193 lines
22 KiB
Markdown
193 lines
22 KiB
Markdown
---
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layout: default
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title: "Sequences"
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parent: Composite
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ancestor: Funcons-beta
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---
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[Funcons-beta] : [Sequences.cbs]
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-----------------------------
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### Sequences of values
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<div class="highlighter-rouge"><pre class="highlight"><code>[
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_length">length</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_index">index</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_is-in">is-in</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_first">first</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_second">second</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_third">third</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_first-n">first-n</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_drop-first-n">drop-first-n</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_reverse">reverse</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_n-of">n-of</a></span>
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<i class="keyword">Funcon</i> <span class="name"><a href="#Name_intersperse">intersperse</a></span>
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]</code></pre></div>
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Sequences of two or more values are not themselves values, nor is the empty
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sequence a value. However, sequences can be provided to funcons as arguments,
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and returned as results. Many operations on composite values can be expressed
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by extracting their components as sequences, operating on the sequences, then
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forming the required composite values from the resulting sequences.
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A sequence with elements <code><i class="var">X<sub class="sub">1</sub></i></code>, ..., <code><i class="var">Xn</i></code> is written <code><i class="var">X<sub class="sub">1</sub></i>,...,<i class="var">Xn</i></code>.
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A sequence with a single element <code><i class="var">X</i></code> is identified with (and written) <code><i class="var">X</i></code>.
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An empty sequence is indicated by the absence of a term.
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Any sequence <code><i class="var">X<sup class="sup">*</sup></i></code> can be enclosed in parentheses <code>(<i class="var">X<sup class="sup">*</sup></i>)</code>, e.g.:
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<code>( )</code>, <code>(1)</code>, <code>(1,2,3)</code>. Superfluous commas are ignored.
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The elements of a type sequence <code><i class="var">T<sub class="sub">1</sub></i>,...,<i class="var">Tn</i></code> are the value sequences <code><i class="var">V<sub class="sub">1</sub></i>,...,<i class="var">Vn</i></code>
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where <code><i class="var">V<sub class="sub">1</sub></i>:<i class="var">T<sub class="sub">1</sub></i></code>, ..., <code><i class="var">Vn</i>:<i class="var">Tn</i></code>. The only element of the empty type sequence <code>( )</code>
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is the empty value sequence <code>( )</code>.
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<code>(<i class="var">T</i>)^<i class="var">N</i></code> is equivalent to <code><i class="var">T</i>,...,<i class="var">T</i></code> with <code><i class="var">N</i></code> occurrences of <code><i class="var">T</i></code>.
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<code>(<i class="var">T</i>)<sup class="sup">*</sup></code> is equivalent to the union of all <code>(<i class="var">T</i>)^<i class="var">N</i></code> for <code><i class="var">N</i></code>>=0,
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<code>(<i class="var">T</i>)<sup class="sup">+</sup></code> is equivalent to the union of all <code>(<i class="var">T</i>)^<i class="var">N</i></code> for <code><i class="var">N</i></code>>=1, and
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<code>(<i class="var">T</i>)<sup class="sup">?</sup></code> is equivalent to <code><i class="var">T</i> | ( )</code>.
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The parentheses around <code><i class="var">T</i></code> above can be omitted when they are not needed for
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disambiguation.
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(Non-trivial) sequence types are not values, so not included in <code><span class="name"><a href="../../Value-Types/index.html#Name_types">types</a></span></code>.
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Meta-variables</i>
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<span id="PartVariable_T"><i class="var">T</i></span>, <span id="PartVariable_T'"><i class="var">T′</i></span> <: <span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span></code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_length">length</span></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-numbers">natural-numbers</a></span></code></pre></div>
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<code><span class="name"><a href="#Name_length">length</a></span>(<i class="var">V<sup class="sup">*</sup></i>)</code> gives the number of elements in <code><i class="var">V<sup class="sup">*</sup></i></code>.
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Rule</i>
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<span class="name"><a href="#Name_length">length</a></span>( ) ~> 0
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_length">length</a></span>(<span id="Variable675_V"><i class="var">V</i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable684_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-successor">natural-successor</a></span>(<span class="name"><a href="#Name_length">length</a></span>(<a href="#Variable684_V*"><i class="var">V<sup class="sup">*</sup></i></a>))</code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_is-in">is-in</span></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span class="name"><a href="../../Primitive/Booleans/index.html#Name_booleans">booleans</a></span>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_is-in">is-in</a></span>(<span id="Variable756_V"><i class="var">V</i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span> ,<span id="Variable765_V'"><i class="var">V′</i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable774_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <span class="name"><a href="../../Primitive/Booleans/index.html#Name_or">or</a></span>(<span class="name"><a href="../../Value-Types/index.html#Name_is-equal">is-equal</a></span>(<a href="#Variable756_V"><i class="var">V</i></a>, <a href="#Variable765_V'"><i class="var">V′</i></a>), <span class="name"><a href="#Name_is-in">is-in</a></span>(<a href="#Variable756_V"><i class="var">V</i></a>, <a href="#Variable774_V*"><i class="var">V<sup class="sup">*</sup></i></a>))
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_is-in">is-in</a></span>(<span id="Variable842_V"><i class="var">V</i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, ( )) ~> <span class="name"><a href="../../Primitive/Booleans/index.html#Name_false">false</a></span></code></pre></div>
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#### Sequence indexing
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_index">index</span></span>(_:<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-numbers">natural-numbers</a></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">?</sup></code></pre></div>
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<code><span class="name"><a href="#Name_index">index</a></span>(<i class="var">N</i>, <i class="var">V<sup class="sup">*</sup></i>)</code> gives the <code><i class="var">N</i></code>th element of <code><i class="var">V<sup class="sup">*</sup></i></code>, if it exists, otherwise <code>( )</code>.
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Rule</i>
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<span class="name"><a href="#Name_index">index</a></span>(1, <span id="Variable988_V"><i class="var">V</i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable997_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <a href="#Variable988_V"><i class="var">V</i></a>
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<i class="keyword">Rule</i>
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<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-predecessor">natural-predecessor</a></span>(<a href="#Variable1047_N"><i class="var">N</i></a>) ~> <span id="Variable1036_N'"><i class="var">N′</i></span>
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-----------------------------------------------------------------
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<span class="name"><a href="#Name_index">index</a></span>(<span id="Variable1047_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable1063_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <span class="name"><a href="#Name_index">index</a></span>(<a href="#Variable1036_N'"><i class="var">N′</i></a>, <a href="#Variable1063_V*"><i class="var">V<sup class="sup">*</sup></i></a>)
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_index">index</a></span>(0, <span id="Variable1108_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> ( )
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_index">index</a></span>(_:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, ( )) ~> ( )</code></pre></div>
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Total indexing funcons:
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_first">first</span></span>(_:<span id="Variable1168_T"><i class="var">T</i></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span id="Variable1193_T"><i class="var">T</i></span>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_first">first</a></span>(<span id="Variable1202_V"><i class="var">V</i></span>:<i class="var">T</i>, <span id="Variable1214_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <a href="#Variable1202_V"><i class="var">V</i></a></code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_second">second</span></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, _:<span id="Variable1251_T"><i class="var">T</i></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span id="Variable1277_T"><i class="var">T</i></span>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_second">second</a></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable1293_V"><i class="var">V</i></span>:<i class="var">T</i>, <span id="Variable1305_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <a href="#Variable1293_V"><i class="var">V</i></a></code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_third">third</span></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, _:<span id="Variable1351_T"><i class="var">T</i></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) : =><span id="Variable1378_T"><i class="var">T</i></span>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_third">third</a></span>(_:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, _:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span>, <span id="Variable1401_V"><i class="var">V</i></span>:<i class="var">T</i>, <span id="Variable1413_V*"><i class="var">V<sup class="sup">*</sup></i></span>:<span class="name"><a href="../../Value-Types/index.html#Name_values">values</a></span><sup class="sup">*</sup>) ~> <a href="#Variable1401_V"><i class="var">V</i></a></code></pre></div>
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#### Homogeneous sequences
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_first-n">first-n</span></span>(_:<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-numbers">natural-numbers</a></span>, _:(<span id="Variable1473_T"><i class="var">T</i></span>)<sup class="sup">*</sup>) : =>(<span id="Variable1495_T"><i class="var">T</i></span>)<sup class="sup">*</sup>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_first-n">first-n</a></span>(0, <span id="Variable1514_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> ( )
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<i class="keyword">Rule</i>
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<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-predecessor">natural-predecessor</a></span>(<a href="#Variable1570_N"><i class="var">N</i></a>) ~> <span id="Variable1559_N'"><i class="var">N′</i></span>
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-----------------------------------------------------------------
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<span class="name"><a href="#Name_first-n">first-n</a></span>(<span id="Variable1570_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, <span id="Variable1578_V"><i class="var">V</i></span>:<i class="var">T</i>, <span id="Variable1590_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> (<a href="#Variable1578_V"><i class="var">V</i></a>,<span class="name"><a href="#Name_first-n">first-n</a></span>(<a href="#Variable1559_N'"><i class="var">N′</i></a>, <a href="#Variable1590_V*"><i class="var">V<sup class="sup">*</sup></i></a>))
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_first-n">first-n</a></span>(<span id="Variable1651_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, ( )) ~> ( )</code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_drop-first-n">drop-first-n</span></span>(_:<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-numbers">natural-numbers</a></span>, _:(<span id="Variable1689_T"><i class="var">T</i></span>)<sup class="sup">*</sup>) : =>(<span id="Variable1711_T"><i class="var">T</i></span>)<sup class="sup">*</sup>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_drop-first-n">drop-first-n</a></span>(0, <span id="Variable1730_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> <a href="#Variable1730_V*"><i class="var">V<sup class="sup">*</sup></i></a>
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<i class="keyword">Rule</i>
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<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-predecessor">natural-predecessor</a></span>(<a href="#Variable1788_N"><i class="var">N</i></a>) ~> <span id="Variable1777_N'"><i class="var">N′</i></span>
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-----------------------------------------------------------------------
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<span class="name"><a href="#Name_drop-first-n">drop-first-n</a></span>(<span id="Variable1788_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, _:<i class="var">T</i>, <span id="Variable1807_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> <span class="name"><a href="#Name_drop-first-n">drop-first-n</a></span>(<a href="#Variable1777_N'"><i class="var">N′</i></a>, <a href="#Variable1807_V*"><i class="var">V<sup class="sup">*</sup></i></a>)
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_drop-first-n">drop-first-n</a></span>(<span id="Variable1857_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, ( )) ~> ( )</code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_reverse">reverse</span></span>(_:(<span id="Variable1887_T"><i class="var">T</i></span>)<sup class="sup">*</sup>) : =>(<span id="Variable1908_T"><i class="var">T</i></span>)<sup class="sup">*</sup>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_reverse">reverse</a></span>( ) ~> ( )
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_reverse">reverse</a></span>(<span id="Variable1937_V"><i class="var">V</i></span>:<i class="var">T</i>, <span id="Variable1949_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> (<span class="name"><a href="#Name_reverse">reverse</a></span>(<a href="#Variable1949_V*"><i class="var">V<sup class="sup">*</sup></i></a>), <a href="#Variable1937_V"><i class="var">V</i></a>)</code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_n-of">n-of</span></span>(<span id="Variable2002_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-numbers">natural-numbers</a></span>, <span id="Variable2011_V"><i class="var">V</i></span>:<span id="Variable2015_T"><i class="var">T</i></span>) : =>(<span id="Variable2030_T"><i class="var">T</i></span>)<sup class="sup">*</sup>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_n-of">n-of</a></span>(0, _:<i class="var">T</i>) ~> ( )
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<i class="keyword">Rule</i>
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<span class="name"><a href="../../Primitive/Integers/index.html#Name_natural-predecessor">natural-predecessor</a></span>(<a href="#Variable2096_N"><i class="var">N</i></a>) ~> <span id="Variable2085_N'"><i class="var">N′</i></span>
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--------------------------------------------------
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<span class="name"><a href="#Name_n-of">n-of</a></span>(<span id="Variable2096_N"><i class="var">N</i></span>:<span class="name"><a href="../../Primitive/Integers/index.html#Name_positive-integers">positive-integers</a></span>, <span id="Variable2104_V"><i class="var">V</i></span>:<i class="var">T</i>) ~> (<a href="#Variable2104_V"><i class="var">V</i></a>, <span class="name"><a href="#Name_n-of">n-of</a></span>(<a href="#Variable2085_N'"><i class="var">N′</i></a>, <a href="#Variable2104_V"><i class="var">V</i></a>))</code></pre></div>
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<div class="highlighter-rouge"><pre class="highlight"><code><i class="keyword">Funcon</i>
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<span class="name"><span id="Name_intersperse">intersperse</span></span>(_:<span id="Variable2160_T'"><i class="var">T′</i></span>, _:(<span id="Variable2171_T"><i class="var">T</i></span>)<sup class="sup">*</sup>) : =>(<span id="Variable2193_T"><i class="var">T</i></span>, (<span id="Variable2199_T'"><i class="var">T′</i></span>, <span id="Variable2204_T"><i class="var">T</i></span>)<sup class="sup">*</sup>)<sup class="sup">?</sup>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_intersperse">intersperse</a></span>(_:<i class="var">T′</i>, ( )) ~> ( )
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_intersperse">intersperse</a></span>(_:<i class="var">T′</i>, <span id="Variable2270_V"><i class="var">V</i></span>) ~> <a href="#Variable2270_V"><i class="var">V</i></a>
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<i class="keyword">Rule</i>
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<span class="name"><a href="#Name_intersperse">intersperse</a></span>(<span id="Variable2292_V'"><i class="var">V′</i></span>:<i class="var">T′</i>, <span id="Variable2305_V1"><i class="var">V<sub class="sub">1</sub></i></span>:<i class="var">T</i>, <span id="Variable2317_V2"><i class="var">V<sub class="sub">2</sub></i></span>:<i class="var">T</i>, <span id="Variable2329_V*"><i class="var">V<sup class="sup">*</sup></i></span>:(<i class="var">T</i>)<sup class="sup">*</sup>) ~> (<a href="#Variable2305_V1"><i class="var">V<sub class="sub">1</sub></i></a>, <a href="#Variable2292_V'"><i class="var">V′</i></a>, <span class="name"><a href="#Name_intersperse">intersperse</a></span>(<a href="#Variable2292_V'"><i class="var">V′</i></a>, <a href="#Variable2317_V2"><i class="var">V<sub class="sub">2</sub></i></a>, <a href="#Variable2329_V*"><i class="var">V<sup class="sup">*</sup></i></a>))</code></pre></div>
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____
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From the [PLanCompS Project] | [CBS-beta issues...] | [Suggest an improvement...]
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[Sequences.cbs]: Sequences.cbs
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"CBS SOURCE FILE"
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[Funcons-beta]: /CBS-beta/docs/Funcons-beta
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"FUNCONS-BETA"
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[Unstable-Funcons-beta]: /CBS-beta/docs/Unstable-Funcons-beta
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"UNSTABLE-FUNCONS-BETA"
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[Languages-beta]: /CBS-beta/docs/Languages-beta
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"LANGUAGES-BETA"
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[Unstable-Languages-beta]: /CBS-beta/docs/Unstable-Languages-beta
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"UNSTABLE-LANGUAGES-BETA"
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[CBS-beta]: /CBS-beta "CBS-BETA"
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[PLanCompS Project]: https://plancomps.github.io
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"PROGRAMMING LANGUAGE COMPONENTS AND SPECIFICATIONS PROJECT HOME PAGE"
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[CBS-beta issues...]: https://github.com/plancomps/CBS-beta/issues
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"CBS-BETA ISSUE REPORTS ON GITHUB"
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[Suggest an improvement...]: mailto:plancomps@gmail.com?Subject=CBS-beta%20-%20comment&Body=Re%3A%20CBS-beta%20specification%20at%20Values/Composite/Sequences/Sequences.cbs%0A%0AComment/Query/Issue/Suggestion%3A%0A%0A%0ASignature%3A%0A
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"GENERATE AN EMAIL TEMPLATE"
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