From 04ec4331a47fb913cd1c979b5555bddc5162a87c Mon Sep 17 00:00:00 2001
From: Gerd Heber <gheber@hdfgroup.org>
Date: Sun, 10 Jan 2016 18:06:15 -0500
Subject: [PATCH] Updated notebook.

---
 examples/Chapter1.ipynb | 17 +++++++++++++----
 1 file changed, 13 insertions(+), 4 deletions(-)

diff --git a/examples/Chapter1.ipynb b/examples/Chapter1.ipynb
index b3c99fe..5fb8e1d 100644
--- a/examples/Chapter1.ipynb
+++ b/examples/Chapter1.ipynb
@@ -165,8 +165,8 @@
     "\n",
     "### Representation of a combination\n",
     "\n",
-    "A combination is represented internally in the system by a list having the following general form:\n",
-    "$$\\texttt{(:cmbn degree}\\quad(\\lambda_1\\cdot\\sigma_1)\\dots(\\lambda_k\\cdot\\sigma_k)\\texttt{)}$$\n",
+    "A combination is represented internally in the system by a structure having the following general form:\n",
+    "$$\\#\\texttt{S(CMBN}\\quad\\texttt{DEGREE P}\\quad\\texttt{LIST (}\\texttt{(}\\lambda_1\\cdot\\sigma_1\\texttt{)}\\dots\\texttt{(}\\lambda_k\\cdot\\sigma_k\\texttt{)))}$$\n",
     "and containing\n",
     "\n",
     "1. The degree of the combination corresponding to the index $p\\in\\mathbb{Z}$ of the group $C_p$ to which this combination belongs.\n",
@@ -174,7 +174,14 @@
     "\n",
     "This choice of representation implaies that only homogeneous combinations will be considered. A type $\\texttt{CMBN}$ and a printing method are provided with this internal representation. The external form of a combination is shown in the examples.\n",
     "\n",
-    "### Ordering the generators"
+    "### Ordering the generators\n",
+    "\n",
+    "In order to speed up the execution of algorithms involving combinations, the list of pairs $(\\lambda_i.\\sigma_i)$ is ordered by an adequate ordering function (ex: the lexicographical ordering n the symbols). For programming convenience, an enumerated type $\\texttt{CMPR}$ has been defined:\n",
+    "```lisp\n",
+    "(deftype cmpr() '(member :less :equal :greater))\n",
+    "```\n",
+    "A number of macros, functions and methods have been defined on usual sets (symbols, numbers, lists, etc.) taking their value in the set $\\texttt{[:less, :equal, :greater]}$. Of course, the user may define its own function for a particular case.\n",
+    "The functions $\\texttt{f-cmpr}$, $\\texttt{s-cmpr}$, and $\\texttt{l-cmpr}$ can be used to compare various couples of common items."
    ]
   },
   {
@@ -329,7 +336,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Functions handling combinations"
+    "### Functions handling combinations\n",
+    "\n",
+    "The software provides a set of functions, methods, and macros to create or modify combinations."
    ]
   },
   {