From 04c39f383a5ff9c6e0a3572a39067b7a719f2ba9 Mon Sep 17 00:00:00 2001
From: gheber <gheber@hdfgroup.org>
Date: Mon, 4 Jul 2016 18:17:38 -0500
Subject: [PATCH] Added Francis' changes.

---
 examples/15_Loop_spaces_fibrations.ipynb | 1402 +++++++++++-----------
 1 file changed, 679 insertions(+), 723 deletions(-)

diff --git a/examples/15_Loop_spaces_fibrations.ipynb b/examples/15_Loop_spaces_fibrations.ipynb
index 88f45ce..e4d2c29 100644
--- a/examples/15_Loop_spaces_fibrations.ipynb
+++ b/examples/15_Loop_spaces_fibrations.ipynb
@@ -8,22 +8,22 @@
     "\n",
     "## The canonical loop space fibration\n",
     "\n",
-    "Let $X$ be a  simplicial set. The software `Kenzo` implements the *canonical* twisted cartesian product of $X$ by its loop space $GX$, denoted $X \\times_\\tau GX$, giving the Kan (**Daniel M. Kan**. *A combinatorial definition of homotopy groups*, Ann. of Math., 1958, vol. 67, pp 282-312.) model of the contractible path space of $X$. The *canonical fibration*\n",
+    "Let $X$ be a  simplicial set. The program `Kenzo` implements the *canonical* twisted cartesian product of $X$ by its loop space $GX$, denoted $X \\times_\\tau GX$, giving the Kan (**Daniel M. Kan**. *A combinatorial definition of homotopy groups*, Ann. of Math., 1958, vol. 67, pp 282-312.) model of the contractible path space of $X$. The *canonical fibration*\n",
     "$$ GX\\quad \\hookrightarrow\\quad X \\times_\\tau GX  \\hskip 1.2em \\longrightarrow \\hskip -1.2em \\longrightarrow  \n",
     "   \\quad X,$$\n",
-    "is defined by the trivial application $\\tau$, where $\\tau(x)$ is the *letter* $x$ in $GX$. This twisted cartesian product has the same simplices as the non-twisted one. So both associated chain complexes are equal as graded modules. The essential difference  resides in the differential morphism. More precisely, the face operator $\\partial_n$ behaves differently in the twisted cartesian product. Let $(x,g) \\in X \\times_\\tau GX$, the rule for the face operators are:\n",
+    "is defined by the application $\\tau$, where $\\tau(x)$ is the *letter* $x$ in $GX$. This twisted cartesian product has the same simplices as the non-twisted one. So both associated chain complexes are equal as graded modules. The essential difference  resides in the differential morphism. More precisely, the face operator $\\partial_n$ behaves differently in the twisted cartesian product. Let $(x,g) \\in (X \\times_\\tau GX)_n$, then the rules for the face operators are:\n",
     "$$\n",
     "\\begin{aligned}\n",
     "\\partial_i(x,g) & =  & (\\partial_i x, \\partial_i g), \\qquad i<n, \\\\\n",
     "\\partial_n(x,g) & =  & (\\partial_n x, \\tau(x).\\partial_n g), \\qquad i=n,\n",
     "\\end{aligned}\n",
     "$$\n",
-    "where $\\tau(x)$ is the *letter* $x$ in $GX$ and '**.**' is the group product in $GX$. Let us recall that, from the mathematical definition of $GX$, if $x$ is $n$-degenerate, for instance $x=\\eta_n x'$, the $\\tau(x)$ is the null element of $GX_n$.\n",
+    "where $\\tau(x)$ is the *letter* $x$ in $GX$ and '**.**' is the group product in $GX$. Let us recall that, from the mathematical definition of $GX$, if $x \\in X_{n+1}$ is $n$-degenerate, for instance $x=\\eta_n x'$, then $\\tau(x)$ is the neutral element of $GX_n$.\n",
     "The behaviour of $\\partial_n$ induces a change in the differential morphism of the chain complex associated to the twisted cartesian product.\n",
     "\n",
     "## The associated tensor twisted product\n",
     "\n",
-    "We recall that around the cartesian product $X \\times GX$,  the Eilenberg-Zilber theorem allows to build the reduction (see the function `ez`):\n",
+    "We recall that about the cartesian product $X \\times GX$,  the Eilenberg-Zilber theorem allows to build the reduction (see the function `ez`):\n",
     "$$\n",
     "\\begin{aligned}\n",
     "{{\\cal C}_*(X \\times GX)} \\quad& \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times GX)} \\cr\n",
@@ -39,7 +39,7 @@
     " \\qquad{{\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)} & & \\cr\n",
     "\\end{aligned}\n",
     "$$\n",
-    "In fact, the interesting object produced by this machinery is the bottom chain complex  ${\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)$ which is nothing but the *twisted tensor product* (**Edgar H. Brown Jr.** *Twisted tensor products, I*,\n",
+    "In fact, the interesting object produced by this machinery is the bottom chain complex ${\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)$, the (algebraic) twisted tensor-product of the \"base space\" $C_\\ast(X)$ by the \"fibre space\" $C_\\ast(GX)$, the twist being defined by a *canonical* twisting cochain $t \\in C^1(X ; C_\\ast(GX))$ (**Edgar H. Brown Jr.** *Twisted tensor products, I*,\n",
     "Ann. of Math., 1959, vol. 69, pp. 223-246.).\n",
     "\n",
     "## Main functions for the twisted products\n",
@@ -49,8 +49,8 @@
     "\n",
     "`dtau-d` *space* *[Function]*\n",
     "> From the simplicial set $X$, here the argument *space*, build the morphism perturbation (degree -1)\n",
-    "$$d_\\tau-d: X \\times GX \\longrightarrow X \\times GX.$$\n",
-    "The work is essentialy done by the  lisp function `dtau-d-intr` which computes the difference between  the differential morphisms taking in account the discrepancy between the last face operator ($\\partial_n$) in both chain complexes, $X \\times_\\tau GX$ and $X \\times GX$.\n",
+    "$$d_\\tau - d : C_\\ast(X \\times GX) \\longrightarrow C_\\ast(X \\times GX).$$\n",
+    "The work is essentialy done by the  lisp function `dtau-d-intr` which computes the difference between  the differential morphisms taking in account the discrepancy between the last face operator ($\\partial_n$) in both chain complexes, $C_\\ast(X \\times_\\tau GX)$ and $C_\\ast(X \\times GX)$.\n",
     "\n",
     "`szczarba` *space* *[Function]*\n",
     "> Starting from the simplicial set $X$, (the argument *space*), return two values:  \n",
@@ -75,11 +75,11 @@
     "This algorithm implements in particular  in a very efficient way the formulas of Szczarba (**R.H. Szczarba.** *The homology of twisted cartesian products.* Transactions of the American Math. Society, 1961, vol. 100, pp. 197-216).\n",
     "\n",
     "`twisted-tnsr-prdc` *space* *[Function]*\n",
-    "> Return the twisted tensor product (a chain complex)  $X \\otimes_t GX$. This is done simply by extraction of the bottom chain complex from the reduction  `(szczarba` *space*`)`.\n",
+    "> Return the twisted tensor product (a chain complex)  $C_\\ast(X) \\otimes_t C_\\ast(GX)$. This is done simply by extraction of the bottom chain complex from the reduction  `(szczarba` *space*`)`.\n",
     "\n",
     "### Examples\n",
     "\n",
-    "For testing the functions above, we shall use a \"soft\" version of $\\bar{\\Delta}$ analogous to `soft-delta` or `soft-delta-infinity`. The input of the simplices will be done in clear with the help of the macro `code`. As this version does not exists in `Kenzo`, we have modified some functions of `soft-delta`, as indicated."
+    "For testing the functions above, we shall use a \"soft\" version of $\\bar{\\Delta}$ analogous to `soft-delta` or `soft-delta-infinity`. The input of the simplices will be done in clear with the help of the macro `code`. As this version does not exist in `Kenzo`, we have modified some functions of `soft-delta`, as indicated."
    ]
   },
   {
@@ -479,7 +479,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The user will have noted that we may obtain the same twisted cartesian product by the fibration theory. Let us define a general function `loop-fbr` with  a reduced simplicial set, *space*, as argument. This function uses the function `fibration-total` that we have seen in the  chapter about fibrations. Here, we see that the lisp function for the simplicial morphism, (keyword `:sintr`), implements straightforwardly the canonical twisting operator which applies a simplex of $X$ to the corresponding 1-letter word of $GX$. Note that the execution time by the fibration function is a little longer than for the direct twisted cartesian product and it requires more memory."
+    "The user will have noted that we may obtain the same twisted cartesian product by the fibration theory. Let us define a general function `loop-fbr` with  a reduced simplicial set, *space*, as argument. This function uses the function `fibration-total` that we have seen in the  chapter about fibrations. Here, we see that the lisp function for the simplicial morphism, (keyword `:sintr`), implements straightforwardly the canonical twisting operator which applies a simplex of $X$ to the corresponding 1-letter word of $GX$."
    ]
   },
   {
@@ -569,50 +569,6 @@
     "         (print(face fb-tt-db j 4 smpx2)))"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": 18,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "NIL"
-      ]
-     },
-     "execution_count": 18,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(time (dotimes (j 1000) (face tw-crts-pr 4 4 smpx2)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 19,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "NIL"
-      ]
-     },
-     "execution_count": 19,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(time (dotimes (j 1000) (face fb-tt-db 4 4 smpx2)))"
-   ]
-  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -622,7 +578,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 20,
+   "execution_count": 18,
    "metadata": {
     "collapsed": false
    },
@@ -633,7 +589,7 @@
        "[K34 Morphism (degree -1): K18 -> K18]"
       ]
      },
-     "execution_count": 20,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -644,7 +600,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 21,
+   "execution_count": 19,
    "metadata": {
     "collapsed": false
    },
@@ -659,7 +615,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 21,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -670,7 +626,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 22,
+   "execution_count": 20,
    "metadata": {
     "collapsed": false
    },
@@ -688,7 +644,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 22,
+     "execution_count": 20,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -699,7 +655,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 23,
+   "execution_count": 21,
    "metadata": {
     "collapsed": false
    },
@@ -717,7 +673,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 23,
+     "execution_count": 21,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -735,7 +691,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 24,
+   "execution_count": 22,
    "metadata": {
     "collapsed": false
    },
@@ -746,7 +702,7 @@
        "[K35 Simplicial-Set]"
       ]
      },
-     "execution_count": 24,
+     "execution_count": 22,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -755,6 +711,50 @@
     "(def pri3 (R-proj-space 3))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K57 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tw-pri3 (twisted-crts-prdc pri3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       ":LOCALLY-EFFECTIVE"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(basis tw-pri3)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 25,
@@ -765,7 +765,7 @@
     {
      "data": {
       "text/plain": [
-       "[K57 Simplicial-Set]"
+       "[K52 Simplicial-Set]"
       ]
      },
      "execution_count": 25,
@@ -774,7 +774,7 @@
     }
    ],
    "source": [
-    "(def tw-pri3 (twisted-crts-prdc pri3))"
+    "(def crt-pri3 (crts-prdc pri3 (loop-space pri3)))"
    ]
   },
   {
@@ -783,50 +783,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       ":LOCALLY-EFFECTIVE"
-      ]
-     },
-     "execution_count": 26,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(basis tw-pri3)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 27,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "[K52 Simplicial-Set]"
-      ]
-     },
-     "execution_count": 27,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def crt-pri3 (crts-prdc pri3 (loop-space pri3)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 28,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -834,7 +790,7 @@
        "<CrPr - 4 - <<Loop[5]>>>"
       ]
      },
-     "execution_count": 28,
+     "execution_count": 26,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -845,7 +801,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 27,
    "metadata": {
     "collapsed": false
    },
@@ -868,7 +824,7 @@
        "NIL"
       ]
      },
-     "execution_count": 29,
+     "execution_count": 27,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -877,6 +833,62 @@
     "(dotimes (i 5) (print (face crt-pri3 i 4 s)))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - 3 - <<Loop[4]>>>> \n",
+      "<AbSm 0 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
+      "<AbSm 1 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
+      "<AbSm 2 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
+      "<AbSm - <CrPr - 3 2-1-0 <<Loop>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (i 5) (print (face tw-pri3 i 4 s)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K63 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def fb-tt-pri3 (loop-fbr pri3))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 30,
@@ -907,62 +919,6 @@
      "output_type": "execute_result"
     }
    ],
-   "source": [
-    "(dotimes (i 5) (print (face tw-pri3 i 4 s)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 31,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "[K63 Simplicial-Set]"
-      ]
-     },
-     "execution_count": 31,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def fb-tt-pri3 (loop-fbr pri3))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 32,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\n",
-      "<AbSm - <CrPr - 3 - <<Loop[4]>>>> \n",
-      "<AbSm 0 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
-      "<AbSm 1 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
-      "<AbSm 2 <CrPr 1-0 0 - <<Loop[3]>>>> \n",
-      "<AbSm - <CrPr - 3 2-1-0 <<Loop>>>> "
-     ]
-    },
-    {
-     "data": {
-      "text/plain": [
-       "NIL"
-      ]
-     },
-     "execution_count": 32,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(dotimes (j 5) (print (face fb-tt-pri3 j 4 s)))"
    ]
@@ -976,7 +932,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 31,
    "metadata": {
     "collapsed": false
    },
@@ -987,7 +943,7 @@
        "[K90 Reduction K23 => K88]"
       ]
      },
-     "execution_count": 33,
+     "execution_count": 31,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -998,7 +954,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 32,
    "metadata": {
     "collapsed": false
    },
@@ -1017,7 +973,7 @@
        "NIL"
       ]
      },
-     "execution_count": 34,
+     "execution_count": 32,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1028,7 +984,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 33,
    "metadata": {
     "collapsed": false
    },
@@ -1042,7 +998,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 35,
+     "execution_count": 33,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1054,7 +1010,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 34,
    "metadata": {
     "collapsed": false
    },
@@ -1070,7 +1026,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 36,
+     "execution_count": 34,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1083,7 +1039,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 35,
    "metadata": {
     "collapsed": false
    },
@@ -1102,7 +1058,7 @@
        "NIL"
       ]
      },
-     "execution_count": 37,
+     "execution_count": 35,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1113,7 +1069,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 36,
    "metadata": {
     "collapsed": false
    },
@@ -1189,7 +1145,7 @@
        "NIL"
       ]
      },
-     "execution_count": 38,
+     "execution_count": 36,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1230,7 +1186,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 39,
+   "execution_count": 37,
    "metadata": {
     "collapsed": false
    },
@@ -1241,7 +1197,7 @@
        "[K112 Simplicial-Set]"
       ]
      },
-     "execution_count": 39,
+     "execution_count": 37,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1252,7 +1208,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 38,
    "metadata": {
     "collapsed": false
    },
@@ -1263,7 +1219,7 @@
        "[K117 Simplicial-Group]"
       ]
      },
-     "execution_count": 40,
+     "execution_count": 38,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1274,7 +1230,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 39,
    "metadata": {
     "collapsed": false
    },
@@ -1285,7 +1241,7 @@
        "[K160 Chain-Complex]"
       ]
      },
-     "execution_count": 41,
+     "execution_count": 39,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1303,7 +1259,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 40,
    "metadata": {
     "collapsed": false
    },
@@ -1314,7 +1270,7 @@
        "[K163 Morphism (degree 0): K112 -> K160]"
       ]
      },
-     "execution_count": 42,
+     "execution_count": 40,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1332,7 +1288,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 41,
    "metadata": {
     "collapsed": false
    },
@@ -1346,7 +1302,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 43,
+     "execution_count": 41,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1357,7 +1313,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 42,
    "metadata": {
     "collapsed": false
    },
@@ -1373,7 +1329,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 44,
+     "execution_count": 42,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1384,7 +1340,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 43,
    "metadata": {
     "collapsed": false
    },
@@ -1395,7 +1351,7 @@
        "[K164 Morphism (degree 0): K160 -> K117]"
       ]
      },
-     "execution_count": 45,
+     "execution_count": 43,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1417,7 +1373,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 46,
+   "execution_count": 44,
    "metadata": {
     "collapsed": false
    },
@@ -1431,7 +1387,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 46,
+     "execution_count": 44,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1445,12 +1401,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The morphism {\\tt cochain} is simply a composition of morphisms. We apply it upon the simplices `(0)`, `(0 1 2)`, `(0 1 2 3)`, ..., coded as the integers $1,7,15,\\ldots$."
+    "The morphism `cochain` is simply a composition of morphisms. We apply it upon the simplices `(0)`, `(0 1 2)`, `(0 1 2 3)`, ..., coded as the integers $1,7,15,\\ldots$."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 45,
    "metadata": {
     "collapsed": false
    },
@@ -1461,7 +1417,7 @@
        "[K166 Morphism (degree -1): K112 -> K117]"
       ]
      },
-     "execution_count": 47,
+     "execution_count": 45,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1472,7 +1428,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 46,
    "metadata": {
     "collapsed": false
    },
@@ -1485,7 +1441,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 48,
+     "execution_count": 46,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1496,7 +1452,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 47,
    "metadata": {
     "collapsed": false
    },
@@ -1510,7 +1466,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 49,
+     "execution_count": 47,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1521,7 +1477,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 48,
    "metadata": {
     "collapsed": false
    },
@@ -1536,7 +1492,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 50,
+     "execution_count": 48,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1547,7 +1503,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 51,
+   "execution_count": 49,
    "metadata": {
     "collapsed": false
    },
@@ -1566,7 +1522,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 51,
+     "execution_count": 49,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1577,7 +1533,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 52,
+   "execution_count": 50,
    "metadata": {
     "collapsed": false
    },
@@ -1601,7 +1557,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 52,
+     "execution_count": 50,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1619,7 +1575,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 53,
+   "execution_count": 51,
    "metadata": {
     "collapsed": false
    },
@@ -1630,7 +1586,7 @@
        "24"
       ]
      },
-     "execution_count": 53,
+     "execution_count": 51,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1641,7 +1597,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 54,
+   "execution_count": 52,
    "metadata": {
     "collapsed": false
    },
@@ -1665,7 +1621,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 54,
+     "execution_count": 52,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1676,7 +1632,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 55,
+   "execution_count": 53,
    "metadata": {
     "collapsed": false
    },
@@ -1687,7 +1643,7 @@
        "120"
       ]
      },
-     "execution_count": 55,
+     "execution_count": 53,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1714,7 +1670,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 56,
+   "execution_count": 54,
    "metadata": {
     "collapsed": false
    },
@@ -1725,7 +1681,7 @@
        "[K167 Chain-Complex]"
       ]
      },
-     "execution_count": 56,
+     "execution_count": 54,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1736,7 +1692,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 55,
    "metadata": {
     "collapsed": false
    },
@@ -1747,7 +1703,7 @@
        "[K169 Chain-Complex]"
       ]
      },
-     "execution_count": 57,
+     "execution_count": 55,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1758,7 +1714,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 58,
+   "execution_count": 56,
    "metadata": {
     "collapsed": false
    },
@@ -1769,7 +1725,7 @@
        "[K171 Morphism (degree 0): K167 -> K169]"
       ]
      },
-     "execution_count": 58,
+     "execution_count": 56,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1794,7 +1750,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 59,
+   "execution_count": 57,
    "metadata": {
     "collapsed": false
    },
@@ -1805,7 +1761,7 @@
        "[K172 Morphism (degree 0): K112 -> K112]"
       ]
      },
-     "execution_count": 59,
+     "execution_count": 57,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1816,7 +1772,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 60,
+   "execution_count": 58,
    "metadata": {
     "collapsed": false
    },
@@ -1827,7 +1783,7 @@
        "[K173 Morphism (degree 0): K117 -> K117]"
       ]
      },
-     "execution_count": 60,
+     "execution_count": 58,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1838,7 +1794,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 61,
+   "execution_count": 59,
    "metadata": {
     "collapsed": false
    },
@@ -1849,7 +1805,7 @@
        "[K185 Morphism (degree -1): K139 -> K139]"
       ]
      },
-     "execution_count": 61,
+     "execution_count": 59,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1872,7 +1828,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 62,
+   "execution_count": 60,
    "metadata": {
     "collapsed": false
    },
@@ -1889,7 +1845,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 62,
+     "execution_count": 60,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -1898,6 +1854,56 @@
     "(? (dffr d-twtp-gd) 2 (tnpr 1 3 1 (loop3 0 7 1)))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <TnPr 1 <<Loop[7]>>>>\n",
+       "<-1 * <TnPr 1 <<Loop[0 3][7]>>>>\n",
+       "<1 * <TnPr 3 <<Loop[3\\-1][5]>>>>\n",
+       "<-1 * <TnPr 3 <<Loop[6]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 61,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? nabla 2 (tnpr 1 3 1 (loop3 0 7 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K186 Morphism (degree -1): K139 -> K139]"
+      ]
+     },
+     "execution_count": 62,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero (sbtr nabla (dffr d-twtp-gd)))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 63,
@@ -1910,10 +1916,6 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 1}\n",
-       "<1 * <TnPr 1 <<Loop[7]>>>>\n",
-       "<-1 * <TnPr 1 <<Loop[0 3][7]>>>>\n",
-       "<1 * <TnPr 3 <<Loop[3\\-1][5]>>>>\n",
-       "<-1 * <TnPr 3 <<Loop[6]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -1923,7 +1925,7 @@
     }
    ],
    "source": [
-    "(? nabla 2 (tnpr 1 3 1 (loop3 0 7 1)))"
+    "(? zero 2 (tnpr 1 3 1 (loop3 0 7 1)))"
    ]
   },
   {
@@ -1932,52 +1934,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "[K186 Morphism (degree -1): K139 -> K139]"
-      ]
-     },
-     "execution_count": 64,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def zero (sbtr nabla (dffr d-twtp-gd)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 65,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 65,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? zero 2 (tnpr 1 3 1 (loop3 0 7 1)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 66,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -1987,7 +1943,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 66,
+     "execution_count": 64,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2015,7 +1971,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 67,
+   "execution_count": 65,
    "metadata": {
     "collapsed": false
    },
@@ -2026,7 +1982,7 @@
        "[K189 Morphism (degree -2): K112 -> K117]"
       ]
      },
-     "execution_count": 67,
+     "execution_count": 65,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2035,6 +1991,50 @@
     "(def t-cup-t (i-cmps (aprd gd) (tnsr-prdc cochain cochain) (cprd d)))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K190 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 66,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def dt (cmps (dffr gd) cochain))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 67,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K191 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 67,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def td (cmps cochain (dffr d)))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 68,
@@ -2045,7 +2045,7 @@
     {
      "data": {
       "text/plain": [
-       "[K190 Morphism (degree -2): K112 -> K117]"
+       "[K193 Morphism (degree -2): K112 -> K117]"
       ]
      },
      "execution_count": 68,
@@ -2054,7 +2054,7 @@
     }
    ],
    "source": [
-    "(def dt (cmps (dffr gd) cochain))"
+    "(def zero-2 (i-add dt td t-cup-t))"
    ]
   },
   {
@@ -2067,7 +2067,9 @@
     {
      "data": {
       "text/plain": [
-       "[K191 Morphism (degree -2): K112 -> K117]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 69,
@@ -2076,7 +2078,7 @@
     }
    ],
    "source": [
-    "(def td (cmps cochain (dffr d)))"
+    "(? zero-2 3 15)"
    ]
   },
   {
@@ -2089,7 +2091,9 @@
     {
      "data": {
       "text/plain": [
-       "[K193 Morphism (degree -2): K112 -> K117]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 70,
@@ -2098,7 +2102,7 @@
     }
    ],
    "source": [
-    "(def zero-2 (i-add dt td t-cup-t))"
+    "(? zero-2 4 31)"
    ]
   },
   {
@@ -2112,7 +2116,7 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2122,7 +2126,7 @@
     }
    ],
    "source": [
-    "(? zero-2 3 15)"
+    "(? zero-2 5 63)"
    ]
   },
   {
@@ -2131,54 +2135,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 2}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 72,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? zero-2 4 31)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 73,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 3}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 73,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? zero-2 5 63)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 74,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -2188,7 +2144,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 74,
+     "execution_count": 72,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2267,7 +2223,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 75,
+   "execution_count": 73,
    "metadata": {
     "collapsed": false
    },
@@ -2278,7 +2234,7 @@
        "[K112 Simplicial-Set]"
       ]
      },
-     "execution_count": 75,
+     "execution_count": 73,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2287,6 +2243,52 @@
     "(def delta (deltab))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 74,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K194 Morphism (degree 1): K134 -> K134]"
+      ]
+     },
+     "execution_count": 74,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def chi-x (crts-contraction delta))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 75,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 75,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 +null-loop+))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 76,
@@ -2297,7 +2299,10 @@
     {
      "data": {
       "text/plain": [
-       "[K194 Morphism (degree 1): K134 -> K134]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <CrPr - 96 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 76,
@@ -2306,7 +2311,7 @@
     }
    ],
    "source": [
-    "(def chi-x (crts-contraction delta))"
+    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 1)))"
    ]
   },
   {
@@ -2315,55 +2320,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 77,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? chi-x 0 (crpr 0 1 0 +null-loop+))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 78,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
-       "<-1 * <CrPr - 96 0 <<Loop>>>>\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 78,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 1)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 79,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -2375,7 +2331,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 79,
+     "execution_count": 77,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2386,7 +2342,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 80,
+   "execution_count": 78,
    "metadata": {
     "collapsed": false
    },
@@ -2402,7 +2358,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 80,
+     "execution_count": 78,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2413,7 +2369,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 81,
+   "execution_count": 79,
    "metadata": {
     "collapsed": false
    },
@@ -2428,7 +2384,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 81,
+     "execution_count": 79,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2439,7 +2395,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 82,
+   "execution_count": 80,
    "metadata": {
     "collapsed": false
    },
@@ -2455,7 +2411,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 82,
+     "execution_count": 80,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2466,7 +2422,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 83,
+   "execution_count": 81,
    "metadata": {
     "collapsed": false
    },
@@ -2484,7 +2440,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 83,
+     "execution_count": 81,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2495,7 +2451,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 84,
+   "execution_count": 82,
    "metadata": {
     "collapsed": false
    },
@@ -2510,7 +2466,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 84,
+     "execution_count": 82,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2519,6 +2475,56 @@
     "(? chi-x 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 83,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 83,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 84,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <CrPr 3-1 7 2-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "<1 * <CrPr 3-2 7 1-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 84,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 85,
@@ -2540,7 +2546,7 @@
     }
    ],
    "source": [
-    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -2555,8 +2561,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 4}\n",
-       "<-1 * <CrPr 3-1 7 2-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
-       "<1 * <CrPr 3-2 7 1-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2566,7 +2571,7 @@
     }
    ],
    "source": [
-    "(? chi-x 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -2590,7 +2595,7 @@
     }
    ],
    "source": [
-    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+    "(? chi-x 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -2605,7 +2610,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 4}\n",
-       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
+       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2615,7 +2620,7 @@
     }
    ],
    "source": [
-    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -2638,55 +2643,6 @@
      "output_type": "execute_result"
     }
    ],
-   "source": [
-    "(? chi-x 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 90,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 4}\n",
-       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 90,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 91,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 4}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 91,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
    "source": [
     "(? chi-x 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
    ]
@@ -2707,7 +2663,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 92,
+   "execution_count": 90,
    "metadata": {
     "collapsed": false
    },
@@ -2718,7 +2674,7 @@
        "T"
       ]
      },
-     "execution_count": 92,
+     "execution_count": 90,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -2727,6 +2683,50 @@
     "(setf *tnpr-with-degrees* t)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K196 Morphism (degree 1): K160 -> K160]"
+      ]
+     },
+     "execution_count": 91,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def chi-t (tnpr-contraction delta))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 92,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K162 Reduction K134 => K160]"
+      ]
+     },
+     "execution_count": 92,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def reduc (szczarba delta))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 93,
@@ -2737,7 +2737,7 @@
     {
      "data": {
       "text/plain": [
-       "[K196 Morphism (degree 1): K160 -> K160]"
+       "[K152 Morphism (degree 0): K134 -> K160]"
       ]
      },
      "execution_count": 93,
@@ -2746,7 +2746,7 @@
     }
    ],
    "source": [
-    "(def chi-t (tnpr-contraction delta))"
+    "(def f (f reduc))"
    ]
   },
   {
@@ -2759,7 +2759,10 @@
     {
      "data": {
       "text/plain": [
-       "[K162 Reduction K134 => K160]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 94,
@@ -2768,7 +2771,7 @@
     }
    ],
    "source": [
-    "(def reduc (szczarba delta))"
+    "(? f 0 (crpr 0 1 0 +null-loop+))"
    ]
   },
   {
@@ -2781,7 +2784,9 @@
     {
      "data": {
       "text/plain": [
-       "[K152 Morphism (degree 0): K134 -> K160]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 95,
@@ -2790,7 +2795,7 @@
     }
    ],
    "source": [
-    "(def f (f reduc))"
+    "(? chi-t *)"
    ]
   },
   {
@@ -2805,7 +2810,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 0}\n",
-       "<1 * <TnPr 0 1 0 <<Loop>>>>\n",
+       "<1 * <TnPr 0 32 0 <<Loop[96]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2815,7 +2820,7 @@
     }
    ],
    "source": [
-    "(? f 0 (crpr 0 1 0 +null-loop+))"
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1)))"
    ]
   },
   {
@@ -2830,6 +2835,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2854,7 +2860,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 0}\n",
-       "<1 * <TnPr 0 32 0 <<Loop[96]>>>>\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96][384]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2864,7 +2870,7 @@
     }
    ],
    "source": [
-    "(? f 0 (crpr 0 1 0 (loop3 0 96 1)))"
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1)))"
    ]
   },
   {
@@ -2879,7 +2885,8 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 1}\n",
-       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[384]>>>>\n",
+       "<-1 * <TnPr 1 384 0 <<Loop>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2904,7 +2911,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 0}\n",
-       "<1 * <TnPr 0 1 0 <<Loop[96][384]>>>>\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96][384][1536]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2914,7 +2921,7 @@
     }
    ],
    "source": [
-    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1)))"
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1 0 (+ 512 1024) 1)))"
    ]
   },
   {
@@ -2929,8 +2936,9 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 1}\n",
-       "<-1 * <TnPr 1 96 0 <<Loop[384]>>>>\n",
-       "<-1 * <TnPr 1 384 0 <<Loop>>>>\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[384][1536]>>>>\n",
+       "<-1 * <TnPr 1 384 0 <<Loop[1536]>>>>\n",
+       "<-1 * <TnPr 1 1536 0 <<Loop>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2955,7 +2963,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 0}\n",
-       "<1 * <TnPr 0 1 0 <<Loop[96][384][1536]>>>>\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -2965,7 +2973,7 @@
     }
    ],
    "source": [
-    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1 0 (+ 512 1024) 1)))"
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 2)))"
    ]
   },
   {
@@ -2980,9 +2988,8 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 1}\n",
-       "<-1 * <TnPr 1 96 0 <<Loop[384][1536]>>>>\n",
-       "<-1 * <TnPr 1 384 0 <<Loop[1536]>>>>\n",
-       "<-1 * <TnPr 1 1536 0 <<Loop>>>>\n",
+       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[96]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3006,8 +3013,9 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 0}\n",
-       "<1 * <TnPr 0 1 0 <<Loop[96\\2]>>>>\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <TnPr 0 1 1 <<Loop[0 3][224]>>>>\n",
+       "<1 * <TnPr 1 3 0 <<Loop[192]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3017,7 +3025,7 @@
     }
    ],
    "source": [
-    "(? f 0 (crpr 0 1 0 (loop3 0 96 2)))"
+    "(? f 1 (crpr 0 3 0 (loop3 0 (+ 32 64 128) 1)))"
    ]
   },
   {
@@ -3031,9 +3039,10 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
-       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
-       "<-1 * <TnPr 1 96 0 <<Loop[96]>>>>\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<-1 * <TnPr 1 3 1 <<Loop[224]>>>>\n",
+       "<1 * <TnPr 1 96 1 <<Loop[224]>>>>\n",
+       "<1 * <TnPr 2 224 0 <<Loop>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3057,9 +3066,11 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 1}\n",
-       "<1 * <TnPr 0 1 1 <<Loop[0 3][224]>>>>\n",
-       "<1 * <TnPr 1 3 0 <<Loop[192]>>>>\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<-1 * <TnPr 0 1 2 <<Loop[1-0 3][0 7][1 416]>>>>\n",
+       "<1 * <TnPr 0 1 2 <<Loop[1-0 3][1 7][480]>>>>\n",
+       "<1 * <TnPr 1 3 1 <<Loop[0 6][448]>>>>\n",
+       "<1 * <TnPr 2 7 0 <<Loop[384]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3069,7 +3080,7 @@
     }
    ],
    "source": [
-    "(? f 1 (crpr 0 3 0 (loop3 0 (+ 32 64 128) 1)))"
+    "(? f 2 (crpr 0 7 0 (loop3 0 (+ 32 64 128 256) 1)))"
    ]
   },
   {
@@ -3083,10 +3094,16 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 2}\n",
-       "<-1 * <TnPr 1 3 1 <<Loop[224]>>>>\n",
-       "<1 * <TnPr 1 96 1 <<Loop[224]>>>>\n",
-       "<1 * <TnPr 2 224 0 <<Loop>>>>\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2 480]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][1 480]>>>>\n",
+       "<-1 * <TnPr 0 32 3 <<Loop[2-1-0 96][1-0 224][2 480]>>>>\n",
+       "<1 * <TnPr 0 32 3 <<Loop[2-1-0 96][2-0 224][1 480]>>>>\n",
+       "<-1 * <TnPr 1 5 2 <<Loop[480]>>>>\n",
+       "<1 * <TnPr 1 160 2 <<Loop[480]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[448]>>>>\n",
+       "<-1 * <TnPr 2 224 1 <<Loop[448]>>>>\n",
+       "<-1 * <TnPr 3 480 0 <<Loop>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3110,11 +3127,8 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 2}\n",
-       "<-1 * <TnPr 0 1 2 <<Loop[1-0 3][0 7][1 416]>>>>\n",
-       "<1 * <TnPr 0 1 2 <<Loop[1-0 3][1 7][480]>>>>\n",
-       "<1 * <TnPr 1 3 1 <<Loop[0 6][448]>>>>\n",
-       "<1 * <TnPr 2 7 0 <<Loop[384]>>>>\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\2][3-0 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3124,7 +3138,7 @@
     }
    ],
    "source": [
-    "(? f 2 (crpr 0 7 0 (loop3 0 (+ 32 64 128 256) 1)))"
+    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -3138,16 +3152,8 @@
      "data": {
       "text/plain": [
        "\n",
-       "----------------------------------------------------------------------{CMBN 3}\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2 480]>>>>\n",
-       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][1 480]>>>>\n",
-       "<-1 * <TnPr 0 32 3 <<Loop[2-1-0 96][1-0 224][2 480]>>>>\n",
-       "<1 * <TnPr 0 32 3 <<Loop[2-1-0 96][2-0 224][1 480]>>>>\n",
-       "<-1 * <TnPr 1 5 2 <<Loop[480]>>>>\n",
-       "<1 * <TnPr 1 160 2 <<Loop[480]>>>>\n",
-       "<1 * <TnPr 2 7 1 <<Loop[448]>>>>\n",
-       "<-1 * <TnPr 2 224 1 <<Loop[448]>>>>\n",
-       "<-1 * <TnPr 3 480 0 <<Loop>>>>\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<1 * <TnPr 2 7 2 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3172,7 +3178,6 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\2][3-0 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3182,7 +3187,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -3197,7 +3202,6 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 4}\n",
-       "<1 * <TnPr 2 7 2 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3222,6 +3226,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\-2][3-0 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3231,7 +3236,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -3246,6 +3251,7 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 4}\n",
+       "<1 * <TnPr 2 7 2 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3270,7 +3276,6 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\-2][3-0 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3280,7 +3285,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -3295,7 +3300,6 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 4}\n",
-       "<1 * <TnPr 2 7 2 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3320,6 +3324,8 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][3-2 224\\-2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][3-1 224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3329,7 +3335,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -3368,8 +3374,9 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
-       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][3-2 224\\-2]>>>>\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][3-1 224\\-2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][2-0 224\\-2]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][1-0 224\\-2]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[0 192\\2][224\\-2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3379,7 +3386,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
    ]
   },
   {
@@ -3418,9 +3425,8 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][2-0 224\\-2]>>>>\n",
-       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][1-0 224\\-2]>>>>\n",
-       "<1 * <TnPr 2 7 1 <<Loop[0 192\\2][224\\-2]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][3-2 224\\2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][3-1 224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3430,7 +3436,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -3469,8 +3475,9 @@
       "text/plain": [
        "\n",
        "----------------------------------------------------------------------{CMBN 3}\n",
-       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][3-2 224\\2]>>>>\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][3-1 224\\2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][2-0 224\\2]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][1-0 224\\2]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[0 192\\-2][224\\2]>>>>\n",
        "------------------------------------------------------------------------------\n"
       ]
      },
@@ -3480,7 +3487,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -3507,6 +3514,13 @@
     "(? chi-t *)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We verify now the required property of the homotopy operator $\\chi_{\\times\\tau}$. As usual, we build by composition a morphism `zero`."
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 124,
@@ -3517,12 +3531,7 @@
     {
      "data": {
       "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 3}\n",
-       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][2-0 224\\2]>>>>\n",
-       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][1-0 224\\2]>>>>\n",
-       "<1 * <TnPr 2 7 1 <<Loop[0 192\\-2][224\\2]>>>>\n",
-       "------------------------------------------------------------------------------\n"
+       "[K135 Morphism (degree -1): K134 -> K134]"
       ]
      },
      "execution_count": 124,
@@ -3531,7 +3540,7 @@
     }
    ],
    "source": [
-    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+    "(def dfr (bndr (twisted-crts-prdc delta)))"
    ]
   },
   {
@@ -3544,9 +3553,7 @@
     {
      "data": {
       "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 4}\n",
-       "------------------------------------------------------------------------------\n"
+       "[K201 Morphism (degree 0): K134 -> K134]"
       ]
      },
      "execution_count": 125,
@@ -3555,14 +3562,8 @@
     }
    ],
    "source": [
-    "(? chi-t *)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We verify now the required property of the homotopy operator $\\chi_{\\times\\tau}$. As usual, we build by composition a morphism `zero`."
+    "(def zero (i-sbtr (idnt-mrph (sorc dfr)) (cmps dfr chi-x) \n",
+    "                   (cmps chi-x dfr)))"
    ]
   },
   {
@@ -3575,7 +3576,7 @@
     {
      "data": {
       "text/plain": [
-       "[K135 Morphism (degree -1): K134 -> K134]"
+       "<CrPr 2 7 1 <<Loop[1 224\\-2][0 224\\2]>>>"
       ]
      },
      "execution_count": 126,
@@ -3584,7 +3585,7 @@
     }
    ],
    "source": [
-    "(def dfr (bndr (twisted-crts-prdc delta)))"
+    "(def simplx (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
    ]
   },
   {
@@ -3593,51 +3594,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "[K201 Morphism (degree 0): K134 -> K134]"
-      ]
-     },
-     "execution_count": 127,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def zero (i-sbtr (idnt-mrph (sorc dfr)) (cmps dfr chi-x) \n",
-    "                   (cmps chi-x dfr)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 128,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "<CrPr 2 7 1 <<Loop[1 224\\-2][0 224\\2]>>>"
-      ]
-     },
-     "execution_count": 128,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def simplx (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 129,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -3647,7 +3603,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 129,
+     "execution_count": 127,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3665,7 +3621,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 130,
+   "execution_count": 128,
    "metadata": {
     "collapsed": false
    },
@@ -3676,7 +3632,7 @@
        "[K160 Chain-Complex]"
       ]
      },
-     "execution_count": 130,
+     "execution_count": 128,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3685,56 +3641,56 @@
     "(def tw (twisted-tnsr-prdc d))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 129,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K206 Morphism (degree 0): K160 -> K160]"
+      ]
+     },
+     "execution_count": 129,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero-2  (i-sbtr (idnt-mrph tw) (cmps tw chi-t) (cmps chi-t tw)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 130,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 0 1 3 <<Loop[31\\2][992\\-1]>>>"
+      ]
+     },
+     "execution_count": 130,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def simpx2 (tnpr 0 1 3 (loop3 0 (mask 5) 2 0 (* 32 (mask 5)) -1)))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 131,
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "[K206 Morphism (degree 0): K160 -> K160]"
-      ]
-     },
-     "execution_count": 131,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def zero-2  (i-sbtr (idnt-mrph tw) (cmps tw chi-t) (cmps chi-t tw)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 132,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "<TnPr 0 1 3 <<Loop[31\\2][992\\-1]>>>"
-      ]
-     },
-     "execution_count": 132,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def simpx2 (tnpr 0 1 3 (loop3 0 (mask 5) 2 0 (* 32 (mask 5)) -1)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 133,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -3744,7 +3700,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 133,
+     "execution_count": 131,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3762,7 +3718,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 134,
+   "execution_count": 132,
    "metadata": {
     "collapsed": false
    },
@@ -3773,7 +3729,7 @@
        "<TnPr 0 1 0 <<Loop>>>"
       ]
      },
-     "execution_count": 134,
+     "execution_count": 132,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3784,7 +3740,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 135,
+   "execution_count": 133,
    "metadata": {
     "collapsed": false
    },
@@ -3798,7 +3754,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 135,
+     "execution_count": 133,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3825,7 +3781,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 136,
+   "execution_count": 134,
    "metadata": {
     "collapsed": false
    },
@@ -3836,7 +3792,7 @@
        "[K209 Cohomology-Class on K160 of degree 0]"
       ]
      },
-     "execution_count": 136,
+     "execution_count": 134,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3856,7 +3812,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 137,
+   "execution_count": 135,
    "metadata": {
     "collapsed": false
    },
@@ -3867,7 +3823,7 @@
        "[K210 Morphism (degree 0): K207 -> K160]"
       ]
      },
-     "execution_count": 137,
+     "execution_count": 135,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3887,7 +3843,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 138,
+   "execution_count": 136,
    "metadata": {
     "collapsed": false
    },
@@ -3898,7 +3854,7 @@
        "[K211 Reduction K160 => K207]"
       ]
      },
-     "execution_count": 138,
+     "execution_count": 136,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3912,7 +3868,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 139,
+   "execution_count": 137,
    "metadata": {
     "collapsed": false
    },
@@ -3931,7 +3887,7 @@
        "NIL"
       ]
      },
-     "execution_count": 139,
+     "execution_count": 137,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3942,7 +3898,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 140,
+   "execution_count": 138,
    "metadata": {
     "collapsed": false
    },
@@ -3956,7 +3912,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 140,
+     "execution_count": 138,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3967,7 +3923,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 141,
+   "execution_count": 139,
    "metadata": {
     "collapsed": false
    },
@@ -3981,7 +3937,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 141,
+     "execution_count": 139,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -3992,7 +3948,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 142,
+   "execution_count": 140,
    "metadata": {
     "collapsed": false
    },
@@ -4066,7 +4022,7 @@
        "NIL"
       ]
      },
-     "execution_count": 142,
+     "execution_count": 140,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4089,7 +4045,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 143,
+   "execution_count": 141,
    "metadata": {
     "collapsed": false
    },
@@ -4100,7 +4056,7 @@
        "[K1 Simplicial-Set]"
       ]
      },
-     "execution_count": 143,
+     "execution_count": 141,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4109,6 +4065,51 @@
     "(def db (soft-deltab))"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 142,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K231 Morphism (degree 1): K88 -> K88]"
+      ]
+     },
+     "execution_count": 142,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tt (tnpr-contraction db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 143,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 3 (DELT . 31) 4 <<Loop[(DELT . 63)\\2]>>>"
+      ]
+     },
+     "execution_count": 143,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gnrt1 (tnpr 3 (code '(0 1 2 3 4)) \n",
+    "                 4 (loop3 0 (code '(0 1 2 3 4 5)) 2)))"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 144,
@@ -4119,7 +4120,9 @@
     {
      "data": {
       "text/plain": [
-       "[K231 Morphism (degree 1): K88 -> K88]"
+       "\n",
+       "----------------------------------------------------------------------{CMBN 8}\n",
+       "------------------------------------------------------------------------------\n"
       ]
      },
      "execution_count": 144,
@@ -4128,7 +4131,7 @@
     }
    ],
    "source": [
-    "(def tt (tnpr-contraction db))"
+    "(? tt 7 gnrt1)"
    ]
   },
   {
@@ -4141,7 +4144,7 @@
     {
      "data": {
       "text/plain": [
-       "<TnPr 3 (DELT . 31) 4 <<Loop[(DELT . 63)\\2]>>>"
+       "<TnPr 4 (DELT . 5460) 5 <<Loop[(DELT . 10922)\\5]>>>"
       ]
      },
      "execution_count": 145,
@@ -4150,8 +4153,8 @@
     }
    ],
    "source": [
-    "(def gnrt1 (tnpr 3 (code '(0 1 2 3 4)) \n",
-    "                 4 (loop3 0 (code '(0 1 2 3 4 5)) 2)))"
+    "(def gnrt2 (tnpr 4 (code '(2 4 6 8 10 12))  \n",
+    "                  5 (loop3 0 (code '(1 3 5 7 9 11 13)) 5)))"
    ]
   },
   {
@@ -4160,53 +4163,6 @@
    "metadata": {
     "collapsed": false
    },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "----------------------------------------------------------------------{CMBN 8}\n",
-       "------------------------------------------------------------------------------\n"
-      ]
-     },
-     "execution_count": 146,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(? tt 7 gnrt1)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 147,
-   "metadata": {
-    "collapsed": false
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "<TnPr 4 (DELT . 5460) 5 <<Loop[(DELT . 10922)\\5]>>>"
-      ]
-     },
-     "execution_count": 147,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "(def gnrt2 (tnpr 4 (code '(2 4 6 8 10 12))  \n",
-    "                  5 (loop3 0 (code '(1 3 5 7 9 11 13)) 5)))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 148,
-   "metadata": {
-    "collapsed": false
-   },
    "outputs": [
     {
      "data": {
@@ -4216,7 +4172,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 148,
+     "execution_count": 146,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4234,7 +4190,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 149,
+   "execution_count": 147,
    "metadata": {
     "collapsed": false
    },
@@ -4252,7 +4208,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 149,
+     "execution_count": 147,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4263,7 +4219,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 150,
+   "execution_count": 148,
    "metadata": {
     "collapsed": false
    },
@@ -4287,7 +4243,7 @@
        "------------------------------------------------------------------------------\n"
       ]
      },
-     "execution_count": 150,
+     "execution_count": 148,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -4298,7 +4254,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 151,
+   "execution_count": 149,
    "metadata": {
     "collapsed": false
    },
@@ -4309,7 +4265,7 @@
        "535"
       ]
      },
-     "execution_count": 151,
+     "execution_count": 149,
      "metadata": {},
      "output_type": "execute_result"
     }