Additive properties of Hahn series

If Γ is ordered and R has zero, then HahnSeries Γ R consists of formal series over Γ with coefficients in R, whose supports are partially well-ordered. With further structure on R and Γ, we can add further structure on HahnSeries Γ R. When R has an addition operation, HahnSeries Γ R also has addition by adding coefficients.

Main Definitions

• If R is a (commutative) additive monoid or group, then so is HahnSeries Γ R.

References

• [J. van der Hoeven, Operators on Generalized Power Series][van_der_hoeven]

instance HahnSeries.instAdd {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] : Add (HahnSeries Γ R)

Equations

• HahnSeries.instAdd = { add := fun (x y : HahnSeries Γ R) => { coeff := x.coeff + y.coeff, isPWO_support' := ⋯ } }

instance HahnSeries.instAddMonoid {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] : AddMonoid (HahnSeries Γ R)

Equations

• HahnSeries.instAddMonoid = AddMonoid.mk ⋯ ⋯ nsmulRec ⋯ ⋯

@[simp]

theorem HahnSeries.coeff_add' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x y : HahnSeries Γ R} : (x + y).coeff = x.coeff + y.coeff

@[deprecated HahnSeries.coeff_add' (since := "2025-01-31")]

theorem HahnSeries.add_coeff' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x y : HahnSeries Γ R} : (x + y).coeff = x.coeff + y.coeff

Alias of HahnSeries.coeff_add'.

theorem HahnSeries.coeff_add {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x y : HahnSeries Γ R} {a : Γ} : (x + y).coeff a = x.coeff a + y.coeff a

@[deprecated HahnSeries.coeff_add (since := "2025-01-31")]

theorem HahnSeries.add_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x y : HahnSeries Γ R} {a : Γ} : (x + y).coeff a = x.coeff a + y.coeff a

Alias of HahnSeries.coeff_add.

@[simp]

theorem HahnSeries.coeff_nsmul {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x : HahnSeries Γ R} {n : ℕ} : (n • x).coeff = n • x.coeff

@[deprecated HahnSeries.coeff_nsmul (since := "2025-01-31")]

theorem HahnSeries.nsmul_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x : HahnSeries Γ R} {n : ℕ} : (n • x).coeff = n • x.coeff

Alias of HahnSeries.coeff_nsmul.

@[simp]

theorem HahnSeries.map_add {Γ : Type u_1} {R : Type u_3} {S : Type u_4} [PartialOrder Γ] [AddMonoid R] [AddMonoid S] (f : R →+ S) {x y : HahnSeries Γ R} : (x + y).map f = x.map f + y.map f

def HahnSeries.addOppositeEquiv {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] : HahnSeries Γ Rᵃᵒᵖ ≃+ (HahnSeries Γ R)ᵃᵒᵖ

addOppositeEquiv is an additive monoid isomorphism between Hahn series over Γ with coefficients in the opposite additive monoid Rᵃᵒᵖ and the additive opposite of Hahn series over Γ with coefficients R.

Equations

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

theorem HahnSeries.addOppositeEquiv_symm_apply_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) (a : Γ) : (addOppositeEquiv.symm x).coeff a = AddOpposite.op ((AddOpposite.unop x).coeff a)

theorem HahnSeries.addOppositeEquiv_apply {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : addOppositeEquiv x = AddOpposite.op { coeff := fun (a : Γ) => AddOpposite.unop (x.coeff a), isPWO_support' := ⋯ }

@[simp]

theorem HahnSeries.addOppositeEquiv_support {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (addOppositeEquiv x)).support = x.support

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_support {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (addOppositeEquiv.symm x).support = (AddOpposite.unop x).support

@[simp]

theorem HahnSeries.addOppositeEquiv_orderTop {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (addOppositeEquiv x)).orderTop = x.orderTop

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_orderTop {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (addOppositeEquiv.symm x).orderTop = (AddOpposite.unop x).orderTop

@[simp]

theorem HahnSeries.addOppositeEquiv_leadingCoeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (addOppositeEquiv x)).leadingCoeff = AddOpposite.unop x.leadingCoeff

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_leadingCoeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (addOppositeEquiv.symm x).leadingCoeff = AddOpposite.op (AddOpposite.unop x).leadingCoeff

theorem HahnSeries.support_add_subset {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] {x y : HahnSeries Γ R} : (x + y).support ⊆ x.support ∪ y.support

theorem HahnSeries.min_le_min_add {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hx : x ≠ 0) (hy : y ≠ 0) (hxy : x + y ≠ 0) : ⋯.min ⋯ ⊓ ⋯.min ⋯ ≤ ⋯.min ⋯

theorem HahnSeries.min_orderTop_le_orderTop_add {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} : x.orderTop ⊓ y.orderTop ≤ (x + y).orderTop

theorem HahnSeries.min_order_le_order_add {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x + y ≠ 0) : x.order ⊓ y.order ≤ (x + y).order

theorem HahnSeries.orderTop_add_eq_left {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop

theorem HahnSeries.orderTop_add_eq_right {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop) : (x + y).orderTop = y.orderTop

theorem HahnSeries.leadingCoeff_add_eq_left {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x + y).leadingCoeff = x.leadingCoeff

theorem HahnSeries.leadingCoeff_add_eq_right {R : Type u_3} [AddMonoid R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop) : (x + y).leadingCoeff = y.leadingCoeff

def HahnSeries.single.addMonoidHom {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (a : Γ) : R →+ HahnSeries Γ R

single as an additive monoid/group homomorphism

Equations

• HahnSeries.single.addMonoidHom a = { toZeroHom := HahnSeries.single a, map_add' := ⋯ }

@[simp]

theorem HahnSeries.single.addMonoidHom_apply_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (a : Γ) (r : R) (j : Γ) : ((addMonoidHom a) r).coeff j = Pi.single a r j

def HahnSeries.coeff.addMonoidHom {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (g : Γ) : HahnSeries Γ R →+ R

coeff g as an additive monoid/group homomorphism

Equations

• HahnSeries.coeff.addMonoidHom g = { toFun := fun (f : HahnSeries Γ R) => f.coeff g, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem HahnSeries.coeff.addMonoidHom_apply {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (g : Γ) (f : HahnSeries Γ R) : (addMonoidHom g) f = f.coeff g

theorem HahnSeries.embDomain_add {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] [PartialOrder Γ'] (f : Γ ↪o Γ') (x y : HahnSeries Γ R) : embDomain f (x + y) = embDomain f x + embDomain f y

instance HahnSeries.instAddCommMonoid {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddCommMonoid R] : AddCommMonoid (HahnSeries Γ R)

Equations

• HahnSeries.instAddCommMonoid = AddCommMonoid.mk ⋯

@[simp]

theorem HahnSeries.coeff_sum {Γ : Type u_1} {R : Type u_3} {α : Type u_7} [PartialOrder Γ] [AddCommMonoid R] {s : Finset α} {x : α → HahnSeries Γ R} (g : Γ) : (∑ i ∈ s, x i).coeff g = ∑ i ∈ s, (x i).coeff g

instance HahnSeries.instNeg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] : Neg (HahnSeries Γ R)

Equations

• HahnSeries.instNeg = { neg := fun (x : HahnSeries Γ R) => { coeff := fun (a : Γ) => -x.coeff a, isPWO_support' := ⋯ } }

instance HahnSeries.instAddGroup {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] : AddGroup (HahnSeries Γ R)

Equations

• HahnSeries.instAddGroup = AddGroup.mk ⋯

@[simp]

theorem HahnSeries.coeff_neg' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).coeff = -x.coeff

@[deprecated HahnSeries.coeff_neg' (since := "2025-01-31")]

theorem HahnSeries.neg_coeff' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).coeff = -x.coeff

Alias of HahnSeries.coeff_neg'.

theorem HahnSeries.coeff_neg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {a : Γ} : (-x).coeff a = -x.coeff a

@[deprecated HahnSeries.coeff_neg (since := "2025-01-31")]

theorem HahnSeries.neg_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {a : Γ} : (-x).coeff a = -x.coeff a

Alias of HahnSeries.coeff_neg.

@[simp]

theorem HahnSeries.support_neg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).support = x.support

@[simp]

theorem HahnSeries.map_neg {Γ : Type u_1} {R : Type u_3} {S : Type u_4} [PartialOrder Γ] [AddGroup R] [AddGroup S] (f : R →+ S) {x : HahnSeries Γ R} : (-x).map f = -x.map f

theorem HahnSeries.orderTop_neg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).orderTop = x.orderTop

@[simp]

theorem HahnSeries.order_neg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] [Zero Γ] {f : HahnSeries Γ R} : (-f).order = f.order

@[simp]

theorem HahnSeries.coeff_sub' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff

@[deprecated HahnSeries.coeff_sub' (since := "2025-01-31")]

theorem HahnSeries.sub_coeff' {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff

Alias of HahnSeries.coeff_sub'.

theorem HahnSeries.coeff_sub {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} {a : Γ} : (x - y).coeff a = x.coeff a - y.coeff a

@[deprecated HahnSeries.coeff_sub (since := "2025-01-31")]

theorem HahnSeries.sub_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} {a : Γ} : (x - y).coeff a = x.coeff a - y.coeff a

Alias of HahnSeries.coeff_sub.

@[simp]

theorem HahnSeries.map_sub {Γ : Type u_1} {R : Type u_3} {S : Type u_4} [PartialOrder Γ] [AddGroup R] [AddGroup S] (f : R →+ S) {x y : HahnSeries Γ R} : (x - y).map f = x.map f - y.map f

theorem HahnSeries.min_orderTop_le_orderTop_sub {R : Type u_3} [AddGroup R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} : x.orderTop ⊓ y.orderTop ≤ (x - y).orderTop

theorem HahnSeries.orderTop_sub {R : Type u_3} [AddGroup R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x - y).orderTop = x.orderTop

theorem HahnSeries.leadingCoeff_sub {R : Type u_3} [AddGroup R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x - y).leadingCoeff = x.leadingCoeff

instance HahnSeries.instAddCommGroup {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddCommGroup R] : AddCommGroup (HahnSeries Γ R)

Equations

• HahnSeries.instAddCommGroup = AddCommGroup.mk ⋯

instance HahnSeries.instSMul {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] : SMul R (HahnSeries Γ V)

Equations

• HahnSeries.instSMul = { smul := fun (r : R) (x : HahnSeries Γ V) => { coeff := r • x.coeff, isPWO_support' := ⋯ } }

@[simp]

theorem HahnSeries.coeff_smul {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] {r : R} {x : HahnSeries Γ V} {a : Γ} : (r • x).coeff a = r • x.coeff a

@[deprecated HahnSeries.coeff_smul (since := "2025-01-31")]

theorem HahnSeries.smul_coeff {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] {r : R} {x : HahnSeries Γ V} {a : Γ} : (r • x).coeff a = r • x.coeff a

Alias of HahnSeries.coeff_smul.

instance HahnSeries.instSMulZeroClass {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] : SMulZeroClass R (HahnSeries Γ V)

Equations

• HahnSeries.instSMulZeroClass = SMulZeroClass.mk ⋯

theorem HahnSeries.orderTop_smul_not_lt {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] (r : R) (x : HahnSeries Γ V) : ¬(r • x).orderTop < x.orderTop

theorem HahnSeries.order_smul_not_lt {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] [Zero Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0) : ¬(r • x).order < x.order

theorem HahnSeries.le_order_smul {R : Type u_3} {V : Type u_8} [Zero V] [SMulZeroClass R V] {Γ : Type u_9} [Zero Γ] [LinearOrder Γ] (r : R) (x : HahnSeries Γ V) (h : r • x ≠ 0) : x.order ≤ (r • x).order

instance HahnSeries.instDistribMulAction {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Monoid R] [AddMonoid V] [DistribMulAction R V] : DistribMulAction R (HahnSeries Γ V)

Equations

• HahnSeries.instDistribMulAction = DistribMulAction.mk ⋯ ⋯

instance HahnSeries.instIsScalarTower {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Monoid R] [AddMonoid V] [DistribMulAction R V] {S : Type u_9} [Monoid S] [DistribMulAction S V] [SMul R S] [IsScalarTower R S V] : IsScalarTower R S (HahnSeries Γ V)

instance HahnSeries.instSMulCommClass {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Monoid R] [AddMonoid V] [DistribMulAction R V] {S : Type u_9} [Monoid S] [DistribMulAction S V] [SMulCommClass R S V] : SMulCommClass R S (HahnSeries Γ V)

instance HahnSeries.instModule {Γ : Type u_1} {R : Type u_3} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] : Module R (HahnSeries Γ V)

Equations

• HahnSeries.instModule = Module.mk ⋯ ⋯

def HahnSeries.single.linearMap {Γ : Type u_1} {R : Type u_3} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] (a : Γ) : V →ₗ[R] HahnSeries Γ V

single as a linear map

Equations

• HahnSeries.single.linearMap a = { toFun := (↑(HahnSeries.single.addMonoidHom a)).toFun, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem HahnSeries.single.linearMap_apply {Γ : Type u_1} {R : Type u_3} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] (a : Γ) (a✝ : V) : (linearMap a) a✝ = (↑(addMonoidHom a)).toFun a✝

def HahnSeries.coeff.linearMap {Γ : Type u_1} {R : Type u_3} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] (g : Γ) : HahnSeries Γ V →ₗ[R] V

coeff g as a linear map

Equations

• HahnSeries.coeff.linearMap g = { toFun := (↑(HahnSeries.coeff.addMonoidHom g)).toFun, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem HahnSeries.coeff.linearMap_apply {Γ : Type u_1} {R : Type u_3} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] (g : Γ) (a✝ : HahnSeries Γ V) : (linearMap g) a✝ = (↑(addMonoidHom g)).toFun a✝

@[simp]

theorem HahnSeries.map_smul {Γ : Type u_1} {R : Type u_3} {U : Type u_5} {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] [AddCommMonoid U] [Module R U] (f : U →ₗ[R] V) {r : R} {x : HahnSeries Γ U} : (r • x).map f = r • x.map f

theorem HahnSeries.embDomain_smul {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} [PartialOrder Γ] [Semiring R] [PartialOrder Γ'] (f : Γ ↪o Γ') (r : R) (x : HahnSeries Γ R) : embDomain f (r • x) = r • embDomain f x

def HahnSeries.embDomainLinearMap {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} [PartialOrder Γ] [Semiring R] [PartialOrder Γ'] (f : Γ ↪o Γ') : HahnSeries Γ R →ₗ[R] HahnSeries Γ' R

Extending the domain of Hahn series is a linear map.

Equations

• HahnSeries.embDomainLinearMap f = { toFun := HahnSeries.embDomain f, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem HahnSeries.embDomainLinearMap_apply {Γ : Type u_1} {Γ' : Type u_2} {R : Type u_3} [PartialOrder Γ] [Semiring R] [PartialOrder Γ'] (f : Γ ↪o Γ') (a✝ : HahnSeries Γ R) : (embDomainLinearMap f) a✝ = embDomain f a✝