Special case extensions to remove checks from /⁼ and F⁼⁼

5 files changed, 18 insertions, 17 deletions

diff --git a/docs/bqn.js b/docs/bqn.js
index a670eedb..6da47847 100644
--- a/docs/bqn.js
+++ b/docs/bqn.js
@@ -401,10 +401,10 @@ let runtime_0 = [
 ];
 
 let [runtime,setPrims,setInv] = run(
-  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↑"),str("𝕨⥊𝕩: 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"),str("+-÷⋆√¬")]
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an array"),str("⊢⊣˜∘○⊸⟜⊘◶"),str("=≠≢"),str("<"),str("≍"),str("↕/»«⊔"),str("⥊↑↓⌽⍉⊏⊑"),str("↑"),str("↓"),str("𝕩: 𝕨 must "),str("have rank at most 1"),str("consist of integers"),str("« or »: 𝕩 must have rank at least 1"),str("« or »: 𝕨 must not have higher rank than 𝕩"),str("« or »: Rank of 𝕨 must be at least rank of 𝕩 minus 1"),str("« or »: 𝕨 must share 𝕩\'s major cell shape"),str("↕𝕩: 𝕩 must consist of natural numbers"),str("↕𝕩: 𝕩 must be a number or list"),str("𝕨↕𝕩: 𝕨 must have rank at most 1"),str("𝕨↕𝕩: Length of 𝕨 must be at most rank of 𝕩"),str("𝕨↕𝕩: 𝕨 must consist of natural numbers"),str("𝕨↕𝕩: Window length 𝕨 must be at most axis length plus one"),str("˘: Argument lengths don\'t agree"),str("˝: 𝕩 must have rank at least 1"),str("˝: Identity does not exist"),str("∘⌊⌽↑"),str("𝕨⥊𝕩: 𝕨 must have rank at most 1"),str("𝕨⥊𝕩: 𝕨 must consist of natural numbers"),str("𝕨⥊𝕩: Can\'t compute axis length when rest of shape is empty"),str("𝕨⥊𝕩: 𝕨 must consist of natural numbers or ∘ ⌊ ⌽ ↑"),str("𝕨⥊𝕩: 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],[1,1,6],[2,1,7],[1,1,8],[0,0,9],[2,1,10],[0,0,11],[1,0,12],[1,1,13],[1,1,14],[0,0,15],[0,0,16],[1,0,17],[0,0,[[],[18]]],[1,1,19],[1,1,20],[1,0,21],[0,0,[[],[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]]],[1,1,35],[0,0,36],[0,0,[[],[37]]],[0,0,[[],[38]]],[0,0,39],[0,0,40],[0,0,[[],[41]]],[0,0,[[],[42]]],[0,1,43],[0,0,44],[0,0,[[],[45]]],[0,0,46],[0,0,47],[0,0,[[],[48]]],[2,1,49],[1,0,50],[1,0,51],[0,0,52],[0,0,[[],[53]]],[2,0,54],[0,0,55],[0,0,[[],[56]]],[0,0,57],[2,1,58],[0,0,59],[0,0,60],[0,0,61],[0,0,62],[0,0,[[],[63]]],[0,0,[[],[64]]],[0,0,65],[0,0,66],[0,0,67],[0,1,68],[1,1,69],[0,0,70],[0,0,71],[0,0,72],[0,0,73],[0,0,[[],[74]]],[0,0,75],[1,0,76],[1,0,77],[0,0,[[],[78]]],[1,0,79],[0,0,80],[0,0,[[],[81]]],[0,0,[[],[82]]],[0,0,[[],[83]]],[1,1,84],[2,0,85],[2,0,86],[2,0,87],[0,0,[[],[88]]],[0,0,[[],[89]]],[0,0,[[],[90]]],[0,0,[[],[91]]],[0,0,[[],[92]]],[0,0,[[],[93]]],[0,0,[[],[94]]],[1,1,95],[0,0,96],[0,0,97],[0,0,[[],[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],[2,0,107],[0,0,108],[0,0,[[],[109]]],[0,0,[[],[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],[0,0,122],[0,0,[[],[123]]],[0,0,[[],[124]]],[0,0,[[],[125]]],[0,0,[[],[126]]],[0,0,[[],[127]]],[0,0,128],[0,0,[[],[129]]],[1,0,130],[2,0,131],[0,0,132],[0,0,133],[0,0,134],[0,0,135],[0,0,136],[1,0,137],[1,0,[[],[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]]],[0,0,[[],[147]]],[0,0,148],[2,0,149],[0,0,[[],[150]]],[0,0,151],[0,0,[[],[152]]],[0,0,[[],[153]]],[0,0,154],[1,1,155],[0,0,156],[0,0,[[],[157]]],[0,0,158],[0,0,[[],[159]]],[0,0,160],[0,0,[[],[161]]],[0,0,162],[0,0,163],[0,0,[[],[164]]],[2,0,165],[0,0,[[],[166]]],[0,0,[[],[167]]],[0,0,168],[0,0,169],[1,0,[[],[170]]],[0,0,[[],[171]]],[1,0,172],[0,0,173],[1,0,174],[2,0,[[],[175]]],[0,0,176],[0,0,177],[0,0,[[],[178]]],[0,0,179],[0,0,180],[0,0,[[],[181]]],[0,0,182],[1,0,[[],[183]]],[1,1,184],[0,0,[[],[185]]],[0,0,[[],[186]]],[0,0,[[],[187]]],[0,0,[[],[188]]],[0,0,[[],[189]]],[0,0,[[],[190]]],[0,0,[[],[191]]],[0,0,192],[0,0,193],[1,1,194],[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],[0,0,204],[0,0,205],[0,0,206],[0,0,[[],[207]]],[0,0,208],[2,1,209],[2,1,210],[1,1,211],[1,1,212],[1,1,213],[0,0,[[],[214]]],[0,0,215],[0,0,216],[0,0,217],[0,0,218],[0,0,[[],[219]]],[0,0,[[],[220]]],[1,1,221],[0,0,222],[0,0,[[],[223]]],[0,0,224],[1,0,225],[0,0,226],[0,0,[[],[227]]],[0,0,228],[0,0,229],[0,0,[[],[230]]],[0,0,231],[0,0,232],[0,0,233],[0,0,234],[0,0,[[],[235]]],[0,0,236],[2,0,237],[0,0,[[],[238]]],[0,0,239],[0,0,240],[0,1,241],[1,0,242],[0,0,243],[0,0,244],[1,0,245],[0,0,[[],[246]]],[0,0,247],[0,0,248],[1,0,249],[0,0,[[],[250]]],[0,0,[[],[251]]],[0,0,252],[0,0,253],[0,0,[[],[254]]],[0,0,255],[0,0,[[],[256]]],[0,0,257],[0,0,258],[1,0,259],[0,0,260],[0,0,261],[0,0,[[],[262]]],[0,0,263],[1,0,264],[0,0,265],[0,0,[[],[266]]],[2,0,267],[1,1,268],[1,1,269],[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],[0,0,[[],[281]]],[0,0,282],[0,0,283],[0,0,[[],[284]]],[0,0,285],[1,1,286],[2,1,287],[0,0,288],[0,0,[[],[289]]],[0,0,290],[0,0,[[],[291]]],[0,0,292],[1,0,293],[0,0,[[],[294]]],[0,0,295],[0,0,296],[0,0,297],[0,0,298],[0,0,[[],[299]]],[0,0,[[],[300]]],[0,0,301],[0,0,302],[0,0,303],[0,0,304]]
+ ,[[0,183],[4014,3],[4022,3],[4026,3],[4039,3],[4054,3],[4066,2],[4095,3],[4127,2],[4144,4],[4206,5],[4242,3],[4249,5],[4307,4],[4373,3],[4428,3],[4439,4],[4504,9],[4573,5],[4650,5],[4717,4],[4773,8],[4945,8],[5102,3],[5147,3],[5196,3],[5242,6],[5327,3],[5351,7],[5465,3],[5527,4],[5598,6],[5662,3],[5702,5],[5779,3],[5796,14],[5976,3],[6022,3],[6042,3],[6059,4],[6117,4],[6193,5],[6267,3],[6326,0],[6337,4],[6404,3],[6414,3],[6421,3],[6435,3],[6443,3],[6481,5],[6518,5],[6545,3],[6549,3],[6563,16],[6858,3],[6883,3],[6908,3],[6935,3],[6994,4],[7094,7],[7290,7],[7364,15],[7597,5],[7676,3],[7684,6],[7711,5],[7732,6],[7759,11],[8111,9],[8268,4],[8344,7],[8456,7],[8573,3],[8680,4],[8742,8],[8854,8],[8928,7],[8990,9],[9148,9],[9296,4],[9407,8],[9621,3],[9665,3],[9802,2],[9823,14],[9948,9],[10129,10],[10202,3],[10228,10],[10636,4],[10740,3],[10814,3],[10865,4],[10992,4],[11108,2],[11116,3],[11124,3],[11132,3],[11152,8],[11227,3],[11235,3],[11243,3],[11251,3],[11259,3],[11267,3],[11278,3],[11299,7],[11340,6],[11559,4],[11659,4],[11719,3],[11762,5],[11900,3],[11947,3],[11973,3],[12022,3],[12051,3],[12059,3],[12092,3],[12136,3],[12184,3],[12196,3],[12269,4],[12305,3],[12320,3],[12335,3],[12353,3],[12367,3],[12377,3],[12429,6],[12452,6],[12480,3],[12492,3],[12523,3],[12531,3],[12539,3],[12547,5],[12582,12],[12755,3],[12766,3],[12786,3],[12806,3],[12817,3],[12828,3],[12843,7],[12914,6],[12960,3],[12978,3],[12985,6],[13020,10],[13197,4],[13253,11],[13454,4],[13521,3],[13546,2],[13561,3],[13588,7],[13729,3],[13758,3],[13768,3],[13787,3],[13797,3],[13806,3],[13816,3],[13833,11],[13906,6],[14006,8],[14084,3],[14092,3],[14134,7],[14160,16],[14674,5],[14690,7],[14826,5],[14859,6],[14888,3],[14898,3],[14908,4],[14947,4],[14970,3],[14978,3],[15003,3],[15015,6],[15111,2],[15161,8],[15342,11],[15547,3],[15606,3],[15633,3],[15650,3],[15667,8],[15849,3],[15863,3],[15894,2],[15928,3],[15963,3],[15984,3],[16005,5],[16076,3],[16096,8],[16326,3],[16394,3],[16424,3],[16455,3],[16489,5],[16556,3],[16565,3],[16578,3],[16582,3],[16590,3],[16600,2],[16614,2],[16629,6],[16723,3],[16733,3],[16740,9],[16874,3],[16900,3],[16924,3],[16945,3],[16958,2],[16962,3],[16998,5],[17100,3],[17147,5],[17170,3],[17187,14],[17471,13],[17668,3],[17711,4],[17794,3],[17830,3],[17859,3],[17870,4],[17906,7],[18041,10],[18231,6],[18266,3],[18331,3],[18363,4],[18386,1],[18412,5],[18423,4],[18434,4],[18445,5],[18465,4],[18540,3],[18573,3],[18603,5],[18628,3],[18694,3],[18708,3],[18719,5],[18818,4],[18947,3],[18971,3],[19017,3],[19050,3],[19079,5],[19111,3],[19147,3],[19173,3],[19201,3],[19223,5],[19233,3],[19266,3],[19286,6],[19301,2],[19309,2],[19317,5],[19355,3],[19365,3],[19381,5],[19432,3],[19457,6],[19566,3],[19606,3],[19647,5],[19697,6],[19834,3],[19842,8],[19899,5],[19928,3],[19943,3],[19993,3],[20009,2],[20027,3],[20048,3],[20066,3],[20095,3],[20114,3],[20133,6],[20212,5],[20227,14],[20391,3],[20412,3],[20428,4],[20484,3],[20516,3],[20599,3],[20633,3],[20662,3],[20679,3],[20696,3]]
 );
 
 // Use high-precision modulus (⚇0)
diff --git a/docs/spec/inferred.html b/docs/spec/inferred.html
index a221ede0..fd247c65 100644
--- a/docs/spec/inferred.html
+++ b/docs/spec/inferred.html
@@ -122,6 +122,7 @@
 <p>Fill elements of iteration modifiers such as <code><span class='Modifier'>¨⌜</span></code> are not specified. It is reasonable to define the fill element of <code><span class='Function'>𝔽</span><span class='Modifier'>⌜</span></code> or <code><span class='Function'>𝔽</span><span class='Modifier'>¨</span></code> to be <code><span class='Function'>𝔽</span></code> applied to the fill elements of the arguments. Regardless of definition, computing the fill element cannot cause side effects or an error.</p>
 <h2 id="undo"><a class="header" href="#undo">Undo</a></h2>
 <p>The Undo 1-modifier <code><span class='Modifier'>⁼</span></code>, given an operand <code><span class='Function'>𝔽</span></code> and argument <code><span class='Value'>𝕩</span></code>, and possibly a left argument <code><span class='Value'>𝕨</span></code>, finds a value <code><span class='Value'>y</span></code> such that <code><span class='Value'>𝕩</span><span class='Function'>≡</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Value'>y</span></code>, that is, an element of the pre-image of <code><span class='Value'>𝕩</span></code> under <code><span class='Function'>𝔽</span></code> or <code><span class='Value'>𝕨</span><span class='Function'>𝔽⊢</span></code>. Thus it satisfies the constraint <code><span class='Value'>𝕩</span> <span class='Function'>≡</span> <span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code> (<code><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Function'>⊢</span></code> is a <em>right inverse</em> of <code><span class='Value'>𝕨</span><span class='Function'>𝔽⊢</span></code>) provided <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> and <code><span class='Function'>𝔽</span></code> both complete without error. <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> should of course give an error if no inverse element exists, and can also fail if no inverse can be found. It is also preferred for <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span></code> to give an error if there are many choices of inverse with no clear way to choose one of them: for example, <code><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Function'>⍉</span><span class='Value'>m</span></code> returns the diagonal of matrix <code><span class='Value'>m</span></code>; <code><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Function'>⍉</span><span class='Modifier'>⁼</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span></code> requires values to be chosen for the off-diagonal elements in its result. It is better to give an error, encouraging the programmer to use a fully-specified approach like <code><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Modifier2'>⌾</span><span class='Paren'>(</span><span class='Number'>0</span><span class='Ligature'>‿</span><span class='Number'>0</span><span class='Modifier2'>⊸</span><span class='Function'>⍉</span><span class='Paren'>)</span></code> applied to a matrix of initial elements, than to return a result that could be very different from other implementations.</p>
+<p>If a value <code><span class='Value'>𝕩</span></code> isn't in the range of <code><span class='Function'>𝔽</span></code>, then no result satisfies the rule for <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code>. As <code><span class='Function'>𝔽</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span></code> would then always give an error by the above rules, a BQN implementation with extensions could define it in any way. However, two such extensions are specified: <code><span class='Function'>F</span><span class='Modifier'>⁼⁼</span></code> and <code><span class='Function'>F</span><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Function'>-</span><span class='Value'>n</span><span class='Paren'>)</span><span class='Modifier'>⁼</span></code> are extended to the full domain of <code><span class='Function'>F</span></code> and <code><span class='Function'>/</span><span class='Modifier'>⁼</span></code> to any list of natural numbers, as noted below.</p>
 <p>When working with limited-precision numbers, it may be difficult or impossible to exactly invert the operand function. Instead, it is generally acceptable to perform a computation that, if done with unlimited precision, would exactly invert <code><span class='Function'>𝔽</span></code> computed with unlimited precision. This principle is the basis for the numeric inverses specified below. It is also acceptable to find an inverse by numeric methods, provided that the error in the inverse value found relative to an unlimited-precision inverse can be kept close to the inherent error in the implementation's number format.</p>
 <p>Regardless of which cases for Undo are supported, the result of a call, and whether it is an error, must depend only on the values of the inputs <code><span class='Function'>𝔽</span></code>, <code><span class='Value'>𝕩</span></code>, and (if present) <code><span class='Value'>𝕨</span></code>.</p>
 <h3 id="required-functions"><a class="header" href="#required-functions">Required functions</a></h3>
@@ -190,7 +191,7 @@
 </tr>
 </tbody>
 </table>
-<p>The following structural functions have unique inverses, except in a few cases. Dyadic <code><span class='Function'>⍉</span></code> with repeated axes is excluded, and monadic <code><span class='Function'>&lt;</span></code> can only be inverted on a rank-0 array. Dyadic <code><span class='Function'>⊣</span></code> is invertible only if the arguments match, and in this case any return value is valid, but in BQN the shared argument value is returned. For <code><span class='Function'>/</span><span class='Modifier'>⁼</span></code> the argument must be a list of non-descending natural numbers, and the result's fill element is 0.</p>
+<p>The following structural functions have unique inverses, except in a few cases. Dyadic <code><span class='Function'>⍉</span></code> with repeated axes is excluded, and monadic <code><span class='Function'>&lt;</span></code> can only be inverted on a rank-0 array. Dyadic <code><span class='Function'>⊣</span></code> is invertible only if the arguments match, and in this case any return value is valid, but in BQN the shared argument value is returned. For <code><span class='Function'>/</span><span class='Modifier'>⁼</span></code> the argument must be a list of natural numbers (it's extended to handle any such list even though the range consists of non-descending ones), and the result's fill element is 0.</p>
 <table>
 <thead>
 <tr>
@@ -410,8 +411,8 @@
 </tr>
 <tr>
 <td><code><span class='Modifier'>⁼</span></code></td>
-<td><code><span class='Brace'>{</span><span class='Value'>r</span><span class='Gets'>←</span><span class='Function'>𝔽</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Function'>!</span><span class='Value'>𝕩</span><span class='Function'>≡𝔽</span><span class='Modifier'>⁼</span><span class='Value'>r</span><span class='Separator'>⋄</span><span class='Value'>r</span><span class='Brace'>}</span></code></td>
-<td></td>
+<td><code><span class='Brace'>{</span><span class='Function'>𝔽</span><span class='Brace'>}</span></code></td>
+<td>Extended: no result check</td>
 </tr>
 <tr>
 <td><code><span class='Modifier2'>⌾</span></code></td>
@@ -421,7 +422,7 @@
 <tr>
 <td><code><span class='Modifier2'>⍟</span><span class='Value'>n</span></code></td>
 <td><code><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Function'>-</span><span class='Value'>n</span><span class='Paren'>)</span></code></td>
-<td>Atomic number <code><span class='Value'>n</span></code></td>
+<td>Atomic number <code><span class='Value'>n</span></code>; no check, like <code><span class='Modifier'>⁼</span></code></td>
 </tr>
 <tr>
 <td><code><span class='Modifier2'>⊘</span></code></td>
diff --git a/spec/inferred.md b/spec/inferred.md
index 5ed454b7..707d70ca 100644
--- a/spec/inferred.md
+++ b/spec/inferred.md
@@ -62,6 +62,8 @@ Fill elements of iteration modifiers such as `¨⌜` are not specified. It is re
 
 The Undo 1-modifier `⁼`, given an operand `𝔽` and argument `𝕩`, and possibly a left argument `𝕨`, finds a value `y` such that `𝕩≡𝕨𝔽y`, that is, an element of the pre-image of `𝕩` under `𝔽` or `𝕨𝔽⊢`. Thus it satisfies the constraint `𝕩 ≡ 𝕨𝔽𝕨𝔽⁼𝕩` (`𝕨𝔽⁼⊢` is a *right inverse* of `𝕨𝔽⊢`) provided `𝔽⁼` and `𝔽` both complete without error. `𝔽⁼` should of course give an error if no inverse element exists, and can also fail if no inverse can be found. It is also preferred for `𝔽⁼` to give an error if there are many choices of inverse with no clear way to choose one of them: for example, `0‿0⍉m` returns the diagonal of matrix `m`; `0‿0⍉⁼2‿3` requires values to be chosen for the off-diagonal elements in its result. It is better to give an error, encouraging the programmer to use a fully-specified approach like `2‿3⌾(0‿0⊸⍉)` applied to a matrix of initial elements, than to return a result that could be very different from other implementations.
 
+If a value `𝕩` isn't in the range of `𝔽`, then no result satisfies the rule for `𝔽⁼𝕩`. As `𝔽⁼𝕩` would then always give an error by the above rules, a BQN implementation with extensions could define it in any way. However, two such extensions are specified: `F⁼⁼` and `F⍟(-n)⁼` are extended to the full domain of `F` and `/⁼` to any list of natural numbers, as noted below.
+
 When working with limited-precision numbers, it may be difficult or impossible to exactly invert the operand function. Instead, it is generally acceptable to perform a computation that, if done with unlimited precision, would exactly invert `𝔽` computed with unlimited precision. This principle is the basis for the numeric inverses specified below. It is also acceptable to find an inverse by numeric methods, provided that the error in the inverse value found relative to an unlimited-precision inverse can be kept close to the inherent error in the implementation's number format.
 
 Regardless of which cases for Undo are supported, the result of a call, and whether it is an error, must depend only on the values of the inputs `𝔽`, `𝕩`, and (if present) `𝕨`.
@@ -88,7 +90,7 @@ Unlike these inverses, the logarithm function—base *e* for `⋆⁼𝕩` and ba
 |-----|-------|-------
 | `⋆` | `Log` | `÷˜○Log`
 
-The following structural functions have unique inverses, except in a few cases. Dyadic `⍉` with repeated axes is excluded, and monadic `<` can only be inverted on a rank-0 array. Dyadic `⊣` is invertible only if the arguments match, and in this case any return value is valid, but in BQN the shared argument value is returned. For `/⁼` the argument must be a list of non-descending natural numbers, and the result's fill element is 0.
+The following structural functions have unique inverses, except in a few cases. Dyadic `⍉` with repeated axes is excluded, and monadic `<` can only be inverted on a rank-0 array. Dyadic `⊣` is invertible only if the arguments match, and in this case any return value is valid, but in BQN the shared argument value is returned. For `/⁼` the argument must be a list of natural numbers (it's extended to handle any such list even though the range consists of non-descending ones), and the result's fill element is 0.
 
 | Fn  | 1                   | 2
 |-----|---------------------|-------
@@ -148,9 +150,9 @@ Inverses of other modifiers and derived functions or modifiers obtained from the
 | `F G`   |                      |
 | `·F G`  |                      |
 | `○`     | `{𝔾⁼(𝔾𝕨)𝔽⁼𝕩}`        |
-| `⁼`     | `{r←𝔽𝕩⋄!𝕩≡𝔽⁼r⋄r}`    |
+| `⁼`     | `{𝔽}`                | Extended: no result check
 | `⌾`     | `{𝔽⁼⌾𝔾}`             | Verify result for computational Under
-| `⍟n`    | `⍟(-n)`              | Atomic number `n`
+| `⍟n`    | `⍟(-n)`              | Atomic number `n`; no check, like `⁼`
 | `⊘`     | `{(𝔽⁼)⊘(𝔾⁼)}`        |
 | `⊸`     | `{𝔽⊸(𝔾⁼)}`           | Dyadic case or constant `𝔽` only
 | `⟜`     | `{𝔾⁼𝔽⁼}`             | Dyadic case
diff --git a/src/r1.bqn b/src/r1.bqn
index a43a5621..9d302118 100644
--- a/src/r1.bqn
+++ b/src/r1.bqn
@@ -212,7 +212,6 @@ Indices←{
 IndicesInv←{
   IA 1==𝕩
   IA 1×´Nat⌜𝕩
-  IX (0⊸≤)◶⟨1,1×´(⊢≤○(⊑⟜𝕩)1⊸+)⌜∘↕⟩1-˜≠𝕩
   GroupLen 𝕩
 }
 SelfClas ← (PermInv∘⍋∘Rep˜⊏˜¯1+`⊢) _self
@@ -861,9 +860,8 @@ SwapInverse ← INF _lookup_ ⟨
 ⟩
 ⌜ ↩ _table
 ¨ ↩ _each
-CheckInv ← (4=0⊸⊑)◶⟨"+-÷⋆√¬"_glyphLookup_(6=↕7)1⊸⊑,¬·'⁼'_isGlyph 2⊸⊑⟩Decompose
 Mod1Inverse ← INF˙ _lookup_ ⟨
-  '⁼', {𝕏_invChk_(𝕏{i←Inverse𝕗⋄𝕨I𝕩})}⍟CheckInv
+  '⁼', ⊢
   '˜', {SwapInverse𝕩}
   '¨', {𝕏⁼¨                     ⊣·IX 0<≡∘⊢}
   '⌜', {𝕏⁼⌜⊘(IA∘0)              ⊣·IX 0<≡∘⊢}
@@ -876,7 +874,7 @@ Mod2Inverse ← INF˙ _lookup_ ⟨
   '∘', AtopInverse
   '○', {Fi←𝕎⁼⋄𝕏⁼ Fi⊘(𝕏⊸Fi)}
   '⌾', {𝕎⁼⌾𝕏}  # Need to verify for computation Under
-  '⍟', Int∘⊢◶⟨IA∘0˙,0⊸≤◶{𝕎⍟(-𝕩)_invChk_(𝕎⍟𝕩)}‿{𝕎⍟(-𝕩)}⟩
+  '⍟', Int∘⊢◶⟨IA∘0˙,{𝕎⍟(-𝕩)}⟩
   '⊘', {(𝕎⁼)⊘(𝕏⁼)}
   '⊸', IsConstant∘⊣ ⊣◶{INF⊘𝕏}‿⊢ {𝕎⊸(𝕏⁼)}
   '⟜', {(𝕨IsConstant∘⊢◶⟨IA∘0˙,{𝕩𝕎{SwapInverse𝕗}⊢}⟩𝕩)⊘(𝕏⁼𝕎⁼)}
diff --git a/test/cases/undo.bqn b/test/cases/undo.bqn
index 73885c96..65a64d13 100644
--- a/test/cases/undo.bqn
+++ b/test/cases/undo.bqn
@@ -70,5 +70,5 @@
 2 % ט⍟3⁼256
 √⊘-{𝕗⁼≡6𝕗⁼⊢} ¯3‿2
 ∧´ {(𝕎≡𝕎⁼⁼)𝕩}⟜2‿4‿1¨ /‿⌽‿<
-! % /⁼⁼2‿3‿0
-! % /⍟¯1⁼2‿3‿0
+(/2‿3) ≡ /⁼⁼2‿3‿0
+(/2‿3) ≡ /⍟¯1⁼2‿3‿0