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Mathlib.RingTheory.HahnSeries.Basic

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, with the most studied case being when Γ is a linearly ordered abelian group and R is a field, in which case HahnSeries Γ R is a valued field, with value group Γ. These generalize Laurent series (with value group ), and Laurent series are implemented that way in the file RingTheory/LaurentSeries.

Main Definitions #

References #

theorem HahnSeries.ext_iff {Γ : Type u_1} {R : Type u_2} :
∀ {inst : PartialOrder Γ} {inst_1 : Zero R} (x y : HahnSeries Γ R), x = y x.coeff = y.coeff
theorem HahnSeries.ext {Γ : Type u_1} {R : Type u_2} :
∀ {inst : PartialOrder Γ} {inst_1 : Zero R} (x y : HahnSeries Γ R), x.coeff = y.coeffx = y
structure HahnSeries (Γ : Type u_1) (R : Type u_2) [PartialOrder Γ] [Zero R] :
Type (max u_1 u_2)

If Γ is linearly ordered and R has zero, then HahnSeries Γ R consists of formal series over Γ with coefficients in R, whose supports are well-founded.

Instances For
    theorem HahnSeries.coeff_injective {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] :
    Function.Injective HahnSeries.coeff
    @[simp]
    theorem HahnSeries.coeff_inj {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} {y : HahnSeries Γ R} :
    x.coeff = y.coeff x = y
    def HahnSeries.support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R) :
    Set Γ

    The support of a Hahn series is just the set of indices whose coefficients are nonzero. Notably, it is well-founded.

    Equations
    Instances For
      @[simp]
      theorem HahnSeries.isPWO_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R) :
      @[simp]
      theorem HahnSeries.isWF_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R) :
      @[simp]
      theorem HahnSeries.mem_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (x : HahnSeries Γ R) (a : Γ) :
      a HahnSeries.support x x.coeff a 0
      instance HahnSeries.instZeroHahnSeries {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] :
      Equations
      • HahnSeries.instZeroHahnSeries = { zero := { coeff := 0, isPWO_support' := } }
      instance HahnSeries.instInhabitedHahnSeries {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] :
      Equations
      • HahnSeries.instInhabitedHahnSeries = { default := 0 }
      Equations
      • =
      @[simp]
      theorem HahnSeries.zero_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} :
      0.coeff a = 0
      @[simp]
      theorem HahnSeries.coeff_fun_eq_zero_iff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} :
      x.coeff = 0 x = 0
      theorem HahnSeries.ne_zero_of_coeff_ne_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g 0) :
      x 0
      @[simp]
      theorem HahnSeries.support_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] :
      @[simp]
      @[simp]
      theorem HahnSeries.support_eq_empty_iff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {x : HahnSeries Γ R} :

      /-- Change a HahnSeries with coefficients in HahnSeries to a HahnSeries on the Lex product. -/ def of_iterate {Γ' : Type*} [PartialOrder Γ'] (x : HahnSeries Γ (HahnSeries Γ' R)) : HahnSeries (Γ ×ₗ Γ') R where coeff := fun g => coeff (coeff x g.1) g.2 isPWO_support' := by intro f hf simp_all only have hf' : ∀ n, (f n).1 ∈ Function.support fun g ↦ x.coeff g := by intro n hn simp_all only [Function.mem_support, ne_eq] specialize hf n rw [hn] at hf exact hf rfl sorry -- See Mathlib.Data.MvPolynomial.Monad for join and bind operations need a monotone pair. have: nonrec theorem IsPWO.exists_monotone_subseq (h : s.IsPWO) (f : ℕ → α) (hf : ∀ n, f n ∈ s) : ∃ g : ℕ ↪o ℕ, Monotone (f ∘ g) := h.exists_monotone_subseq f hf #align set.is_pwo.exists_monotone_subseq Set.IsPWO.exists_monotone_subseq map sequence to Γ, get monotone subsequence (use ext property) if stationary at a ∈ Γ, look inside {a} × Γ', get monotone subsequence. if not stationary, use lex order. /-- Change a Hahn series on a lex product to a Hahn series with coefficients in a Hahn series. -/ def to_iterate {Γ' : Type*} [PartialOrder Γ'] (x : HahnSeries (Γ ×ₗ Γ') R) : HahnSeries Γ (HahnSeries Γ' R) where coeff := fun g => { coeff := fun g' => coeff x (g, g') isPWO_support' := sorry } isPWO_support' := sorry

      def HahnSeries.single {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (a : Γ) :

      single a r is the Hahn series which has coefficient r at a and zero otherwise.

      Equations
      Instances For
        @[simp]
        theorem HahnSeries.single_coeff_same {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] (a : Γ) (r : R) :
        ((HahnSeries.single a) r).coeff a = r
        @[simp]
        theorem HahnSeries.single_coeff_of_ne {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {b : Γ} {r : R} (h : b a) :
        ((HahnSeries.single a) r).coeff b = 0
        theorem HahnSeries.single_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {b : Γ} {r : R} :
        ((HahnSeries.single a) r).coeff b = if b = a then r else 0
        @[simp]
        theorem HahnSeries.support_single_of_ne {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} (h : r 0) :
        theorem HahnSeries.support_single_subset {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} :
        theorem HahnSeries.eq_of_mem_support_single {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} {b : Γ} (h : b HahnSeries.support ((HahnSeries.single a) r)) :
        b = a
        theorem HahnSeries.single_eq_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} :
        theorem HahnSeries.single_ne_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} (h : r 0) :
        @[simp]
        theorem HahnSeries.single_eq_zero_iff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} :
        (HahnSeries.single a) r = 0 r = 0
        Equations
        • =
        def HahnSeries.order {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] (x : HahnSeries Γ R) :
        Γ

        The order of a nonzero Hahn series x is a minimal element of Γ where x has a nonzero coefficient, the order of 0 is 0.

        Equations
        Instances For
          @[simp]
          theorem HahnSeries.order_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] :
          theorem HahnSeries.order_of_ne {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x 0) :
          theorem HahnSeries.coeff_order_ne_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} (hx : x 0) :
          x.coeff (HahnSeries.order x) 0
          theorem HahnSeries.order_le_of_coeff_ne_zero {R : Type u_2} [Zero R] {Γ : Type u_3} [LinearOrderedCancelAddCommMonoid Γ] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g 0) :
          @[simp]
          theorem HahnSeries.order_single {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {a : Γ} {r : R} [Zero Γ] (h : r 0) :
          theorem HahnSeries.coeff_eq_zero_of_lt_order {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] [Zero Γ] {x : HahnSeries Γ R} {i : Γ} (hi : i < HahnSeries.order x) :
          x.coeff i = 0
          def HahnSeries.embDomain {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] (f : Γ ↪o Γ') :
          HahnSeries Γ RHahnSeries Γ' R

          Extends the domain of a HahnSeries by an OrderEmbedding.

          Equations
          Instances For
            @[simp]
            theorem HahnSeries.embDomain_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} {x : HahnSeries Γ R} {a : Γ} :
            (HahnSeries.embDomain f x).coeff (f a) = x.coeff a
            @[simp]
            theorem HahnSeries.embDomain_mk_coeff {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : ΓΓ'} (hfi : Function.Injective f) (hf : ∀ (g g' : Γ), f g f g' g g') {x : HahnSeries Γ R} {a : Γ} :
            (HahnSeries.embDomain { toEmbedding := { toFun := f, inj' := hfi }, map_rel_iff' := } x).coeff (f a) = x.coeff a
            theorem HahnSeries.embDomain_notin_image_support {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} {x : HahnSeries Γ R} {b : Γ'} (hb : bf '' HahnSeries.support x) :
            (HahnSeries.embDomain f x).coeff b = 0
            theorem HahnSeries.support_embDomain_subset {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} {x : HahnSeries Γ R} :
            theorem HahnSeries.embDomain_notin_range {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} {x : HahnSeries Γ R} {b : Γ'} (hb : bSet.range f) :
            (HahnSeries.embDomain f x).coeff b = 0
            @[simp]
            theorem HahnSeries.embDomain_zero {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} :
            @[simp]
            theorem HahnSeries.embDomain_single {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} {g : Γ} {r : R} :
            theorem HahnSeries.embDomain_injective {Γ : Type u_1} {R : Type u_2} [PartialOrder Γ] [Zero R] {Γ' : Type u_3} [PartialOrder Γ'] {f : Γ ↪o Γ'} :
            theorem HahnSeries.forallLTEqZero_supp_BddBelow {Γ : Type u_1} {R : Type u_2} [Zero R] [LinearOrder Γ] (f : ΓR) (n : Γ) (hn : m < n, f m = 0) :
            @[simp]
            theorem HahnSeries.ofSuppBddBelow_coeff {Γ : Type u_1} {R : Type u_2} [Zero R] [LinearOrder Γ] [LocallyFiniteOrder Γ] (f : ΓR) (hf : BddBelow (Function.support f)) :
            ∀ (a : Γ), (HahnSeries.ofSuppBddBelow f hf).coeff a = f a
            def HahnSeries.ofSuppBddBelow {Γ : Type u_1} {R : Type u_2} [Zero R] [LinearOrder Γ] [LocallyFiniteOrder Γ] (f : ΓR) (hf : BddBelow (Function.support f)) :

            Construct a Hahn series from any function whose support is bounded below.

            Equations
            Instances For
              theorem HahnSeries.order_ofForallLtEqZero {Γ : Type u_1} {R : Type u_2} [Zero R] [LinearOrder Γ] [LocallyFiniteOrder Γ] [Zero Γ] (f : ΓR) (hf : f 0) (n : Γ) (hn : m < n, f m = 0) :