Additive properties of Hahn series

If Γ is ordered and R has zero, then R⟦Γ⟧ 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 R⟦Γ⟧. When R has an addition operation, R⟦Γ⟧ also has addition by adding coefficients.

Main Definitions

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

References

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

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)

theorem HahnSeries.support_smul_subset {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).support ⊆ x.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) : (r • x).coeff = r • x.coeff

@[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

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)

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.orderTop_le_orderTop_smul {R : Type u_3} {V : Type u_8} [Zero V] [SMulZeroClass R V] {Γ : Type u_9} [LinearOrder Γ] (r : R) (x : HahnSeries Γ V) : x.orderTop ≤ (r • 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

theorem HahnSeries.truncLT_smul {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] {V : Type u_8} [Zero V] [SMulZeroClass R V] [DecidableLT Γ] (c : Γ) (r : R) (x : HahnSeries Γ V) : (truncLT c) (r • x) = r • (truncLT c) x

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

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

@[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

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

@[simp]

theorem HahnSeries.single_add {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] (a : Γ) (r s : R) : (single a) (r + s) = (single a) r + (single a) s

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

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

@[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.

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.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 ⋯) ≤ ⋯.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} : min 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) : min 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

theorem HahnSeries.ne_zero_of_eq_add_single {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] [Zero Γ] {x y : HahnSeries Γ R} (hxy : x = y + (single x.order) x.leadingCoeff) (hy : y ≠ 0) : x ≠ 0

theorem HahnSeries.coeff_order_of_eq_add_single {Γ : Type u_1} [PartialOrder Γ] {R : Type u_8} [AddCancelCommMonoid R] [Zero Γ] {x y : HahnSeries Γ R} (hxy : x = y + (single x.order) x.leadingCoeff) : y.coeff x.order = 0

theorem HahnSeries.order_lt_order_of_eq_add_single {R : Type u_8} {Γ : Type u_9} [LinearOrder Γ] [Zero Γ] [AddCancelCommMonoid R] {x y : HahnSeries Γ R} (hxy : x = y + (single x.order) x.leadingCoeff) (hy : y ≠ 0) : x.order < y.order

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

@[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

@[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

theorem HahnSeries.truncLT_add {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddMonoid R] [DecidableLT Γ] (c : Γ) (x y : HahnSeries Γ R) : (truncLT c) (x + y) = (truncLT c) x + (truncLT c) y

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

@[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] [NegZeroClass R] : Neg (HahnSeries Γ R)

theorem HahnSeries.support_neg_subset {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [NegZeroClass R] (x : HahnSeries Γ R) : (-x).support ⊆ x.support

@[simp]

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

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

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

theorem HahnSeries.support_sub_subset {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] (x y : HahnSeries Γ R) : (x - y).support ⊆ x.support ∪ y.support

@[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

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

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

@[simp]

theorem HahnSeries.single_sub {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] (a : Γ) (r s : R) : (single a) (r - s) = (single a) r - (single a) s

@[simp]

theorem HahnSeries.single_neg {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] (a : Γ) (r : R) : (single a) (-r) = -(single a) r

@[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

@[simp]

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

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

@[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} : min 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

theorem HahnSeries.orderTop_sub_ne {Γ : Type u_1} {R : Type u_3} [PartialOrder Γ] [AddGroup R] {x y : HahnSeries Γ R} {g : Γ} (hxg : x.orderTop = ↑g) (hyg : y.orderTop = ↑g) (hxyc : x.leadingCoeff = y.leadingCoeff) : (x - y).orderTop ≠ ↑g

theorem HahnSeries.le_orderTop_of_leadingCoeff_eq {R : Type u_3} [AddGroup R] {Γ : Type u_8} [LinearOrder Γ] {x y : HahnSeries Γ R} {g : Γ} (hxg : x.orderTop = ↑g) (hyg : y.orderTop = ↑g) (hxyc : x.leadingCoeff = y.leadingCoeff) : ↑g < (x - y).orderTop

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

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)

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)

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

@[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

@[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

def HahnSeries.ofFinsuppLinearMap {Γ : Type u_1} (R : Type u_3) {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] : (Γ →₀ V) →ₗ[R] HahnSeries Γ V

ofFinsupp as a linear map.

@[simp]

theorem HahnSeries.coeff_ofFinsuppLinearMap {Γ : Type u_1} (R : Type u_3) {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] (f : Γ →₀ V) (a : Γ) : ((ofFinsuppLinearMap R) f).coeff a = f a

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.

@[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✝

def HahnSeries.truncLTLinearMap {Γ : Type u_1} (R : Type u_3) {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] [DecidableLT Γ] (c : Γ) : HahnSeries Γ V →ₗ[R] HahnSeries Γ V

HahnSeries.truncLT as a linear map.

@[simp]

theorem HahnSeries.coe_truncLTLinearMap {Γ : Type u_1} (R : Type u_3) {V : Type u_6} [PartialOrder Γ] [Semiring R] [AddCommMonoid V] [Module R V] [DecidableLT Γ] (c : Γ) : ⇑(truncLTLinearMap R c) = ⇑(truncLT c)