Different lattice figures

11 files changed, 4226 insertions, 312 deletions

diff --git a/doc/talks/2023-01-18-tocatta/assets/lattice8.svg b/doc/talks/2023-01-18-tocatta/assets/lattice8.svg
index 3bada791..c94a69b2 100644
--- a/doc/talks/2023-01-18-tocatta/assets/lattice8.svg
+++ b/doc/talks/2023-01-18-tocatta/assets/lattice8.svg
@@ -25,8 +25,8 @@
      inkscape:document-units="mm"
      showgrid="false"
      inkscape:zoom="1.4734708"
-     inkscape:cx="451.65469"
-     inkscape:cy="272.14655"
+     inkscape:cx="399.39712"
+     inkscape:cy="248.39311"
      inkscape:window-width="1920"
      inkscape:window-height="999"
      inkscape:window-x="0"
@@ -433,8 +433,8 @@
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663"
-       cx="147.35568"
-       cy="95.24971"
+       cx="132.33408"
+       cy="68.007973"
        r="2.7302806" />
     <circle
        style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
@@ -480,6 +480,17 @@
     <text
        xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="65.261436"
+       id="text3707-1"><tspan
+         sodipodi:role="line"
+         id="tspan3705-5"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="65.261436">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="23.457415"
        y="24.042637"
        id="text3750"><tspan
@@ -501,15 +512,32 @@
          y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="23.457415"
        y="41.972523"
        id="text3750-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6"
-         style="fill:#999999;stroke-width:0.264583"
+         style="fill:#000000;stroke-width:0.264583"
          x="23.457415"
-         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
+         y="41.972523">$\sqsupseteq \{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-4"
+       cx="18.004833"
+       cy="70.942116"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="73.643356"
+       id="text3750-7"><tspan
+         sodipodi:role="line"
+         id="tspan3748-65"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="73.643356">$\to \{\}$</tspan></text>
     <text
        xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
@@ -521,67 +549,33 @@
          style="fill:#000000;stroke-width:0.264583"
          x="14.395845"
          y="52.022549">return OK</tspan></text>
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     <text
        xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="6.9525447"
-       y="65.279381"
-       id="text3707-5"><tspan
+       x="14.395845"
+       y="92.005798"
+       id="text4280-3"><tspan
          sodipodi:role="line"
-         id="tspan3705-4"
+         id="tspan4278-7"
          style="fill:#000000;stroke-width:0.264583"
-         x="6.9525447"
-         y="65.279381">$write(\{b\})$:</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
-       x="23.457415"
-       y="75.619804"
-       id="text3750-7"><tspan
-         sodipodi:role="line"
-         id="tspan3748-4"
-         style="fill:#999999;stroke-width:0.264583"
-         x="23.457415"
-         y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
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-         style="fill:#999999;stroke-width:0.264583"
-         x="23.457415"
-         y="84.598038">$\not\sqsupseteq \{b\}$</tspan></text>
+         x="14.395845"
+         y="92.005798">return $\{\}\sqcup\{a\}=\{a\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1-5"
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     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="23.457415"
-       y="93.54969"
-       id="text3750-0-0"><tspan
+       y="83.577797"
+       id="text3750-0-6"><tspan
          sodipodi:role="line"
-         id="tspan3748-6-7"
-         style="fill:#999999;stroke-width:0.264583"
+         id="tspan3748-6-9"
+         style="fill:#000000;stroke-width:0.264583"
          x="23.457415"
-         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+         y="83.577797">$\to \{a\}$</tspan></text>
   </g>
 </svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/latticeB_1.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_1.svg
new file mode 100644
index 00000000..92232a1b
--- /dev/null
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_1.svg
@@ -0,0 +1,576 @@
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+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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+         xml:space="preserve"
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+       transform="translate(51.996784,1.0317046)"
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+           sodipodi:role="line"
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+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-93-1)"
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+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-7-2)"
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+       d="m 101.90418,61.091809 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
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+       sodipodi:nodetypes="cc" />
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+       d="m 165.29305,89.571762 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
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+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75-7-1)"
+       d="m 165.29305,32.445235 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2-6-9"
+       sodipodi:nodetypes="cc" />
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663"
+       cx="147.35568"
+       cy="95.24971"
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+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3"
+       cx="176.97627"
+       cy="68.155472"
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+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1"
+       cx="18.004833"
+       cy="39.402473"
+       r="2.7302806" />
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0"
+       cx="18.004833"
+       cy="30.371933"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6"
+       cx="18.004833"
+       cy="21.341394"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="13.702217"
+       id="text3707"><tspan
+         sodipodi:role="line"
+         id="tspan3705"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="13.702217">$write(\{a\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="24.042637"
+       id="text3750"><tspan
+         sodipodi:role="line"
+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="33.02087"
+       id="text3750-3"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="41.972523"
+       id="text3750-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1-4"
+       cx="18.004833"
+       cy="90.979645"
+       r="2.7302806" />
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-5"
+       cx="18.004833"
+       cy="81.949104"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-2"
+       cx="18.004833"
+       cy="72.918564"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="65.279381"
+       id="text3707-5"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="65.279381">$write(\{b\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="59.340172"
+       y="118.94403"
+       id="text3707-5-3"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6"
+         style="fill:#000000;stroke-width:0.264583"
+         x="59.340172"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="75.619804"
+       id="text3750-7"><tspan
+         sodipodi:role="line"
+         id="tspan3748-4"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="84.598038"
+       id="text3750-3-4"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-3"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="93.54969"
+       id="text3750-0-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6-7"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6"
+       cx="118.63729"
+       cy="67.630196"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-0"
+       cx="67.201363"
+       cy="124.91125"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="72.653946"
+       y="127.6125"
+       id="text3750-9"><tspan
+         sodipodi:role="line"
+         id="tspan3748-3"
+         style="fill:#000000;stroke-width:0.264583"
+         x="72.653946"
+         y="127.6125">$\to \{a\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="62.764446"
+       y="146.3129"
+       id="text4280-6"><tspan
+         sodipodi:role="line"
+         style="fill:#000000;stroke-width:0.264583"
+         x="62.764446"
+         y="146.3129"
+         id="tspan2630">return $\{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-7"
+       cx="67.201363"
+       cy="133.94179"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="72.653946"
+       y="136.59073"
+       id="text3750-3-6"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-0"
+         style="fill:#000000;stroke-width:0.264583"
+         x="72.653946"
+         y="136.59073">$\to \{\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-0-6"
+       cx="174.36296"
+       cy="124.91125"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="179.81555"
+       y="127.6125"
+       id="text3750-9-1"><tspan
+         sodipodi:role="line"
+         id="tspan3748-3-8"
+         style="fill:#000000;stroke-width:0.264583"
+         x="179.81555"
+         y="127.6125">$\to \{\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="162.7999"
+       y="118.94403"
+       id="text3707-5-3-7"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6-5"
+         style="fill:#000000;stroke-width:0.264583"
+         x="162.7999"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583"
+       x="134.97845"
+       y="132.58472"
+       id="text3899"><tspan
+         sodipodi:role="line"
+         id="tspan3897"
+         style="stroke-width:0.264583"
+         x="134.97845"
+         y="132.58472">;</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="169.92604"
+       y="146.3129"
+       id="text4280-6-7"><tspan
+         sodipodi:role="line"
+         style="fill:#000000;stroke-width:0.264583"
+         x="169.92604"
+         y="146.3129"
+         id="tspan2630-9">return $\{b\}$</tspan></text>
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-7-2"
+       cx="174.36296"
+       cy="133.94179"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="179.81555"
+       y="136.59073"
+       id="text3750-3-6-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-0-2"
+         style="fill:#000000;stroke-width:0.264583"
+         x="179.81555"
+         y="136.59073">$\to \{b\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:end;text-anchor:end;fill:#ff0000;stroke-width:0.264583"
+       x="259.75"
+       y="135.53111"
+       id="text9043"><tspan
+         sodipodi:role="line"
+         id="tspan9041"
+         style="fill:#ff0000;stroke-width:0.264583"
+         x="259.75"
+         y="135.53111">${\Large\textbf{??!}}$~~~~~</tspan><tspan
+         sodipodi:role="line"
+         style="fill:#ff0000;stroke-width:0.264583"
+         x="259.75"
+         y="146.11446"
+         id="tspan9372">$\{a\} \not\sqsubseteq \{b\}$</tspan></text>
+  </g>
+</svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/latticeC.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_2.svg
index c815af94..c07cba2b 100644
--- a/doc/talks/2023-01-18-tocatta/assets/latticeC.svg
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_2.svg
@@ -3,12 +3,12 @@
 
 <svg
    width="1000"
-   height="400"
-   viewBox="0 0 264.58333 105.83333"
+   height="600"
+   viewBox="0 0 264.58333 158.75"
    version="1.1"
    id="svg5"
    inkscape:version="1.2.2 (b0a8486541, 2022-12-01)"
-   sodipodi:docname="latticeC.svg"
+   sodipodi:docname="latticeB_2.svg"
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
    xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
    xmlns="http://www.w3.org/2000/svg"
@@ -25,8 +25,8 @@
      inkscape:document-units="mm"
      showgrid="false"
      inkscape:zoom="1.4734708"
-     inkscape:cx="450.97602"
-     inkscape:cy="272.14655"
+     inkscape:cx="586.70996"
+     inkscape:cy="321.01077"
      inkscape:window-width="1920"
      inkscape:window-height="999"
      inkscape:window-x="0"
@@ -270,10 +270,10 @@
      id="layer1"
      sodipodi:insensitive="true">
     <rect
-       style="fill:#ffffff;stroke:none;stroke-width:1;stop-color:#000000"
+       style="fill:#ffffff;stroke:none;stroke-width:0.999998;stop-color:#000000"
        id="rect288"
-       width="209.84705"
-       height="104.42732"
+       width="262.58151"
+       height="156.82782"
        x="0.77790999"
        y="0.93738818" />
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@@ -283,24 +283,24 @@
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     <text
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-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="99.307442"
        id="text951"><tspan
          sodipodi:role="line"
          id="tspan949"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="99.307442">$\{\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="13.151893"
        id="text1005"><tspan
          sodipodi:role="line"
          id="tspan1003"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="13.151893">$\{a,b,c\}$</tspan></text>
     <g
@@ -308,35 +308,35 @@
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       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="67.008698"
          id="text1009"><tspan
            sodipodi:role="line"
            id="tspan1007"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="67.008698">$\{a\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
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          id="text1009-3"><tspan
            sodipodi:role="line"
            id="tspan1007-6"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="67.017166">$\{b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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          id="text1009-7"><tspan
            sodipodi:role="line"
            id="tspan1007-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
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            y="67.008698">$\{c\}$</tspan></text>
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@@ -346,35 +346,35 @@
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       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
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          id="text1117"><tspan
            sodipodi:role="line"
            id="tspan1115"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="40.841526">$\{a,c\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
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          id="text1117-3"><tspan
            sodipodi:role="line"
            id="tspan1115-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="40.841526">$\{a,b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="176.20593"
          y="40.841526"
          id="text1117-6"><tspan
            sodipodi:role="line"
            id="tspan1115-2"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="40.841526">$\{b,c\}$</tspan></text>
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@@ -433,20 +433,20 @@
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        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663"
-       cx="117.08417"
-       cy="38.792969"
+       cx="147.35568"
+       cy="95.24971"
        r="2.7302806" />
     <circle
        style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-3"
-       cx="123.57843"
-       cy="39.100433"
+       cx="141.05479"
+       cy="95.316383"
        r="2.7302806" />
     <circle
        style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-6"
-       cx="130.27556"
-       cy="39.441635"
+       cx="118.63729"
+       cy="67.630196"
        r="2.7302806" />
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
@@ -468,13 +468,13 @@
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     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="6.9525447"
        y="13.702217"
        id="text3707"><tspan
          sodipodi:role="line"
          id="tspan3705"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="6.9525447"
          y="13.702217">$write(\{a\})$:</tspan></text>
     <text
@@ -490,48 +490,26 @@
          y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="33.02087"
        id="text3750-3"><tspan
          sodipodi:role="line"
          id="tspan3748-2"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="41.972523"
        id="text3750-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="41.972523">$\sqsupseteq \{a\}$</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="52.022549"
-       id="text4280"><tspan
-         sodipodi:role="line"
-         id="tspan4278"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="52.022549">return OK</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="102.48464"
-       id="text4280-8"><tspan
-         sodipodi:role="line"
-         id="tspan4278-8"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="102.48464">return OK</tspan></text>
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
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@@ -563,36 +541,36 @@
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     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="75.619804"
        id="text3750-7"><tspan
          sodipodi:role="line"
          id="tspan3748-4"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="75.619804">$\sqsupseteq \{b\}$</tspan></text>
+         y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="84.598038"
        id="text3750-3-4"><tspan
          sodipodi:role="line"
          id="tspan3748-2-3"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+         y="84.598038">$\not\sqsupseteq \{b\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
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        y="93.54969"
        id="text3750-0-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6-7"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="93.54969">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
   </g>
 </svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/latticeA.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_3.svg
index 400ccff8..198d1f5d 100644
--- a/doc/talks/2023-01-18-tocatta/assets/latticeA.svg
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_3.svg
@@ -3,12 +3,12 @@
 
 <svg
    width="1000"
-   height="400"
-   viewBox="0 0 264.58333 105.83333"
+   height="600"
+   viewBox="0 0 264.58333 158.75"
    version="1.1"
    id="svg5"
    inkscape:version="1.2.2 (b0a8486541, 2022-12-01)"
-   sodipodi:docname="latticeA.svg"
+   sodipodi:docname="latticeB_3.svg"
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
    xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
    xmlns="http://www.w3.org/2000/svg"
@@ -24,9 +24,9 @@
      inkscape:deskcolor="#d1d1d1"
      inkscape:document-units="mm"
      showgrid="false"
-     inkscape:zoom="1.4734708"
-     inkscape:cx="450.97602"
-     inkscape:cy="272.14655"
+     inkscape:zoom="2.0838024"
+     inkscape:cx="465.25525"
+     inkscape:cy="269.21938"
      inkscape:window-width="1920"
      inkscape:window-height="999"
      inkscape:window-x="0"
@@ -270,10 +270,10 @@
      id="layer1"
      sodipodi:insensitive="true">
     <rect
-       style="fill:#ffffff;stroke:none;stroke-width:1;stop-color:#000000"
+       style="fill:#ffffff;stroke:none;stroke-width:0.999998;stop-color:#000000"
        id="rect288"
-       width="209.84705"
-       height="104.42732"
+       width="262.58151"
+       height="156.82782"
        x="0.77790999"
        y="0.93738818" />
   </g>
@@ -283,24 +283,24 @@
      inkscape:label="Layer 2">
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="99.307442"
        id="text951"><tspan
          sodipodi:role="line"
          id="tspan949"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="99.307442">$\{\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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        y="13.151893"
        id="text1005"><tspan
          sodipodi:role="line"
          id="tspan1003"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="13.151893">$\{a,b,c\}$</tspan></text>
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@@ -308,35 +308,35 @@
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       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="67.008698"
          id="text1009"><tspan
            sodipodi:role="line"
            id="tspan1007"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="67.008698">$\{a\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
          y="67.017166"
          id="text1009-3"><tspan
            sodipodi:role="line"
            id="tspan1007-6"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="67.017166">$\{b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="176.20593"
          y="67.008698"
          id="text1009-7"><tspan
            sodipodi:role="line"
            id="tspan1007-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="67.008698">$\{c\}$</tspan></text>
     </g>
@@ -346,35 +346,35 @@
        style="fill:#000000">
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-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
          y="40.841526"
          id="text1117"><tspan
            sodipodi:role="line"
            id="tspan1115"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="40.841526">$\{a,c\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="40.841526"
          id="text1117-3"><tspan
            sodipodi:role="line"
            id="tspan1115-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="40.841526">$\{a,b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="176.20593"
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          id="text1117-6"><tspan
            sodipodi:role="line"
            id="tspan1115-2"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="40.841526">$\{b,c\}$</tspan></text>
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@@ -433,20 +433,14 @@
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        id="path3663"
-       cx="117.08417"
-       cy="38.792969"
+       cx="147.35568"
+       cy="95.24971"
        r="2.7302806" />
     <circle
        style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-3"
-       cx="126.08154"
-       cy="67.968384"
-       r="2.7302806" />
-    <circle
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-       id="path3663-6"
-       cx="119.58919"
-       cy="67.645035"
+       cx="176.97627"
+       cy="68.155472"
        r="2.7302806" />
     <circle
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@@ -468,13 +462,13 @@
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        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="6.9525447"
        y="13.702217"
        id="text3707"><tspan
          sodipodi:role="line"
          id="tspan3705"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="6.9525447"
          y="13.702217">$write(\{a\})$:</tspan></text>
     <text
@@ -490,37 +484,26 @@
          y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="33.02087"
        id="text3750-3"><tspan
          sodipodi:role="line"
          id="tspan3748-2"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="41.972523"
        id="text3750-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="41.972523">$\sqsupseteq \{a\}$</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="52.022549"
-       id="text4280"><tspan
-         sodipodi:role="line"
-         id="tspan4278"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="52.022549">return OK</tspan></text>
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-1-4"
@@ -563,25 +546,31 @@
          y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="23.457415"
        y="84.598038"
        id="text3750-3-4"><tspan
          sodipodi:role="line"
          id="tspan3748-2-3"
-         style="fill:#999999;stroke-width:0.264583"
+         style="fill:#000000;stroke-width:0.264583"
          x="23.457415"
-         y="84.598038">$\not\sqsupseteq \{b\}$</tspan></text>
+         y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="93.54969"
        id="text3750-0-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6-7"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="93.54969">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6"
+       cx="118.63729"
+       cy="67.630196"
+       r="2.7302806" />
   </g>
 </svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/lattice9.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_4.svg
index 8b3c6585..c5f6148d 100644
--- a/doc/talks/2023-01-18-tocatta/assets/lattice9.svg
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_4.svg
@@ -3,12 +3,12 @@
 
 <svg
    width="1000"
-   height="400"
-   viewBox="0 0 264.58333 105.83333"
+   height="600"
+   viewBox="0 0 264.58333 158.75"
    version="1.1"
    id="svg5"
    inkscape:version="1.2.2 (b0a8486541, 2022-12-01)"
-   sodipodi:docname="lattice9.svg"
+   sodipodi:docname="latticeB_4.svg"
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
    xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
    xmlns="http://www.w3.org/2000/svg"
@@ -24,9 +24,9 @@
      inkscape:deskcolor="#d1d1d1"
      inkscape:document-units="mm"
      showgrid="false"
-     inkscape:zoom="1.4734708"
-     inkscape:cx="451.65469"
-     inkscape:cy="272.14655"
+     inkscape:zoom="1.0419012"
+     inkscape:cx="505.32622"
+     inkscape:cy="316.72869"
      inkscape:window-width="1920"
      inkscape:window-height="999"
      inkscape:window-x="0"
@@ -270,10 +270,10 @@
      id="layer1"
      sodipodi:insensitive="true">
     <rect
-       style="fill:#ffffff;stroke:none;stroke-width:1;stop-color:#000000"
+       style="fill:#ffffff;stroke:none;stroke-width:0.999998;stop-color:#000000"
        id="rect288"
-       width="209.84705"
-       height="104.42732"
+       width="262.58151"
+       height="156.82782"
        x="0.77790999"
        y="0.93738818" />
   </g>
@@ -283,24 +283,24 @@
      inkscape:label="Layer 2">
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="99.307442"
        id="text951"><tspan
          sodipodi:role="line"
          id="tspan949"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="99.307442">$\{\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="13.151893"
        id="text1005"><tspan
          sodipodi:role="line"
          id="tspan1003"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="13.151893">$\{a,b,c\}$</tspan></text>
     <g
@@ -308,35 +308,35 @@
        transform="translate(51.996784,3.5774043)">
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="67.008698"
          id="text1009"><tspan
            sodipodi:role="line"
            id="tspan1007"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="67.008698">$\{a\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
          y="67.017166"
          id="text1009-3"><tspan
            sodipodi:role="line"
            id="tspan1007-6"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="67.017166">$\{b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="176.20593"
          y="67.008698"
          id="text1009-7"><tspan
            sodipodi:role="line"
            id="tspan1007-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="67.008698">$\{c\}$</tspan></text>
     </g>
@@ -346,35 +346,35 @@
        style="fill:#000000">
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="112.90984"
          y="40.841526"
          id="text1117"><tspan
            sodipodi:role="line"
            id="tspan1115"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="40.841526">$\{a,c\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="40.841526"
          id="text1117-3"><tspan
            sodipodi:role="line"
            id="tspan1115-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="40.841526">$\{a,b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="176.20593"
          y="40.841526"
          id="text1117-6"><tspan
            sodipodi:role="line"
            id="tspan1115-2"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="40.841526">$\{b,c\}$</tspan></text>
     </g>
@@ -433,20 +433,14 @@
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663"
-       cx="177.03145"
-       cy="67.991631"
+       cx="147.35568"
+       cy="95.24971"
        r="2.7302806" />
     <circle
        style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-3"
-       cx="126.08154"
-       cy="67.968384"
-       r="2.7302806" />
-    <circle
-       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
-       id="path3663-6"
-       cx="119.58919"
-       cy="67.645035"
+       cx="176.97627"
+       cy="68.155472"
        r="2.7302806" />
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
@@ -468,13 +462,13 @@
        r="2.7302806" />
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="6.9525447"
        y="13.702217"
        id="text3707"><tspan
          sodipodi:role="line"
          id="tspan3705"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="6.9525447"
          y="13.702217">$write(\{a\})$:</tspan></text>
     <text
@@ -490,15 +484,15 @@
          y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="33.02087"
        id="text3750-3"><tspan
          sodipodi:role="line"
          id="tspan3748-2"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
     <text
        xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
@@ -510,17 +504,6 @@
          style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
          y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="52.022549"
-       id="text4280"><tspan
-         sodipodi:role="line"
-         id="tspan4278"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="52.022549">return OK</tspan></text>
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-1-4"
@@ -552,6 +535,17 @@
          y="65.279381">$write(\{b\})$:</tspan></text>
     <text
        xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="59.340172"
+       y="118.94403"
+       id="text3707-5-3"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6"
+         style="fill:#000000;stroke-width:0.264583"
+         x="59.340172"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="75.619804"
@@ -563,25 +557,31 @@
          y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="23.457415"
        y="84.598038"
        id="text3750-3-4"><tspan
          sodipodi:role="line"
          id="tspan3748-2-3"
-         style="fill:#999999;stroke-width:0.264583"
+         style="fill:#000000;stroke-width:0.264583"
          x="23.457415"
-         y="84.598038">$\not\sqsupseteq \{b\}$</tspan></text>
+         y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="93.54969"
        id="text3750-0-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6-7"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="93.54969">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6"
+       cx="118.63729"
+       cy="67.630196"
+       r="2.7302806" />
   </g>
 </svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/latticeB.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_5.svg
index 06725d75..c2b668be 100644
--- a/doc/talks/2023-01-18-tocatta/assets/latticeB.svg
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_5.svg
@@ -3,12 +3,12 @@
 
 <svg
    width="1000"
-   height="400"
-   viewBox="0 0 264.58333 105.83333"
+   height="600"
+   viewBox="0 0 264.58333 158.75"
    version="1.1"
    id="svg5"
    inkscape:version="1.2.2 (b0a8486541, 2022-12-01)"
-   sodipodi:docname="latticeB.svg"
+   sodipodi:docname="latticeB_5.svg"
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
    xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
    xmlns="http://www.w3.org/2000/svg"
@@ -25,8 +25,8 @@
      inkscape:document-units="mm"
      showgrid="false"
      inkscape:zoom="1.4734708"
-     inkscape:cx="450.97602"
-     inkscape:cy="272.14655"
+     inkscape:cx="408.21983"
+     inkscape:cy="319.65343"
      inkscape:window-width="1920"
      inkscape:window-height="999"
      inkscape:window-x="0"
@@ -270,10 +270,10 @@
      id="layer1"
      sodipodi:insensitive="true">
     <rect
-       style="fill:#ffffff;stroke:none;stroke-width:1;stop-color:#000000"
+       style="fill:#ffffff;stroke:none;stroke-width:0.999998;stop-color:#000000"
        id="rect288"
-       width="209.84705"
-       height="104.42732"
+       width="262.58151"
+       height="156.82782"
        x="0.77790999"
        y="0.93738818" />
   </g>
@@ -283,24 +283,24 @@
      inkscape:label="Layer 2">
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="99.307442"
        id="text951"><tspan
          sodipodi:role="line"
          id="tspan949"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="99.307442">$\{\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
        x="164.56372"
        y="13.151893"
        id="text1005"><tspan
          sodipodi:role="line"
          id="tspan1003"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="164.56372"
          y="13.151893">$\{a,b,c\}$</tspan></text>
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@@ -308,35 +308,35 @@
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-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
          y="67.008698"
          id="text1009"><tspan
            sodipodi:role="line"
            id="tspan1007"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="67.008698">$\{a\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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          id="text1009-3"><tspan
            sodipodi:role="line"
            id="tspan1007-6"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="67.017166">$\{b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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          id="text1009-7"><tspan
            sodipodi:role="line"
            id="tspan1007-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="67.008698">$\{c\}$</tspan></text>
     </g>
@@ -346,35 +346,35 @@
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-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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          id="text1117"><tspan
            sodipodi:role="line"
            id="tspan1115"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="112.90984"
            y="40.841526">$\{a,c\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
          x="49.27084"
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          id="text1117-3"><tspan
            sodipodi:role="line"
            id="tspan1115-5"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="49.27084"
            y="40.841526">$\{a,b\}$</tspan></text>
       <text
          xml:space="preserve"
-         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;stroke-width:0.264583;fill:#000000"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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          id="text1117-6"><tspan
            sodipodi:role="line"
            id="tspan1115-2"
-           style="stroke-width:0.264583;fill:#000000"
+           style="fill:#000000;stroke-width:0.264583"
            x="176.20593"
            y="40.841526">$\{b,c\}$</tspan></text>
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@@ -433,20 +433,14 @@
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        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663"
-       cx="117.08417"
-       cy="38.792969"
+       cx="147.35568"
+       cy="95.24971"
        r="2.7302806" />
     <circle
        style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-3"
-       cx="123.57843"
-       cy="39.100433"
-       r="2.7302806" />
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-       id="path3663-6"
-       cx="119.58919"
-       cy="67.645035"
+       cx="176.97627"
+       cy="68.155472"
        r="2.7302806" />
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
@@ -468,13 +462,13 @@
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     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583;fill:#000000"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
        x="6.9525447"
        y="13.702217"
        id="text3707"><tspan
          sodipodi:role="line"
          id="tspan3705"
-         style="stroke-width:0.264583;fill:#000000"
+         style="fill:#000000;stroke-width:0.264583"
          x="6.9525447"
          y="13.702217">$write(\{a\})$:</tspan></text>
     <text
@@ -490,48 +484,26 @@
          y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="33.02087"
        id="text3750-3"><tspan
          sodipodi:role="line"
          id="tspan3748-2"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="33.02087">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="41.972523"
        id="text3750-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="41.972523">$\sqsupseteq \{a\}$</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="52.022549"
-       id="text4280"><tspan
-         sodipodi:role="line"
-         id="tspan4278"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="52.022549">return OK</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
-       x="14.395845"
-       y="102.48464"
-       id="text4280-8"><tspan
-         sodipodi:role="line"
-         id="tspan4278-8"
-         style="fill:#000000;stroke-width:0.264583"
-         x="14.395845"
-         y="102.48464">return OK</tspan></text>
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
     <circle
        style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
        id="path3663-1-4"
@@ -563,6 +535,17 @@
          y="65.279381">$write(\{b\})$:</tspan></text>
     <text
        xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="59.340172"
+       y="118.94403"
+       id="text3707-5-3"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6"
+         style="fill:#000000;stroke-width:0.264583"
+         x="59.340172"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
        style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="75.619804"
@@ -585,14 +568,37 @@
          y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
     <text
        xml:space="preserve"
-       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
        x="23.457415"
        y="93.54969"
        id="text3750-0-0"><tspan
          sodipodi:role="line"
          id="tspan3748-6-7"
-         style="fill:#000000;stroke-width:0.264583"
+         style="fill:#999999;stroke-width:0.264583"
          x="23.457415"
-         y="93.54969">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6"
+       cx="118.63729"
+       cy="67.630196"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-0"
+       cx="67.201363"
+       cy="124.91125"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="72.653946"
+       y="127.6125"
+       id="text3750-9"><tspan
+         sodipodi:role="line"
+         id="tspan3748-3"
+         style="fill:#000000;stroke-width:0.264583"
+         x="72.653946"
+         y="127.6125">$\to \{a\}$</tspan></text>
   </g>
 </svg>
diff --git a/doc/talks/2023-01-18-tocatta/assets/latticeB_6.svg b/doc/talks/2023-01-18-tocatta/assets/latticeB_6.svg
new file mode 100644
index 00000000..980823fc
--- /dev/null
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_6.svg
@@ -0,0 +1,632 @@
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+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
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+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
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+         id="tspan3748-2-0"
+         style="fill:#000000;stroke-width:0.264583"
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new file mode 100644
index 00000000..154c0b7d
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index 00000000..21766415
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+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
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+         x="23.457415"
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+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
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+         x="72.653946"
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+         id="tspan3748-2-0"
+         style="fill:#000000;stroke-width:0.264583"
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+         id="tspan3748-3-8"
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+         id="tspan3705-4-6-5"
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+    <text
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new file mode 100644
index 00000000..b60f8afe
--- /dev/null
+++ b/doc/talks/2023-01-18-tocatta/assets/latticeB_9.svg
@@ -0,0 +1,699 @@
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+           style="fill:#000000;stroke-width:0.264583"
+           x="112.90984"
+           y="67.017166">$\{b\}$</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
+         x="176.20593"
+         y="67.008698"
+         id="text1009-7"><tspan
+           sodipodi:role="line"
+           id="tspan1007-5"
+           style="fill:#000000;stroke-width:0.264583"
+           x="176.20593"
+           y="67.008698">$\{c\}$</tspan></text>
+    </g>
+    <g
+       id="g1183"
+       transform="translate(51.996784,1.0317046)"
+       style="fill:#000000">
+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
+         x="112.90984"
+         y="40.841526"
+         id="text1117"><tspan
+           sodipodi:role="line"
+           id="tspan1115"
+           style="fill:#000000;stroke-width:0.264583"
+           x="112.90984"
+           y="40.841526">$\{a,c\}$</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
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+         id="text1117-3"><tspan
+           sodipodi:role="line"
+           id="tspan1115-5"
+           style="fill:#000000;stroke-width:0.264583"
+           x="49.27084"
+           y="40.841526">$\{a,b\}$</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;text-align:center;text-anchor:middle;fill:#000000;stroke-width:0.264583"
+         x="176.20593"
+         y="40.841526"
+         id="text1117-6"><tspan
+           sodipodi:role="line"
+           id="tspan1115-2"
+           style="fill:#000000;stroke-width:0.264583"
+           x="176.20593"
+           y="40.841526">$\{b,c\}$</tspan></text>
+    </g>
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2)"
+       d="M 153.33622,90.367682 118.34198,73.428915"
+       id="path1300" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9)"
+       d="M 177.46016,90.367682 212.4544,73.428915"
+       id="path1300-2" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-93)"
+       d="M 153.33622,61.655656 118.34198,44.716889"
+       id="path1300-0" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-7)"
+       d="M 177.46016,61.655656 212.4544,44.716889"
+       id="path1300-2-6" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-93-1)"
+       d="M 118.34198,61.655656 153.33622,44.716889"
+       id="path1300-0-7" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-7-2)"
+       d="M 212.4544,61.655656 177.46016,44.716889"
+       id="path1300-2-6-9" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-93-1-4)"
+       d="M 118.34198,34.227412 153.33622,17.288645"
+       id="path1300-0-7-5" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-7-2-8)"
+       d="M 212.4544,34.227412 177.46016,17.288645"
+       id="path1300-2-6-9-0" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75)"
+       d="m 228.52843,61.091809 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75-2)"
+       d="m 101.90418,61.091809 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2-9"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75-7)"
+       d="m 165.29305,89.571762 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2-6"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow2-9-75-7-1)"
+       d="m 165.29305,32.445235 0.33313,-12.554874 m -0.33313,12.554874 0.33313,-12.554874"
+       id="path1300-2-2-6-9"
+       sodipodi:nodetypes="cc" />
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663"
+       cx="147.35568"
+       cy="95.24971"
+       r="2.7302806" />
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3"
+       cx="176.97627"
+       cy="68.155472"
+       r="2.7302806" />
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1"
+       cx="18.004833"
+       cy="39.402473"
+       r="2.7302806" />
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0"
+       cx="18.004833"
+       cy="30.371933"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6"
+       cx="18.004833"
+       cy="21.341394"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="13.702217"
+       id="text3707"><tspan
+         sodipodi:role="line"
+         id="tspan3705"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="13.702217">$write(\{a\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="24.042637"
+       id="text3750"><tspan
+         sodipodi:role="line"
+         id="tspan3748"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="24.042637">$\sqsupseteq \{a\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="33.02087"
+       id="text3750-3"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="33.02087">$\not\sqsupseteq \{a\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="41.972523"
+       id="text3750-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="41.972523">$\not\sqsupseteq \{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-1-4"
+       cx="18.004833"
+       cy="90.979645"
+       r="2.7302806" />
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-5"
+       cx="18.004833"
+       cy="81.949104"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-2"
+       cx="18.004833"
+       cy="72.918564"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="6.9525447"
+       y="65.279381"
+       id="text3707-5"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4"
+         style="fill:#000000;stroke-width:0.264583"
+         x="6.9525447"
+         y="65.279381">$write(\{b\})$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="59.340172"
+       y="118.94403"
+       id="text3707-5-3"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6"
+         style="fill:#000000;stroke-width:0.264583"
+         x="59.340172"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="75.619804"
+       id="text3750-7"><tspan
+         sodipodi:role="line"
+         id="tspan3748-4"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="75.619804">$\not\sqsupseteq \{b\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="23.457415"
+       y="84.598038"
+       id="text3750-3-4"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-3"
+         style="fill:#000000;stroke-width:0.264583"
+         x="23.457415"
+         y="84.598038">$\sqsupseteq \{b\} \to$ OK</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#999999;stroke-width:0.264583"
+       x="23.457415"
+       y="93.54969"
+       id="text3750-0-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-6-7"
+         style="fill:#999999;stroke-width:0.264583"
+         x="23.457415"
+         y="93.54969">$\not\sqsupseteq \{b\}$</tspan></text>
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6"
+       cx="118.63729"
+       cy="67.630196"
+       r="2.7302806" />
+    <circle
+       style="fill:#800080;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-0"
+       cx="67.201363"
+       cy="124.91125"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="72.653946"
+       y="127.6125"
+       id="text3750-9"><tspan
+         sodipodi:role="line"
+         id="tspan3748-3"
+         style="fill:#000000;stroke-width:0.264583"
+         x="72.653946"
+         y="127.6125">$\to \{a\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="62.764446"
+       y="146.3129"
+       id="text4280-6"><tspan
+         sodipodi:role="line"
+         style="fill:#000000;stroke-width:0.264583"
+         x="62.764446"
+         y="146.3129"
+         id="tspan2630">return $\{a\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-7"
+       cx="67.201363"
+       cy="133.94179"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="72.653946"
+       y="136.59073"
+       id="text3750-3-6"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-0"
+         style="fill:#000000;stroke-width:0.264583"
+         x="72.653946"
+         y="136.59073">$\to \{\}$</tspan></text>
+    <circle
+       style="fill:#ff0000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-6-6-0-6"
+       cx="174.36296"
+       cy="124.91125"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="179.81555"
+       y="127.6125"
+       id="text3750-9-1"><tspan
+         sodipodi:role="line"
+         id="tspan3748-3-8"
+         style="fill:#000000;stroke-width:0.264583"
+         x="179.81555"
+         y="127.6125">$\to \{\}$</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="162.7999"
+       y="118.94403"
+       id="text3707-5-3-7"><tspan
+         sodipodi:role="line"
+         id="tspan3705-4-6-5"
+         style="fill:#000000;stroke-width:0.264583"
+         x="162.7999"
+         y="118.94403">$read()$:</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;stroke-width:0.264583"
+       x="134.97845"
+       y="132.58472"
+       id="text3899"><tspan
+         sodipodi:role="line"
+         id="tspan3897"
+         style="stroke-width:0.264583"
+         x="134.97845"
+         y="132.58472">;</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="169.92604"
+       y="146.3129"
+       id="text4280-6-7"><tspan
+         sodipodi:role="line"
+         style="fill:#000000;stroke-width:0.264583"
+         x="169.92604"
+         y="146.3129"
+         id="tspan2630-9">return $\{b\}$</tspan></text>
+    <circle
+       style="fill:#008000;stroke:none;stroke-width:0.499999;stroke-dasharray:none;stop-color:#000000"
+       id="path3663-3-0-7-2"
+       cx="174.36296"
+       cy="133.94179"
+       r="2.7302806" />
+    <text
+       xml:space="preserve"
+       style="font-size:8.46667px;line-height:1.25;font-family:sans-serif;fill:#000000;stroke-width:0.264583"
+       x="179.81555"
+       y="136.59073"
+       id="text3750-3-6-0"><tspan
+         sodipodi:role="line"
+         id="tspan3748-2-0-2"
+         style="fill:#000000;stroke-width:0.264583"
+         x="179.81555"
+         y="136.59073">$\to \{b\}$</tspan></text>
+  </g>
+</svg>