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authorMarshall Lochbaum <mwlochbaum@gmail.com>2021-08-07 20:41:31 -0400
committerMarshall Lochbaum <mwlochbaum@gmail.com>2021-08-07 20:41:31 -0400
commit7bf2aa4054b8378a76dff63acdccbcdad91f68e6 (patch)
tree1bff7a2a2b11fa259bc80722fdcb0609768c7718 /docs
parent532796eb397c8374e0546de9e4ab70c2955349f3 (diff)
BREAKING: Don't allow First of empty or reshaping empty to non-empty
Diffstat (limited to 'docs')
-rw-r--r--docs/bqn.js8
-rw-r--r--docs/doc/fill.html21
-rw-r--r--docs/doc/find.html6
-rw-r--r--docs/doc/pick.html13
-rw-r--r--docs/doc/reshape.html8
-rw-r--r--docs/doc/types.html2
-rw-r--r--docs/spec/primitive.html4
7 files changed, 28 insertions, 34 deletions
diff --git a/docs/bqn.js b/docs/bqn.js
index 4e686bd3..54b69dce 100644
--- a/docs/bqn.js
+++ b/docs/bqn.js
@@ -375,10 +375,10 @@ let runtime_0 = [
];
let [runtime,setPrims] = run(
- 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- ,[provide[0],provide[1],provide[2],provide[3],provide[4],provide[5],provide[6],provide[7],provide[8],provide[9],provide[10],runtime_0[0],provide[12],provide[13],provide[14],provide[15],provide[16],provide[17],runtime_0[1],runtime_0[2],runtime_0[3],runtime_0[4],runtime_0[5],runtime_0[6],runtime_0[7],runtime_0[8],runtime_0[9],runtime_0[10],runtime_0[11],runtime_0[12],provide[18],provide[19],runtime_0[13],runtime_0[14],runtime_0[15],runtime_0[16],provide[20],provide[21],provide[22],runtime_0[17],runtime_0[18],runtime_0[19],runtime_0[20],runtime_0[21],runtime_0[22],0,1,2,Infinity,-Infinity,-1,3,8,5,4,'∾','+','-','×','÷','⋆','¬','⌊','⌈','∨','∧','≠','=','>','≥','\0','˙','√','<','⊢','⊣','≍','↑','↓','↕','⌽','⍉','/','⊔','˜','¨','⌜','˘','`','∘','○','⌾','⍟','⊘','⊸','⟜',str("+-×÷⋆√⌊⌈|¬∧∨<>≠=≤≥≡≢⊣⊢⥊∾≍↑↓↕«»⌽⍉/⍋⍒⊏⊑⊐⊒∊⍷⊔!˙˜˘¨⌜⁼´˝`∘○⊸⟜⌾⊘◶⎉⚇⍟⎊%"),str("´: Identity not found"),str("´: 𝕩 must be a list"),str("Mapping: Equal-rank argument shapes don\'t agree"),str("Mapping: Argument shape prefixes don\'t 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Shape must be exact when reshaping with ∘"),str("⊔: Grouping argument must consist of integers"),str("⊔: Grouping argument values cannot be less than ¯1"),str("⊔𝕩: 𝕩 must be a list"),str("𝕨⊔𝕩: Rank of simple 𝕨 must be at most rank of 𝕩"),str("𝕨⊔𝕩: Lengths of 𝕨 must equal to 𝕩, or one more only in a rank-1 component"),str("𝕨⊔𝕩: 𝕩 must be an array"),str("⎉ or ⚇: 𝔾 result must have rank at most 1"),str("⎉ or ⚇: 𝔾 result must have 1 to 3 elements"),str("⎉ or ⚇: 𝔾 result must consist of integers"),str("⍟: 𝕨𝔾𝕩 must consist of integers"),str("≥: Needs two arguments"),str("≤: Needs two arguments"),str("𝕨⊔𝕩: Compound 𝕨 must be a list"),str("𝕨⊔𝕩: Total rank of 𝕨 must be at most rank of 𝕩"),str("⊒: Rank of 𝕨 must be at least 1"),str("⊒: Rank of 𝕩 must be at least cell rank of 𝕨"),str("𝕨⍉𝕩: 𝕨 must have rank at most 1"),str("𝕨⍉𝕩: Length of 𝕨 must not exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must consist of natural numbers"),str("𝕨⍉𝕩: Skipped result axis"),str("Cannot currently invert blocks"),str("Cannot invert modifier"),str("⁼: Inverse failed"),str("⁼: Inverse does not exist"),str("⁼: Inverse not found")]
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Shape must be exact when reshaping with ∘"),str("𝕨⥊𝕩: Can\'t produce non-empty array from empty 𝕩"),str("⊔: Grouping argument must consist of integers"),str("⊔: Grouping argument values cannot be less than ¯1"),str("⊔𝕩: 𝕩 must be a list"),str("𝕨⊔𝕩: Rank of simple 𝕨 must be at most rank of 𝕩"),str("𝕨⊔𝕩: Lengths of 𝕨 must equal to 𝕩, or one more only in a rank-1 component"),str("𝕨⊔𝕩: 𝕩 must be an array"),str("⎉ or ⚇: 𝔾 result must have rank at most 1"),str("⎉ or ⚇: 𝔾 result must have 1 to 3 elements"),str("⎉ or ⚇: 𝔾 result must consist of integers"),str("⍟: 𝕨𝔾𝕩 must consist of integers"),str("≥: Needs two arguments"),str("≤: Needs two arguments"),str("𝕨⊔𝕩: Compound 𝕨 must be a list"),str("𝕨⊔𝕩: Total rank of 𝕨 must be at most rank of 𝕩"),str("⊒: Rank of 𝕨 must be at least 1"),str("⊒: Rank of 𝕩 must be at least cell rank of 𝕨"),str("𝕨⍉𝕩: 𝕨 must have rank at most 1"),str("𝕨⍉𝕩: Length of 𝕨 must not exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must consist of natural numbers"),str("𝕨⍉𝕩: Skipped result axis"),str("Cannot currently invert blocks"),str("Cannot invert modifier"),str("⁼: Inverse failed"),str("⁼: Inverse does not exist"),str("⁼: Inverse not found")]
+ ,[[0,1,0],[0,0,1],[0,0,2],[0,0,3],[0,0,4],[0,0,[[],[5]]],[0,0,6],[1,1,7],[2,1,8],[1,1,9],[0,0,10],[2,1,11],[0,0,12],[1,0,13],[1,1,14],[1,1,15],[0,0,16],[0,0,17],[1,0,18],[0,0,[[],[19]]],[1,1,20],[1,1,21],[1,0,22],[0,0,[[],[23]]],[0,0,24],[0,0,25],[0,0,26],[0,0,27],[0,0,28],[0,0,29],[0,0,30],[0,0,31],[0,0,32],[0,0,[[],[33]]],[0,0,34],[0,0,[[],[35]]],[1,1,36],[0,0,37],[0,0,[[],[38]]],[0,0,[[],[39]]],[0,0,40],[0,0,41],[0,0,[[],[42]]],[0,0,[[],[43]]],[0,1,44],[0,0,45],[0,0,[[],[46]]],[0,0,47],[0,0,48],[0,0,[[],[49]]],[2,1,50],[1,0,51],[1,0,52],[0,0,53],[0,0,[[],[54]]],[2,0,55],[0,0,56],[0,0,[[],[57]]],[0,0,58],[2,1,59],[0,0,60],[0,0,61],[0,0,62],[0,0,63],[0,0,[[],[64]]],[0,0,[[],[65]]],[0,0,66],[0,0,67],[0,0,68],[0,1,69],[1,1,70],[0,0,71],[0,0,72],[0,0,73],[0,0,74],[0,0,[[],[75]]],[0,0,76],[1,0,77],[1,0,78],[0,0,[[],[79]]],[1,0,80],[0,0,81],[0,0,[[],[82]]],[0,0,[[],[83]]],[0,0,[[],[84]]],[1,1,85],[2,0,86],[2,0,87],[2,0,88],[0,0,[[],[89]]],[0,0,[[],[90]]],[0,0,[[],[91]]],[0,0,[[],[92]]],[0,0,[[],[93]]],[0,0,[[],[94]]],[0,0,[[],[95]]],[1,1,96],[2,0,97],[1,1,98],[0,0,99],[0,0,100],[0,0,[[],[101]]],[0,0,102],[0,0,103],[0,0,104],[0,0,105],[0,0,[[],[106]]],[0,0,107],[0,0,[[],[108]]],[0,0,109],[2,0,110],[0,0,111],[0,0,[[],[112]]],[0,0,113],[0,0,[[],[114]]],[0,0,115],[0,0,116],[0,0,117],[0,0,118],[0,0,119],[0,0,120],[0,0,121],[1,1,122],[0,0,[[],[123]]],[0,0,[[],[124]]],[0,0,[[],[125]]],[0,0,[[],[126]]],[0,0,[[],[127]]],[0,0,[[],[128]]],[0,0,129],[0,0,[[],[130]]],[2,0,131],[0,0,132],[0,0,133],[0,0,134],[1,0,135],[1,0,[[],[136]]],[0,0,137],[0,0,[[],[138]]],[0,0,[[],[139]]],[0,0,140],[0,0,141],[0,0,142],[0,0,143],[0,0,[[],[144]]],[0,0,[[],[145]]],[0,0,146],[2,0,147],[0,0,[[],[148]]],[0,0,149],[0,0,[[],[150]]],[0,0,[[],[151]]],[0,0,152],[1,1,153],[0,0,154],[0,0,[[],[155]]],[0,0,156],[0,0,[[],[157]]],[0,0,158],[0,0,[[],[159]]],[0,0,160],[0,0,161],[0,0,[[],[162]]],[2,0,163],[0,0,[[],[164]]],[0,0,[[],[165]]],[0,0,166],[0,0,167],[1,0,[[],[168]]],[0,0,[[],[169]]],[1,0,170],[0,0,171],[1,0,172],[2,0,[[],[173]]],[0,0,174],[0,0,175],[0,0,[[],[176]]],[0,0,177],[0,0,178],[0,0,[[],[179]]],[1,0,[[],[180]]],[1,1,181],[0,0,[[],[182]]],[0,0,[[],[183]]],[0,0,[[],[184]]],[0,0,[[],[185]]],[0,0,[[],[186]]],[0,0,[[],[187]]],[0,0,[[],[188]]],[0,0,189],[0,0,190],[1,1,191],[0,0,192],[0,0,[[],[193]]],[0,0,[[],[194]]],[0,0,195],[0,0,196],[0,0,197],[0,0,198],[0,0,199],[0,0,200],[0,0,201],[0,0,202],[0,0,203],[2,1,204],[2,1,205],[1,1,206],[1,1,207],[1,1,208],[0,0,[[],[209]]],[0,0,210],[0,0,211],[0,0,212],[0,0,[[],[213]]],[0,0,214],[0,0,[[],[215]]],[0,0,[[],[216]]],[0,0,[[],[217]]],[1,1,218],[0,0,219],[0,0,[[],[220]]],[0,0,221],[1,0,222],[0,0,223],[0,0,[[],[224]]],[0,0,225],[0,0,226],[0,0,[[],[227]]],[0,0,228],[0,0,229],[0,0,230],[0,0,231],[0,0,[[],[232]]],[0,0,233],[2,0,234],[0,0,[[],[235]]],[0,0,236],[0,0,237],[1,0,238],[0,0,239],[0,0,240],[1,0,241],[0,0,[[],[242]]],[0,0,243],[0,0,244],[1,0,245],[0,0,[[],[246]]],[0,0,[[],[247]]],[0,0,248],[0,0,249],[0,0,[[],[250]]],[0,0,251],[0,0,[[],[252]]],[0,0,253],[0,0,254],[1,0,255],[0,0,256],[0,0,257],[0,0,[[],[258]]],[0,0,259],[1,0,260],[0,0,261],[0,0,[[],[262]]],[2,0,263],[1,1,264],[1,1,265],[0,0,266],[0,0,267],[0,0,268],[0,0,269],[0,0,270],[0,0,271],[0,0,272],[0,0,273],[0,0,274],[0,0,[[],[275]]],[0,0,276],[0,0,[[],[277]]],[0,0,278],[0,0,279],[0,0,[[],[280]]],[1,1,281],[2,1,282],[0,0,283],[0,0,[[],[284]]],[0,0,285],[0,0,[[],[286]]],[0,0,287],[1,0,288],[0,0,[[],[289]]],[0,0,290],[0,0,291],[0,0,292],[0,0,293],[0,0,[[],[294]]],[0,0,[[],[295]]],[0,0,296],[0,0,297],[0,0,298],[0,0,299]]
+ ,[[0,184],[4073,3],[4081,3],[4085,3],[4098,3],[4110,3],[4119,3],[4125,2],[4154,3],[4186,2],[4203,4],[4265,5],[4301,3],[4308,5],[4366,4],[4432,3],[4487,3],[4498,4],[4563,9],[4632,5],[4709,5],[4776,4],[4832,8],[5004,8],[5161,3],[5206,3],[5255,3],[5372,6],[5457,3],[5481,7],[5595,3],[5657,4],[5728,6],[5792,3],[5832,5],[5909,3],[5926,14],[6106,3],[6152,3],[6172,3],[6189,4],[6247,4],[6323,5],[6397,3],[6456,0],[6467,4],[6547,3],[6557,3],[6564,3],[6578,3],[6586,3],[6624,5],[6661,5],[6688,3],[6692,3],[6706,16],[7001,3],[7026,3],[7051,3],[7078,3],[7137,4],[7237,7],[7434,7],[7480,14],[7704,5],[7783,3],[7791,6],[7818,5],[7839,6],[7866,11],[8199,9],[8356,4],[8432,7],[8544,7],[8661,3],[8769,4],[8831,8],[8943,8],[9017,7],[9079,9],[9237,9],[9385,4],[9496,8],[9710,3],[9754,3],[9891,2],[9912,13],[10025,9],[10207,10],[10280,3],[10306,10],[10714,4],[10819,3],[10893,3],[10944,4],[11071,4],[11187,2],[11195,6],[11210,2],[11269,3],[11277,3],[11285,3],[11305,8],[11380,3],[11388,3],[11396,3],[11404,3],[11412,3],[11420,3],[11431,3],[11452,7],[11493,6],[11712,4],[11812,3],[11855,5],[11993,3],[12040,3],[12066,3],[12099,3],[12143,3],[12191,3],[12203,3],[12276,2],[12302,4],[12338,3],[12353,3],[12382,3],[12397,3],[12415,3],[12429,3],[12439,3],[12491,6],[12519,3],[12547,3],[12555,3],[12563,5],[12598,12],[12771,3],[12782,3],[12799,3],[12816,3],[12827,3],[12838,3],[12853,7],[12926,6],[12972,3],[12990,3],[12997,6],[13032,10],[13209,4],[13265,11],[13466,4],[13533,3],[13558,2],[13573,3],[13600,7],[13741,3],[13770,3],[13780,3],[13799,3],[13809,3],[13818,3],[13828,3],[13845,11],[13918,6],[14021,8],[14099,3],[14107,3],[14149,7],[14175,9],[14421,5],[14437,6],[14559,5],[14592,6],[14621,3],[14631,3],[14641,4],[14680,4],[14703,3],[14711,3],[14736,6],[14832,2],[14882,8],[15063,11],[15268,3],[15327,3],[15354,3],[15371,3],[15388,8],[15570,3],[15584,3],[15615,2],[15649,3],[15684,3],[15702,3],[15720,5],[15791,3],[15811,8],[16041,3],[16109,3],[16139,3],[16170,3],[16204,5],[16271,3],[16280,3],[16288,3],[16298,2],[16312,2],[16327,6],[16421,3],[16431,3],[16438,9],[16572,3],[16598,3],[16616,3],[16640,3],[16661,3],[16672,3],[16685,2],[16689,3],[16725,5],[16827,3],[16874,5],[16897,3],[16914,14],[17198,13],[17395,3],[17438,4],[17521,3],[17557,3],[17586,3],[17597,4],[17633,7],[17770,10],[17960,6],[17995,3],[18060,3],[18088,4],[18111,5],[18122,4],[18133,4],[18144,5],[18164,4],[18239,3],[18272,3],[18302,5],[18327,3],[18393,3],[18407,3],[18418,5],[18517,4],[18646,3],[18670,3],[18716,3],[18749,3],[18778,5],[18810,3],[18846,3],[18871,3],[18896,3],[18918,5],[18928,3],[18961,3],[18981,6],[18996,2],[19004,2],[19012,5],[19050,3],[19060,3],[19076,5],[19127,3],[19152,6],[19261,3],[19301,3],[19342,5],[19392,6],[19529,3],[19537,8],[19594,5],[19623,3],[19638,3],[19688,2],[19706,3],[19727,3],[19745,3],[19774,3],[19793,3],[19812,6],[19891,5],[19906,14],[20070,3],[20091,3],[20107,4],[20163,3],[20195,3],[20278,3],[20312,3],[20341,3],[20358,3],[20375,3]]
);
let rtAssert = runtime[42] = assertFn("!");
diff --git a/docs/doc/fill.html b/docs/doc/fill.html
index 5ef9baf1..843e80e8 100644
--- a/docs/doc/fill.html
+++ b/docs/doc/fill.html
@@ -6,7 +6,7 @@
<div class="nav"><a href="https://github.com/mlochbaum/BQN">BQN</a> / <a href="../index.html">main</a> / <a href="index.html">doc</a></div>
<h1 id="fill-elements">Fill elements</h1>
<p>A few array operations need an array element to use when no existing element applies. BQN tries to maintain a &quot;default&quot; element for every array, known as a fill element, for this purpose. If it's known, the fill element is a nested array structure where each atom is either <code><span class='Number'>0</span></code> or <code><span class='String'>' '</span></code>. If no fill is known, a function that requests it results in an error.</p>
-<p>Fills are used by <a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>) when a value in <code><span class='Value'>𝕨</span></code> is larger than the corresponding length in <code><span class='Value'>𝕩</span></code>, by the two <a href="shift.html">Nudge</a> functions (<code><span class='Function'>»«</span></code>) when <code><span class='Value'>𝕩</span></code> is non-empty, and by <a href="pick.html">First</a> (<code><span class='Function'>⊑</span></code>) and <a href="reshape.html">Reshape</a> (<code><span class='Function'>⥊</span></code>) when <code><span class='Value'>𝕩</span></code> is empty. Except for these specific cases, the fill value an array has can't affect the program. The result of <a href="match.html">Match</a> (<code><span class='Function'>≡</span></code>) doesn't depend on fills, and any attempt to compute a fill can't cause side effects.</p>
+<p>Fills are used by <a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>) when a value in <code><span class='Value'>𝕨</span></code> is larger than the corresponding length in <code><span class='Value'>𝕩</span></code>, by the two <a href="shift.html">Nudge</a> functions (<code><span class='Function'>»«</span></code>) when <code><span class='Value'>𝕩</span></code> is non-empty, and by <a href="reshape.html">Reshape</a> (<code><span class='Function'>⥊</span></code>) when <code><span class='Value'>𝕨</span></code> contains <code><span class='Function'>↑</span></code>. Except for these specific cases, the fill value an array has can't affect the program. The result of <a href="match.html">Match</a> (<code><span class='Function'>≡</span></code>) doesn't depend on fills, and any attempt to compute a fill can't cause side effects.</p>
<h2 id="using-fills">Using fills</h2>
<p>For the examples in this section we'll use the fact that an all-number array usually has <code><span class='Number'>0</span></code> as a fill while a string has <code><span class='String'>' '</span></code> (BQN maintains fills alongside array values rather than deriving them from arrays, so it's possible to construct arrays where this isn't true, but this probably wouldn't happen in ordinary code).</p>
<p><a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>) and <a href="shift.html">Nudge</a> (<code><span class='Function'>»«</span></code>) in either direction use the fill for padding, to extend the array past its boundary. For example, <code><span class='Value'>𝕨</span><span class='Function'>↑</span><span class='Value'>𝕩</span></code> will add elements to one side when a number in <code><span class='Function'>|</span><span class='Value'>𝕨</span></code> is larger than the corresponding length in <code><span class='Function'>≢</span><span class='Value'>𝕩</span></code>.</p>
@@ -26,18 +26,15 @@
<span class='Function'>»</span><span class='Bracket'>⟨⟩</span> <span class='Comment'># Fill not needed
</span>⟨⟩
</pre>
-<p><a href="pick.html">First</a> (<code><span class='Function'>⊑</span></code>) and <a href="reshape.html">Reshape</a> (<code><span class='Function'>⥊</span></code>) use the fill when <code><span class='Value'>𝕩</span></code> is empty, and in the case of Reshape only when the result needs to be non-empty.</p>
-<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqRICIiCgo0IOKlisKoIOKfqOKGlTAsICIi4p+pCgow4oC/MyDipYog4p+o4p+pICAjIEZpbGwgbm90IG5lZWRlZA==">↗️</a><pre> <span class='Function'>⊑</span> <span class='String'>&quot;&quot;</span>
-' '
-
- <span class='Number'>4</span> <span class='Function'>⥊</span><span class='Modifier'>¨</span> <span class='Bracket'>⟨</span><span class='Function'>↕</span><span class='Number'>0</span><span class='Separator'>,</span> <span class='String'>&quot;&quot;</span><span class='Bracket'>⟩</span>
-⟨ ⟨ 0 0 0 0 ⟩ " " ⟩
-
- <span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>3</span> <span class='Function'>⥊</span> <span class='Bracket'>⟨⟩</span> <span class='Comment'># Fill not needed
-</span>↕0‿3
+<p><a href="reshape.html#computed-lengths">Reshape</a> (<code><span class='Function'>⥊</span></code>) uses the fill when <code><span class='Value'>𝕨</span></code> contains <code><span class='Function'>↑</span></code> and the product of the rest of <code><span class='Value'>𝕨</span></code> doesn't evenly divide the number of elements in <code><span class='Value'>𝕩</span></code>.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oaR4oC/OCDipYogImNvbXBsZXRlcGFydCI=">↗️</a><pre> <span class='Function'>↑</span><span class='Ligature'>‿</span><span class='Number'>8</span> <span class='Function'>⥊</span> <span class='String'>&quot;completepart&quot;</span>
+┌─
+╵"complete
+ part "
+ ┘
</pre>
-<p>If for some reason you need to find an array's fill element, the easiest way is <code><span class='Function'>⊑</span><span class='Number'>0</span><span class='Function'>⥊</span><span class='Value'>a</span></code>.</p>
-<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqRMOKliiJzdHJpbmci">↗️</a><pre> <span class='Function'>⊑</span><span class='Number'>0</span><span class='Function'>⥊</span><span class='String'>&quot;string&quot;</span>
+<p>If for some reason you need to find an array's fill element, the easiest general way is probably <code><span class='Function'>⊑»</span><span class='Number'>1</span><span class='Function'>↑⥊</span><span class='Value'>a</span></code>.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqRwrsx4oaR4qWKInN0cmluZyI=">↗️</a><pre> <span class='Function'>⊑»</span><span class='Number'>1</span><span class='Function'>↑⥊</span><span class='String'>&quot;string&quot;</span>
' '
</pre>
<h2 id="how-fills-are-computed">How fills are computed</h2>
diff --git a/docs/doc/find.html b/docs/doc/find.html
index bc883db2..1421500e 100644
--- a/docs/doc/find.html
+++ b/docs/doc/find.html
@@ -32,14 +32,14 @@
<span class='String'>&quot;string&quot;</span> <span class='Paren'>(</span><span class='Function'>≢</span><span class='Modifier2'>∘</span><span class='Function'>⊢↑⍷</span><span class='Paren'>)</span> <span class='String'>&quot;substring&quot;</span> <span class='Comment'># APL style
</span>⟨ 0 0 0 1 0 0 0 0 0 ⟩
</pre>
-<p>If <code><span class='Value'>𝕨</span></code> is larger than <code><span class='Value'>𝕩</span></code>, the result is empty, and there's no error even in cases where Windows would fail. One place this tends to come up is when applying <a href="pick.html">First</a> (<code><span class='Function'>⊑</span></code>) the result: <code><span class='Function'>⊑⍷</span></code> tests whether <code><span class='Value'>𝕨</span></code> appears in <code><span class='Value'>𝕩</span></code> at the first position, that is, whether it's a prefix of <code><span class='Value'>𝕩</span></code>. If <code><span class='Value'>𝕨</span></code> is longer than <code><span class='Value'>𝕩</span></code> it shouldn't be a prefix, so 0 is appropriate.</p>
-<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=Imxvb29vb29uZyIg4o23ICJzaG9ydCIKCjkg4oaVICJzaG9ydCIKCuKKkSAibG9vb29vb25nIiDijbcgInNob3J0Ig==">↗️</a><pre> <span class='String'>&quot;loooooong&quot;</span> <span class='Function'>⍷</span> <span class='String'>&quot;short&quot;</span>
+<p>If <code><span class='Value'>𝕨</span></code> is larger than <code><span class='Value'>𝕩</span></code>, the result is empty, and there's no error even in cases where Windows would fail. One place this tends to come up is when applying <a href="pick.html#first">First</a> (<code><span class='Function'>⊑</span></code>) the result: <code><span class='Function'>⊑⍷</span></code> tests whether <code><span class='Value'>𝕨</span></code> appears in <code><span class='Value'>𝕩</span></code> at the first position, that is, whether it's a prefix of <code><span class='Value'>𝕩</span></code>. If <code><span class='Value'>𝕨</span></code> is longer than <code><span class='Value'>𝕩</span></code> it shouldn't be a prefix. First will fail but using a <a href="fold.html">fold</a> <code><span class='Number'>0</span><span class='Function'>⊣</span><span class='Modifier'>´</span><span class='Function'>⥊</span><span class='Modifier2'>∘</span><span class='Function'>⍷</span></code> instead gives a 0 in this case.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=Imxvb29vb29uZyIg4o23ICJzaG9ydCIKCjkg4oaVICJzaG9ydCIKCjAg4oqjwrQgImxvb29vb29uZyIg4o23ICJzaG9ydCI=">↗️</a><pre> <span class='String'>&quot;loooooong&quot;</span> <span class='Function'>⍷</span> <span class='String'>&quot;short&quot;</span>
⟨⟩
<span class='Number'>9</span> <span class='Function'>↕</span> <span class='String'>&quot;short&quot;</span>
ERROR
- <span class='Function'>⊑</span> <span class='String'>&quot;loooooong&quot;</span> <span class='Function'>⍷</span> <span class='String'>&quot;short&quot;</span>
+ <span class='Number'>0</span> <span class='Function'>⊣</span><span class='Modifier'>´</span> <span class='String'>&quot;loooooong&quot;</span> <span class='Function'>⍷</span> <span class='String'>&quot;short&quot;</span>
0
</pre>
<p>This pattern also works in the high-rank case discussed below, testing whether <code><span class='Value'>𝕨</span></code> is a multi-dimensional prefix starting at the lowest-index corner of <code><span class='Value'>𝕩</span></code>.</p>
diff --git a/docs/doc/pick.html b/docs/doc/pick.html
index b0e99195..a296d6c0 100644
--- a/docs/doc/pick.html
+++ b/docs/doc/pick.html
@@ -6,7 +6,7 @@
<div class="nav"><a href="https://github.com/mlochbaum/BQN">BQN</a> / <a href="../index.html">main</a> / <a href="index.html">doc</a></div>
<h1 id="pick">Pick</h1>
<p>Pick (<code><span class='Function'>⊑</span></code>) chooses elements from <code><span class='Value'>𝕩</span></code> based on <a href="indices.html">index</a> lists from <code><span class='Value'>𝕨</span></code>. <code><span class='Value'>𝕨</span></code> can be a plain list, or even one number if <code><span class='Value'>𝕩</span></code> is a list, in order to get one element from <code><span class='Value'>𝕩</span></code>. It can also be an array of index lists, or have deeper array structure: each index list will be replaced with the element of <code><span class='Value'>𝕩</span></code> at that index, effectively applying to <code><span class='Value'>𝕨</span></code> at <a href="depth.html#the-depth-modifier">depth</a> 1.</p>
-<p>With no <code><span class='Value'>𝕨</span></code>, monadic <code><span class='Function'>⊑</span><span class='Value'>𝕩</span></code> takes the first element of <code><span class='Value'>𝕩</span></code> in index order, or its <a href="fill.html">fill element</a> if <code><span class='Value'>𝕩</span></code> is empty (causing an error if no fill is known).</p>
+<p>With no <code><span class='Value'>𝕨</span></code>, monadic <code><span class='Function'>⊑</span><span class='Value'>𝕩</span></code> takes the first element of <code><span class='Value'>𝕩</span></code> in index order, with an error if <code><span class='Value'>𝕩</span></code> is empty.</p>
<p>While sometimes &quot;scatter-point&quot; indexing is necessary, using Pick to select multiple elements from <code><span class='Value'>𝕩</span></code> is less array-oriented than <a href="select.html">Select</a> (<code><span class='Function'>⊏</span></code>), and probably slower. Consider rearranging your data so that you can select along axes instead of picking out elements.</p>
<h2 id="one-element">One element</h2>
<p>When the left argument is a number, Pick gets an element from a list:</p>
@@ -55,15 +55,12 @@
<span class='Function'>⊑</span> <span class='Function'>↕</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>1</span>
⟨ 0 0 0 0 ⟩
</pre>
-<p>If <code><span class='Value'>𝕩</span></code> is empty then Pick always results in an error. First never gives an error: instead it returns the <a href="fill.html">fill element</a> for <code><span class='Value'>𝕩</span></code>.</p>
-<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqRICIiCuKKkSDiiaLPgAriipEgMOKGkTzin6giICAiLOKGlTTin6k=">↗️</a><pre> <span class='Function'>⊑</span> <span class='String'>&quot;&quot;</span>
-' '
+<p>If <code><span class='Value'>𝕩</span></code> is empty then First results in an error, like Pick.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqRICIiCuKKkSDiiaLPgA==">↗️</a><pre> <span class='Function'>⊑</span> <span class='String'>&quot;&quot;</span>
+ERROR
<span class='Function'>⊑</span> <span class='Function'>≢</span><span class='Number'>π</span>
-0
- <span class='Function'>⊑</span> <span class='Number'>0</span><span class='Function'>↑&lt;</span><span class='Bracket'>⟨</span><span class='String'>&quot; &quot;</span><span class='Separator'>,</span><span class='Function'>↕</span><span class='Number'>4</span><span class='Bracket'>⟩</span>
-⟨ " " ⟨ 0 0 0 0 ⟩ ⟩
+ERROR
</pre>
-<p>So one way to find the fill element for an array <code><span class='Value'>𝕩</span></code> of any shape is <code><span class='Function'>⊑</span><span class='Number'>0</span><span class='Function'>⥊</span><span class='Value'>𝕩</span></code>.</p>
<p>In APL it's common to get the last element of a list with an idiom that translates to <code><span class='Function'>⊑⌽</span></code>, or First-<a href="reverse.html">Reverse</a>. In BQN the most straightforward way is to select with index <code><span class='Number'>¯1</span></code> instead. I also sometimes use <a href="fold.html">Fold</a> with the Right <a href="identity.html">identity function</a>.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=4oqR4oy9ICJsYXN0IgrCrzHiipEgImxhc3QiCuKKosK0ICJsYXN0Ig==">↗️</a><pre> <span class='Function'>⊑⌽</span> <span class='String'>&quot;last&quot;</span>
't'
diff --git a/docs/doc/reshape.html b/docs/doc/reshape.html
index 23e77e9c..b9d16cc5 100644
--- a/docs/doc/reshape.html
+++ b/docs/doc/reshape.html
@@ -142,12 +142,12 @@
235 236 237
</pre>
-<p>If the left argument implies a larger number of elements, then the argument elements are reused cyclically. Below, we reach the last element <code><span class='Number'>247</span></code> and start over at <code><span class='Number'>135</span></code>. If the array doesn't have any elements to start with, its <a href="fill.html">fill element</a> is used instead, but it's probably best not to invoke this case!</p>
-<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=MTUg4qWKIGEKCjQg4qWKIOKGlTAgICMgRmlsbCBmb3Ig4oaVMCBpcyAw">↗️</a><pre> <span class='Number'>15</span> <span class='Function'>⥊</span> <span class='Value'>a</span>
+<p>If the left argument implies a larger number of elements, then the argument elements are reused cyclically. Below, we reach the last element <code><span class='Number'>247</span></code> and start over at <code><span class='Number'>135</span></code>. If the array doesn't have any elements to start with, you'll get an error as there aren't any elements available.</p>
+<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=MTUg4qWKIGEKCjQg4qWKIOKGlTA=">↗️</a><pre> <span class='Number'>15</span> <span class='Function'>⥊</span> <span class='Value'>a</span>
⟨ 135 136 137 145 146 147 235 236 237 245 246 247 135 136 137 ⟩
- <span class='Number'>4</span> <span class='Function'>⥊</span> <span class='Function'>↕</span><span class='Number'>0</span> <span class='Comment'># Fill for ↕0 is 0
-</span>⟨ 0 0 0 0 ⟩
+ <span class='Number'>4</span> <span class='Function'>⥊</span> <span class='Function'>↕</span><span class='Number'>0</span>
+ERROR
</pre>
<p>Reshape is the idiomatic way to make an array filled with a constant value (that is, where all elements are the same). For an atom element, just reshape it directly; for an arbitrary element, first enclose it to create a unit, and then reshape it.</p>
<a class="replLink" title="Open in the REPL" target="_blank" href="https://mlochbaum.github.io/BQN/try.html#code=M+KAvzQg4qWKIDAKCjUg4qWKIDwgInN0cmluZyI=">↗️</a><pre> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'>⥊</span> <span class='Number'>0</span>
diff --git a/docs/doc/types.html b/docs/doc/types.html
index 46356766..bcc119f6 100644
--- a/docs/doc/types.html
+++ b/docs/doc/types.html
@@ -59,7 +59,7 @@
<h3 id="arrays">Arrays</h3>
<p><em>Full documentation <a href="array.html">here</a>.</em></p>
<p>A BQN array is a multidimensional arrangement of data. This means it has a certain <a href="shape.html"><em>shape</em></a>, which is a finite list of natural numbers giving the length along each axis, and it contains an <em>element</em> for each possible <a href="indices.html"><em>index</em></a>, which is a choice of one natural number that's less than each axis length in the shape. The total number of elements, or <em>bound</em>, is then the product of all the lengths in the shape. The shape may have any length including zero, and this shape is known as the array's <em>rank</em>. An array of rank 0, which always contains exactly one element, is called a <em>unit</em>, while an array of rank 1 is called a <em>list</em> and an array of rank 2 is called a <em>table</em>.</p>
-<p>Each array—empty or nonempty—has an inferred property called a <em>fill</em>. The fill either indicates what element should be used to pad an array, or that such an element is not known and an error should result. Fills can be used by <a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>), the two <a href="shift.html">Nudge</a> functions (<code><span class='Function'>»«</span></code>), <a href="pick.html">First</a> (<code><span class='Function'>⊑</span></code>), and <a href="reshape.html">Reshape</a> (<code><span class='Function'>⥊</span></code>).</p>
+<p>Each array—empty or nonempty—has an inferred property called a <a href="fill.html"><em>fill</em></a>. The fill either indicates what element should be used to pad an array, or that such an element is not known and an error should result. Fills can be used by <a href="take.html">Take</a> (<code><span class='Function'>↑</span></code>), the two <a href="shift.html">Nudge</a> functions (<code><span class='Function'>»«</span></code>), and <a href="reshape.html">Reshape</a> (<code><span class='Function'>⥊</span></code>).</p>
<p>Arrays are value types (or immutable), so that there is no way to &quot;change&quot; the shape or elements of an array. An array with different properties is a different array. As a consequence, arrays are an inductive type, and it's not possible for an array to contain itself, or contain an array that contains itself, and so on. However, it is possible for an array to contain a function or other operation that has access to the array through a variable, and in this sense an array can &quot;know about&quot; itself.</p>
<p>Different elements of an array should not influence each other. While some APLs force numbers placed in the same array to a common representation, which may have different precision properties, BQN values must not change behavior when placed in an array. However, this doesn't preclude changing the storage type of an array for better performance: for example, in a BQN implementation using 64-bit floats, an array whose elements are all integers that fit in 32-bit int range might be represented as an array of 32-bit ints.</p>
<h2 id="operation-types">Operation types</h2>
diff --git a/docs/spec/primitive.html b/docs/spec/primitive.html
index fc60460d..b2e127c6 100644
--- a/docs/spec/primitive.html
+++ b/docs/spec/primitive.html
@@ -122,13 +122,13 @@
<p><strong>Enclose</strong> (<code><span class='Function'>&lt;</span></code>) forms a unit array that contains its argument.</p>
<p><strong>Merge</strong> (<code><span class='Function'>&gt;</span></code>) combines the outer axes of an array of arrays with inner axes: it requires that all elements of its argument have the same shape, and creates an array such that <code><span class='Paren'>(</span><span class='Value'>i</span><span class='Function'>∾</span><span class='Value'>j</span><span class='Paren'>)</span><span class='Function'>⊑&gt;</span><span class='Value'>𝕩</span></code> is <code><span class='Value'>i</span><span class='Function'>⊑</span><span class='Value'>j</span><span class='Function'>⊑</span><span class='Value'>𝕩</span></code>. It also accepts atom elements of <code><span class='Value'>𝕩</span></code>, converting them to unit arrays, or an atom argument, which is returned unchanged. <strong>Solo</strong> and <strong>Couple</strong> (<code><span class='Function'>≍</span></code>) turn one or two arguments into major cells of the result and can be defined easily in terms of Merge.</p>
<p><strong>Join To</strong> (<code><span class='Function'>∾</span></code>) combines its two arguments along an existing initial axis, unless both arguments are units, in which case it creates an axis and is identical to Couple (<code><span class='Function'>≍</span></code>). The arguments must differ in rank by at most 1, and the result rank is equal to the maximum of 1 and the higher argument rank. Each argument with rank less than the result, and each major cell of an argument with rank equal to it, becomes a major cell of the result, with cells from the left argument placed before those from the right. <strong>Join</strong> (<code><span class='Function'>∾</span></code>) generalizes the equal-rank subset of this behavior to an array of values instead of just two. The argument must be an array (unlike Merge), and its elements must all the same rank, which is at least the argument rank. Atom elements are treated as unit arrays. Then &quot;outer&quot; argument axes are matched up with leading &quot;inner&quot; element axes, and elements are joined along these axes. In order to allow this, the length of an element along a particular axis must depend only on the position along the corresponding axis in the argument. An empty argument to Join is return unchanged, as though the element rank is equal to the argument rank.</p>
-<p><strong>Deshape</strong> (<code><span class='Function'>⥊</span></code>) differs from the provided function (which returns the element list of an array) only in that it accepts an atom, returning a one-element list containing it. <strong>Reshape</strong> (<code><span class='Function'>⥊</span></code>) is extended in numerous ways. It accepts any list of natural numbers (including as a unit array or atom) for the left argument and any right argument; <code><span class='Value'>𝕩</span></code> is deshaped first so that it is treated as a list of elements. These elements are repeated cyclically to fill the result array in ravel order. If <code><span class='Value'>𝕩</span></code> is empty but the result is not, then the result consists of fill elements for <code><span class='Value'>𝕩</span></code>. Furthermore, at most one element of <code><span class='Value'>𝕨</span></code> can be a &quot;length code&quot;: one of the primitives <code><span class='Modifier2'>∘</span><span class='Function'>⌊⌽↑</span></code>. In this case, a target length is computed from the number of elements in <code><span class='Value'>𝕩</span></code> divided by the product of the other elements of <code><span class='Value'>𝕨</span></code> (which must not be zero). If the target length is an integer then it is used directly for the length code. Otherwise, an error is given if the length code is <code><span class='Modifier2'>∘</span></code>, and the target length is rounded down if the code is <code><span class='Function'>⌊</span></code> and up if it's <code><span class='Function'>⌽</span></code> or <code><span class='Function'>↑</span></code>. With code <code><span class='Function'>⌽</span></code>, elements are repeated cyclically as usual, but with code <code><span class='Function'>↑</span></code>, the extra elements after each argument element is used are fill values for <code><span class='Value'>𝕩</span></code>.</p>
+<p><strong>Deshape</strong> (<code><span class='Function'>⥊</span></code>) differs from the provided function (which returns the element list of an array) only in that it accepts an atom, returning a one-element list containing it. <strong>Reshape</strong> (<code><span class='Function'>⥊</span></code>) is extended in numerous ways. It accepts any list of natural numbers (including as a unit array or atom) for the left argument and any right argument; <code><span class='Value'>𝕩</span></code> is deshaped first so that it is treated as a list of elements. These elements are repeated cyclically to fill the result array in ravel order. If <code><span class='Value'>𝕩</span></code> is empty then a non-empty requested result shape causes an error. Furthermore, at most one element of <code><span class='Value'>𝕨</span></code> can be a &quot;length code&quot;: one of the primitives <code><span class='Modifier2'>∘</span><span class='Function'>⌊⌽↑</span></code>. In this case, a target length is computed from the number of elements in <code><span class='Value'>𝕩</span></code> divided by the product of the other elements of <code><span class='Value'>𝕨</span></code> (which must not be zero). If the target length is an integer then it is used directly for the length code. Otherwise, an error is given if the length code is <code><span class='Modifier2'>∘</span></code>, and the target length is rounded down if the code is <code><span class='Function'>⌊</span></code> and up if it's <code><span class='Function'>⌽</span></code> or <code><span class='Function'>↑</span></code>. With code <code><span class='Function'>⌽</span></code>, elements are repeated cyclically as usual, but with code <code><span class='Function'>↑</span></code>, the extra elements after each argument element is used are fill values for <code><span class='Value'>𝕩</span></code>.</p>
<p><strong>Transpose</strong> (<code><span class='Function'>⍉</span></code>) reorders axes of its argument to place the first axis last; if the argument has one or fewer axes then it's enclosed if it's an atom and otherwise returned unchanged. <strong>Reorder Axes</strong> (<code><span class='Function'>⍉</span></code>) requires the left argument to be a list or unit of natural numbers, with length at most the rank of the right argument. This list is extended to match the right argument rank exactly by repeatedly appending the least unused natural number (for example, given <code><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span></code>, <code><span class='Number'>2</span></code> is appended). After extension, it specifies a result axis for each axis of the right argument. There must be no gaps in the list: that is, with the result rank equal to one plus the greatest value present, every result axis must appear at least once. Now each argument axis is &quot;sent to&quot; the specified result axis: in terms of indices, <code><span class='Value'>i</span><span class='Function'>⊑</span><span class='Value'>𝕨</span><span class='Function'>⍉</span><span class='Value'>𝕩</span></code> is <code><span class='Paren'>(</span><span class='Value'>𝕨</span><span class='Function'>⊏</span><span class='Value'>i</span><span class='Paren'>)</span><span class='Function'>⊑</span><span class='Value'>𝕩</span></code> if <code><span class='Value'>𝕨</span></code> is complete. If multiple argument axes correspond to the same result axis, then a diagonal is taken, and it's as long as the shortest of those argument axes. Like Transpose, Reorder Axes encloses <code><span class='Value'>𝕩</span></code> if it's an atom, so that its result is always an array.</p>
<h3 id="indices-and-selection">Indices and selection</h3>
<p>Each element in an array <code><span class='Value'>s</span><span class='Function'>⥊</span><span class='Value'>e</span></code> is associated with an <em>index</em>, which is a list of natural numbers <code><span class='Value'>i</span></code> such that <code><span class='Function'>∧</span><span class='Modifier'>´</span><span class='Value'>i</span><span class='Function'>&lt;</span><span class='Value'>s</span></code>. The list of all indices, which corresponds to the element list <code><span class='Value'>e</span></code>, contains all such lists <code><span class='Value'>i</span></code> in lexicographic order. That is, index <code><span class='Value'>i</span></code> comes before <code><span class='Value'>j</span></code> exactly when the two indices are not the same, and <code><span class='Value'>i</span></code> has the smaller value at the first position where they are unequal. The index of an element along a particular axis <code><span class='Value'>a</span></code> is the value <code><span class='Value'>a</span><span class='Function'>⊑</span><span class='Value'>i</span></code>.</p>
<p><strong>Range</strong> (<code><span class='Function'>↕</span></code>) is extended to apply to a list of natural numbers, in addition to the provided case of a single natural number (an enclosed natural number <code><span class='Value'>𝕩</span></code> should still result in an error). For a list <code><span class='Value'>𝕩</span></code>, the result is an array of shape <code><span class='Value'>𝕩</span></code> in which the value at a given index is that index, as a list of natural numbers. That is, <code><span class='Value'>i</span><span class='Function'>≡</span><span class='Value'>i</span><span class='Function'>⊑↕</span><span class='Value'>𝕩</span></code> for any list of natural numbers <code><span class='Value'>i</span></code> with <code><span class='Function'>∧</span><span class='Modifier'>´</span><span class='Value'>i</span><span class='Function'>&lt;</span><span class='Value'>𝕩</span></code>.</p>
<p><strong>Pick</strong> (<code><span class='Function'>⊑</span></code>) is extended to array left arguments. In this case, it requires every depth-1 array in the nested structure of <code><span class='Value'>𝕨</span></code> to be a valid index list for <code><span class='Value'>𝕩</span></code>, and every atom to be contained in one of these lists. The result is <code><span class='Value'>𝕨</span></code> with each index list replaced by the element of <code><span class='Value'>𝕩</span></code> at that index. In the simple case where <code><span class='Value'>𝕨</span></code> itself is an index list, the result is the element of <code><span class='Value'>𝕩</span></code> at index <code><span class='Value'>𝕨</span></code>.</p>
-<p><strong>First</strong> (<code><span class='Function'>⊑</span></code>) simply takes the first element of its argument in index order, or the fill element if <code><span class='Value'>𝕩</span></code> is empty.</p>
+<p><strong>First</strong> (<code><span class='Function'>⊑</span></code>) simply takes the first element of its argument in index order, with an error if <code><span class='Value'>𝕩</span></code> is empty.</p>
<p>For <strong>Select</strong> (<code><span class='Function'>⊏</span></code>), <code><span class='Value'>𝕨</span></code> is an array of natural numbers, or a list of such arrays; if it's an empty list, it's interpreted as the former. The given arrays are matched with leading axes of <code><span class='Value'>𝕩</span></code> and used to select from those axes. Their shape is retained, so that the final shape is the combined shapes of each array of natural numbers in <code><span class='Value'>𝕨</span></code> in order, followed by the trailing (unmatched) shape of <code><span class='Value'>𝕩</span></code>. This means that a single axis in <code><span class='Value'>𝕩</span></code> can correspond to any number of axes in <code><span class='Value'>𝕨</span><span class='Function'>⊏</span><span class='Value'>𝕩</span></code>, depending on the rank of that portion of <code><span class='Value'>𝕨</span></code>. More precisely, the value of the result at an index <code><span class='Value'>j</span></code> is obtained by splitting <code><span class='Value'>j</span></code> into one index into each array of <code><span class='Value'>𝕨</span></code> followed by a partial index into <code><span class='Value'>𝕩</span></code>. An index <code><span class='Value'>i</span></code> for <code><span class='Value'>𝕩</span></code> comes from selecting from each array of <code><span class='Value'>𝕨</span></code> and appending the results to the partial index from <code><span class='Value'>j</span></code>, and the value <code><span class='Value'>i</span><span class='Function'>⊑</span><span class='Value'>𝕩</span></code> is <code><span class='Value'>j</span><span class='Function'>⊑</span><span class='Value'>𝕨</span><span class='Function'>⊏</span><span class='Value'>𝕩</span></code>.</p>
<p><strong>First Cell</strong> (<code><span class='Function'>⊏</span></code>) selects the initial major cell of <code><span class='Value'>𝕩</span></code>, giving an error if <code><span class='Value'>𝕩</span></code> has rank 0 or length 0.</p>
<p><strong>Group</strong> (<code><span class='Function'>⊔</span></code>) performs an opposite operation to Select, so that <code><span class='Value'>𝕨</span></code> specifies not the argument index that result values come from, but the result index that argument values go to. The general case is that <code><span class='Value'>𝕨</span></code> is a list of arrays of numbers; if it has depth less than 2 it's converted to this form by first enclosing it if it's an atom, then placing it in a length-1 list. After this transformation, the result rank is <code><span class='Function'>≠</span><span class='Value'>𝕨</span></code>, and each result element has rank <code><span class='Paren'>(</span><span class='Function'>≠</span><span class='Value'>𝕨</span><span class='Paren'>)</span><span class='Function'>+</span><span class='Paren'>(</span><span class='Function'>=</span><span class='Value'>𝕩</span><span class='Paren'>)</span><span class='Function'>-+</span><span class='Modifier'>´</span><span class='Function'>=</span><span class='Modifier'>¨</span><span class='Value'>𝕨</span></code>, with the initial <code><span class='Function'>≠</span><span class='Value'>𝕨</span></code> axes corresponding to elements of <code><span class='Value'>𝕨</span></code> and the remainder to trailing axes of <code><span class='Value'>𝕩</span></code>. Each atom in <code><span class='Value'>𝕨</span></code> can be either a natural number or <code><span class='Number'>¯1</span></code> (which indicates the corresponding position in <code><span class='Value'>𝕩</span></code> will be omitted). If <code><span class='Number'>¯1</span></code> doesn't appear, the result has the property that each cell of <code><span class='Value'>𝕩</span></code> appears in the corresponding element of <code><span class='Value'>𝕨</span><span class='Function'>⊏</span><span class='Value'>𝕨</span><span class='Function'>⊔</span><span class='Value'>𝕩</span></code>. More concretely, the length of the result along axis <code><span class='Value'>a</span></code> is the maximum value in <code><span class='Value'>a</span><span class='Function'>⊑</span><span class='Value'>𝕨</span></code> plus one, or zero if <code><span class='Value'>a</span><span class='Function'>⊑</span><span class='Value'>𝕨</span></code> is empty. Axis <code><span class='Value'>a</span></code> corresponds to <code><span class='Function'>=</span><span class='Value'>a</span><span class='Function'>⊑</span><span class='Value'>𝕨</span></code> axes in <code><span class='Value'>𝕩</span></code>, and an element of the result at position <code><span class='Value'>i</span></code> along this axis contains all positions in <code><span class='Value'>𝕩</span></code> where <code><span class='Value'>i</span><span class='Function'>=</span><span class='Value'>a</span><span class='Function'>⊑</span><span class='Value'>𝕨</span></code>. There may be multiple such positions, and they're arranged along axis <code><span class='Value'>a</span></code> of that result element according to their index order in <code><span class='Value'>𝕩</span></code>. The shapes of components of <code><span class='Value'>𝕨</span></code> must match the corresponding axes of <code><span class='Value'>𝕩</span></code>, except for rank-1 components of <code><span class='Value'>𝕨</span></code>, which can match or have an extra element. This element, which like the others is either a natural number or <code><span class='Number'>¯1</span></code>, gives the minimum length of the result axis corresponding to the component of <code><span class='Value'>𝕨</span></code> in question, but otherwise does not affect the result. <strong>Group Indices</strong> treats its argument <code><span class='Value'>𝕩</span></code> as a left argument for Group and uses a right argument made up of indices, which is <code><span class='Function'>↕≠</span><span class='Value'>𝕩</span></code> if <code><span class='Value'>𝕩</span></code> has depth 1 and <code><span class='Function'>↕∾≢</span><span class='Modifier'>¨</span><span class='Value'>𝕩</span></code> if it has depth 2. Because the depth-1 case uses atomic indices, <code><span class='Value'>𝕩</span></code> is required to be a list (and it can't be an atom). Much like Range, the result has depth one higher than the argument.</p>