{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "To load \"kenzo\":\n",
      "  Load 1 ASDF system:\n",
      "    kenzo\n",
      "; Loading \"kenzo\"\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(:KENZO)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(ql:quickload :kenzo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "---done---"
     ]
    },
    {
     "data": {
      "text/plain": [
       "NIL"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:cat-init)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2.1 Representation of a reduction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"----------------------------------------------------------------[class-doc]\n",
       "REDUCTION\n",
       "Slots: (tcc bcc f g h idnm orgn)\n",
       "Instances of this class represent reductions of chain complexes.\n",
       "\n",
       "     ^    h     s ^                 f ° g = 1\n",
       "     C  ----->    C                          C\n",
       "      ^                     h ° d + d ° h = 1  - g ° f\n",
       "    | |                                      ^\n",
       "  f | | g                                    C\n",
       "    | |                             f ° h = 0\n",
       "    v                               h ° g = 0\n",
       "     C                              h ° h = 0\n",
       "\n",
       "The class has 7 slots:\n",
       "                                                               ^\n",
       "1. TCC, an object of type CHAIN-COMPLEX, the top chain complex C. Its reader\n",
       "   function is TCC1.\n",
       "\n",
       "2. BCC, an object of type CHAIN-COMPLEX, the bottom chain complex C. Its reader\n",
       "   function is BCC1.\n",
       "\n",
       "3. F, an object of type MORPHISM representing the morphism f. Its reader\n",
       "   function is F1.\n",
       "\n",
       "4. G, an object of type MORPHISM representing the morphism g. Its reader\n",
       "   function is G1.\n",
       "\n",
       "5. H, an object of type MORPHISM representing the morphism h. Its reader\n",
       "   function is H1.\n",
       "\n",
       "9. IDNM, an integer, a system-generated identifier for this object. Its reader\n",
       "   function is IDNM.\n",
       "\n",
       "10. ORGN, a list containg a comment indicating the origin of the object. The\n",
       "    should be unique (per session), since it is used in the implementation for\n",
       "    caching purposes. Its reader function is ORGN.\n",
       "-----------------------------------------------------------------------------\""
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(documentation 'cat:reduction 'type)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2.2 The function $\\texttt{build-rdct}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"--------------------------------------------------------------[function-doc]\n",
       "BUILD-RDCT\n",
       "Args: (&key f g h orgn)\n",
       "Returns an instance of the class REDUCTION. The keyword arguments are as\n",
       "follows:\n",
       "\n",
       ":F F, an object of type morphism representing the morphism f of a reduction\n",
       "\n",
       ":G G, an object of type morphism representing the morphism g of a reduction\n",
       "\n",
       ":H H, an object of type morphism representing the morphism h of a reduction\n",
       "\n",
       ":ORGN ORGN, a list containing a relevant and carefully chosen comment about\n",
       "            the origin of the chain complex. This comment should be unique\n",
       "            for a Kenzo session (between calls of CAT-INIT), as it is used\n",
       "            for caching purposes.\n",
       "\n",
       "Use this function instead of creating instances via the standard constructor\n",
       "MAKE-INSTANCE.\n",
       "------------------------------------------------------------------------------\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(documentation 'cat:build-rdct 'function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2.4 Verification functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"--------------------------------------------------------------[function-doc]\n",
       "PRE-CHECK-RDCT\n",
       "Args: (rdct)\n",
       "Assigns the following Lisp global variables, using morphisms from RDCT and the\n",
       "differentials and identity morphisms of the underlying chain complexes,\n",
       "according to these formulas:\n",
       "\n",
       "*TDD*         = d  ° d\n",
       "                 ^    ^\n",
       "                 C    C\n",
       "\n",
       "*BDD*         = d  ° d\n",
       "                 C    C\n",
       "\n",
       "*ID-FG*       = Id  - f ° g\n",
       "                  C\n",
       "\n",
       "*ID-GF-DH-HD* = Id  - g ° f - (d ° h + h ° d )\n",
       "                  ^             ^           ^\n",
       "                  C             C           C\n",
       "\n",
       "*HH*          = h ° h\n",
       "*FH*          = f ° h\n",
       "*HG*          = h ° g\n",
       "\n",
       "*DF-FD*       = d  ° f - f ° d\n",
       "                 C            ^\n",
       "                              C\n",
       "\n",
       "*DG-GD*       = d  ° g - g ° d\n",
       "                 ^            C\n",
       "                 C\n",
       "\n",
       "------------------------------------------------------------------------------\""
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(documentation 'cat:pre-check-rdct 'function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"--------------------------------------------------------------[function-doc]\n",
       "CHECK-RDCT-NO-WAIT\n",
       "Args: ()\n",
       "Maps combinations of the top and bottom chain complexes of a reduction using\n",
       "the morphisms created by PRE-CHECK-RDCT. Having no parameters, the\n",
       "combinations must be provided via the Lisp global variables *TC* and *BC*,\n",
       "for the top and bottom chain complex, respectively. If the morphisms are\n",
       "coherent, the result of each mapping is a null combination.\n",
       "------------------------------------------------------------------------------\""
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(documentation 'cat:check-rdct-no-wait 'function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "CDELTA"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(defun cdelta (dmns)\n",
    "    (cat:build-chcm\n",
    "     :cmpr #'cat:l-cmpr\n",
    "     :basis :locally-effective\n",
    "     :bsgn '(0)\n",
    "     :intr-dffr\n",
    "         #'(lambda (degr gmsm)\n",
    "            (cat:make-cmbn\n",
    "             :degr (1- degr)\n",
    "             :list (do ((rslt cat:+empty-list+\n",
    "                              (cons (cons sign\n",
    "                                          (append\n",
    "                                           (subseq gmsm 0 nark)\n",
    "                                           (subseq gmsm (1+ nark))))\n",
    "                                    rslt))\n",
    "                        (sign 1 (- sign))\n",
    "                        (nark 0 (1+ nark)))\n",
    "                       ((> nark degr) rslt))))\n",
    "     :strt :gnrt\n",
    "     :orgn `(locally effective version of C_* delta ,dmns)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MAKE-F"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(defun make-f (tdmns bdmns)\n",
    "    (cat:build-mrph\n",
    "     :sorc (cdelta tdmns) :trgt (cdelta bdmns) :degr 0\n",
    "     :intr #'(lambda (degr gmsm)\n",
    "                (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns))\n",
    "                                        gmsm)))\n",
    "                     (if pos\n",
    "                         (if (< pos degr)\n",
    "                             (cat:zero-cmbn degr)\n",
    "                             (cat:cmbn degr 1 (nconc (butlast gmsm) (list bdmns))))\n",
    "                         (cat:cmbn degr 1 gmsm))))\n",
    "     :strt :gnrt\n",
    "     :orgn `(projection delta ,tdmns => delta ,bdmns)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MAKE-G"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(defun make-g (tdmns bdmns)\n",
    "    (cat:build-mrph\n",
    "     :sorc (cdelta bdmns) :trgt (cdelta tdmns) :degr 0\n",
    "     :intr #'identity\n",
    "     :strt :cmbn\n",
    "     :orgn `(injection delta ,bdmns => delta ,tdmns)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MAKE-H"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(defun make-h (tdmns bdmns)\n",
    "    (cat:build-mrph\n",
    "     :sorc (cdelta tdmns) :trgt (cdelta tdmns) :degr +1\n",
    "     :intr #'(lambda (degr gmsm)\n",
    "                (let ((pos (position-if #'(lambda (vertex) (>= vertex bdmns))\n",
    "                                        gmsm)))\n",
    "                     (if pos\n",
    "                         (if (member bdmns gmsm)\n",
    "                             (cat:zero-cmbn (1+ degr))\n",
    "                             (cat:cmbn (1+ degr) (cat:-1-expt-n pos)\n",
    "                                       (append (subseq gmsm 0 pos) (list bdmns)\n",
    "                                               (subseq gmsm pos))))\n",
    "                         (cat:zero-cmbn (1+ degr)))))\n",
    "     :strt :gnrt\n",
    "     :orgn `(homotopy for delta ,tdmns => ,bdmns)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "MAKE-RDCT"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(defun make-rdct (tdmns bdmns)\n",
    "    (let ((rdct (cat:build-rdct\n",
    "                 :f (make-f tdmns bdmns)\n",
    "                 :g (make-g tdmns bdmns)\n",
    "                 :h (make-h tdmns bdmns)\n",
    "                 :orgn `(reduction delta ,tdmns ,bdmns))))\n",
    "         rdct))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[K8 Reduction K1 => K3]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:def rdct (make-rdct 6 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(REDUCTION DELTA 6 3)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:orgn rdct)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "---done---"
     ]
    },
    {
     "data": {
      "text/plain": [
       "NIL"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:pre-check-rdct rdct)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "----------------------------------------------------------------------{CMBN 2}\n",
       "<1 * (0 1 2)>\n",
       "<10 * (1 2 3)>\n",
       "<100 * (1 2 4)>\n",
       "<1000 * (2 3 4)>\n",
       "------------------------------------------------------------------------------\n"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(setf cat:*tc* (cat:cmbn 2 1 '(0 1 2) 10 '(1 2 3) 100 '(1 2 4) 1000 '(2 3 4)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "----------------------------------------------------------------------{CMBN 3}\n",
       "<4 * (0 1 2 3)>\n",
       "------------------------------------------------------------------------------\n"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(setf cat:*bc* (cat:cmbn 3 4 '(0 1 2 3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "*TC* => \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "<1 * (0 1 2)>\n",
      "<10 * (1 2 3)>\n",
      "<100 * (1 2 4)>\n",
      "<1000 * (2 3 4)>\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "*BC* => \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "<4 * (0 1 2 3)>\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *TDD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 0}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *BDD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 1}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *DF-FD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 1}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *DG-GD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *ID-FG* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *ID-GF-DH-HD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *HH* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 4}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *FH* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *HG* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 4}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "---done---"
     ]
    },
    {
     "data": {
      "text/plain": [
       "NIL"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:check-rdct-no-wait)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[K35 Reduction K1 => K30]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:def trdct (make-rdct 6 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[K39 Reduction K30 => K3]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:def brdct (make-rdct 4 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[K45 Reduction K1 => K3]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(setf rdct (cat:cmps brdct trdct))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "---done---"
     ]
    },
    {
     "data": {
      "text/plain": [
       "NIL"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:pre-check-rdct rdct)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "*TC* => \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "<1 * (0 1 2)>\n",
      "<10 * (1 2 3)>\n",
      "<100 * (1 2 4)>\n",
      "<1000 * (2 3 4)>\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "*BC* => \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "<4 * (0 1 2 3)>\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *TDD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 0}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *BDD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 1}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *DF-FD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 1}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *DG-GD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *ID-FG* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *ID-GF-DH-HD* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 2}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *HH* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 4}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *FH* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 3}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "Checking *HG* = 0\n",
      "Result: \n",
      "----------------------------------------------------------------------{CMBN 4}\n",
      "------------------------------------------------------------------------------\n",
      "\n",
      "---done---"
     ]
    },
    {
     "data": {
      "text/plain": [
       "NIL"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cat:check-rdct-no-wait)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "CCL Lisp",
   "language": "lisp",
   "name": "lisp"
  },
  "language_info": {
   "codemirror_mode": "text/x-common-lisp",
   "mimetype": "text/x-common-lisp",
   "name": "common-lisp",
   "pygments_lexer": "common-lisp",
   "version": "X3J13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}