doc-src/TutorialI/Types/numerics.tex
 author paulson Tue Jul 17 13:46:21 2001 +0200 (2001-07-17) changeset 11428 332347b9b942 parent 11416 91886738773a child 11480 0fba0357c04c permissions -rw-r--r--
tidying the index
 paulson@10794  1 % $Id$  paulson@11389  2 paulson@11389  3 \section{Numbers}  paulson@11389  4 \label{sec:numbers}  paulson@11389  5 paulson@11174  6 Until now, our numerical examples have used the type of \textbf{natural  paulson@11174  7 numbers},  paulson@10594  8 \isa{nat}. This is a recursive datatype generated by the constructors  paulson@10594  9 zero and successor, so it works well with inductive proofs and primitive  paulson@11174  10 recursive function definitions. HOL also provides the type  paulson@10794  11 \isa{int} of \textbf{integers}, which lack induction but support true  paulson@11174  12 subtraction. The integers are preferable to the natural numbers for reasoning about  paulson@11174  13 complicated arithmetic expressions, even for some expressions whose  paulson@11174  14 value is non-negative. The logic HOL-Real also has the type  paulson@11174  15 \isa{real} of real numbers. Isabelle has no subtyping, so the numeric  paulson@11174  16 types are distinct and there are functions to convert between them.  paulson@11174  17 Fortunately most numeric operations are overloaded: the same symbol can be  paulson@11174  18 used at all numeric types. Table~\ref{tab:overloading} in the appendix  paulson@11174  19 shows the most important operations, together with the priorities of the  paulson@11174  20 infix symbols.  paulson@10594  21 paulson@11416  22 \index{linear arithmetic}%  paulson@10594  23 Many theorems involving numeric types can be proved automatically by  paulson@10594  24 Isabelle's arithmetic decision procedure, the method  paulson@11416  25 \methdx{arith}. Linear arithmetic comprises addition, subtraction  paulson@10594  26 and multiplication by constant factors; subterms involving other operators  paulson@10594  27 are regarded as variables. The procedure can be slow, especially if the  paulson@10594  28 subgoal to be proved involves subtraction over type \isa{nat}, which  paulson@10594  29 causes case splits.  paulson@10594  30 paulson@10594  31 The simplifier reduces arithmetic expressions in other  paulson@10594  32 ways, such as dividing through by common factors. For problems that lie  paulson@10881  33 outside the scope of automation, HOL provides hundreds of  paulson@10594  34 theorems about multiplication, division, etc., that can be brought to  paulson@10881  35 bear. You can locate them using Proof General's Find  paulson@10881  36 button. A few lemmas are given below to show what  paulson@10794  37 is available.  paulson@10594  38 paulson@10594  39 \subsection{Numeric Literals}  nipkow@10779  40 \label{sec:numerals}  paulson@10594  41 paulson@11416  42 \index{numeric literals|(}%  paulson@10594  43 Literals are available for the types of natural numbers, integers  paulson@10594  44 and reals and denote integer values of arbitrary size.  paulson@10594  45 They begin  paulson@10594  46 with a number sign (\isa{\#}), have an optional minus sign (\isa{-}) and  paulson@10594  47 then one or more decimal digits. Examples are \isa{\#0}, \isa{\#-3}  paulson@10594  48 and \isa{\#441223334678}.  paulson@10594  49 paulson@10594  50 Literals look like constants, but they abbreviate  paulson@10594  51 terms, representing the number in a two's complement binary notation.  paulson@10794  52 Isabelle performs arithmetic on literals by rewriting rather  paulson@10594  53 than using the hardware arithmetic. In most cases arithmetic  paulson@10594  54 is fast enough, even for large numbers. The arithmetic operations  paulson@10794  55 provided for literals include addition, subtraction, multiplication,  paulson@10794  56 integer division and remainder. Fractions of literals (expressed using  paulson@10794  57 division) are reduced to lowest terms.  paulson@10594  58 paulson@11416  59 \begin{warn}\index{overloading!and arithmetic}  paulson@10794  60 The arithmetic operators are  paulson@10594  61 overloaded, so you must be careful to ensure that each numeric  paulson@10594  62 expression refers to a specific type, if necessary by inserting  paulson@10594  63 type constraints. Here is an example of what can go wrong:  paulson@10794  64 \par  paulson@10594  65 \begin{isabelle}  paulson@10594  66 \isacommand{lemma}\ "\#2\ *\ m\ =\ m\ +\ m"  paulson@10594  67 \end{isabelle}  paulson@10594  68 %  paulson@10594  69 Carefully observe how Isabelle displays the subgoal:  paulson@10594  70 \begin{isabelle}  paulson@10594  71 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m  paulson@10594  72 \end{isabelle}  paulson@10594  73 The type \isa{'a} given for the literal \isa{\#2} warns us that no numeric  paulson@10594  74 type has been specified. The problem is underspecified. Given a type  paulson@10594  75 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.  paulson@10794  76 \end{warn}  paulson@10794  77 paulson@10881  78 \begin{warn}  paulson@11428  79 \index{recdef@\isacommand {recdef} (command)!and numeric literals}  paulson@11416  80 Numeric literals are not constructors and therefore  paulson@11416  81 must not be used in patterns. For example, this declaration is  paulson@11416  82 rejected:  paulson@10881  83 \begin{isabelle}  paulson@10881  84 \isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline  nipkow@11148  85 "h\ \#3\ =\ \#2"\isanewline  nipkow@11148  86 "h\ i\ \ =\ i"  paulson@10881  87 \end{isabelle}  paulson@10881  88 paulson@10881  89 You should use a conditional expression instead:  paulson@10881  90 \begin{isabelle}  paulson@10881  91 "h\ i\ =\ (if\ i\ =\ \#3\ then\ \#2\ else\ i)"  paulson@10881  92 \end{isabelle}  paulson@11416  93 \index{numeric literals|)}  paulson@10881  94 \end{warn}  paulson@10881  95 paulson@10594  96 paulson@10594  97 nipkow@11216  98 \subsection{The Type of Natural Numbers, {\tt\slshape nat}}  paulson@10594  99 paulson@11416  100 \index{natural numbers|(}\index{*nat (type)|(}%  paulson@10594  101 This type requires no introduction: we have been using it from the  paulson@10794  102 beginning. Hundreds of theorems about the natural numbers are  paulson@10594  103 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}. Only  paulson@10594  104 in exceptional circumstances should you resort to induction.  paulson@10594  105 paulson@10594  106 \subsubsection{Literals}  paulson@11416  107 \index{numeric literals!for type \protect\isa{nat}}%  paulson@11416  108 The notational options for the natural numbers are confusing. The  paulson@11416  109 constant \cdx{0} is overloaded to serve as the neutral value  paulson@11416  110 in a variety of additive types. The symbols \sdx{1} and \sdx{2} are  paulson@10594  111 not constants but abbreviations for \isa{Suc 0} and \isa{Suc(Suc 0)},  paulson@10594  112 respectively. The literals \isa{\#0}, \isa{\#1} and \isa{\#2} are  paulson@10794  113 syntactically different from \isa{0}, \isa{1} and \isa{2}. You will  paulson@10594  114 sometimes prefer one notation to the other. Literals are obviously  paulson@10794  115 necessary to express large values, while \isa{0} and \isa{Suc} are needed  paulson@10794  116 in order to match many theorems, including the rewrite rules for primitive  paulson@10794  117 recursive functions. The following default simplification rules replace  paulson@10794  118 small literals by zero and successor:  paulson@10594  119 \begin{isabelle}  paulson@10594  120 \#0\ =\ 0  paulson@10594  121 \rulename{numeral_0_eq_0}\isanewline  paulson@10594  122 \#1\ =\ 1  paulson@10594  123 \rulename{numeral_1_eq_1}\isanewline  paulson@10594  124 \#2\ +\ n\ =\ Suc\ (Suc\ n)  paulson@10594  125 \rulename{add_2_eq_Suc}\isanewline  paulson@10594  126 n\ +\ \#2\ =\ Suc\ (Suc\ n)  paulson@10594  127 \rulename{add_2_eq_Suc'}  paulson@10594  128 \end{isabelle}  paulson@10594  129 In special circumstances, you may wish to remove or reorient  paulson@10594  130 these rules.  paulson@10594  131 paulson@10594  132 \subsubsection{Typical lemmas}  paulson@10594  133 Inequalities involving addition and subtraction alone can be proved  paulson@10594  134 automatically. Lemmas such as these can be used to prove inequalities  paulson@10594  135 involving multiplication and division:  paulson@10594  136 \begin{isabelle}  paulson@10594  137 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%  paulson@10594  138 \rulename{mult_le_mono}\isanewline  paulson@10594  139 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\  paulson@10594  140 *\ k\ <\ j\ *\ k%  paulson@10594  141 \rulename{mult_less_mono1}\isanewline  paulson@10594  142 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%  paulson@10594  143 \rulename{div_le_mono}  paulson@10594  144 \end{isabelle}  paulson@10594  145 %  paulson@10594  146 Various distributive laws concerning multiplication are available:  paulson@10594  147 \begin{isabelle}  paulson@10594  148 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%  paulson@11416  149 \rulenamedx{add_mult_distrib}\isanewline  paulson@10594  150 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%  paulson@11416  151 \rulenamedx{diff_mult_distrib}\isanewline  paulson@10594  152 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)  paulson@11416  153 \rulenamedx{mod_mult_distrib}  paulson@10594  154 \end{isabelle}  paulson@10594  155 paulson@10594  156 \subsubsection{Division}  paulson@11416  157 \index{division!for type \protect\isa{nat}}%  paulson@10881  158 The infix operators \isa{div} and \isa{mod} are overloaded.  paulson@10881  159 Isabelle/HOL provides the basic facts about quotient and remainder  paulson@10881  160 on the natural numbers:  paulson@10594  161 \begin{isabelle}  paulson@10594  162 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)  paulson@10594  163 \rulename{mod_if}\isanewline  paulson@10594  164 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%  paulson@11416  165 \rulenamedx{mod_div_equality}  paulson@10594  166 \end{isabelle}  paulson@10594  167 paulson@10594  168 Many less obvious facts about quotient and remainder are also provided.  paulson@10594  169 Here is a selection:  paulson@10594  170 \begin{isabelle}  paulson@10594  171 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%  paulson@10594  172 \rulename{div_mult1_eq}\isanewline  paulson@10594  173 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%  paulson@10594  174 \rulename{mod_mult1_eq}\isanewline  paulson@10594  175 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%  paulson@10594  176 \rulename{div_mult2_eq}\isanewline  paulson@10594  177 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%  paulson@10594  178 \rulename{mod_mult2_eq}\isanewline  paulson@10594  179 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%  paulson@10594  180 \rulename{div_mult_mult1}  paulson@10594  181 \end{isabelle}  paulson@10594  182 paulson@10594  183 Surprisingly few of these results depend upon the  paulson@11416  184 divisors' being nonzero.  paulson@11416  185 \index{division!by zero}%  paulson@11416  186 That is because division by  paulson@10794  187 zero yields zero:  paulson@10594  188 \begin{isabelle}  paulson@10594  189 a\ div\ 0\ =\ 0  paulson@10594  190 \rulename{DIVISION_BY_ZERO_DIV}\isanewline  paulson@10594  191 a\ mod\ 0\ =\ a%  paulson@10594  192 \rulename{DIVISION_BY_ZERO_MOD}  paulson@10594  193 \end{isabelle}  paulson@10594  194 As a concession to convention, these equations are not installed as default  paulson@11174  195 simplification rules. In \isa{div_mult_mult1} above, one of  nipkow@11161  196 the two divisors (namely~\isa{c}) must still be nonzero.  paulson@10594  197 paulson@11416  198 The \textbf{divides} relation\index{divides relation}  paulson@11416  199 has the standard definition, which  paulson@10594  200 is overloaded over all numeric types:  paulson@10594  201 \begin{isabelle}  paulson@10594  202 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k  paulson@11416  203 \rulenamedx{dvd_def}  paulson@10594  204 \end{isabelle}  paulson@10594  205 %  paulson@10594  206 Section~\ref{sec:proving-euclid} discusses proofs involving this  paulson@10594  207 relation. Here are some of the facts proved about it:  paulson@10594  208 \begin{isabelle}  paulson@10594  209 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%  paulson@11416  210 \rulenamedx{dvd_anti_sym}\isanewline  paulson@10594  211 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)  paulson@11416  212 \rulenamedx{dvd_add}  paulson@10594  213 \end{isabelle}  paulson@10594  214 nipkow@11216  215 \subsubsection{Simplifier Tricks}  paulson@10594  216 The rule \isa{diff_mult_distrib} shown above is one of the few facts  paulson@10594  217 about \isa{m\ -\ n} that is not subject to  paulson@10594  218 the condition \isa{n\ \isasymle \ m}. Natural number subtraction has few  paulson@10794  219 nice properties; often you should remove it by simplifying with this split  paulson@10794  220 rule:  paulson@10594  221 \begin{isabelle}  paulson@10594  222 `P(a-b)\ =\ ((a