From f4d45e6721784c23419045b0ec496d8460efc871 Mon Sep 17 00:00:00 2001
From: Marshall Lochbaum <mwlochbaum@gmail.com>
Date: Thu, 11 Feb 2021 15:19:25 -0500
Subject: Dyadic scan inverse

---
 docs/spec/inferred.html | 19 ++++++++++++++-----
 1 file changed, 14 insertions(+), 5 deletions(-)

(limited to 'docs/spec')

diff --git a/docs/spec/inferred.html b/docs/spec/inferred.html
index 79d6e00d..dd7de0f7 100644
--- a/docs/spec/inferred.html
+++ b/docs/spec/inferred.html
@@ -389,11 +389,6 @@
 <td></td>
 </tr>
 <tr>
-<td><code><span class='Modifier'>`</span></code></td>
-<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'>&lt;=</span><span class='Value'>𝕩</span><span class='Separator'>⋄</span><span class='Paren'>(</span><span class='Function'>⊏∾</span><span class='Number'>¯1</span><span class='Modifier2'>⊸</span><span class='Function'>↓𝔽</span><span class='Modifier'>⁼¨</span><span class='Number'>1</span><span class='Modifier2'>⊸</span><span class='Function'>↓</span><span class='Paren'>)</span><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>&lt;≠</span><span class='Paren'>)</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
-<td></td>
-</tr>
-<tr>
 <td><code><span class='Function'>F</span><span class='Modifier2'>∘</span><span class='Function'>G</span></code></td>
 <td><code><span class='Brace'>{</span><span class='Value'>𝕨</span><span class='Function'>G</span><span class='Modifier'>⁼</span><span class='Function'>F</span><span class='Modifier'>⁼</span><span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
 <td></td>
@@ -460,6 +455,20 @@
 </tr>
 </tbody>
 </table>
+<table>
+<thead>
+<tr>
+<th>Mod</th>
+<th>Inverse</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td><code><span class='Modifier'>`</span></code></td>
+<td><code><span class='Brace'>{</span><span class='Function'>!</span><span class='Number'>0</span><span class='Function'>&lt;=</span><span class='Value'>𝕩</span> <span class='Separator'>⋄</span> <span class='Value'>𝕨</span> <span class='Paren'>(</span><span class='Function'>»𝔽</span><span class='Modifier'>⁼¨</span><span class='Function'>⊢</span><span class='Paren'>)</span><span class='Brace'>{</span><span class='Paren'>(</span><span class='Function'>⊏∾⊏𝔽</span><span class='Number'>1</span><span class='Modifier2'>⊸</span><span class='Function'>↓</span><span class='Paren'>)</span><span class='Modifier2'>⍟</span><span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>&lt;≠</span><span class='Paren'>)</span><span class='Modifier2'>⊘</span><span class='Function'>𝔽</span><span class='Brace'>}</span> <span class='Value'>𝕩</span><span class='Brace'>}</span></code></td>
+</tr>
+</tbody>
+</table>
 <h2 id="under">Under</h2>
 <p>The Under 2-modifier <code><span class='Modifier2'>⌾</span></code> conceptually applies its left operand under the action of its right operand. Setting <code><span class='Value'>z</span><span class='Gets'>←</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier2'>⌾</span><span class='Function'>𝔾</span><span class='Value'>𝕩</span></code>, it satisfies <code><span class='Paren'>(</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier2'>○</span><span class='Function'>𝔾</span><span class='Value'>𝕩</span><span class='Paren'>)</span> <span class='Function'>≡</span> <span class='Function'>𝔾</span><span class='Value'>z</span></code>. We might say that <code><span class='Function'>𝔾</span></code> transforms values to a new domain, and <code><span class='Modifier2'>⌾</span><span class='Function'>𝔾</span></code> lifts actions <code><span class='Function'>𝔽</span></code> performed in this domain to the original domain of values. For example, addition in the logarithmic domain corresponds to multiplication in the linear domain: <code><span class='Function'>+</span><span class='Modifier2'>⌾</span><span class='Paren'>(</span><span class='Function'>⋆</span><span class='Modifier'>⁼</span><span class='Paren'>)</span></code> is <code><span class='Function'>×</span></code> (but less precise if computed in floating point).</p>
 <p>Let <code><span class='Value'>v</span><span class='Gets'>←</span><span class='Value'>𝕨</span><span class='Function'>𝔽</span><span class='Modifier2'>○</span><span class='Function'>𝔾</span><span class='Value'>𝕩</span></code>, so that <code><span class='Value'>v</span><span class='Function'>≡𝔾</span><span class='Value'>z</span></code>. <code><span class='Value'>v</span></code> is of course well-defined, so the inference step is to find <code><span class='Value'>z</span></code> based on <code><span class='Value'>v</span></code> and possibly the original inputs. We distinguish three cases for Under:</p>
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