diff --git a/darc_substitution.ipynb b/darc_substitution.ipynb
new file mode 100644
index 0000000..433094c
--- /dev/null
+++ b/darc_substitution.ipynb
@@ -0,0 +1,92 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "id": "initial_id",
+   "metadata": {
+    "collapsed": true,
+    "ExecuteTime": {
+     "end_time": "2024-08-02T22:22:16.261963Z",
+     "start_time": "2024-08-02T22:22:15.861856Z"
+    }
+   },
+   "source": "from src.models import SubstitutionKey\n",
+   "outputs": [],
+   "execution_count": 2
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2024-08-02T22:22:16.442689Z",
+     "start_time": "2024-08-02T22:22:16.434147Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "width = 7\n",
+    "height = 10\n",
+    "s1 = SubstitutionKey.init_matrix(width, height)\n",
+    "s2 = SubstitutionKey.init_matrix(width, height)\n",
+    "s3 = SubstitutionKey.init_matrix(width, height)\n",
+    "\n",
+    "c1 = SubstitutionKey.init_matrix(width, height)\n",
+    "c2 = SubstitutionKey.init_matrix(width, height)\n",
+    "c3 = SubstitutionKey.init_matrix(width, height)\n",
+    "\n",
+    "m1 = SubstitutionKey.init_matrix(width, height)\n",
+    "m2 = SubstitutionKey.init_matrix(width, height)\n",
+    "m3 = SubstitutionKey.init_matrix(width, height)\n",
+    "m4 = SubstitutionKey.init_matrix(width, height)\n",
+    "m5 = SubstitutionKey.init_matrix(width, height)\n",
+    "\n",
+    "s_m1 = s1 ^ m4\n",
+    "c_m1 = c1 ^ m4\n",
+    "s_m4 = s1 ^ m5\n",
+    "c_m4 = c1 ^ m5\n",
+    "\n",
+    "m_c1 = SubstitutionKey.init_matrix(width, height)\n",
+    "m_s1 = SubstitutionKey.init_matrix(width, height) \n",
+    "\n",
+    "m_s2 = m1 ^ m_c1\n",
+    "m_c2 = m1 ^ m_s1\n",
+    "\n",
+    "s_m3 = m_s2 ^ s2 ^ m3\n",
+    "s_m2 = m1 ^ s2 ^ m2 ^ m_s1\n",
+    "c_m2 = m_c2 ^ c2 ^ m2\n",
+    "c_m3 = m1 ^ m_c1 ^ c2 ^ m3\n"
+   ],
+   "id": "58883d386ec57cc4",
+   "outputs": [],
+   "execution_count": 3
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "outputs": [],
+   "execution_count": null,
+   "source": "",
+   "id": "2e1b925a48925228"
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/docs/spn_algebra_template.md b/docs/spn_algebra_template.md
new file mode 100644
index 0000000..843ba4b
--- /dev/null
+++ b/docs/spn_algebra_template.md
@@ -0,0 +1,278 @@
+# Permutation Algebra
+This notebook is a prerequisite to following the DARC tutorial
+
+
+
+## Key Generation Parameters
+- block size: The number of characters we encode in a block. Block size isn't using in this notebook
+- key height: The alphabet length. If we want to encode bytes, our alphabet length is 256.
+  If we want to encode lowercase letters a-z our alphabet length is 26. NKodes {{ height }} key {{ width }} attribute alphabet is {{ total_attr }}.
+- key width: The number of bytes an encrypted charter is in our alphabet.
+
+```
+height = {{ height }}
+width = {{ width }} 
+```
+
+## Operand Types
+### Inner Permutation Key
+An inner permutation key (inner key for short) is a list of `height` rows, each a random permutation of an identity array of length `width`.
+
+```
+i0 = InnerKey.init_matrix(width, height)
+i1 = InnerKey.init_matrix(width, height)
+i2 = InnerKey.init_matrix(width, height)
+i_identity = InnerKey.init_identity_matrix(width, height)
+
+i0.matrix:
+{% for row in i0.matrix -%}
+{{ row }}
+{% endfor %}
+
+i1.matrix:
+{% for row in i1.matrix -%}
+{{ row }}
+{% endfor %}
+
+i2.matrix:
+{% for row in i2.matrix -%}
+{{ row }}
+{% endfor %}
+
+i_identity.matrix:
+{% for row in i_identity.matrix -%}
+{{ row }}
+{% endfor %}
+```
+
+
+### Outer Permutation Key
+An outer key is a list of `height` columns, each a random permutation of an identity array of length `height`.
+It is used to permute the rows of inner, substitution and outer keys.
+
+```
+o0 = OuterKey.init_matrix(height)
+o1 = OuterKey.init_matrix(height)
+o2 = OuterKey.init_matrix(height)
+o_identity = OuterKey.init_identity_matrix(height)
+
+o0.matrix:
+{{ o0.matrix }}
+
+o1.matrix:
+{{ o1.matrix }}
+
+o2.matrix:
+{{ o2.matrix }}
+
+o_identity.matrix:
+{{ o_identity.matrix }}
+```
+
+### Substitution Key
+Substitution key is a matrix of `height` rows and `width` columns. Each row is a list of randomly generated bytes.
+
+```
+s0 = SubstitutionKey.init_matrix(width, height)
+s1 = SubstitutionKey.init_matrix(width, height)
+s2 = SubstitutionKey.init_matrix(width, height)
+s_identity = SubstitutionKey.init_identity_matrix(width, height)
+
+s0.matrix:
+{% for row in s0.matrix -%}
+{{ row }}
+{% endfor %}
+
+s1.matrix:
+{% for row in s1.matrix -%}
+{{ row }}
+{% endfor %}
+
+s2.matrix:
+{% for row in s2.matrix -%}
+{{ row }}
+{% endfor %}
+
+s_identity.matrix:
+{% for row in s_identity.matrix -%}
+{{ row }}
+{% endfor %}
+```
+
+
+## Operators Types
+### < Outer Permutation
+#### Substitution Key < Outer Key
+```
+s0_o0 = s0 < o0
+```
+
+| idx |s0|s0| s0_o0 |
+|-----|-|-|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ s0.matrix[idx] }}|{{ o0.matrix[0][idx] }}|{{ s0_o0.matrix[idx] }}|
+{% endfor %}
+
+#### Inner Key < Outer Key
+i0_o0 = i0 < o0
+
+| idx | i0 | o0 | i0_o0 |
+|-----|----|----|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ i0.matrix[idx] }}|{{ o0.matrix[0][idx] }}|{{ i0_o0.matrix[idx] }}|
+{% endfor %}
+
+### << Inner Permutation
+#### Outer Key << Outer Key
+o0_o1 = o0 << o1
+
+| idx | o0 | o1 | o0_o1 |
+|-----|----|----|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ o0.matrix[0][idx] }}|{{ o1.matrix[0][idx] }}|{{ o0_o1.matrix[0][idx] }}|
+{% endfor %}
+
+#### Inner Key << Inner Key
+i0_i1 = i0 << i1
+
+| idx | i0                                                       | i1 | i0_i1 |
+|-----|----------------------------------------------------------|----|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ i0.matrix[idx] }}|{{ i1.matrix[idx] }}|{{ i0_i1.matrix[idx] }}|
+{% endfor %}
+
+#### Substitution Key << Inner Key
+s0_i0 = s0 << i0
+
+| idx | s0 | i0 | s0_i0 |
+|-----|----|----|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ s0.matrix[idx] }}|{{ i0.matrix[idx] }}|{{ s0_i0.matrix[idx] }}|
+{% endfor %}
+
+### ~Permutation Inversion
+#### ~Outer Key
+inv_o0 = ~o0
+
+| idx | o0 | ~o0 |
+|-----|----|-----|
+{% for idx in range(height) -%}
+|{{idx}}|{{ o0.matrix[0][idx] }}|{{ inv_o0.matrix[0][idx] }}|
+{% endfor %}
+
+
+#### ~Inner Key
+inv_i0 = ~i0
+
+| idx | i0 | ~i0 |
+|-----|----|-----|
+{% for idx in range(height) -%}
+|{{idx}}|{{ i0.matrix[idx] }}|{{ inv_i0.matrix[idx] }}|
+{% endfor %}
+
+### ^ Substitution
+#### Substitution Key ^ Substitution Key
+s0_s1 = s0 ^ s1
+
+| idx | s0 | s1 | s0_s1 |
+|-----|----|----|-------|
+{% for idx in range(height) -%}
+|{{idx}}|{{ s0.matrix[idx] }}|{{ s1.matrix[idx] }}|{{ s0_s1.matrix[idx] }}|
+{% endfor %}
+
+## Substitution Permutation Algebra
+#### properites:
+- associative: (a + b) + c = a + (b + c)
+- commutative: a + b = b + a
+- identity: a + 0 = a or a * 1 = a
+- inverse: a + (-a) = 0 or a * 1/a = 1 (for all a ~= 0)
+- distributive: a(b + c) = ab + ac
+
+### Associative
+#### Outer Key
+```
+(o0 << o1 << o2) == ((o0 << o1) << o2) == (o0 << (o1 << o2))
+```
+#### Inner Key
+```
+(i0 << i1 << i2) == ((i0 << i1) << i2) == (i0 << (i1 << i2))
+```
+
+#### Substitution Key
+```
+(s0 ^ s1 ^ s2) == ((s0 ^ s1) ^ s2) == (s0 ^ (s1 ^ s2))
+```
+
+### Commutative Property
+Substitution is the only key type that is commutative
+
+#### Substitution Key
+```
+(s0 ^ s1) == (s1 ^ s0)
+```
+
+### Identity Property
+#### Outer Key
+```
+(o0 << o_identity) == o0
+```
+
+#### Inner Key
+```
+(i0 << i_identity) == i0
+```
+
+#### Substitution Key
+```
+(s0 ^ s_identity) == s0
+```
+
+### Inverse Property
+#### Outer Key
+```
+(o0 << ~o0) == o_identity
+```
+
+#### Inner Key
+```
+(i0 << ~i0) == i_identity
+```
+
+#### Substitution Key
+```
+(s0 ^ s0) == s_identity
+```
+
+### Distributive
+#### Inner Key/Outer Key
+```
+((i0 << i1) < o0) == ((i0 < o0) << (i1 < o0))
+```
+
+#### Substitution Key/Outer Key
+```
+((s0 ^ s1) < o0) == ((s0 < o0) ^ (s1 < o0))
+```
+#### Substitution Key/Inner Key
+```
+((s0 ^ s1) << i0) == ((s0 << i0) ^ (s1 << i0))
+```
+#### Substitution Key/Inner Key/Outer Key
+```
+((s0 << i0) < o0) == ((s0 < o0) << (i0 < o0))
+```
+### Other Examples
+```
+(s0 << (i0 < o0)) == (((s0 < ~o0) << i0) < o0)
+((s0 < o0) << i0) == ((s0 << (i0 < ~o0)) < o0)
+(~(i0 << i1)) == (~i1 << ~i0)
+(~(o0 << o1)) == (~o1 << ~o0)
+i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)
+```
+
+### Becareful about your order of operation
+***Always use parenthesis to control the order of operation***
+
+```
+i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!
+```
diff --git a/render_markdown.py b/render_markdown.py
new file mode 100644
index 0000000..5f0ad03
--- /dev/null
+++ b/render_markdown.py
@@ -0,0 +1,83 @@
+from jinja2 import Environment, FileSystemLoader
+from src.models import OuterKey, InnerKey, SubstitutionKey
+
+
+def render_spn_algebra(data: dict):
+    # Set up the Jinja2 environment and template loader
+    file_loader = FileSystemLoader('./docs')
+    env = Environment(loader=file_loader)
+
+    # Load the template
+    template = env.get_template('spn_algebra_template.md')
+    # Render the template with the data
+    output = template.render(data)
+
+    # Print or save the output
+    output_file = "/Users/donov/Desktop/NKode_documentation/nkode/docs/spn_algebra.md"
+    with open(output_file, 'w') as fp:
+        fp.write(output)
+        print("File written successfully")
+
+
+if __name__ == "__main__":
+    height = 10
+    width = 7
+    total_attr = height * width
+    i0 = InnerKey.init_matrix(width, height)
+    i1 = InnerKey.init_matrix(width, height)
+    i2 = InnerKey.init_matrix(width, height)
+    i_identity = InnerKey.init_identity_matrix(width, height)
+
+    o0 = OuterKey.init_matrix(height)
+    o1 = OuterKey.init_matrix(height)
+    o2 = OuterKey.init_matrix(height)
+    o_identity = OuterKey.init_identity_matrix(height)
+
+    s0 = SubstitutionKey.init_matrix(width, height)
+    s1 = SubstitutionKey.init_matrix(width, height)
+    s2 = SubstitutionKey.init_matrix(width, height)
+    s_identity = SubstitutionKey.init_identity_matrix(width, height)
+
+    s0_o0 = s0 < o0
+
+    i0_o0 = i0 < o0
+
+    o0_o1 = o0 << o1
+
+    i0_i1 = i0 << i1
+
+    s0_i0 = s0 << i0
+
+    inv_o0 = ~o0
+
+    inv_i0 = ~i0
+
+    s0_s1 = s0 ^ s1
+
+    data = {
+        'height': height,
+        'width': width,
+        'total_attr': total_attr,
+        'i0': i0,
+        'i1': i1,
+        'i2': i2,
+        'i_identity': i_identity,
+        'o0': o0,
+        'o1': o1,
+        'o2': o2,
+        'o_identity': o_identity,
+        's0': s0,
+        's1': s1,
+        's2': s2,
+        's_identity': s_identity,
+        's0_o0': s0_o0,
+        'i0_o0': i0_o0,
+        'o0_o1': o0_o1,
+        'i0_i1': i0_i1,
+        's0_i0': s0_i0,
+        'inv_o0': inv_o0,
+        'inv_i0': inv_i0,
+        's0_s1': s0_s1
+    }
+
+    render_spn_algebra(data)
diff --git a/sp_network_algebra_tutorial.ipynb b/sp_network_algebra_tutorial.ipynb
index 132cfed..493f0e9 100644
--- a/sp_network_algebra_tutorial.ipynb
+++ b/sp_network_algebra_tutorial.ipynb
@@ -12,18 +12,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 69,
-   "outputs": [],
    "source": [
     "from src.models import OuterKey, InnerKey, SubstitutionKey"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.685220Z",
-     "start_time": "2024-07-04T14:12:16.446987Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -40,19 +36,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 70,
-   "outputs": [],
    "source": [
     "height = 4\n",
     "width = 3"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.686123Z",
-     "start_time": "2024-07-04T14:12:16.453332Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -67,17 +59,6 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 71,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]"
-     },
-     "execution_count": 71,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "i0 = InnerKey.init_matrix(width, height)\n",
     "i1 = InnerKey.init_matrix(width, height)\n",
@@ -87,12 +68,10 @@
     "i0.matrix"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.830392Z",
-     "start_time": "2024-07-04T14:12:16.460887Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -106,17 +85,6 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 72,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "[[2, 0, 1, 3]]"
-     },
-     "execution_count": 72,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "o0 = OuterKey.init_matrix(height)\n",
     "o1 = OuterKey.init_matrix(height)\n",
@@ -126,17 +94,15 @@
     "o0.matrix"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.831572Z",
-     "start_time": "2024-07-04T14:12:16.469310Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
    "source": [
-    "### Substitution Key (formerly Alphabet Key or alpha_key)\n",
+    "### Substitution Key\n",
     "Sub key is ..."
    ],
    "metadata": {
@@ -145,32 +111,19 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 73,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]"
-     },
-     "execution_count": 73,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
-    "a0 = SubstitutionKey.init_matrix(width, height)\n",
-    "a1 = SubstitutionKey.init_matrix(width, height)\n",
-    "a2 = SubstitutionKey.init_matrix(width, height)\n",
-    "a_identity = SubstitutionKey.init_identity_matrix(width, height)\n",
+    "s0 = SubstitutionKey.init_matrix(width, height)\n",
+    "s1 = SubstitutionKey.init_matrix(width, height)\n",
+    "s2 = SubstitutionKey.init_matrix(width, height)\n",
+    "s_identity = SubstitutionKey.init_identity_matrix(width, height)\n",
     "\n",
-    "a0.matrix"
+    "s0.matrix"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.832461Z",
-     "start_time": "2024-07-04T14:12:16.476713Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -185,31 +138,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 74,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[2, 0, 1, 3]]\n",
-      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
-      "[[73, 204, 152], [51, 14, 207], [3, 171, 4], [142, 175, 64]]\n"
-     ]
-    }
-   ],
    "source": [
-    "a0_o0 = a0 < o0\n",
+    "s0_o0 = s0 < o0\n",
     "print(o0.matrix)\n",
-    "print(a0.matrix)\n",
-    "print(a0_o0.matrix)"
+    "print(s0.matrix)\n",
+    "print(s0_o0.matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.833320Z",
-     "start_time": "2024-07-04T14:12:16.484335Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -222,18 +161,6 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 75,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[2, 0, 1, 3]]\n",
-      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
-      "[[1, 0, 2], [0, 1, 2], [1, 0, 2], [2, 1, 0]]\n"
-     ]
-    }
-   ],
    "source": [
     "i0_o0 = i0 < o0\n",
     "print(o0.matrix)\n",
@@ -241,12 +168,10 @@
     "print(i0_o0.matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.834157Z",
-     "start_time": "2024-07-04T14:12:16.492114Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -260,18 +185,6 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 76,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[0, 1, 3, 2]]\n",
-      "[[2, 0, 1, 3]]\n",
-      "[[2, 0, 3, 1]]\n"
-     ]
-    }
-   ],
    "source": [
     "o0_o1 = o0 << o1\n",
     "print(o1.matrix)\n",
@@ -279,12 +192,10 @@
     "print(o0_o1.matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.834973Z",
-     "start_time": "2024-07-04T14:12:16.499677Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -297,18 +208,6 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 77,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[1, 0, 2], [0, 1, 2], [0, 2, 1], [2, 1, 0]]\n",
-      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
-      "[[1, 0, 2], [1, 0, 2], [1, 2, 0], [0, 1, 2]]\n"
-     ]
-    }
-   ],
    "source": [
     "i0_i1 = i0 << i1\n",
     "print(i1.matrix)\n",
@@ -316,12 +215,10 @@
     "print(i0_i1.matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.835789Z",
-     "start_time": "2024-07-04T14:12:16.507413Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -334,31 +231,17 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 78,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
-      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
-      "[[51, 14, 207], [171, 3, 4], [204, 73, 152], [64, 175, 142]]\n"
-     ]
-    }
-   ],
    "source": [
-    "a0_i0 = a0 << i0\n",
+    "s0_i0 = s0 << i0\n",
     "print(i0.matrix)\n",
-    "print(a0.matrix)\n",
-    "print(a0_i0.matrix)"
+    "print(s0.matrix)\n",
+    "print(s0_i0.matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.854289Z",
-     "start_time": "2024-07-04T14:12:16.515159Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -372,28 +255,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 79,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[2, 0, 1, 3]]\n",
-      "[[1, 2, 0, 3]]\n"
-     ]
-    }
-   ],
    "source": [
     "print(o0.matrix)\n",
     "print((~o0).matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.855384Z",
-     "start_time": "2024-07-04T14:12:16.522867Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -406,28 +276,15 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 80,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n",
-      "[[0, 1, 2], [1, 0, 2], [1, 0, 2], [2, 1, 0]]\n"
-     ]
-    }
-   ],
    "source": [
     "print(i0.matrix)\n",
     "print((~i0).matrix)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.856225Z",
-     "start_time": "2024-07-04T14:12:16.530564Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -441,33 +298,18 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 81,
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "[[51, 14, 207], [3, 171, 4], [73, 204, 152], [142, 175, 64]]\n",
-      "[[1, 41, 28], [228, 126, 163], [255, 48, 188], [53, 120, 13]]\n",
-      "[[50, 39, 211], [231, 213, 167], [182, 252, 36], [187, 215, 77]]\n",
-      "50\n"
-     ]
-    }
-   ],
    "source": [
-    "a0_a1 = a0 ^ a1\n",
-    "print(a0.matrix)\n",
-    "print(a1.matrix)\n",
-    "print(a0_a1.matrix)\n",
-    "print(a0.matrix[0][0] ^ a1.matrix[0][0])"
+    "s0_s1 = s0 ^ s1\n",
+    "print(s0.matrix)\n",
+    "print(s1.matrix)\n",
+    "print(s0_s1.matrix)\n",
+    "print(s0.matrix[0][0] ^ s1.matrix[0][0])"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.857040Z",
-     "start_time": "2024-07-04T14:12:16.538722Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -489,27 +331,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 82,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 82,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(o0 << o1 << o2) == ((o0 << o1) << o2) == (o0 << (o1 << o2))"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.858322Z",
-     "start_time": "2024-07-04T14:12:16.546322Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -522,27 +351,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 83,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 83,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(i0 << i1 << i2) == ((i0 << i1) << i2) == (i0 << (i1 << i2))"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.863346Z",
-     "start_time": "2024-07-04T14:12:16.554187Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -555,27 +371,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 84,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 84,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(a0 ^ a1 ^ a2) == ((a0 ^ a1) ^ a2) == (a0 ^ (a1 ^ a2))"
-   ],
+   "source": "(s0 ^ s1 ^ s2) == ((s0 ^ s1) ^ s2) == (s0 ^ (s1 ^ s2))",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.864336Z",
-     "start_time": "2024-07-04T14:12:16.562025Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -590,27 +391,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 85,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 85,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(a0 ^ a1) == (a1 ^ a0)"
-   ],
+   "source": "(s0 ^ s1) == (s1 ^ s0)",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.865178Z",
-     "start_time": "2024-07-04T14:12:16.569781Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -624,27 +410,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 86,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 86,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(o0 << o_identity) == o0"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.866342Z",
-     "start_time": "2024-07-04T14:12:16.578046Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -657,27 +430,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 87,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 87,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(i0 << i_identity) == i0"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.867309Z",
-     "start_time": "2024-07-04T14:12:16.589133Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -690,27 +450,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 88,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 88,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(a0 ^ a_identity) == a0"
-   ],
+   "source": "(s0 ^ s_identity) == s0",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.868143Z",
-     "start_time": "2024-07-04T14:12:16.599974Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -724,27 +469,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 89,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 89,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(o0 << ~o0) == o_identity"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.868953Z",
-     "start_time": "2024-07-04T14:12:16.608436Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -757,27 +489,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 90,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 90,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(i0 << ~i0) == i_identity"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.869884Z",
-     "start_time": "2024-07-04T14:12:16.619030Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -790,27 +509,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 91,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 91,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(a0 ^ a0) == a_identity"
-   ],
+   "source": "(s0 ^ s0) == s_identity",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.870708Z",
-     "start_time": "2024-07-04T14:12:16.629750Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -824,27 +528,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 92,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 92,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "((i0 << i1) < o0) == ((i0 < o0) << (i1 < o0))"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.871520Z",
-     "start_time": "2024-07-04T14:12:16.638198Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -857,27 +548,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 93,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 93,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "((a0 ^ a1) < o0) == ((a0 < o0) ^ (a1 < o0))"
-   ],
+   "source": "((s0 ^ s1) < o0) == ((s0 < o0) ^ (s1 < o0))",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.872334Z",
-     "start_time": "2024-07-04T14:12:16.646323Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -890,27 +566,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 94,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 94,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "((a0 ^ a1) << i0) == ((a0 << i0) ^ (a1 << i0))"
-   ],
+   "source": "((s0 ^ s1) << i0) == ((s0 << i0) ^ (s1 << i0))",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.873134Z",
-     "start_time": "2024-07-04T14:12:16.654286Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -923,27 +584,12 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 95,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 95,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "((a0 << i0) < o0) == ((a0 < o0) << (i0 < o0))"
-   ],
+   "source": "((s0 << i0) < o0) == ((s0 < o0) << (i0 < o0))",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.874410Z",
-     "start_time": "2024-07-04T14:12:16.662231Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -956,123 +602,54 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 96,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 96,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(a0 << (i0 < o0)) == (((a0 < ~o0) << i0) < o0)"
-   ],
+   "source": "(s0 << (i0 < o0)) == (((s0 < ~o0) << i0) < o0)",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.882191Z",
-     "start_time": "2024-07-04T14:12:16.670307Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "code",
-   "execution_count": 97,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 97,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "((a0 < o0) << i0) == ((a0 << (i0 < ~o0)) < o0)"
-   ],
+   "source": "((s0 < o0) << i0) == ((s0 << (i0 < ~o0)) < o0)",
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.882308Z",
-     "start_time": "2024-07-04T14:12:16.674096Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "code",
-   "execution_count": 98,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 98,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(~(i0 << i1)) == (~i1 << ~i0)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.882394Z",
-     "start_time": "2024-07-04T14:12:16.677061Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "code",
-   "execution_count": 99,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 99,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(~(o0 << o1)) == (~o1 << ~o0)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.883495Z",
-     "start_time": "2024-07-04T14:12:16.683278Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "code",
-   "execution_count": 100,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 100,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "i0 == ((i0 << i2 << i1) << ~(i2 << i1)) == ((i0 << i2 << i1) << ~i1 << ~i2)"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.883574Z",
-     "start_time": "2024-07-04T14:12:16.687331Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   },
   {
    "cell_type": "markdown",
@@ -1086,27 +663,14 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 101,
-   "outputs": [
-    {
-     "data": {
-      "text/plain": "True"
-     },
-     "execution_count": 101,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "i0 < (o0 << o1 << o2) != (i0 < (o0 << o1 << o2)) # not equal !!!!!!"
    ],
    "metadata": {
-    "collapsed": false,
-    "ExecuteTime": {
-     "end_time": "2024-07-04T14:12:16.883645Z",
-     "start_time": "2024-07-04T14:12:16.690509Z"
-    }
-   }
+    "collapsed": false
+   },
+   "outputs": [],
+   "execution_count": null
   }
  ],
  "metadata": {