SEQUENCES

DETAILS

 
Introduction

      Enumerated Sequences

      Formal Sequences

      Compatibility

 
Creating Sequences

      The Formal Sequence Constructor
            [! x in F | P(x) !]

      The Enumerated Sequence Constructor
            [ ] : Null -> ESeqEnum
            [ U | ] : Struct -> SeqEnum
            [ e_1, e_2, ..., e_n ] : Elt, ..., Elt -> SeqEnum
            [ U | e_1, e_2, ..., e_m ] : Struct, Elt, ..., Elt -> SeqEnum
            [ e(x) : x in E | P(x) ]
            [ U | e(x) : x in E | P(x) ]
            [ e(x_1,...,x_k) : x_1 in E_1, ..., x_kin E_k | P(x_1, ..., x_k) ]
            [ U | e(x_1,...,x_k) : x_1 in E_1, ...,x_k in E_k | P(x_1, ..., x_k) ]

      The Arithmetic Progression Constructors
            [ i..j ] : RngIntElt, RngIntElt -> SeqEnum
            [ i .. j by k ] : RngIntElt, RngIntElt, RngIntElt -> SeqEnum
            Example Seq_Progression (H10E1)

      Literal Sequences
            \[ m_1, ..., m_n ] : RngIntElt, ..., RngIntElt -> [ RngIntElt ]

 
Power Sequences
      PowerSequence(R) : Struct -> PowSeqEnum
      S in P : SeqEnum, PowSeqEnum -> BoolElt
      P ! S : PowSeqEnum, SeqEnum -> SeqEnum
      Example Seq_PowerSequence (H10E2)

 
Operators on Sequences

      Access Functions
            # S : SeqEnum -> RngIntElt
            Parent(S) : Seq -> Struct
            Universe(S) : Seq -> Struct
            S[i] : SeqEnum, RngIntElt -> Elt

      Selection Operators on Enumerated Sequences
            S[I] : SeqEnum, [RngIntElt] -> SeqEnum
            Minimum(S) : SeqEnum -> Elt, RngIntElt
            Maximum(S) : SeqEnum -> Elt, RngIntElt
            Index(S, x) : SeqEnum, Elt -> RngIntElt
            Representative(R) : SeqEnum -> Elt
            Random(R) : SeqEnum -> Elt
            Explode(R) : SeqEnum -> List
            Eltseq(R) : SeqEnum -> SeqEnum

      Modifying Enumerated Sequences
            Append(~S, x) : SeqEnum, Elt ->
            Exclude(~S, x) : SeqEnum, Elt ->
            Include(~S, x) : SeqEnum, Elt ->
            Insert(~S, i, x) : SeqEnum, RngIntElt, Elt ->
            Insert(~S, k, m, T) : SeqEnum, RngIntElt, RngIntElt, SeqEnum ->
            Prune(~S) : SeqEnum ->
            Remove(~S, i) : SeqEnum, RngIntElt ->
            Reverse(~S) : SeqEnum ->
            Rotate(~S, p) : SeqEnum, RngIntElt ->
            Sort(~S) : SeqEnum ->
            Sort(~S, C) : SeqEnum, UserProgram ->
            Undefine(~S, i) : SeqEnum, RngIntElt ->
            ChangeUniverse(S, V) : SeqEnum, Str ->
            CanChangeUniverse(S, V) : SeqEnum, Str -> Bool, SeqEnum
            Example Seq_Farey (H10E3)

      Creating New Enumerated Sequences from Existing Ones
            S cat T : SeqEnum, SeqEnum -> SeqEnum
            Partition(S, p) : SeqEnum, RngIntElt -> SeqEnum(SeqEnum)
            Partition(S, P) : SeqEnum, [RngIntElt] -> SeqEnum(SeqEnum)
            Setseq(S) : SetEnum -> SeqEnum
            Seqset(S) : SeqEnum -> SetEnum
            Example Seq_EgyptianFractions (H10E4)

            Operations on Sequences of Booleans
                  And(S, T) : [ BoolElt ], [ BoolElt ] -> [BoolElt]
                  Or(S, T) : [ BoolElt ], [ BoolElt ] -> [ BoolElt ]
                  Xor(S, T) : [ BoolElt ], [ BoolElt ] -> [ BoolElt ]
                  Not(S) : [ BoolElt ] -> [ BoolElt ]

 
Predicates on Sequences
      IsComplete(S) : SeqEnum -> BoolElt
      IsDefined(S, i) : SeqEnum, RngIntElt -> BoolElt
      IsEmpty(S) : SeqEnum -> BoolElt
      IsNull(S) : SeqEnum -> BoolElt

      Membership Testing
            x in S : Elt, Seq -> BoolElt
            x notin S : Elt, Seq -> BoolElt
            IsSubsequence(S, T) : SeqEnum, SeqEnum -> BoolElt
            S eq T : SeqEnum, SeqEnum -> BoolElt
            S ne T : SeqEnum, SeqEnum -> BoolElt

      Testing Order Relations
            S lt T : SeqEnum, SeqEnum -> BoolElt
            S le T : SeqEnum, SeqEnum -> BoolElt
            S ge T : SeqEnum, SeqEnum -> BoolElt
            S gt T : SeqEnum, SeqEnum -> BoolElt

 
Recursion, Reduction, and Iteration

      Recursion
            Self(n) : RngIntElt -> Elt
            Example Seq_Self (H10E5)

      Reduction
            & S : Op, SeqEnum -> Elt

 
Iteration
      for x in S do statements; end for;

 
Bibliography