<h4>10.1 <span class="Heading">Gap categories for Z functions</span></h4>
<p>A <span class="Math">\mathbb{Z}</span>-function is an enumerated collection of objects in which repetitions are allowed and order does matter. The reason behind calling it a <span class="Math">\mathbb{Z}</span>-function rather than simply a sequence, is to avoid possible conflicts with other packages that use the terms <em>Sequence</em> and <em>IsSequence</em>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VoidZFunction</code>( <var class="Arg">func</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an integer</p>
<p>The global function has no arguments and the output is an empty <span class="Math">\mathbb{Z}</span>-function. That means, it can not be evaluated yet.</p>
<p>The argument is a function <var class="Arg">func</var> that can be applied on integers. The output is a <span class="Math">\mathbb{Z}</span>-function <code class="code">z_func</code>. We call <varclass="Arg">func</var> the <code class="code">UnderlyingFunction</code> of <code class="code">z_func</code>.</p>
<p>The argument is a <var class="Arg">z_func</var>. The output is its <code class="code">UnderlyingFunction</code> function. I.e., the function that will be applied on index <code class="code">i</code> whenever we call <var class="Arg">z_func</var>[<code class="code">i</code>].</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZFunctionValue</code>( <var class="Arg">z_func</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a Gap object</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> and an integer <var class="Arg">i</var>. The output is <var class="Arg">z_func</var>[<var class="Arg">i</var>].</p>
<p>The arguments are an integer <var class="Arg">n</var>, a Gap object <var class="Arg">val_n</var>, a function <var class="Arg">lower_func</var>, a function <var class="Arg">upper_func</var> and a function <var class="Arg">compare_func</var>. The output is the <span class="Math">\mathbb{Z}</span>-function <code class="code">z_func</code> defined as follows: <code class="code">z_func</code>[<code class="code">i</code>] is equal to <var class="Arg">lower_func</var>(<code class="code">z_func</code>[<code class="code">i+1</code>]) if <code class="code">i</code><code class="code"><</code><var class="Arg">n</var>; and is equal to <var class="Arg">val_n</var> if <code class="code">i</code>=<varclass="Arg">n</var>; and is equal to <var class="Arg">upper_func</var>(<code class="code">z_func</code>[<code class="code">i-1</code>]) otherwise. At each call, the method compares the computed value to the previous or next value via the function <var class="Arg">compare_func</var>; and in the affermative case, the method sets a upper or lower stable values.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StableUpperValue</code>( <var class="Arg">z_func</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a Gap object</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>. We say that <var class="Arg">z_func</var> has a stable upper value <code class="code">val</code>, if there is an index <code class="code">n</code> such that <var class="Arg">z_func</var>[<code class="code">i</code>] is equal to <code class="code">val</code> for all indices <code class="code">i</code>'s greater or equal to n. In that case, the output is the value val.
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> with a stable upper value <code class="code">val</code>. The output is some index where <var class="Arg">z_func</var> starts to take values equal to <code class="code">val</code>.</p>
<p>The arguments are a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>, an integer <var class="Arg">n</var> and an object <var class="Arg">val</var>. The operation sets <var class="Arg">val</var> as a stable upper value for <var class="Arg">z_func</var> at the index <var class="Arg">n</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StableLowerValue</code>( <var class="Arg">z_func</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a Gap object</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>. We say that <var class="Arg">z_func</var> has a stable lower value <code class="code">val</code>, if there is an index <code class="code">n</code> such that <var class="Arg">z_func</var>[<code class="code">i</code>] is equal to <code class="code">val</code> for all indices <code class="code">i</code>'s less or equal to n. In that case, the output is the value val.
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> with a stable lower value <code class="code">val</code>. The output is some index where <var class="Arg">z_func</var> starts to take values equal to <code class="code">val</code>.</p>
<p>The arguments are a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>, an integer <var class="Arg">n</var> and an object <var class="Arg">val</var>. The operation sets <var class="Arg">val</var> as a stable lower value for <var class="Arg">z_func</var> at the index <var class="Arg">n</var>.</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>. The output is another <span class="Math">\mathbb{Z}</span>-function <code class="code">ref_z_func</code> such that <code class="code">ref_z_func</code>[<code class="code">i</code>] is equal to <var class="Arg">z_func</var>[<code class="code">-i</code>] for all <code class="code">i</code>'s in\mathbb{Z}.
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> and an integer <var class="Arg">n</var>. The output is another <span class="Math">\mathbb{Z}</span>-function <code class="code">m</code> such that <code class="code">m</code>[<code class="code">i</code>] is equal to <var class="Arg">z_func</var>[<code class="code">n+i</code>].</p>
<p>The arguments are a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> and a function <var class="Arg">F</var> that can be applied on one argument. The output is another <span class="Math">\mathbb{Z}</span>-function <code class="code">m</code> such that <code class="code">m</code>[<code class="code">i</code>] is equal to <var class="Arg">F</var>(<var class="Arg">z_func</var>[<code class="code">i</code>]).</p>
<p>The arguments are a list of <span class="Math">\mathbb{Z}</span>-functions <var class="Arg">L</var> and a function <var class="Arg">F</var> with <code class="code">Length</code>(<var class="Arg">L</var>) arguments. The output is another <span class="Math">\mathbb{Z}</span>-function <code class="code">m</code> such that <code class="code">m</code>[<code class="code">i</code>] is equal to <var class="Arg">F</var>(<var class="Arg">L</var>[1][<code class="code">i</code>],..., <var class="Arg">L</var>[<code class="code">Length</code>(<var class="Arg">L</var>)][<code class="code">i</code>]). We call the list <var class="Arg">L</var> the <code class="code">BaseZFunctions</code> of <code class="code">m</code> and <var class="Arg">F</var> the <code class="code">AppliedMap</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaseZFunctions</code>( <var class="Arg">z_func</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a list of <span class="Math">\mathbb{Z}</span>-functions</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> that has been defined by applying a map <code class="code">F</code> on a list <code class="code">L</code> of <span class="Math">\mathbb{Z}</span>-functions. The output is the list <code class="code">L</code>.</p>
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var> that has been defined by applying a map <code class="code">F</code> on a list <code class="code">L</code> of <span class="Math">\mathbb{Z}</span>-functions. The output is the function <code class="code">F</code>.</p>
<p>The argument is a dense list <var class="Arg">L</var> of <span class="Math">\mathbb{Z}</span>-functions. The output is another <span class="Math">\mathbb{Z}</span>-function <code class="code">m</code> such that <code class="code">m</code>[<code class="code">i</code>] is equal to [<var class="Arg">L</var>[1][<code class="code">i</code>],..., <var class="Arg">L</var>[<code class="code">Length</code>(<var class="Arg">L</var>)][<code class="code">i</code>]] for all indices <code class="code">i</code>'s in \mathbb{Z}.
<p>The argument is a <span class="Math">\mathbb{Z}</span>-function <var class="Arg">z_func</var>, an integer <var class="Arg">n</var> and a dense list <var class="Arg">L</var>. The output is a new <span class="Math">\mathbb{Z}</span>-function whose values between <var class="Arg">n</var> and <var class="Arg">n</var>+<code class="code">Length</code>(<var class="Arg">L</var>)-1 are the entries of <var class="Arg">L</var>.</p>
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