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Withe () is a primitive operator which forms a hook. Called dyadically with function operands f and g, it uses g monadically to pre-processes the right argument before applying f to the given left argument and pre-processed right argument. Called monadically with operands f and g, it uses g monadically to pre-processes the argument before applying f to the argument and pre-processed argument.

In usage, X f⍩g Y is equivalent to X f g Y, and f⍩g Y is equivalent to Y f g Y. Thus, Withe can be defined as the dop {⍺←⍵ ⋄ ⍺ ⍺⍺ ⍵⍵ ⍵} and is equivalent to the derived function f∘g⍨⍨.

Withe first appeared in A Dictionary of APL,[1] but substituted by the 2-train in J after the proposal in Iverson and McDonnell's paper Phrasal Forms.[2] It is present in Kap as Compose () and in BQN as After ().[3]

See also


  1. Ken Iverson. A Dictionary of APL. Withe. APL Quote Quad, Volume 18, Number 1, 1987-09.
  2. Ken Iverson and Eugene McDonnell. Phrasal forms at APL89.
  3. Marshall Lochbaum. Tutorial: Combinators. Before and After. Accessed 2021-09-01.

APL built-ins [edit]
Primitives (Timeline) Functions
Monadic ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
Structural ShapeReshapeTallyDepthRavelEnlistTableCatenateReverseRotateTransposeRazeMixSplitEncloseNestCut (K)PairLinkPartitioned EnclosePartition
Selection FirstPickTakeDropUniqueIdentityStopSelectReplicateExpandSet functions (IntersectionUnionWithout) ∙ Bracket indexingIndexCartesian ProductSort
Selector Index generatorGradeIndex OfInterval IndexIndicesDealPrefix and suffix vectors
Computational MatchNot MatchMembershipFindNub SieveEncodeDecodeMatrix InverseMatrix DivideFormatExecuteMaterialiseRange
Operators Monadic EachCommuteConstantReplicateExpandReduceWindowed ReduceScanOuter ProductKeyI-BeamSpawnFunction axis
Dyadic BindCompositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner ProductDeterminantPowerAtUnderRankDepthVariantStencilCutDirect definition (operator)
Quad names Index originComparison toleranceMigration levelAtomic vector