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_2} [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_2} [PartialOrder Γ] [AddMonoid R] : AddMonoid (HahnSeries Γ R)

Equations

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

@[simp]

theorem HahnSeries.add_coeff' {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} : (x + y).coeff = x.coeff + y.coeff

theorem HahnSeries.add_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} {a : Γ} : (x + y).coeff a = x.coeff a + y.coeff a

theorem HahnSeries.addOppositeEquiv_apply {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : HahnSeries.addOppositeEquiv x = AddOpposite.op { coeff := fun (a : Γ) => AddOpposite.unop (x.coeff a), isPWO_support' := ⋯ }

theorem HahnSeries.addOppositeEquiv_symm_apply_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) (a : Γ) : (HahnSeries.addOppositeEquiv.symm x).coeff a = AddOpposite.op ((AddOpposite.unop x).coeff a)

def HahnSeries.addOppositeEquiv {Γ : Type u_1} {R : Type u_2} [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.

@[simp]

theorem HahnSeries.addOppositeEquiv_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (HahnSeries.addOppositeEquiv x)).support = x.support

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (HahnSeries.addOppositeEquiv.symm x).support = (AddOpposite.unop x).support

@[simp]

theorem HahnSeries.addOppositeEquiv_orderTop {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (HahnSeries.addOppositeEquiv x)).orderTop = x.orderTop

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_orderTop {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (HahnSeries.addOppositeEquiv.symm x).orderTop = (AddOpposite.unop x).orderTop

@[simp]

theorem HahnSeries.addOppositeEquiv_leadingCoeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : HahnSeries Γ Rᵃᵒᵖ) : (AddOpposite.unop (HahnSeries.addOppositeEquiv x)).leadingCoeff = AddOpposite.unop x.leadingCoeff

@[simp]

theorem HahnSeries.addOppositeEquiv_symm_leadingCoeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (x : (HahnSeries Γ R)ᵃᵒᵖ) : (HahnSeries.addOppositeEquiv.symm x).leadingCoeff = AddOpposite.op (AddOpposite.unop x).leadingCoeff

theorem HahnSeries.support_add_subset {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} : (x + y).support ⊆ x.support ∪ y.support

theorem HahnSeries.min_le_min_add {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hx : x ≠ 0) (hy : y ≠ 0) (hxy : x + y ≠ 0) : min (⋯.min ⋯) (⋯.min ⋯) ≤ ⋯.min ⋯

theorem HahnSeries.min_orderTop_le_orderTop_add {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} : min x.orderTop y.orderTop ≤ (x + y).orderTop

theorem HahnSeries.min_order_le_order_add {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [Zero Γ] [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order

theorem HahnSeries.orderTop_add_eq_left {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop

theorem HahnSeries.orderTop_add_eq_right {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop) : (x + y).orderTop = y.orderTop

theorem HahnSeries.leadingCoeff_add_eq_left {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x + y).leadingCoeff = x.leadingCoeff

theorem HahnSeries.leadingCoeff_add_eq_right {R : Type u_2} [AddMonoid R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : y.orderTop < x.orderTop) : (x + y).leadingCoeff = y.leadingCoeff

@[simp]

theorem HahnSeries.single.addMonoidHom_apply_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (a : Γ) (r : R) (j : Γ) : ((HahnSeries.single.addMonoidHom a) r).coeff j = Pi.single a r j

def HahnSeries.single.addMonoidHom {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (a : Γ) : R →+ HahnSeries Γ R

single as an additive monoid/group homomorphism

Equations

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

@[simp]

theorem HahnSeries.coeff.addMonoidHom_apply {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddMonoid R] (g : Γ) (f : HahnSeries Γ R) : (HahnSeries.coeff.addMonoidHom g) f = f.coeff g

def HahnSeries.coeff.addMonoidHom {Γ : Type u_1} {R : Type u_2} [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' := ⋯ }

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

instance HahnSeries.instAddCommMonoid {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddCommMonoid R] : AddCommMonoid (HahnSeries Γ R)

Equations

• HahnSeries.instAddCommMonoid = let __src := inferInstanceAs (AddMonoid (HahnSeries Γ R)); AddCommMonoid.mk ⋯

instance HahnSeries.instNeg {Γ : Type u_1} {R : Type u_2} [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_2} [PartialOrder Γ] [AddGroup R] : AddGroup (HahnSeries Γ R)

Equations

• HahnSeries.instAddGroup = let __src := inferInstanceAs (AddMonoid (HahnSeries Γ R)); AddGroup.mk ⋯

@[simp]

theorem HahnSeries.neg_coeff' {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).coeff = -x.coeff

theorem HahnSeries.neg_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {a : Γ} : (-x).coeff a = -x.coeff a

@[simp]

theorem HahnSeries.support_neg {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).support = x.support

@[simp]

theorem HahnSeries.sub_coeff' {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} : (x - y).coeff = x.coeff - y.coeff

theorem HahnSeries.sub_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} {a : Γ} : (x - y).coeff a = x.coeff a - y.coeff a

theorem HahnSeries.orderTop_neg {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] {x : HahnSeries Γ R} : (-x).orderTop = x.orderTop

@[simp]

theorem HahnSeries.order_neg {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddGroup R] [Zero Γ] {f : HahnSeries Γ R} : (-f).order = f.order

theorem HahnSeries.min_orderTop_le_orderTop_sub {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} : min x.orderTop y.orderTop ≤ (x - y).orderTop

theorem HahnSeries.orderTop_sub {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x - y).orderTop = x.orderTop

theorem HahnSeries.leadingCoeff_sub {R : Type u_2} [AddGroup R] {Γ : Type u_3} [LinearOrder Γ] {x : HahnSeries Γ R} {y : HahnSeries Γ R} (hxy : x.orderTop < y.orderTop) : (x - y).leadingCoeff = x.leadingCoeff

instance HahnSeries.instAddCommGroup {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [AddCommGroup R] : AddCommGroup (HahnSeries Γ R)

Equations

• HahnSeries.instAddCommGroup = let __src := inferInstanceAs (AddCommMonoid (HahnSeries Γ R)); let __src_1 := inferInstanceAs (AddGroup (HahnSeries Γ R)); AddCommGroup.mk ⋯

instance HahnSeries.instSMul {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [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.smul_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Zero V] [SMulZeroClass R V] {r : R} {x : HahnSeries Γ V} {a : Γ} : (r • x).coeff a = r • x.coeff a

instance HahnSeries.instSMulZeroClass {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Zero V] [SMulZeroClass R V] : SMulZeroClass R (HahnSeries Γ V)

Equations

• HahnSeries.instSMulZeroClass = let __src := inferInstanceAs (SMul R (HahnSeries Γ V)); SMulZeroClass.mk ⋯

theorem HahnSeries.orderTop_smul_not_lt {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [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_2} [PartialOrder Γ] {V : Type u_3} [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_2} {V : Type u_3} [Zero V] [SMulZeroClass R V] {Γ : Type u_4} [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_2} [PartialOrder Γ] {V : Type u_3} [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_2} [PartialOrder Γ] {V : Type u_3} [Monoid R] [AddMonoid V] [DistribMulAction R V] {S : Type u_4} [Monoid S] [DistribMulAction S V] [SMul R S] [IsScalarTower R S V] : IsScalarTower R S (HahnSeries Γ V)

Equations

• ⋯ = ⋯

instance HahnSeries.instSMulCommClass {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] {V : Type u_3} [Monoid R] [AddMonoid V] [DistribMulAction R V] {S : Type u_4} [Monoid S] [DistribMulAction S V] [SMulCommClass R S V] : SMulCommClass R S (HahnSeries Γ V)

Equations

• ⋯ = ⋯

instance HahnSeries.instModule {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] : Module R (HahnSeries Γ V)

Equations

• HahnSeries.instModule = let __src := inferInstanceAs (DistribMulAction R (HahnSeries Γ V)); Module.mk ⋯ ⋯

@[simp]

theorem HahnSeries.single.linearMap_apply {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (a : Γ) : ∀ (a_1 : V), (HahnSeries.single.linearMap a) a_1 = (↑(HahnSeries.single.addMonoidHom a)).toFun a_1

def HahnSeries.single.linearMap {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (a : Γ) : V →ₗ[R] HahnSeries Γ V

single as a linear map

Equations

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

@[simp]

theorem HahnSeries.coeff.linearMap_apply {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (g : Γ) : ∀ (a : HahnSeries Γ V), (HahnSeries.coeff.linearMap g) a = (↑(HahnSeries.coeff.addMonoidHom g)).toFun a

def HahnSeries.coeff.linearMap {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {V : Type u_3} [AddCommMonoid V] [Module R V] (g : Γ) : HahnSeries Γ V →ₗ[R] V

coeff g as a linear map

Equations

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

theorem HahnSeries.embDomain_smul {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {Γ' : Type u_4} [PartialOrder Γ'] (f : Γ ↪o Γ') (r : R) (x : HahnSeries Γ R) : HahnSeries.embDomain f (r • x) = r • HahnSeries.embDomain f x

@[simp]

theorem HahnSeries.embDomainLinearMap_apply {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {Γ' : Type u_4} [PartialOrder Γ'] (f : Γ ↪o Γ') : ∀ (a : HahnSeries Γ R), (HahnSeries.embDomainLinearMap f) a = HahnSeries.embDomain f a

def HahnSeries.embDomainLinearMap {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Semiring R] {Γ' : Type u_4} [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' := ⋯ }