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<h1 class="post-title">Assignment1: TSPL Assignment 1</h1>
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<p style="text-align:center;">
<a alt="Source code" href="https://github.com/plfa/plfa.github.io/blob/dev-20.07/courses/tspl/2018/Assignment1.lagda.md">Source</a>
</p>
<div class="post-content">
<pre class="Agda"><a id="112" class="Keyword">module</a> <a id="119" href="/20.07/TSPL/2018/Assignment1/" class="Module">Assignment1</a> <a id="131" class="Keyword">where</a>
</pre>
<h2 id="your-name-and-email-goes-here">YOUR NAME AND EMAIL GOES HERE</h2>
<h2 id="introduction">Introduction</h2>
<!-- This assignment is due **1pm Friday 26 April**. -->
<p>You must do <em>all</em> the exercises labelled “(recommended)”.</p>
<p>Exercises labelled “(stretch)” are there to provide an extra challenge.
You don’t need to do all of these, but should attempt at least a few.</p>
<p>Exercises without a label are optional, and may be done if you want
some extra practice.</p>
<!-- Submit your homework using the "submit" command. -->
<p>Please ensure your files execute correctly under Agda!</p>
<h2 id="imports">Imports</h2>
<pre class="Agda"><a id="673" class="Keyword">import</a> <a id="680" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.html" class="Module">Relation.Binary.PropositionalEquality</a> <a id="718" class="Symbol">as</a> <a id="721" class="Module">Eq</a>
<a id="724" class="Keyword">open</a> <a id="729" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.html" class="Module">Eq</a> <a id="732" class="Keyword">using</a> <a id="738" class="Symbol">(</a><a id="739" href="Agda.Builtin.Equality.html#125" class="Datatype Operator">_≡_</a><a id="742" class="Symbol">;</a> <a id="744" href="Agda.Builtin.Equality.html#182" class="InductiveConstructor">refl</a><a id="748" class="Symbol">;</a> <a id="750" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#1090" class="Function">cong</a><a id="754" class="Symbol">;</a> <a id="756" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#939" class="Function">sym</a><a id="759" class="Symbol">)</a>
<a id="761" class="Keyword">open</a> <a id="766" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#2499" class="Module">Eq.≡-Reasoning</a> <a id="781" class="Keyword">using</a> <a id="787" class="Symbol">(</a><a id="788" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#2597" class="Function Operator">begin_</a><a id="794" class="Symbol">;</a> <a id="796" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#2655" class="Function Operator">_≡⟨⟩_</a><a id="801" class="Symbol">;</a> <a id="803" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#2714" class="Function Operator">_≡⟨_⟩_</a><a id="809" class="Symbol">;</a> <a id="811" href="https://agda.github.io/agda-stdlib/v1.1/Relation.Binary.PropositionalEquality.Core.html#2892" class="Function Operator">_∎</a><a id="813" class="Symbol">)</a>
<a id="815" class="Keyword">open</a> <a id="820" class="Keyword">import</a> <a id="827" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.html" class="Module">Data.Nat</a> <a id="836" class="Keyword">using</a> <a id="842" class="Symbol">(</a><a id="843" href="Agda.Builtin.Nat.html#165" class="Datatype">ℕ</a><a id="844" class="Symbol">;</a> <a id="846" href="Agda.Builtin.Nat.html#183" class="InductiveConstructor">zero</a><a id="850" class="Symbol">;</a> <a id="852" href="Agda.Builtin.Nat.html#196" class="InductiveConstructor">suc</a><a id="855" class="Symbol">;</a> <a id="857" href="Agda.Builtin.Nat.html#298" class="Primitive Operator">_+_</a><a id="860" class="Symbol">;</a> <a id="862" href="Agda.Builtin.Nat.html#501" class="Primitive Operator">_*_</a><a id="865" class="Symbol">;</a> <a id="867" href="Agda.Builtin.Nat.html#388" class="Primitive Operator">_∸_</a><a id="870" class="Symbol">;</a> <a id="872" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Base.html#895" class="Datatype Operator">_≤_</a><a id="875" class="Symbol">;</a> <a id="877" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Base.html#918" class="InductiveConstructor">z≤n</a><a id="880" class="Symbol">;</a> <a id="882" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Base.html#960" class="InductiveConstructor">s≤s</a><a id="885" class="Symbol">)</a>
<a id="887" class="Keyword">open</a> <a id="892" class="Keyword">import</a> <a id="899" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html" class="Module">Data.Nat.Properties</a> <a id="919" class="Keyword">using</a> <a id="925" class="Symbol">(</a><a id="926" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#11578" class="Function">+-assoc</a><a id="933" class="Symbol">;</a> <a id="935" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#11734" class="Function">+-identityʳ</a><a id="946" class="Symbol">;</a> <a id="948" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#11370" class="Function">+-suc</a><a id="953" class="Symbol">;</a> <a id="955" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#11911" class="Function">+-comm</a><a id="961" class="Symbol">;</a>
<a id="965" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#3632" class="Function">≤-refl</a><a id="971" class="Symbol">;</a> <a id="973" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#3815" class="Function">≤-trans</a><a id="980" class="Symbol">;</a> <a id="982" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#3682" class="Function">≤-antisym</a><a id="991" class="Symbol">;</a> <a id="993" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#3927" class="Function">≤-total</a><a id="1000" class="Symbol">;</a> <a id="1002" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#15619" class="Function">+-monoʳ-≤</a><a id="1011" class="Symbol">;</a> <a id="1013" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#15529" class="Function">+-monoˡ-≤</a><a id="1022" class="Symbol">;</a> <a id="1024" href="https://agda.github.io/agda-stdlib/v1.1/Data.Nat.Properties.html#15373" class="Function">+-mono-≤</a><a id="1032" class="Symbol">)</a>
<a id="1034" class="Keyword">open</a> <a id="1039" class="Keyword">import</a> <a id="1046" href="/20.07/Relations/" class="Module">plfa.part1.Relations</a> <a id="1067" class="Keyword">using</a> <a id="1073" class="Symbol">(</a><a id="1074" href="/20.07/Relations/#18905" class="Datatype Operator">_<_</a><a id="1077" class="Symbol">;</a> <a id="1079" href="/20.07/Relations/#18932" class="InductiveConstructor">z<s</a><a id="1082" class="Symbol">;</a> <a id="1084" href="/20.07/Relations/#18989" class="InductiveConstructor">s<s</a><a id="1087" class="Symbol">;</a> <a id="1089" href="/20.07/Relations/#21658" class="InductiveConstructor">zero</a><a id="1093" class="Symbol">;</a> <a id="1095" href="/20.07/Relations/#21700" class="InductiveConstructor">suc</a><a id="1098" class="Symbol">;</a> <a id="1100" href="/20.07/Relations/#21603" class="Datatype">even</a><a id="1104" class="Symbol">;</a> <a id="1106" href="/20.07/Relations/#21623" class="Datatype">odd</a><a id="1109" class="Symbol">)</a>
</pre>
<h2 id="naturals">Naturals</h2>
<h4 id="seven">Exercise <code class="language-plaintext highlighter-rouge">seven</code></h4>
<p>Write out <code class="language-plaintext highlighter-rouge">7</code> in longhand.</p>
<h4 id="plus-example">Exercise <code class="language-plaintext highlighter-rouge">+-example</code></h4>
<p>Compute <code class="language-plaintext highlighter-rouge">3 + 4</code>, writing out your reasoning as a chain of equations.</p>
<h4 id="times-example">Exercise <code class="language-plaintext highlighter-rouge">*-example</code></h4>
<p>Compute <code class="language-plaintext highlighter-rouge">3 * 4</code>, writing out your reasoning as a chain of equations.</p>
<h4 id="power">Exercise <code class="language-plaintext highlighter-rouge">_^_</code> (recommended)</h4>
<p>Define exponentiation, which is given by the following equations.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>n ^ 0 = 1
n ^ (1 + m) = n * (n ^ m)
</code></pre></div></div>
<p>Check that <code class="language-plaintext highlighter-rouge">3 ^ 4</code> is <code class="language-plaintext highlighter-rouge">81</code>.</p>
<h4 id="monus-examples">Exercise <code class="language-plaintext highlighter-rouge">∸-examples</code> (recommended)</h4>
<p>Compute <code class="language-plaintext highlighter-rouge">5 ∸ 3</code> and <code class="language-plaintext highlighter-rouge">3 ∸ 5</code>, writing out your reasoning as a chain of equations.</p>
<h4 id="Bin">Exercise <code class="language-plaintext highlighter-rouge">Bin</code> (stretch)</h4>
<p>A more efficient representation of natural numbers uses a binary
rather than a unary system. We represent a number as a bitstring.</p>
<pre class="Agda"><a id="1932" class="Keyword">data</a> <a id="Bin"></a><a id="1937" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a> <a id="1941" class="Symbol">:</a> <a id="1943" class="PrimitiveType">Set</a> <a id="1947" class="Keyword">where</a>
<a id="Bin.nil"></a><a id="1955" href="/20.07/TSPL/2018/Assignment1/#1955" class="InductiveConstructor">nil</a> <a id="1959" class="Symbol">:</a> <a id="1961" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a>
<a id="Bin.x0_"></a><a id="1967" href="/20.07/TSPL/2018/Assignment1/#1967" class="InductiveConstructor Operator">x0_</a> <a id="1971" class="Symbol">:</a> <a id="1973" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a> <a id="1977" class="Symbol">→</a> <a id="1979" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a>
<a id="Bin.x1_"></a><a id="1985" href="/20.07/TSPL/2018/Assignment1/#1985" class="InductiveConstructor Operator">x1_</a> <a id="1989" class="Symbol">:</a> <a id="1991" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a> <a id="1995" class="Symbol">→</a> <a id="1997" href="/20.07/TSPL/2018/Assignment1/#1937" class="Datatype">Bin</a>
</pre>
<p>For instance, the bitstring</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>1011
</code></pre></div></div>
<p>standing for the number eleven is encoded, right to left, as</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>x1 x1 x0 x1 nil
</code></pre></div></div>
<p>Representations are not unique due to leading zeros.
Hence, eleven is also represented by <code class="language-plaintext highlighter-rouge">001011</code>, encoded as</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>x1 x1 x0 x1 x0 x0 nil
</code></pre></div></div>
<p>Define a function</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>inc : Bin → Bin
</code></pre></div></div>
<p>that converts a bitstring to the bitstring for the next higher
number. For example, since <code class="language-plaintext highlighter-rouge">1100</code> encodes twelve, we should have</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>inc (x1 x1 x0 x1 nil) ≡ x0 x0 x1 x1 nil
</code></pre></div></div>
<p>Confirm that this gives the correct answer for the bitstrings
encoding zero through four.</p>
<p>Using the above, define a pair of functions to convert
between the two representations.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>to : ℕ → Bin
from : Bin → ℕ
</code></pre></div></div>
<p>For the former, choose the bitstring to have no leading zeros if it
represents a positive natural, and represent zero by <code class="language-plaintext highlighter-rouge">x0 nil</code>.
Confirm that these both give the correct answer for zero through four.</p>
<h2 id="induction">Induction</h2>
<h4 id="operators">Exercise <code class="language-plaintext highlighter-rouge">operators</code></h4>
<p>Give another example of a pair of operators that have an identity
and are associative, commutative, and distribute over one another.</p>
<p>Give an example of an operator that has an identity and is
associative but is not commutative.</p>
<h4 id="finite-plus-assoc">Exercise <code class="language-plaintext highlighter-rouge">finite-+-assoc</code> (stretch)</h4>
<p>Write out what is known about associativity of addition on each of the first four
days using a finite story of creation, as
[earlier][plfa.Naturals#finite-creation]</p>
<h4 id="plus-swap">Exercise <code class="language-plaintext highlighter-rouge">+-swap</code> (recommended)</h4>
<p>Show</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>m + (n + p) ≡ n + (m + p)
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">m</code>, <code class="language-plaintext highlighter-rouge">n</code>, and <code class="language-plaintext highlighter-rouge">p</code>. No induction is needed,
just apply the previous results which show addition
is associative and commutative. You may need to use
the following function from the standard library:</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>sym : ∀ {m n : ℕ} → m ≡ n → n ≡ m
</code></pre></div></div>
<h4 id="times-distrib-plus">Exercise <code class="language-plaintext highlighter-rouge">*-distrib-+</code> (recommended)</h4>
<p>Show multiplication distributes over addition, that is,</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>(m + n) * p ≡ m * p + n * p
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">m</code>, <code class="language-plaintext highlighter-rouge">n</code>, and <code class="language-plaintext highlighter-rouge">p</code>.</p>
<h4 id="times-assoc">Exercise <code class="language-plaintext highlighter-rouge">*-assoc</code> (recommended)</h4>
<p>Show multiplication is associative, that is,</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>(m * n) * p ≡ m * (n * p)
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">m</code>, <code class="language-plaintext highlighter-rouge">n</code>, and <code class="language-plaintext highlighter-rouge">p</code>.</p>
<h4 id="times-comm">Exercise <code class="language-plaintext highlighter-rouge">*-comm</code></h4>
<p>Show multiplication is commutative, that is,</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>m * n ≡ n * m
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">m</code> and <code class="language-plaintext highlighter-rouge">n</code>. As with commutativity of addition,
you will need to formulate and prove suitable lemmas.</p>
<h4 id="zero-monus">Exercise <code class="language-plaintext highlighter-rouge">0∸n≡0</code></h4>
<p>Show</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>zero ∸ n ≡ zero
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">n</code>. Did your proof require induction?</p>
<h4 id="monus-plus-assoc">Exercise <code class="language-plaintext highlighter-rouge">∸-+-assoc</code></h4>
<p>Show that monus associates with addition, that is,</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>m ∸ n ∸ p ≡ m ∸ (n + p)
</code></pre></div></div>
<p>for all naturals <code class="language-plaintext highlighter-rouge">m</code>, <code class="language-plaintext highlighter-rouge">n</code>, and <code class="language-plaintext highlighter-rouge">p</code>.</p>
<h4 id="Bin-laws">Exercise <code class="language-plaintext highlighter-rouge">Bin-laws</code> (stretch)</h4>
<p>Recall that
Exercise [Bin][plfa.Naturals#Bin]
defines a datatype <code class="language-plaintext highlighter-rouge">Bin</code> of bitstrings representing natural numbers
and asks you to define functions</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>inc : Bin → Bin
to : ℕ → Bin
from : Bin → ℕ
</code></pre></div></div>
<p>Consider the following laws, where <code class="language-plaintext highlighter-rouge">n</code> ranges over naturals and <code class="language-plaintext highlighter-rouge">x</code>
over bitstrings.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>from (inc x) ≡ suc (from x)
to (from n) ≡ n
from (to x) ≡ x
</code></pre></div></div>
<p>For each law: if it holds, prove; if not, give a counterexample.</p>
<h2 id="relations">Relations</h2>
<h4 id="orderings">Exercise <code class="language-plaintext highlighter-rouge">orderings</code></h4>
<p>Give an example of a preorder that is not a partial order.</p>
<p>Give an example of a partial order that is not a preorder.</p>
<h4 id="leq-antisym-cases">Exercise <code class="language-plaintext highlighter-rouge">≤-antisym-cases</code></h4>
<p>The above proof omits cases where one argument is <code class="language-plaintext highlighter-rouge">z≤n</code> and one
argument is <code class="language-plaintext highlighter-rouge">s≤s</code>. Why is it ok to omit them?</p>
<h4 id="exercise--mono--stretch">Exercise <code class="language-plaintext highlighter-rouge">*-mono-≤</code> (stretch)</h4>
<p>Show that multiplication is monotonic with regard to inequality.</p>
<h4 id="less-trans">Exercise <code class="language-plaintext highlighter-rouge"><-trans</code> (recommended)</h4>
<p>Show that strict inequality is transitive.</p>
<h4 id="trichotomy">Exercise <code class="language-plaintext highlighter-rouge">trichotomy</code></h4>
<p>Show that strict inequality satisfies a weak version of trichotomy, in
the sense that for any <code class="language-plaintext highlighter-rouge">m</code> and <code class="language-plaintext highlighter-rouge">n</code> that one of the following holds:</p>
<ul>
<li><code class="language-plaintext highlighter-rouge">m < n</code>,</li>
<li><code class="language-plaintext highlighter-rouge">m ≡ n</code>, or</li>
<li><code class="language-plaintext highlighter-rouge">m > n</code>
Define <code class="language-plaintext highlighter-rouge">m > n</code> to be the same as <code class="language-plaintext highlighter-rouge">n < m</code>.
You will need a suitable data declaration,
similar to that used for totality.
(We will show that the three cases are exclusive after we introduce
[negation][plfa.Negation].)</li>
</ul>
<h4 id="plus-mono-less">Exercise <code class="language-plaintext highlighter-rouge">+-mono-<</code></h4>
<p>Show that addition is monotonic with respect to strict inequality.
As with inequality, some additional definitions may be required.</p>
<h4 id="leq-iff-less">Exercise <code class="language-plaintext highlighter-rouge">≤-iff-<</code> (recommended)</h4>
<p>Show that <code class="language-plaintext highlighter-rouge">suc m ≤ n</code> implies <code class="language-plaintext highlighter-rouge">m < n</code>, and conversely.</p>
<h4 id="less-trans-revisited">Exercise <code class="language-plaintext highlighter-rouge"><-trans-revisited</code></h4>
<p>Give an alternative proof that strict inequality is transitive,
using the relating between strict inequality and inequality and
the fact that inequality is transitive.</p>
<h4 id="odd-plus-odd">Exercise <code class="language-plaintext highlighter-rouge">o+o≡e</code> (stretch)</h4>
<p>Show that the sum of two odd numbers is even.</p>
<h4 id="Bin-predicates">Exercise <code class="language-plaintext highlighter-rouge">Bin-predicates</code> (stretch)</h4>
<p>Recall that
Exercise [Bin][plfa.Naturals#Bin]
defines a datatype <code class="language-plaintext highlighter-rouge">Bin</code> of bitstrings representing natural numbers.
Representations are not unique due to leading zeros.
Hence, eleven may be represented by both of the following</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>x1 x1 x0 x1 nil
x1 x1 x0 x1 x0 x0 nil
</code></pre></div></div>
<p>Define a predicate</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Can : Bin → Set
</code></pre></div></div>
<p>over all bitstrings that holds if the bitstring is canonical, meaning
it has no leading zeros; the first representation of eleven above is
canonical, and the second is not. To define it, you will need an
auxiliary predicate</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>One : Bin → Set
</code></pre></div></div>
<p>that holds only if the bistring has a leading one. A bitstring is
canonical if it has a leading one (representing a positive number) or
if it consists of a single zero (representing zero).</p>
<p>Show that increment preserves canonical bitstrings.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Can x
------------
Can (inc x)
</code></pre></div></div>
<p>Show that converting a natural to a bitstring always yields a
canonical bitstring.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>----------
Can (to n)
</code></pre></div></div>
<p>Show that converting a canonical bitstring to a natural
and back is the identity.</p>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>Can x
---------------
to (from x) ≡ x
</code></pre></div></div>
<p>(Hint: For each of these, you may first need to prove related
properties of <code class="language-plaintext highlighter-rouge">One</code>.)</p>
</div>
<p style="text-align:center;">
<a alt="Source code" href="https://github.com/plfa/plfa.github.io/blob/dev-20.07/courses/tspl/2018/Assignment1.lagda.md">Source</a>
</p>
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