Documentation

Mathlib.Data.Finsupp.AList

Connections between Finsupp and AList #

Main definitions #

source
noncomputable def Finsupp.toAList {α : Type u_1} {M : Type u_2} [Zero M] (f : α →₀ M) :
AList fun (_x : α) => M

Produce an association list for the finsupp over its support using choice.

Equations
  • f.toAList = { entries := List.map Prod.toSigma f.graph.toList, nodupKeys := ⋯ }
Instances For
    source
    @[simp]
    theorem Finsupp.toAList_entries {α : Type u_1} {M : Type u_2} [Zero M] (f : α →₀ M) :
    f.toAList.entries = List.map Prod.toSigma f.graph.toList
    source
    @[simp]
    theorem Finsupp.toAList_keys_toFinset {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] (f : α →₀ M) :
    f.toAList.keys.toFinset = f.support
    source
    @[simp]
    theorem Finsupp.mem_toAlist {α : Type u_1} {M : Type u_2} [Zero M] {f : α →₀ M} {x : α} :
    x ∈ f.toAList ↔ f x ≠ 0
    source
    noncomputable def AList.lookupFinsupp {α : Type u_1} {M : Type u_2} [Zero M] (l : AList fun (_x : α) => M) :
    α →₀ M

    Converts an association list into a finitely supported function via AList.lookup, sending absent keys to zero.

    Equations
    • l.lookupFinsupp = { support := (List.filter (fun (x : (_ : α) × M) => decide (x.snd ≠ 0)) l.entries).keys.toFinset, toFun := fun (a : α) => (AList.lookup a l).getD 0, mem_support_toFun := ⋯ }
    Instances For
      source
      @[simp]
      theorem AList.lookupFinsupp_apply {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] (l : AList fun (_x : α) => M) (a : α) :
      l.lookupFinsupp a = (AList.lookup a l).getD 0
      source
      @[simp]
      theorem AList.lookupFinsupp_support {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] [DecidableEq M] (l : AList fun (_x : α) => M) :
      l.lookupFinsupp.support = (List.filter (fun (x : (_ : α) × M) => decide (x.snd ≠ 0)) l.entries).keys.toFinset
      source
      theorem AList.lookupFinsupp_eq_iff_of_ne_zero {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] {l : AList fun (_x : α) => M} {a : α} {x : M} (hx : x ≠ 0) :
      l.lookupFinsupp a = x ↔ x ∈ AList.lookup a l
      source
      theorem AList.lookupFinsupp_eq_zero_iff {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] {l : AList fun (_x : α) => M} {a : α} :
      l.lookupFinsupp a = 0 ↔ a ∉ l ∨ 0 ∈ AList.lookup a l
      source
      @[simp]
      theorem AList.empty_lookupFinsupp {α : Type u_1} {M : Type u_2} [Zero M] :
      ∅.lookupFinsupp = 0
      source
      @[simp]
      theorem AList.insert_lookupFinsupp {α : Type u_1} {M : Type u_2} [Zero M] [DecidableEq α] (l : AList fun (_x : α) => M) (a : α) (m : M) :
      (AList.insert a m l).lookupFinsupp = l.lookupFinsupp.update a m
      source
      @[simp]
      theorem AList.singleton_lookupFinsupp {α : Type u_1} {M : Type u_2} [Zero M] (a : α) (m : M) :
      (AList.singleton a m).lookupFinsupp = Finsupp.single a m
      source
      @[simp]
      theorem Finsupp.toAList_lookupFinsupp {α : Type u_1} {M : Type u_2} [Zero M] (f : α →₀ M) :
      f.toAList.lookupFinsupp = f
      source
      theorem AList.lookupFinsupp_surjective {α : Type u_1} {M : Type u_2} [Zero M] :
      Function.Surjective AList.lookupFinsupp