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<h1>Chapter 4: Multiple StepTypes</h1>
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<section class="tex2jax_ignore mathjax_ignore" id="chapter-4-multiple-steptypes">
<h1>Chapter 4: Multiple StepTypes<a class="headerlink" href="#chapter-4-multiple-steptypes" title="Link to this heading">#</a></h1>
<p>To avoid the soundness error from <code class="docutils literal notranslate"><span class="pre">evil_witness</span></code> in Chapter 3, we need to constrain that forward signals “a” and “b” are both 1 in the first step instance. Because this isn’t true for all step instances, we need to create a new step type, called <code class="docutils literal notranslate"><span class="pre">FiboFirstStep</span></code>, which is identical to <code class="docutils literal notranslate"><span class="pre">FiboStep</span></code> except two additional constraints for “a” and “b”:</p>
<div class="pst-scrollable-table-container"><table class="table">
<thead>
<tr class="row-odd"><th class="head text-center"><p>Step Type</p></th>
<th class="head text-center"><p>Step Instance Index</p></th>
<th class="head text-center"><p></p></th>
<th class="head text-center"><p>Signals</p></th>
<th class="head text-center"><p></p></th>
<th class="head text-center"><p></p></th>
<th class="head text-center"><p></p></th>
<th class="head text-center"><p>Setups</p></th>
<th class="head text-center"><p></p></th>
<th class="head text-center"><p></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td class="text-center"><p></p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p>a</p></td>
<td class="text-center"><p>b</p></td>
<td class="text-center"><p>c</p></td>
<td class="text-center"><p>constraint 1</p></td>
<td class="text-center"><p>constraint 2</p></td>
<td class="text-center"><p>constraint 3</p></td>
<td class="text-center"><p>constraint 4</p></td>
<td class="text-center"><p>constraint 5</p></td>
</tr>
<tr class="row-odd"><td class="text-center"><p>fibo first step</p></td>
<td class="text-center"><p>0</p></td>
<td class="text-center"><p>1</p></td>
<td class="text-center"><p>1</p></td>
<td class="text-center"><p>2</p></td>
<td class="text-center"><p>a + b == c</p></td>
<td class="text-center"><p>b == a.next</p></td>
<td class="text-center"><p>c == b.next</p></td>
<td class="text-center"><p>a == 1</p></td>
<td class="text-center"><p>b == 1</p></td>
</tr>
<tr class="row-even"><td class="text-center"><p>fibo step</p></td>
<td class="text-center"><p>1</p></td>
<td class="text-center"><p>1</p></td>
<td class="text-center"><p>2</p></td>
<td class="text-center"><p>3</p></td>
<td class="text-center"><p>a + b == c</p></td>
<td class="text-center"><p>b == a.next</p></td>
<td class="text-center"><p>c == b.next</p></td>
<td class="text-center"><p>n.a.</p></td>
<td class="text-center"><p>n.a.</p></td>
</tr>
<tr class="row-odd"><td class="text-center"><p>fibo step</p></td>
<td class="text-center"><p>2</p></td>
<td class="text-center"><p>2</p></td>
<td class="text-center"><p>3</p></td>
<td class="text-center"><p>5</p></td>
<td class="text-center"><p>a + b == c</p></td>
<td class="text-center"><p>b == a.next</p></td>
<td class="text-center"><p>c == b.next</p></td>
<td class="text-center"><p>n.a.</p></td>
<td class="text-center"><p>n.a.</p></td>
</tr>
<tr class="row-even"><td class="text-center"><p>fibo step</p></td>
<td class="text-center"><p>3</p></td>
<td class="text-center"><p>3</p></td>
<td class="text-center"><p>5</p></td>
<td class="text-center"><p>8</p></td>
<td class="text-center"><p>a + b == c</p></td>
<td class="text-center"><p>b == a.next</p></td>
<td class="text-center"><p>c == b.next</p></td>
<td class="text-center"><p>n.a.</p></td>
<td class="text-center"><p>n.a.</p></td>
</tr>
<tr class="row-odd"><td class="text-center"><p>…</p></td>
<td class="text-center"><p>…</p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p>…</p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p>…</p></td>
<td class="text-center"><p></p></td>
<td class="text-center"><p></p></td>
</tr>
</tbody>
</table>
</div>
<p>The following shows the incremental code of <code class="docutils literal notranslate"><span class="pre">FiboFirstStep</span></code> compared to <code class="docutils literal notranslate"><span class="pre">FiboStep</span></code>:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">FiboFirstStep</span><span class="p">(</span><span class="n">StepType</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># ...</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">wg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="c1"># ...</span>
</pre></div>
</div>
<p>Next, in circuit <code class="docutils literal notranslate"><span class="pre">setup</span></code>, we need to append this step type and constrain that the first step instance is <code class="docutils literal notranslate"><span class="pre">FiboFirstStep</span></code>. Otherwise <code class="docutils literal notranslate"><span class="pre">setup</span></code> is identical to that in Chapter 3:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Fibonacci</span><span class="p">(</span><span class="n">Circuit</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># ...</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">step_type</span><span class="p">(</span><span class="n">FiboFirstStep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s2">&quot;fibo_first_step&quot;</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">pragma_first_step</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="c1"># TODO</span>
</pre></div>
</div>
<p>Finally, in circuit <code class="docutils literal notranslate"><span class="pre">trace</span></code>, we need to instantiate <code class="docutils literal notranslate"><span class="pre">FiboFirstStep</span></code> for the first step instance. Otherwise <code class="docutils literal notranslate"><span class="pre">trace</span></code> is identical to that in Chapter 3:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Fibonacci</span><span class="p">(</span><span class="n">Circuit</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="c1"># ...</span>
<span class="k">def</span> <span class="nf">trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="c1"># ...</span>
</pre></div>
</div>
<p>Putting everything together, we have the following circuit with two step types:</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">chiquito.dsl</span> <span class="kn">import</span> <span class="n">Circuit</span><span class="p">,</span> <span class="n">StepType</span>
<span class="kn">from</span> <span class="nn">chiquito.cb</span> <span class="kn">import</span> <span class="n">eq</span>
<span class="kn">from</span> <span class="nn">chiquito.util</span> <span class="kn">import</span> <span class="n">F</span>
<span class="k">class</span> <span class="nc">FiboFirstStep</span><span class="p">(</span><span class="n">StepType</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">internal</span><span class="p">(</span><span class="s2">&quot;c&quot;</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">transition</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="o">.</span><span class="n">next</span><span class="p">()))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">transition</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="o">.</span><span class="n">next</span><span class="p">()))</span>
<span class="k">def</span> <span class="nf">wg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="n">a_value</span><span class="p">,</span> <span class="n">b_value</span> <span class="o">=</span> <span class="n">args</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">a_value</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">b_value</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">a_value</span> <span class="o">+</span> <span class="n">b_value</span><span class="p">))</span>
<span class="k">class</span> <span class="nc">FiboStep</span><span class="p">(</span><span class="n">StepType</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">c</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">internal</span><span class="p">(</span><span class="s2">&quot;c&quot;</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">constr</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">transition</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="o">.</span><span class="n">next</span><span class="p">()))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">transition</span><span class="p">(</span><span class="n">eq</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="o">.</span><span class="n">next</span><span class="p">()))</span>
<span class="k">def</span> <span class="nf">wg</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="n">a_value</span><span class="p">,</span> <span class="n">b_value</span> <span class="o">=</span> <span class="n">args</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">a</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">a_value</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">circuit</span><span class="o">.</span><span class="n">b</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">b_value</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">assign</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">c</span><span class="p">,</span> <span class="n">F</span><span class="p">(</span><span class="n">a_value</span> <span class="o">+</span> <span class="n">b_value</span><span class="p">))</span>
<span class="k">class</span> <span class="nc">Fibonacci</span><span class="p">(</span><span class="n">Circuit</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">setup</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">a</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="s2">&quot;a&quot;</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">forward</span><span class="p">(</span><span class="s2">&quot;b&quot;</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">step_type</span><span class="p">(</span><span class="n">FiboFirstStep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s2">&quot;fibo_first_step&quot;</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fibo_step</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">step_type</span><span class="p">(</span><span class="n">FiboStep</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s2">&quot;fibo_step&quot;</span><span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">pragma_num_steps</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">pragma_first_step</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">trace</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">args</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fibo_first_step</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">a</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fibo_step</span><span class="p">,</span> <span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">))</span>
<span class="n">prev_a</span> <span class="o">=</span> <span class="n">a</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">b</span>
<span class="n">b</span> <span class="o">+=</span> <span class="n">prev_a</span>
<span class="n">fibo</span> <span class="o">=</span> <span class="n">Fibonacci</span><span class="p">()</span>
<span class="n">fibo_witness</span> <span class="o">=</span> <span class="n">fibo</span><span class="o">.</span><span class="n">gen_witness</span><span class="p">(</span><span class="kc">None</span><span class="p">)</span>
<span class="n">fibo</span><span class="o">.</span><span class="n">halo2_mock_prover</span><span class="p">(</span><span class="n">fibo_witness</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>------ Visiting map -------
key = annotations
key = exposed
key = first_step
key = fixed_assignments
key = fixed_signals
key = forward_signals
key = id
key = last_step
key = num_steps
key = q_enable
key = shared_signals
key = step_types
------ Visiting step_types -------
step_types = Some(
{
107394379366394054393668600732442036746: StepType {
id: 107394379366394054393668600732442036746,
signals: [
InternalSignal {
id: 107394386100787868107202960445504358922,
annotation: &quot;c&quot;,
},
],
constraints: [
Constraint {
annotation: &quot;(a == 1)&quot;,
expr: (a + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)),
},
Constraint {
annotation: &quot;(b == 1)&quot;,
expr: (b + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)),
},
Constraint {
annotation: &quot;((a + b) == c)&quot;,
expr: (a + b + (-c)),
},
],
transition_constraints: [
TransitionConstraint {
annotation: &quot;(b == next(a))&quot;,
expr: (b + (-next(a))),
},
TransitionConstraint {
annotation: &quot;(c == next(b))&quot;,
expr: (c + (-next(b))),
},
],
lookups: [],
},
107394432686947426494166216988007467530: StepType {
id: 107394432686947426494166216988007467530,
signals: [
InternalSignal {
id: 107394439262884915179504041986500463114,
annotation: &quot;c&quot;,
},
],
constraints: [
Constraint {
annotation: &quot;((a + b) == c)&quot;,
expr: (a + b + (-c)),
},
],
transition_constraints: [
TransitionConstraint {
annotation: &quot;(b == next(a))&quot;,
expr: (b + (-next(a))),
},
TransitionConstraint {
annotation: &quot;(c == next(b))&quot;,
expr: (c + (-next(b))),
},
],
lookups: [],
},
},
)
Ok(
(),
)
</pre></div>
</div>
</div>
</div>
<p>We construct and pass over the same <code class="docutils literal notranslate"><span class="pre">evil_witness</span></code> from Chapter 3. This time <code class="docutils literal notranslate"><span class="pre">evil_witness</span></code> fails at the two new constraints we wrote, e.g. <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">==</span> <span class="pre">1</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">==</span> <span class="pre">1</span></code> from <code class="docutils literal notranslate"><span class="pre">FiboFirstStep</span></code>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">evil_witness</span> <span class="o">=</span> <span class="n">fibo_witness</span><span class="o">.</span><span class="n">evil_witness_test</span><span class="p">(</span><span class="n">step_instance_indices</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">assignment_indices</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">rhs</span><span class="o">=</span><span class="p">[</span><span class="n">F</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">2</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">4</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">6</span><span class="p">),</span> <span class="n">F</span><span class="p">(</span><span class="mi">10</span><span class="p">)])</span>
<span class="n">fibo</span><span class="o">.</span><span class="n">halo2_mock_prover</span><span class="p">(</span><span class="n">evil_witness</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Err(
[
ConstraintCaseDebug {
constraint: Constraint {
gate: Gate {
index: 0,
name: &quot;main&quot;,
},
index: 0,
name: &quot;fibo_first_step::(a == 1) =&gt; (a + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)) =&gt; Product(Fixed { query_index: 0, column_index: 0, rotation: Rotation(0) }, Product(Advice { query_index: 0, column_index: 3, rotation: Rotation(0) }, Sum(Advice { query_index: 1, column_index: 0, rotation: Rotation(0) }, Negated(Constant(0x0e0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)))))&quot;,
},
location: InRegion {
region: Region 0 (&#39;circuit&#39;),
offset: 0,
},
cell_values: [
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Advice,
index: 0,
annotation: &quot;srcm forward a&quot;,
},
rotation: 0,
},
&quot;0&quot;,
),
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Advice,
index: 3,
annotation: &quot;&#39;step selector for fibo_first_step&#39;&quot;,
},
rotation: 0,
},
&quot;1&quot;,
),
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Fixed,
index: 0,
annotation: &quot;q_enable&quot;,
},
rotation: 0,
},
&quot;1&quot;,
),
],
},
ConstraintCaseDebug {
constraint: Constraint {
gate: Gate {
index: 0,
name: &quot;main&quot;,
},
index: 1,
name: &quot;fibo_first_step::(b == 1) =&gt; (b + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)) =&gt; Product(Fixed { query_index: 0, column_index: 0, rotation: Rotation(0) }, Product(Advice { query_index: 0, column_index: 3, rotation: Rotation(0) }, Sum(Advice { query_index: 2, column_index: 1, rotation: Rotation(0) }, Negated(Constant(0x0e0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)))))&quot;,
},
location: InRegion {
region: Region 0 (&#39;circuit&#39;),
offset: 0,
},
cell_values: [
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Advice,
index: 1,
annotation: &quot;srcm forward b&quot;,
},
rotation: 0,
},
&quot;0x1c14ef83340fbe5eccdd46def0f28c5c6df8ed2b3ec19a53592c68389ffffff6&quot;,
),
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Advice,
index: 3,
annotation: &quot;&#39;step selector for fibo_first_step&#39;&quot;,
},
rotation: 0,
},
&quot;1&quot;,
),
(
DebugVirtualCell {
name: &quot;&quot;,
column: DebugColumn {
column_type: Fixed,
index: 0,
annotation: &quot;q_enable&quot;,
},
rotation: 0,
},
&quot;1&quot;,
),
],
},
],
)
Constraint 0 (&#39;fibo_first_step::(a == 1) =&gt; (a + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)) =&gt; Product(Fixed { query_index: 0, column_index: 0, rotation: Rotation(0) }, Product(Advice { query_index: 0, column_index: 3, rotation: Rotation(0) }, Sum(Advice { query_index: 1, column_index: 0, rotation: Rotation(0) }, Negated(Constant(0x0e0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)))))&#39;) in gate 0 (&#39;main&#39;) is not satisfied in Region 0 (&#39;circuit&#39;) at offset 0
- Column(&#39;Advice&#39;, 0 - srcm forward a)@0 = 0
- Column(&#39;Advice&#39;, 3 - &#39;step selector for fibo_first_step&#39;)@0 = 1
- Column(&#39;Fixed&#39;, 0 - q_enable)@0 = 1
Constraint 1 (&#39;fibo_first_step::(b == 1) =&gt; (b + (-0xe0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)) =&gt; Product(Fixed { query_index: 0, column_index: 0, rotation: Rotation(0) }, Product(Advice { query_index: 0, column_index: 3, rotation: Rotation(0) }, Sum(Advice { query_index: 2, column_index: 1, rotation: Rotation(0) }, Negated(Constant(0x0e0a77c19a07df2f666ea36f7879462e36fc76959f60cd29ac96341c4ffffffb)))))&#39;) in gate 0 (&#39;main&#39;) is not satisfied in Region 0 (&#39;circuit&#39;) at offset 0
- Column(&#39;Advice&#39;, 1 - srcm forward b)@0 = 0x1c14ef83340fbe5eccdd46def0f28c5c6df8ed2b3ec19a53592c68389ffffff6
- Column(&#39;Advice&#39;, 3 - &#39;step selector for fibo_first_step&#39;)@0 = 1
- Column(&#39;Fixed&#39;, 0 - q_enable)@0 = 1
</pre></div>
</div>
</div>
</div>
<p>By adding new constraints to eliminate soundness errors, this chapter demonstrated an iterative way of developing zero knowledge circuits. Circuit engineer can proactively generate witnesses in edge cases and if a false positive passes the circuit, add more constraints to prevent these edge cases.</p>
<p>Besides, we learned that with multiple step types, Chiquito can create step instances with different signals and constraints, thereby allowing great versatility in circuit design.</p>
<p>Finally, this example shows again that circuit setup and witness generation are separate processes. There can be multiple witnesses for the same circuit setup. As a great analogy, a circuit can be considered a piece of “hardware” whereas its witnesses “software” that are compatible with the hardware.</p>
</section>
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