SEQUENCES
Acknowledgements
Introduction
Enumerated Sequences
Formal Sequences
Compatibility
Creating Sequences
The Formal Sequence Constructor
The Enumerated Sequence Constructor
The Arithmetic Progression Constructors
Literal Sequences
Power Sequences
Operators on Sequences
Access Functions
Selection Operators on Enumerated Sequences
Modifying Enumerated Sequences
Creating New Enumerated Sequences from Existing Ones
Operations on Sequences of Booleans
Predicates on Sequences
Membership Testing
Testing Order Relations
Recursion, Reduction, and Iteration
Recursion
Reduction
Iteration
Bibliography
Introduction
Enumerated Sequences
Formal Sequences
Compatibility
Creating Sequences
The Formal Sequence Constructor
[! x in F | P(x) !]
The Enumerated Sequence Constructor
[ ] : Null -> ESeqEnum
[ U | ] : Str -> SeqEnum
[ e1, e2, ..., en ] : Elt, ..., Elt -> SeqEnum
[ U | e1, e2, ..., em ] : Str, Elt, ..., Elt -> SeqEnum
[ e(x) : x in E | P(x) ]
[ U | e(x) : x in E | P(x) ]
[ e(x1,...,xk) : x1 in E1, ..., xkin Ek | P(x1, ..., xk) ]
[ U | e(x1,...,xk) : x1 in E1, ...,xk in Ek | P(x1, ..., xk) ]
The Arithmetic Progression Constructors
[ i..j ] : RngIntElt, RngIntElt -> SeqEnum
[ i .. j by k ] : RngIntElt, RngIntElt, RngIntElt -> SeqEnum
Example Seq_Progression (H11E1)
Literal Sequences
\[ m1, ..., mn ] : RngIntElt, ..., RngIntElt -> [ RngIntElt ]
Power Sequences
PowerSequence(R) : Str -> PowSeqEnum
S in P : SeqEnum, PowSeqEnum -> BoolElt
P ! S : PowSeqEnum, SeqEnum -> SeqEnum
Example Seq_PowerSequence (H11E2)
Operators on Sequences
Access Functions
# S : SeqEnum -> RngIntElt
Parent(S) : SeqEnum -> Str
Universe(S) : SeqEnum -> Str
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 ->
ParallelSort(~S, ~T) : SeqEnum, SeqEnum ->
Undefine(~S, i) : SeqEnum, RngIntElt ->
ChangeUniverse(S, V) : SeqEnum, Str ->
CanChangeUniverse(S, V) : SeqEnum, Str -> Bool, SeqEnum
Example Seq_Farey (H11E3)
Creating New Enumerated Sequences from Existing Ones
S cat T : SeqEnum, SeqEnum -> SeqEnum
S cat:= T : 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 (H11E4)
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, SeqEnum -> BoolElt
x notin S : Elt, SeqEnum -> 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 (H11E5)
Reduction
&  S : Op, SeqEnum -> Elt
Iteration
x in S
Example Seq_NestedIteration (H11E6)
Example Seq_DualIteration (H11E7)
Bibliography
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