (hs : ∀ {p : M × M}, p ∈ s ↔ p.1 + p.2 = m) : (x * y).coeff m = ∑ p ∈ s, x.coeff p.1 * y.coeff p.2
• AddMonoidAlgebra.mul_apply_antidiagonal 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [Add M] (x y : AddMonoidAlgebra R M) (m : M) (s : Finset (M × M)) (hs : ∀ {p : M × M}, p ∈ s ↔ p.1 + p.2 = m) : (x * y).coeff m = ∑ p ∈ s, x.coeff p.1 * y.coeff p.2
• AddMonoidAlgebra.uniqueRingEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} (M : Type u_4) [Semiring R] [AddMonoid M] [Subsingleton M] (a✝ : AddMonoidAlgebra R M) : (AddMonoidAlgebra.uniqueRingEquiv M) a✝ = a✝.coeff 0
• AddMonoidAlgebra.coeff_uniqueRingEquiv_symm 📋 Mathlib.Algebra.MonoidAlgebra.Defs
  {R : Type u_1} {M : Type u_4} [Semiring R] [AddMonoid M] [Subsingleton M] (r : R) (m : M) : ((AddMonoidAlgebra.uniqueRingEquiv M).symm r).coeff m = r
• AddMonoidAlgebra.coeff_mapDomain 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} [Semiring R] (f : M → N) (x : AddMonoidAlgebra R M) : (AddMonoidAlgebra.mapDomain f x).coeff = Finsupp.mapDomain f x.coeff
• AddMonoidAlgebra.mapDomain_comapDomain 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} [Semiring R] {f : M → N} {x : AddMonoidAlgebra R N} (hx : ↑x.coeff.support ⊆ Set.range f) (hf : Function.Injective f) : AddMonoidAlgebra.mapDomain f (AddMonoidAlgebra.comapDomain f hf x) = x
• AddMonoidAlgebra.coeff_comapDomain 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} [Semiring R] (f : M → N) (hf : Function.Injective f) (x : AddMonoidAlgebra R N) : (AddMonoidAlgebra.comapDomain f hf x).coeff = Finsupp.comapDomain f x.coeff ⋯
• AddMonoidAlgebra.mapDomain_sum 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} {N : Type u_7} [Semiring R] [Semiring S] (f : M → N) (x : AddMonoidAlgebra S M) (v : M → S → AddMonoidAlgebra R M) : AddMonoidAlgebra.mapDomain f (x.coeff.sum v) = x.coeff.sum fun a b => AddMonoidAlgebra.mapDomain f (v a b)
• AddMonoidAlgebra.range_map 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) : Set.range (AddMonoidAlgebra.map f) = {x | ∀ (i : M), x.coeff i ∈ Set.range ⇑f}
• AddMonoidAlgebra.coeff_mapRingHom 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [AddMonoid M] (f : R →+* S) (x : AddMonoidAlgebra R M) (m : M) : ((AddMonoidAlgebra.mapRingHom M f) x).coeff m = f (x.coeff m)
• AddMonoidAlgebra.coeff_mapDomainAddEquiv 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} [Semiring R] [Add M] [Add N] (e : M ≃ N) (x : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.mapDomainAddEquiv R e) x).coeff = Finsupp.equivMapDomain e x.coeff
• AddMonoidAlgebra.coeff_map 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] (f : R →+ S) (x : AddMonoidAlgebra R M) : (AddMonoidAlgebra.map f x).coeff = Finsupp.mapRange ⇑f ⋯ x.coeff
• AddMonoidAlgebra.coeff_toMultiplicative_apply 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  (R : Type u_3) (M : Type u_6) [Semiring R] [Add M] (x : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.toMultiplicative R M) x).coeff = Finsupp.mapDomain (⇑Multiplicative.ofAdd) x.coeff
• MonoidAlgebra.coeff_toAdditive_apply 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  (R : Type u_3) (M : Type u_6) [Semiring R] [Mul M] (x : MonoidAlgebra R M) : ((MonoidAlgebra.toAdditive R M) x).coeff = Finsupp.mapDomain (⇑Additive.ofMul) x.coeff
• AddMonoidAlgebra.coeff_mapAddEquiv 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [Add M] (e : R ≃+ S) (x : AddMonoidAlgebra R M) (m : M) : ((AddMonoidAlgebra.mapAddEquiv M e) x).coeff m = e (x.coeff m)
• AddMonoidAlgebra.coeff_mapDomainRingEquiv 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {M : Type u_6} {N : Type u_7} [Semiring R] [AddMonoid M] [AddMonoid N] (e : M ≃+ N) (x : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.mapDomainRingEquiv R e) x).coeff = Finsupp.equivMapDomain (↑e) x.coeff
• AddMonoidAlgebra.coeff_toMultiplicative_symm_apply 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  (R : Type u_3) (M : Type u_6) [Semiring R] [Add M] (x : MonoidAlgebra R (Multiplicative M)) : ((AddMonoidAlgebra.toMultiplicative R M).symm x).coeff = Finsupp.mapDomain (⇑Multiplicative.toAdd) x.coeff
• MonoidAlgebra.coeff_toAdditive_symm_apply 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  (R : Type u_3) (M : Type u_6) [Semiring R] [Mul M] (x : AddMonoidAlgebra R (Additive M)) : ((MonoidAlgebra.toAdditive R M).symm x).coeff = Finsupp.mapDomain (⇑Additive.toMul) x.coeff
• AddMonoidAlgebra.coeff_mapRingEquiv 📋 Mathlib.Algebra.MonoidAlgebra.MapDomain
  {R : Type u_3} {S : Type u_4} {M : Type u_6} [Semiring R] [Semiring S] [AddMonoid M] (e : R ≃+* S) (x : AddMonoidAlgebra R M) (m : M) : ((AddMonoidAlgebra.mapRingEquiv M e) x).coeff m = e (x.coeff m)
• AddMonoidAlgebra.liftNC_mul 📋 Mathlib.Algebra.MonoidAlgebra.Lift
  {k : Type u₁} {G : Type u₂} {R : Type u_1} [Semiring k] [Add G] [Semiring R] {g_hom : Type u_2} [FunLike g_hom (Multiplicative G) R] [MulHomClass g_hom (Multiplicative G) R] (f : k →+* R) (g : g_hom) (a b : AddMonoidAlgebra k G) (h_comm : ∀ {x y : G}, y ∈ a.coeff.support → Commute (f (b.coeff x)) (g (Multiplicative.ofAdd y))) : (AddMonoidAlgebra.liftNC ↑f ⇑g) (a * b) = (AddMonoidAlgebra.liftNC ↑f ⇑g) a * (AddMonoidAlgebra.liftNC ↑f ⇑g) b
• AddMonoidAlgebra.mem_supported 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] {s : Set M} {x : AddMonoidAlgebra S M} : x ∈ AddMonoidAlgebra.supported R S s ↔ ↑x.coeff.support ⊆ s
• AddMonoidAlgebra.mem_supported' 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] {s : Set M} {x : AddMonoidAlgebra S M} : x ∈ AddMonoidAlgebra.supported R S s ↔ ∀ m ∉ s, x.coeff m = 0
• AddMonoidAlgebra.coeff_mapDomainLinearMap 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} [Semiring R] [Semiring S] [Module R S] (f : M → N) (x : AddMonoidAlgebra S M) : ((AddMonoidAlgebra.mapDomainLinearMap R S f) x).coeff = Finsupp.mapDomain f x.coeff
• AddMonoidAlgebra.uniqueLinearEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Module
  (R : Type u_1) {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] [Zero M] [Subsingleton M] (x : AddMonoidAlgebra S M) : (AddMonoidAlgebra.uniqueLinearEquiv R M) x = x.coeff 0
• AddMonoidAlgebra.coeff_mapDomainLinearEquiv 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} [Semiring R] [Semiring S] [Module R S] (e : M ≃ N) (x : AddMonoidAlgebra S M) : ((AddMonoidAlgebra.mapDomainLinearEquiv R S e) x).coeff = Finsupp.equivMapDomain e x.coeff
• AddMonoidAlgebra.coeffLinearEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Module
  (R : Type u_1) {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] (a✝ : AddMonoidAlgebra S M) : (AddMonoidAlgebra.coeffLinearEquiv R) a✝ = a✝.coeff
• AddMonoidAlgebra.supportedEquivFinsupp_apply_apply 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] (s : Set M) (x : ↥(AddMonoidAlgebra.supported R S s)) (a✝ : { x // x ∈ s }) : ((AddMonoidAlgebra.supportedEquivFinsupp s) x) a✝ = (↑x).coeff ↑a✝
• AddMonoidAlgebra.coeff_supportedEquivFinsupp_symm_apply_coe_support_val 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] (s : Set M) (a✝ : ↑s →₀ S) : (↑((AddMonoidAlgebra.supportedEquivFinsupp s).symm a✝)).coeff.support.val = Multiset.map Subtype.val a✝.support.val
• AddMonoidAlgebra.coeff_supportedEquivFinsupp_symm_apply_coe_apply 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] (s : Set M) (a✝ : ↑s →₀ S) (a : M) : (↑((AddMonoidAlgebra.supportedEquivFinsupp s).symm a✝)).coeff a = if h : a ∈ s then a✝ ⟨a, h⟩ else 0
• AddMonoidAlgebra.supportedEquivFinsupp_apply_support_val 📋 Mathlib.Algebra.MonoidAlgebra.Module
  {R : Type u_1} {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [Module R S] (s : Set M) (x : ↥(AddMonoidAlgebra.supported R S s)) : ((AddMonoidAlgebra.supportedEquivFinsupp s) x).support.val = Multiset.map (fun x_1 => ⟨↑x_1, ⋯⟩) (Multiset.filter (fun x => x ∈ s) (↑x).coeff.support.val).attach
• AddMonoidAlgebra.coeff_mul_add_of_uniqueAdd 📋 Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors
  {R : Type u_1} {A : Type u_2} [Semiring R] [Add A] {f g : AddMonoidAlgebra R A} {a0 b0 : A} (h : UniqueAdd f.coeff.support g.coeff.support a0 b0) : (f * g).coeff (a0 + b0) = f.coeff a0 * g.coeff b0
• Polynomial.coeff_ofFinsupp 📋 Mathlib.Algebra.Polynomial.Basic
  {R : Type u} [Semiring R] (p : AddMonoidAlgebra R ℕ) : { toFinsupp := p }.coeff = p.coeff
• Polynomial.support_toFinsupp 📋 Mathlib.Algebra.Polynomial.Basic
  {R : Type u} [Semiring R] (p : Polynomial R) : p.toFinsupp.coeff.support = p.support
• Polynomial.support_ofFinsupp 📋 Mathlib.Algebra.Polynomial.Basic
  {R : Type u} [Semiring R] (p : AddMonoidAlgebra R ℕ) : { toFinsupp := p }.support = p.coeff.support
• Polynomial.toFinsupp_apply 📋 Mathlib.Algebra.Polynomial.Basic
  {R : Type u} [Semiring R] (f : Polynomial R) (i : ℕ) : f.toFinsupp.coeff i = f.coeff i
• AddMonoidAlgebra.coeff_uniqueAlgEquiv_symm 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] [Subsingleton M] (a : A) (m : M) : ((AddMonoidAlgebra.uniqueAlgEquiv R M).symm a).coeff m = a
• AddMonoidAlgebra.coeff_toMultiplicativeAlgEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  (R : Type u_1) (A : Type u_4) (M : Type u_7) [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] (x : AddMonoidAlgebra A M) : ((AddMonoidAlgebra.toMultiplicativeAlgEquiv R A M) x).coeff = Finsupp.mapDomain (⇑Multiplicative.ofAdd) x.coeff
• MonoidAlgebra.coeff_toAdditiveAlgEquiv_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  (R : Type u_1) (A : Type u_4) (M : Type u_7) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] (x : MonoidAlgebra A M) : ((MonoidAlgebra.toAdditiveAlgEquiv R A M) x).coeff = Finsupp.mapDomain (⇑Additive.ofMul) x.coeff
• AddMonoidAlgebra.domCongr_support 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} {N : Type u_8} [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] [AddMonoid N] (e : M ≃+ N) (x : AddMonoidAlgebra A M) : ((AddMonoidAlgebra.domCongr R A e) x).coeff.support = Finset.map (↑e).toEmbedding x.coeff.support
• AddMonoidAlgebra.coeff_mapAlgHom 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {B : Type u_5} {M : Type u_7} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddMonoid M] (f : A →ₐ[R] B) (x : AddMonoidAlgebra A M) (m : M) : ((AddMonoidAlgebra.mapAlgHom M f) x).coeff m = f (x.coeff m)
• AddMonoidAlgebra.coeff_toMultiplicativeAlgEquiv_symm_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  (R : Type u_1) (A : Type u_4) (M : Type u_7) [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] (x : MonoidAlgebra A (Multiplicative M)) : ((AddMonoidAlgebra.toMultiplicativeAlgEquiv R A M).symm x).coeff = Finsupp.mapDomain (⇑Multiplicative.toAdd) x.coeff
• MonoidAlgebra.coeff_toAdditiveAlgEquiv_symm_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  (R : Type u_1) (A : Type u_4) (M : Type u_7) [CommSemiring R] [Semiring A] [Algebra R A] [Monoid M] (x : AddMonoidAlgebra A (Additive M)) : ((MonoidAlgebra.toAdditiveAlgEquiv R A M).symm x).coeff = Finsupp.mapDomain (⇑Additive.toMul) x.coeff
• AddMonoidAlgebra.coeff_domCongr 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} {N : Type u_8} [CommSemiring R] [Semiring A] [Algebra R A] [AddMonoid M] [AddMonoid N] (e : M ≃+ N) (f : AddMonoidAlgebra A M) (n : N) : ((AddMonoidAlgebra.domCongr R A e) f).coeff n = f.coeff (e.symm n)
• AddMonoidAlgebra.lift_unique 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : AddMonoidAlgebra R M →ₐ[R] A) (f : AddMonoidAlgebra R M) : F f = f.coeff.sum fun m r => r • F (AddMonoidAlgebra.single m 1)
• AddMonoidAlgebra.lift_apply 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : Multiplicative M →* A) (f : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => b • F (Multiplicative.ofAdd a)
• AddMonoidAlgebra.lift_apply' 📋 Mathlib.Algebra.MonoidAlgebra.Basic
  {R : Type u_1} {A : Type u_4} {M : Type u_7} [CommSemiring R] [AddMonoid M] [Semiring A] [Algebra R A] (F : Multiplicative M →* A) (f : AddMonoidAlgebra R M) : ((AddMonoidAlgebra.lift R A M) F) f = f.coeff.sum fun a b => (algebraMap R A) b * F (Multiplicative.ofAdd a)
• AddMonoidAlgebra.support_coeff_one_subset 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Zero G] : (AddMonoidAlgebra.coeff 1).support ⊆ 0
• AddMonoidAlgebra.support_coeff_one 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Zero G] [NeZero 1] : (AddMonoidAlgebra.coeff 1).support = 0
• AddMonoidAlgebra.support_coeff_mul_single_subset 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [DecidableEq G] (x : AddMonoidAlgebra k G) (r : k) (a : G) : (x * AddMonoidAlgebra.single a r).coeff.support ⊆ Finset.image (fun x => x + a) x.coeff.support
• AddMonoidAlgebra.support_coeff_single_mul_subset 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [DecidableEq G] (x : AddMonoidAlgebra k G) (r : k) (a : G) : (AddMonoidAlgebra.single a r * x).coeff.support ⊆ Finset.image (fun x => a + x) x.coeff.support
• AddMonoidAlgebra.support_coeff_mul_subset 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [DecidableEq G] (x y : AddMonoidAlgebra k G) : (x * y).coeff.support ⊆ x.coeff.support + y.coeff.support
• AddMonoidAlgebra.support_coeff_mul_single 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [IsRightCancelAdd G] (f : AddMonoidAlgebra k G) (r : k) (hr : ∀ (y : k), y * r = 0 ↔ y = 0) (x : G) : (f * AddMonoidAlgebra.single x r).coeff.support = Finset.map (addRightEmbedding x) f.coeff.support
• AddMonoidAlgebra.support_coeff_single_mul 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [IsLeftCancelAdd G] (f : AddMonoidAlgebra k G) (r : k) (hr : ∀ (y : k), r * y = 0 ↔ y = 0) (x : G) : (AddMonoidAlgebra.single x r * f).coeff.support = Finset.map (addLeftEmbedding x) f.coeff.support
• AddMonoidAlgebra.mem_span_support 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] (f : AddMonoidAlgebra k G) : f ∈ Submodule.span k (AddMonoidAlgebra.of' k G '' ↑f.coeff.support)
• AddMonoidAlgebra.mem_span_support_coeff 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] (f : AddMonoidAlgebra k G) : f ∈ Submodule.span k (AddMonoidAlgebra.of' k G '' ↑f.coeff.support)
• AddMonoidAlgebra.support_coeff_mul_single_eq_image 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [DecidableEq G] (f : AddMonoidAlgebra k G) {r : k} (hr : ∀ (y : k), y * r = 0 ↔ y = 0) {x : G} (rx : IsAddRightRegular x) : (f * AddMonoidAlgebra.single x r).coeff.support = Finset.image (fun x_1 => x_1 + x) f.coeff.support
• AddMonoidAlgebra.support_coeff_single_mul_eq_image 📋 Mathlib.Algebra.MonoidAlgebra.Support
  {k : Type u₁} {G : Type u₂} [Semiring k] [Add G] [DecidableEq G] (f : AddMonoidAlgebra k G) {r : k} (hr : ∀ (y : k), r * y = 0 ↔ y = 0) {x : G} (lx : IsAddLeftRegular x) : (AddMonoidAlgebra.single x r * f).coeff.support = Finset.image (fun x_1 => x + x_1) f.coeff.support
• AddMonoidAlgebra.infDegree_withTop_some_comp 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [Semiring R] [SemilatticeInf T] [OrderTop T] {D : A → T} {s : AddMonoidAlgebra R A} (hs : s.coeff.support.Nonempty) : AddMonoidAlgebra.infDegree (WithTop.some ∘ D) s = ↑(AddMonoidAlgebra.infDegree D s)
• AddMonoidAlgebra.supDegree_withBot_some_comp 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {s : AddMonoidAlgebra R A} (hs : s.coeff.support.Nonempty) : AddMonoidAlgebra.supDegree (WithBot.some ∘ D) s = ↑(AddMonoidAlgebra.supDegree D s)
• AddMonoidAlgebra.apply_eq_zero_of_not_le_supDegree 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p : AddMonoidAlgebra R A} {a : A} (hlt : ¬D a ≤ AddMonoidAlgebra.supDegree D p) : p.coeff a = 0
• AddMonoidAlgebra.coeff_eq_zero_of_not_le_supDegree 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p : AddMonoidAlgebra R A} {a : A} (hlt : ¬D a ≤ AddMonoidAlgebra.supDegree D p) : p.coeff a = 0
• AddMonoidAlgebra.supDegree_eq_of_isMaxOn 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p : AddMonoidAlgebra R A} {a : A} (hmem : a ∈ p.coeff.support) (hmax : IsMaxOn D (↑p.coeff.support) a) : AddMonoidAlgebra.supDegree D p = D a
• AddMonoidAlgebra.exists_supDegree_mem_support 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [LinearOrder B] [OrderBot B] {p : AddMonoidAlgebra R A} (D : A → B) (hp : p ≠ 0) : ∃ a ∈ p.coeff.support, AddMonoidAlgebra.supDegree D p = D a
• AddMonoidAlgebra.supDegree_mem_support 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [LinearOrder B] [OrderBot B] {p : AddMonoidAlgebra R A} {D : A → B} [AddZeroClass A] (hD : Function.Injective D) (hp : p ≠ 0) : Function.invFun D (AddMonoidAlgebra.supDegree D p) ∈ p.coeff.support
• AddMonoidAlgebra.supDegree_eq_of_max 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p : AddMonoidAlgebra R A} [AddZeroClass A] {b : B} (hb : b ∈ Set.range D) (hmem : Function.invFun D b ∈ p.coeff.support) (hmax : ∀ a ∈ p.coeff.support, D a ≤ b) : AddMonoidAlgebra.supDegree D p = b
• AddMonoidAlgebra.le_inf_support_add 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] (degt : A → T) (f g : AddMonoidAlgebra R A) : f.coeff.support.inf degt ⊓ g.coeff.support.inf degt ≤ (f + g).coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_coeff_add 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] (degt : A → T) (f g : AddMonoidAlgebra R A) : f.coeff.support.inf degt ⊓ g.coeff.support.inf degt ≤ (f + g).coeff.support.inf degt
• AddMonoidAlgebra.sup_support_add_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] (degb : A → B) (f g : AddMonoidAlgebra R A) : (f + g).coeff.support.sup degb ≤ f.coeff.support.sup degb ⊔ g.coeff.support.sup degb
• AddMonoidAlgebra.sup_support_coeff_add_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] (degb : A → B) (f g : AddMonoidAlgebra R A) : (f + g).coeff.support.sup degb ≤ f.coeff.support.sup degb ⊔ g.coeff.support.sup degb
• AddMonoidAlgebra.le_inf_support_coeff_mul 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] [Add A] [Add T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (f g : AddMonoidAlgebra R A) : f.coeff.support.inf degt + g.coeff.support.inf degt ≤ (f * g).coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_mul 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] [Add A] [Add T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (f g : AddMonoidAlgebra R A) : f.coeff.support.inf degt + g.coeff.support.inf degt ≤ (f * g).coeff.support.inf degt
• AddMonoidAlgebra.sup_support_coeff_mul_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] [Add A] [Add B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (f g : AddMonoidAlgebra R A) : (f * g).coeff.support.sup degb ≤ f.coeff.support.sup degb + g.coeff.support.sup degb
• AddMonoidAlgebra.sup_support_mul_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] [Add A] [Add B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (f g : AddMonoidAlgebra R A) : (f * g).coeff.support.sup degb ≤ f.coeff.support.sup degb + g.coeff.support.sup degb
• AddMonoidAlgebra.le_inf_support_list_prod 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] [AddMonoid A] [AddMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (l : List (AddMonoidAlgebra R A)) : (List.map (fun f => f.coeff.support.inf degt) l).sum ≤ l.prod.coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_pow 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [Semiring R] [AddMonoid A] [AddMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (n : ℕ) (f : AddMonoidAlgebra R A) : n • f.coeff.support.inf degt ≤ (f ^ n).coeff.support.inf degt
• AddMonoidAlgebra.sup_support_list_prod_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] [AddMonoid A] [AddMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (l : List (AddMonoidAlgebra R A)) : l.prod.coeff.support.sup degb ≤ (List.map (fun f => f.coeff.support.sup degb) l).sum
• AddMonoidAlgebra.sup_support_pow_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [Semiring R] [AddMonoid A] [AddMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (n : ℕ) (f : AddMonoidAlgebra R A) : (f ^ n).coeff.support.sup degb ≤ n • f.coeff.support.sup degb
• AddMonoidAlgebra.le_inf_support_coeff_finsetProd 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} {ι : Type u_6} [SemilatticeInf T] [OrderTop T] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : ∑ i ∈ s, (f i).coeff.support.inf degt ≤ (∏ i ∈ s, f i).coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_finsetProd 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} {ι : Type u_6} [SemilatticeInf T] [OrderTop T] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : ∑ i ∈ s, (f i).coeff.support.inf degt ≤ (∏ i ∈ s, f i).coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_finset_prod 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} {ι : Type u_6} [SemilatticeInf T] [OrderTop T] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : ∑ i ∈ s, (f i).coeff.support.inf degt ≤ (∏ i ∈ s, f i).coeff.support.inf degt
• AddMonoidAlgebra.sup_support_coeff_finsetProd_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} {ι : Type u_6} [SemilatticeSup B] [OrderBot B] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : (∏ i ∈ s, f i).coeff.support.sup degb ≤ ∑ i ∈ s, (f i).coeff.support.sup degb
• AddMonoidAlgebra.sup_support_finsetProd_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} {ι : Type u_6} [SemilatticeSup B] [OrderBot B] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : (∏ i ∈ s, f i).coeff.support.sup degb ≤ ∑ i ∈ s, (f i).coeff.support.sup degb
• AddMonoidAlgebra.sup_support_finset_prod_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} {ι : Type u_6} [SemilatticeSup B] [OrderBot B] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (s : Finset ι) (f : ι → AddMonoidAlgebra R A) : (∏ i ∈ s, f i).coeff.support.sup degb ≤ ∑ i ∈ s, (f i).coeff.support.sup degb
• AddMonoidAlgebra.le_inf_support_coeff_multisetProd 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (m : Multiset (AddMonoidAlgebra R A)) : (Multiset.map (fun f => f.coeff.support.inf degt) m).sum ≤ m.prod.coeff.support.inf degt
• AddMonoidAlgebra.le_inf_support_multiset_prod 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {T : Type u_4} [SemilatticeInf T] [OrderTop T] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid T] [AddLeftMono T] [AddRightMono T] {degt : A → T} (degt0 : 0 ≤ degt 0) (degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)) (m : Multiset (AddMonoidAlgebra R A)) : (Multiset.map (fun f => f.coeff.support.inf degt) m).sum ≤ m.prod.coeff.support.inf degt
• AddMonoidAlgebra.sup_support_coeff_multisetProd_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (m : Multiset (AddMonoidAlgebra R A)) : m.prod.coeff.support.sup degb ≤ (Multiset.map (fun f => f.coeff.support.sup degb) m).sum
• AddMonoidAlgebra.sup_support_multiset_prod_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [SemilatticeSup B] [OrderBot B] [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B] [AddLeftMono B] [AddRightMono B] {degb : A → B} (degb0 : degb 0 ≤ 0) (degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b) (m : Multiset (AddMonoidAlgebra R A)) : m.prod.coeff.support.sup degb ≤ (Multiset.map (fun f => f.coeff.support.sup degb) m).sum
• AddMonoidAlgebra.apply_supDegree_add_supDegree 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [LinearOrder B] [OrderBot B] {p q : AddMonoidAlgebra R A} {D : A → B} [AddZeroClass A] [Add B] [AddLeftStrictMono B] [AddRightStrictMono B] (hD : Function.Injective D) (hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2) : (p * q).coeff (Function.invFun D (AddMonoidAlgebra.supDegree D p + AddMonoidAlgebra.supDegree D q)) = AddMonoidAlgebra.leadingCoeff D p * AddMonoidAlgebra.leadingCoeff D q
• AddMonoidAlgebra.coeff_supDegree_add_supDegree 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [LinearOrder B] [OrderBot B] {p q : AddMonoidAlgebra R A} {D : A → B} [AddZeroClass A] [Add B] [AddLeftStrictMono B] [AddRightStrictMono B] (hD : Function.Injective D) (hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2) : (p * q).coeff (Function.invFun D (AddMonoidAlgebra.supDegree D p + AddMonoidAlgebra.supDegree D q)) = AddMonoidAlgebra.leadingCoeff D p * AddMonoidAlgebra.leadingCoeff D q
• AddMonoidAlgebra.apply_add_of_supDegree_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p q : AddMonoidAlgebra R A} [AddZeroClass A] [Add B] (hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2) [AddLeftStrictMono B] [AddRightStrictMono B] (hD : Function.Injective D) {ap aq : A} (hp : AddMonoidAlgebra.supDegree D p ≤ D ap) (hq : AddMonoidAlgebra.supDegree D q ≤ D aq) : (p * q).coeff (ap + aq) = p.coeff ap * q.coeff aq
• AddMonoidAlgebra.coeff_add_of_supDegree_le 📋 Mathlib.Algebra.MonoidAlgebra.Degree
  {R : Type u_1} {A : Type u_3} {B : Type u_5} [Semiring R] [SemilatticeSup B] [OrderBot B] {D : A → B} {p q : AddMonoidAlgebra R A} [AddZeroClass A] [Add B] (hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2) [AddLeftStrictMono B] [AddRightStrictMono B] (hD : Function.Injective D) {ap aq : A} (hp : AddMonoidAlgebra.supDegree D p ≤ D ap) (hq : AddMonoidAlgebra.supDegree D q ≤ D aq) : (p * q).coeff (ap + aq) = p.coeff ap * q.coeff aq
• MvPolynomial.finsupp_support_eq_support 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : p.coeff.support = p.support
• MvPolynomial.coeff_X_same 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) : ((MvPolynomial.X i).coeff fun₀ | i => 1) = 1
• MvPolynomial.coeff_zero_X 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) : (MvPolynomial.X i).coeff 0 = 0
• MvPolynomial.coeff_zero 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] : AddMonoidAlgebra.coeff 0 = 0
• MvPolynomial.coeff_zero_one 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] : (AddMonoidAlgebra.coeff 1) 0 = 1
• MvPolynomial.mem_support_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {m : σ →₀ ℕ} : m ∈ p.support ↔ p.coeff m ≠ 0
• MvPolynomial.eq_zero_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} : p = 0 ↔ ∀ (d : σ →₀ ℕ), p.coeff d = 0
• MvPolynomial.notMem_support_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {m : σ →₀ ℕ} : m ∉ p.support ↔ p.coeff m = 0
• MvPolynomial.sum_def 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {p : MvPolynomial σ R} {b : (σ →₀ ℕ) → R → A} : p.coeff.sum b = ∑ m ∈ p.support, b m (p.coeff m)
• MvPolynomial.exists_coeff_ne_zero 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} (h : p ≠ 0) : ∃ d, p.coeff d ≠ 0
• MvPolynomial.ne_zero_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} : p ≠ 0 ↔ ∃ d, p.coeff d ≠ 0
• MvPolynomial.ext 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (p q : MvPolynomial σ R) : (∀ (m : σ →₀ ℕ), p.coeff m = q.coeff m) → p = q
• MvPolynomial.mem_coeffs_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {c : R} : c ∈ p.coeffs ↔ ∃ n ∈ p.support, c = p.coeff n
• MvPolynomial.ext_iff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} : p = q ↔ ∀ (m : σ →₀ ℕ), p.coeff m = q.coeff m
• MvPolynomial.coeff_single_X 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (s s' : σ) (n : ℕ) : ((MvPolynomial.X s).coeff fun₀ | s' => n) = if n = 1 ∧ s = s' then 1 else 0
• MvPolynomial.coeff_X 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (i : σ) (m : σ →₀ ℕ) : (MvPolynomial.X i).coeff m = if (fun₀ | i => 1) = m then 1 else 0
• MvPolynomial.coeff_X' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (i : σ) (m : σ →₀ ℕ) : (MvPolynomial.X i).coeff m = if (fun₀ | i => 1) = m then 1 else 0
• MvPolynomial.coeff_zero_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (a : R) : (MvPolynomial.C a).coeff 0 = a
• MvPolynomial.coeff_mem_coeffs 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} (m : σ →₀ ℕ) (h : p.coeff m ≠ 0) : p.coeff m ∈ p.coeffs
• MvPolynomial.constantCoeff_eq 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : MvPolynomial.constantCoeff p = p.coeff 0
• MvPolynomial.eq_C_of_isEmpty 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [IsEmpty σ] (p : MvPolynomial σ R) : p = MvPolynomial.C (p.coeff 0)
• MvPolynomial.coeff_sum 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {X : Type u_2} (s : Finset X) (f : X → MvPolynomial σ R) (m : σ →₀ ℕ) : (∑ x ∈ s, f x).coeff m = ∑ x ∈ s, (f x).coeff m
• MvPolynomial.coeffAddMonoidHom_apply 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (x✝ : MvPolynomial σ R) : (MvPolynomial.coeffAddMonoidHom m) x✝ = x✝.coeff m
• MvPolynomial.coeff_C_of_ne_zero 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {m : σ →₀ ℕ} (h : m ≠ 0) (a : R) : (MvPolynomial.C a).coeff m = 0
• MvPolynomial.coeff_X_pow 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (i : σ) (m : σ →₀ ℕ) (k : ℕ) : (MvPolynomial.X i ^ k).coeff m = if (fun₀ | i => k) = m then 1 else 0
• MvPolynomial.coeff_one 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (m : σ →₀ ℕ) : (AddMonoidAlgebra.coeff 1) m = if 0 = m then 1 else 0
• MvPolynomial.sum_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} {a : R} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {b : (σ →₀ ℕ) → R → A} (w : b 0 0 = 0) : (MvPolynomial.C a).coeff.sum b = b 0 a
• MvPolynomial.coeff_add_single_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {n : ℕ} [NeZero n] {m : σ →₀ ℕ} (a : R) (i : σ) : (MvPolynomial.C a).coeff (m + fun₀ | i => n) = 0
• MvPolynomial.C_dvd_iff_dvd_coeff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (r : R) (φ : MvPolynomial σ R) : MvPolynomial.C r ∣ φ ↔ ∀ (i : σ →₀ ℕ), r ∣ φ.coeff i
• MvPolynomial.coeff_C 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (m : σ →₀ ℕ) (a : R) : (MvPolynomial.C a).coeff m = if 0 = m then a else 0
• MvPolynomial.lcoeff_apply 📋 Mathlib.Algebra.MvPolynomial.Basic
  (R : Type u) {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (x✝ : MvPolynomial σ R) : (MvPolynomial.lcoeff R m) x✝ = x✝.coeff m
• MvPolynomial.coeff_X_mul 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (s : σ) (p : MvPolynomial σ R) : (MvPolynomial.X s * p).coeff ((fun₀ | s => 1) + m) = p.coeff m
• MvPolynomial.coeff_mul_X 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (s : σ) (p : MvPolynomial σ R) : (p * MvPolynomial.X s).coeff (m + fun₀ | s => 1) = p.coeff m
• MvPolynomial.sum_monomial_eq 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {A : Type u_2} [AddCommMonoid A] {u : σ →₀ ℕ} {r : R} {b : (σ →₀ ℕ) → R → A} (w : b u 0 = 0) : ((MvPolynomial.monomial u) r).coeff.sum b = b u r
• MvPolynomial.coe_coeffsIn 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u_2} {S : Type u_3} (σ : Type u_4) [CommSemiring R] [CommSemiring S] [Module R S] (M : Submodule R S) : ↑(MvPolynomial.coeffsIn σ M) = {p | ∀ (i : σ →₀ ℕ), p.coeff i ∈ M}
• MvPolynomial.as_sum 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : p = ∑ v ∈ p.support, (MvPolynomial.monomial v) (p.coeff v)
• MvPolynomial.support_sum_monomial_coeff 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : ∑ v ∈ p.support, (MvPolynomial.monomial v) (p.coeff v) = p
• MvPolynomial.coeff_prod_X_pow 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (d : σ →₀ ℕ) (x : σ → ℕ) (s : Finset σ) : (∏ y ∈ s, MvPolynomial.X y ^ x y).coeff d = if d = Finsupp.indicator s fun i x_1 => x i then 1 else 0
• MvPolynomial.coeff_smul 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {S₁ : Type u_2} [SMulZeroClass S₁ R] (m : σ →₀ ℕ) (C : S₁) (p : MvPolynomial σ R) : (C • p).coeff m = C • p.coeff m
• MvPolynomial.coeff_C_mul 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (a : R) (p : MvPolynomial σ R) : (MvPolynomial.C a * p).coeff m = a * p.coeff m
• MvPolynomial.coeff_add 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m : σ →₀ ℕ) (p q : MvPolynomial σ R) : (p + q).coeff m = p.coeff m + q.coeff m
• MvPolynomial.coeff_monomial 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (m n : σ →₀ ℕ) (a : R) : ((MvPolynomial.monomial n) a).coeff m = if n = m then a else 0
• MvPolynomial.coeff_addMonoidAlgebraMap 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (g : S₁ →+ R) (φ : MvPolynomial σ S₁) (m : σ →₀ ℕ) : (AddMonoidAlgebra.map g φ).coeff m = g (φ.coeff m)
• MvPolynomial.coeff_mapRange 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (g : S₁ →+ R) (φ : MvPolynomial σ S₁) (m : σ →₀ ℕ) : (AddMonoidAlgebra.map g φ).coeff m = g (φ.coeff m)
• MvPolynomial.coeff_single_X_pow 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (s s' : σ) (n n' : ℕ) : ((MvPolynomial.X s ^ n).coeff fun₀ | s' => n') = if s = s' ∧ n = n' ∨ n = 0 ∧ n' = 0 then 1 else 0
• MvPolynomial.coeff_X_mul' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (m : σ →₀ ℕ) (s : σ) (p : MvPolynomial σ R) : (MvPolynomial.X s * p).coeff m = if s ∈ m.support then p.coeff (m - fun₀ | s => 1) else 0
• MvPolynomial.coeff_mul_X' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (m : σ →₀ ℕ) (s : σ) (p : MvPolynomial σ R) : (p * MvPolynomial.X s).coeff m = if s ∈ m.support then p.coeff (m - fun₀ | s => 1) else 0
• MvPolynomial.mem_coeffsIn 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u_2} {S : Type u_3} {σ : Type u_4} [CommSemiring R] [CommSemiring S] [Module R S] {M : Submodule R S} {p : MvPolynomial σ S} : p ∈ MvPolynomial.coeffsIn σ M ↔ ∀ (i : σ →₀ ℕ), p.coeff i ∈ M
• MvPolynomial.eq_monomial_of_support_subset_singleton 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {φ : MvPolynomial σ R} {d₀ : σ →₀ ℕ} (h : ∀ d ∈ φ.support, d = d₀) : φ = (MvPolynomial.monomial d₀) (φ.coeff d₀)
• MvPolynomial.coeff_mul 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (p q : MvPolynomial σ R) (n : σ →₀ ℕ) : (p * q).coeff n = ∑ x ∈ Finset.HasAntidiagonal.antidiagonal n, p.coeff x.1 * q.coeff x.2
• MvPolynomial.mul_def 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} : p * q = p.coeff.sum fun m a => q.coeff.sum fun n b => (MvPolynomial.monomial (m + n)) (a * b)
• MvPolynomial.coeff_monomial_mul 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) : ((MvPolynomial.monomial s) r * p).coeff (s + m) = r * p.coeff m
• MvPolynomial.coeff_mul_monomial 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) : (p * (MvPolynomial.monomial s) r).coeff (m + s) = p.coeff m * r
• MvPolynomial.monomial_add_induction_on 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {motive : MvPolynomial σ R → Prop} (p : MvPolynomial σ R) (C : ∀ (a : R), motive (MvPolynomial.C a)) (monomial_add : ∀ (a : σ →₀ ℕ) (b : R) (f : MvPolynomial σ R), a ∉ f.coeff.support → b ≠ 0 → motive f → motive ((MvPolynomial.monomial a) b + f)) : motive p
• MvPolynomial.coeff_monomial_mul' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) : ((MvPolynomial.monomial s) r * p).coeff m = if s ≤ m then r * p.coeff (m - s) else 0
• MvPolynomial.coeff_mul_monomial' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] (m s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) : (p * (MvPolynomial.monomial s) r).coeff m = if s ≤ m then p.coeff (m - s) * r else 0
• MvPolynomial.induction_on'' 📋 Mathlib.Algebra.MvPolynomial.Basic
  {R : Type u} {σ : Type u_1} [CommSemiring R] {motive : MvPolynomial σ R → Prop} (p : MvPolynomial σ R) (C : ∀ (a : R), motive (MvPolynomial.C a)) (monomial_add : ∀ (a : σ →₀ ℕ) (b : R) (f : MvPolynomial σ R), a ∉ f.coeff.support → b ≠ 0 → motive f → motive ((MvPolynomial.monomial a) b) → motive ((MvPolynomial.monomial a) b + f)) (mul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * MvPolynomial.X n)) : motive p
• MvPolynomial.eval₂_congr 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {p : MvPolynomial σ R} (f : R →+* S₁) (g₁ g₂ : σ → S₁) (h : ∀ {i : σ} {c : σ →₀ ℕ}, i ∈ c.support → p.coeff c ≠ 0 → g₁ i = g₂ i) : MvPolynomial.eval₂ f g₁ p = MvPolynomial.eval₂ f g₂ p
• MvPolynomial.eval₂_eq' 📋 Mathlib.Algebra.MvPolynomial.Eval
  {R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [Fintype σ] (g : R →+* S₁) (X : σ → S₁) (f : MvPolynomial σ R) : MvPolynomial.eval₂ g X f = ∑ d ∈ f.support, g (f.coeff d) * ∏ i, X i ^ d i

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c