doc-src/TutorialI/Types/numerics.tex
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 paulson@10794  1 % $Id$  paulson@11389  2 paulson@11389  3 \section{Numbers}  paulson@11389  4 \label{sec:numbers}  paulson@11389  5 paulson@11494  6 \index{numbers|(}%  paulson@11174  7 Until now, our numerical examples have used the type of \textbf{natural  paulson@11174  8 numbers},  paulson@10594  9 \isa{nat}. This is a recursive datatype generated by the constructors  paulson@10594  10 zero and successor, so it works well with inductive proofs and primitive  paulson@11174  11 recursive function definitions. HOL also provides the type  paulson@10794  12 \isa{int} of \textbf{integers}, which lack induction but support true  paulson@11174  13 subtraction. The integers are preferable to the natural numbers for reasoning about  paulson@11174  14 complicated arithmetic expressions, even for some expressions whose  paulson@11174  15 value is non-negative. The logic HOL-Real also has the type  paulson@11174  16 \isa{real} of real numbers. Isabelle has no subtyping, so the numeric  paulson@11174  17 types are distinct and there are functions to convert between them.  paulson@11174  18 Fortunately most numeric operations are overloaded: the same symbol can be  paulson@11174  19 used at all numeric types. Table~\ref{tab:overloading} in the appendix  paulson@11174  20 shows the most important operations, together with the priorities of the  paulson@11174  21 infix symbols.  paulson@10594  22 paulson@11416  23 \index{linear arithmetic}%  paulson@10594  24 Many theorems involving numeric types can be proved automatically by  paulson@10594  25 Isabelle's arithmetic decision procedure, the method  paulson@11416  26 \methdx{arith}. Linear arithmetic comprises addition, subtraction  paulson@10594  27 and multiplication by constant factors; subterms involving other operators  paulson@10594  28 are regarded as variables. The procedure can be slow, especially if the  paulson@10594  29 subgoal to be proved involves subtraction over type \isa{nat}, which  paulson@10594  30 causes case splits.  paulson@10594  31 paulson@10594  32 The simplifier reduces arithmetic expressions in other  paulson@10594  33 ways, such as dividing through by common factors. For problems that lie  paulson@10881  34 outside the scope of automation, HOL provides hundreds of  paulson@10594  35 theorems about multiplication, division, etc., that can be brought to  paulson@10881  36 bear. You can locate them using Proof General's Find  paulson@10881  37 button. A few lemmas are given below to show what  paulson@10794  38 is available.  paulson@10594  39 paulson@10594  40 \subsection{Numeric Literals}  nipkow@10779  41 \label{sec:numerals}  paulson@10594  42 paulson@11416  43 \index{numeric literals|(}%  paulson@12156  44 The constants \cdx{0} and \cdx{1} are overloaded. They denote zero and one,  paulson@12156  45 respectively, for all numeric types. Other values are expressed by numeric  paulson@12156  46 literals, which consist of one or more decimal digits optionally preceeded by  paulson@12156  47 a minus sign (\isa{-}). Examples are \isa{2}, \isa{-3} and  paulson@12156  48 \isa{441223334678}. Literals are available for the types of natural numbers,  paulson@12156  49 integers and reals; they denote integer values of arbitrary size.  paulson@10594  50 paulson@10594  51 Literals look like constants, but they abbreviate  paulson@12156  52 terms representing the number in a two's complement binary notation.  paulson@10794  53 Isabelle performs arithmetic on literals by rewriting rather  paulson@10594  54 than using the hardware arithmetic. In most cases arithmetic  paulson@10594  55 is fast enough, even for large numbers. The arithmetic operations  paulson@10794  56 provided for literals include addition, subtraction, multiplication,  paulson@10794  57 integer division and remainder. Fractions of literals (expressed using  paulson@10794  58 division) are reduced to lowest terms.  paulson@10594  59 paulson@11416  60 \begin{warn}\index{overloading!and arithmetic}  paulson@10794  61 The arithmetic operators are  paulson@10594  62 overloaded, so you must be careful to ensure that each numeric  paulson@10594  63 expression refers to a specific type, if necessary by inserting  paulson@10594  64 type constraints. Here is an example of what can go wrong:  paulson@10794  65 \par  paulson@10594  66 \begin{isabelle}  paulson@12156  67 \isacommand{lemma}\ "2\ *\ m\ =\ m\ +\ m"  paulson@10594  68 \end{isabelle}  paulson@10594  69 %  paulson@10594  70 Carefully observe how Isabelle displays the subgoal:  paulson@10594  71 \begin{isabelle}  paulson@12156  72 \ 1.\ (2::'a)\ *\ m\ =\ m\ +\ m  paulson@10594  73 \end{isabelle}  paulson@12156  74 The type \isa{'a} given for the literal \isa{2} warns us that no numeric  paulson@10594  75 type has been specified. The problem is underspecified. Given a type  paulson@10594  76 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.  paulson@10794  77 \end{warn}  paulson@10794  78 paulson@10881  79 \begin{warn}  paulson@11428  80 \index{recdef@\isacommand {recdef} (command)!and numeric literals}  paulson@11416  81 Numeric literals are not constructors and therefore  paulson@11416  82 must not be used in patterns. For example, this declaration is  paulson@11416  83 rejected:  paulson@10881  84 \begin{isabelle}  paulson@10881  85 \isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline  paulson@12156  86 "h\ 3\ =\ 2"\isanewline  nipkow@11148  87 "h\ i\ \ =\ i"  paulson@10881  88 \end{isabelle}  paulson@10881  89 paulson@10881  90 You should use a conditional expression instead:  paulson@10881  91 \begin{isabelle}  paulson@12156  92 "h\ i\ =\ (if\ i\ =\ 3\ then\ 2\ else\ i)"  paulson@10881  93 \end{isabelle}  paulson@11416  94 \index{numeric literals|)}  paulson@10881  95 \end{warn}  paulson@10881  96 paulson@10594  97 paulson@10594  98 nipkow@11216  99 \subsection{The Type of Natural Numbers, {\tt\slshape nat}}  paulson@10594  100 paulson@11416  101 \index{natural numbers|(}\index{*nat (type)|(}%  paulson@10594  102 This type requires no introduction: we have been using it from the  paulson@10794  103 beginning. Hundreds of theorems about the natural numbers are  paulson@10594  104 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}. Only  paulson@10594  105 in exceptional circumstances should you resort to induction.  paulson@10594  106 paulson@10594  107 \subsubsection{Literals}  paulson@11416  108 \index{numeric literals!for type \protect\isa{nat}}%  paulson@12156  109 The notational options for the natural numbers are confusing. Recall that an  paulson@12156  110 overloaded constant can be defined independently for each type; the definition  paulson@12156  111 of \cdx{1} for type \isa{nat} is  paulson@12156  112 \begin{isabelle}  paulson@12156  113 1\ \isasymequiv\ Suc\ 0  paulson@12156  114 \rulename{One_nat_def}  paulson@12156  115 \end{isabelle}  paulson@12156  116 This is installed as a simplification rule, so the simplifier will replace  paulson@12156  117 every occurrence of \isa{1::nat} by \isa{Suc\ 0}. Literals are obviously  paulson@12156  118 better than nested \isa{Suc}s at expressing large values. But many theorems,  paulson@12156  119 including the rewrite rules for primitive recursive functions, can only be  paulson@12156  120 applied to terms of the form \isa{Suc\ $n$}.  paulson@12156  121 paulson@12156  122 The following default simplification rules replace  paulson@10794  123 small literals by zero and successor:  paulson@10594  124 \begin{isabelle}  paulson@12156  125 2\ +\ n\ =\ Suc\ (Suc\ n)  paulson@10594  126 \rulename{add_2_eq_Suc}\isanewline  paulson@12156  127 n\ +\ 2\ =\ Suc\ (Suc\ n)  paulson@10594  128 \rulename{add_2_eq_Suc'}  paulson@10594  129 \end{isabelle}  paulson@12156  130 It is less easy to transform \isa{100} into \isa{Suc\ 99} (for example), and  paulson@12156  131 the simplifier will normally reverse this transformation. Novices should  paulson@12156  132 express natural numbers using \isa{0} and \isa{Suc} only.  paulson@10594  133 paulson@10594  134 \subsubsection{Typical lemmas}  paulson@10594  135 Inequalities involving addition and subtraction alone can be proved  paulson@10594  136 automatically. Lemmas such as these can be used to prove inequalities  paulson@10594  137 involving multiplication and division:  paulson@10594  138 \begin{isabelle}  paulson@10594  139 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%  paulson@10594  140 \rulename{mult_le_mono}\isanewline  paulson@10594  141 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\  paulson@10594  142 *\ k\ <\ j\ *\ k%  paulson@10594  143 \rulename{mult_less_mono1}\isanewline  paulson@10594  144 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%  paulson@10594  145 \rulename{div_le_mono}  paulson@10594  146 \end{isabelle}  paulson@10594  147 %  paulson@10594  148 Various distributive laws concerning multiplication are available:  paulson@10594  149 \begin{isabelle}  paulson@10594  150 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%  paulson@11416  151 \rulenamedx{add_mult_distrib}\isanewline  paulson@10594  152 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%  paulson@11416  153 \rulenamedx{diff_mult_distrib}\isanewline  paulson@10594  154 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)  paulson@11416  155 \rulenamedx{mod_mult_distrib}  paulson@10594  156 \end{isabelle}  paulson@10594  157 paulson@10594  158 \subsubsection{Division}  paulson@11416  159 \index{division!for type \protect\isa{nat}}%  paulson@10881  160 The infix operators \isa{div} and \isa{mod} are overloaded.  paulson@10881  161 Isabelle/HOL provides the basic facts about quotient and remainder  paulson@10881  162 on the natural numbers:  paulson@10594  163 \begin{isabelle}  paulson@10594  164 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)  paulson@10594  165 \rulename{mod_if}\isanewline  paulson@10594  166 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%  paulson@11416  167 \rulenamedx{mod_div_equality}  paulson@10594  168 \end{isabelle}  paulson@10594  169 paulson@10594  170 Many less obvious facts about quotient and remainder are also provided.  paulson@10594  171 Here is a selection:  paulson@10594  172 \begin{isabelle}  paulson@10594  173 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%  paulson@10594  174 \rulename{div_mult1_eq}\isanewline  paulson@10594  175 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%  paulson@10594  176 \rulename{mod_mult1_eq}\isanewline  paulson@10594  177 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%  paulson@10594  178 \rulename{div_mult2_eq}\isanewline  paulson@10594  179 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%  paulson@10594  180 \rulename{mod_mult2_eq}\isanewline  paulson@10594  181 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%  paulson@10594  182 \rulename{div_mult_mult1}  paulson@10594  183 \end{isabelle}  paulson@10594  184 paulson@10594  185 Surprisingly few of these results depend upon the  paulson@11416  186 divisors' being nonzero.  paulson@11416  187 \index{division!by zero}%  paulson@11416  188 That is because division by  paulson@10794  189 zero yields zero:  paulson@10594  190 \begin{isabelle}  paulson@10594  191 a\ div\ 0\ =\ 0  paulson@10594  192 \rulename{DIVISION_BY_ZERO_DIV}\isanewline  paulson@10594  193 a\ mod\ 0\ =\ a%  paulson@10594  194 \rulename{DIVISION_BY_ZERO_MOD}  paulson@10594  195 \end{isabelle}  paulson@10594  196 As a concession to convention, these equations are not installed as default  paulson@11174  197 simplification rules. In \isa{div_mult_mult1} above, one of  nipkow@11161  198 the two divisors (namely~\isa{c}) must still be nonzero.  paulson@10594  199 paulson@11416  200 The \textbf{divides} relation\index{divides relation}  paulson@11416  201 has the standard definition, which  paulson@10594  202 is overloaded over all numeric types:  paulson@10594  203 \begin{isabelle}  paulson@10594  204 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k  paulson@11416  205 \rulenamedx{dvd_def}  paulson@10594  206 \end{isabelle}  paulson@10594  207 %  paulson@10594  208 Section~\ref{sec:proving-euclid} discusses proofs involving this  paulson@10594  209 relation. Here are some of the facts proved about it:  paulson@10594  210 \begin{isabelle}  paulson@10594  211 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%  paulson@11416  212 \rulenamedx{dvd_anti_sym}\isanewline  paulson@10594  213 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)  paulson@11416  214 \rulenamedx{dvd_add}  paulson@10594  215 \end{isabelle}  paulson@10594  216 nipkow@11216  217 \subsubsection{Simplifier Tricks}  paulson@10594  218 The rule \isa{diff_mult_distrib} shown above is one of the few facts  paulson@10594  219 about \isa{m\ -\ n} that is not subject to  paulson@10594  220 the condition \isa{n\ \isasymle \ m}. Natural number subtraction has few  paulson@10794  221 nice properties; often you should remove it by simplifying with this split  paulson@10794  222 rule:  paulson@10594  223 \begin{isabelle}  paulson@10594  224 `P(a-b)\ =\ ((a