doc-src/TutorialI/Types/numerics.tex
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 paulson@10794  1 % $Id$  paulson@10794  2 Until now, our numerical have used the type of \textbf{natural numbers},  paulson@10594  3 \isa{nat}. This is a recursive datatype generated by the constructors  paulson@10594  4 zero and successor, so it works well with inductive proofs and primitive  paulson@10794  5 recursive function definitions. Isabelle/HOL also provides the type  paulson@10794  6 \isa{int} of \textbf{integers}, which lack induction but support true  paulson@10881  7 subtraction. The logic HOL-Real also has the type \isa{real} of real  paulson@10777  8 numbers. Isabelle has no subtyping, so the numeric types are distinct and  nipkow@10978  9 there are functions to convert between them. Fortunately most numeric  nipkow@10978  10 operations are overloaded: the same symbol can be used at all numeric types.  nipkow@10978  11 Table~\ref{tab:overloading} in the appendix shows the most important operations,  nipkow@10978  12 together with the priorities of the infix symbols.  paulson@10594  13 paulson@10594  14 The integers are preferable to the natural numbers for reasoning about  paulson@10594  15 complicated arithmetic expressions. For example, a termination proof  paulson@10594  16 typically involves an integer metric that is shown to decrease at each  paulson@10794  17 loop iteration. Even if the metric cannot become negative, proofs  paulson@10794  18 may be easier if you use the integers instead of the natural  paulson@10594  19 numbers.  paulson@10594  20 paulson@10594  21 Many theorems involving numeric types can be proved automatically by  paulson@10594  22 Isabelle's arithmetic decision procedure, the method  paulson@10594  23 \isa{arith}. Linear arithmetic comprises addition, subtraction  paulson@10594  24 and multiplication by constant factors; subterms involving other operators  paulson@10594  25 are regarded as variables. The procedure can be slow, especially if the  paulson@10594  26 subgoal to be proved involves subtraction over type \isa{nat}, which  paulson@10594  27 causes case splits.  paulson@10594  28 paulson@10594  29 The simplifier reduces arithmetic expressions in other  paulson@10594  30 ways, such as dividing through by common factors. For problems that lie  paulson@10881  31 outside the scope of automation, HOL provides hundreds of  paulson@10594  32 theorems about multiplication, division, etc., that can be brought to  paulson@10881  33 bear. You can locate them using Proof General's Find  paulson@10881  34 button. A few lemmas are given below to show what  paulson@10794  35 is available.  paulson@10594  36 paulson@10594  37 \subsection{Numeric Literals}  nipkow@10779  38 \label{sec:numerals}  paulson@10594  39 paulson@10594  40 Literals are available for the types of natural numbers, integers  paulson@10594  41 and reals and denote integer values of arbitrary size.  paulson@10594  42 They begin  paulson@10594  43 with a number sign (\isa{\#}), have an optional minus sign (\isa{-}) and  paulson@10594  44 then one or more decimal digits. Examples are \isa{\#0}, \isa{\#-3}  paulson@10594  45 and \isa{\#441223334678}.  paulson@10594  46 paulson@10594  47 Literals look like constants, but they abbreviate  paulson@10594  48 terms, representing the number in a two's complement binary notation.  paulson@10794  49 Isabelle performs arithmetic on literals by rewriting rather  paulson@10594  50 than using the hardware arithmetic. In most cases arithmetic  paulson@10594  51 is fast enough, even for large numbers. The arithmetic operations  paulson@10794  52 provided for literals include addition, subtraction, multiplication,  paulson@10794  53 integer division and remainder. Fractions of literals (expressed using  paulson@10794  54 division) are reduced to lowest terms.  paulson@10594  55 paulson@10794  56 \begin{warn}  paulson@10794  57 The arithmetic operators are  paulson@10594  58 overloaded, so you must be careful to ensure that each numeric  paulson@10594  59 expression refers to a specific type, if necessary by inserting  paulson@10594  60 type constraints. Here is an example of what can go wrong:  paulson@10794  61 \par  paulson@10594  62 \begin{isabelle}  paulson@10594  63 \isacommand{lemma}\ "\#2\ *\ m\ =\ m\ +\ m"  paulson@10594  64 \end{isabelle}  paulson@10594  65 %  paulson@10594  66 Carefully observe how Isabelle displays the subgoal:  paulson@10594  67 \begin{isabelle}  paulson@10594  68 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m  paulson@10594  69 \end{isabelle}  paulson@10594  70 The type \isa{'a} given for the literal \isa{\#2} warns us that no numeric  paulson@10594  71 type has been specified. The problem is underspecified. Given a type  paulson@10594  72 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.  paulson@10794  73 \end{warn}  paulson@10794  74 paulson@10881  75 \begin{warn}  paulson@10881  76 Numeric literals are not constructors and therefore must not be used in  paulson@10881  77 patterns. For example, this declaration is rejected:  paulson@10881  78 \begin{isabelle}  paulson@10881  79 \isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline  nipkow@11148  80 "h\ \#3\ =\ \#2"\isanewline  nipkow@11148  81 "h\ i\ \ =\ i"  paulson@10881  82 \end{isabelle}  paulson@10881  83 paulson@10881  84 You should use a conditional expression instead:  paulson@10881  85 \begin{isabelle}  paulson@10881  86 "h\ i\ =\ (if\ i\ =\ \#3\ then\ \#2\ else\ i)"  paulson@10881  87 \end{isabelle}  paulson@10881  88 \end{warn}  paulson@10881  89 paulson@10594  90 paulson@10594  91 paulson@10594  92 \subsection{The type of natural numbers, {\tt\slshape nat}}  paulson@10594  93 paulson@10594  94 This type requires no introduction: we have been using it from the  paulson@10794  95 beginning. Hundreds of theorems about the natural numbers are  paulson@10594  96 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}. Only  paulson@10594  97 in exceptional circumstances should you resort to induction.  paulson@10594  98 paulson@10594  99 \subsubsection{Literals}  paulson@10594  100 The notational options for the natural numbers can be confusing. The  paulson@10594  101 constant \isa{0} is overloaded to serve as the neutral value  paulson@10594  102 in a variety of additive types. The symbols \isa{1} and \isa{2} are  paulson@10594  103 not constants but abbreviations for \isa{Suc 0} and \isa{Suc(Suc 0)},  paulson@10594  104 respectively. The literals \isa{\#0}, \isa{\#1} and \isa{\#2} are  paulson@10794  105 syntactically different from \isa{0}, \isa{1} and \isa{2}. You will  paulson@10594  106 sometimes prefer one notation to the other. Literals are obviously  paulson@10794  107 necessary to express large values, while \isa{0} and \isa{Suc} are needed  paulson@10794  108 in order to match many theorems, including the rewrite rules for primitive  paulson@10794  109 recursive functions. The following default simplification rules replace  paulson@10794  110 small literals by zero and successor:  paulson@10594  111 \begin{isabelle}  paulson@10594  112 \#0\ =\ 0  paulson@10594  113 \rulename{numeral_0_eq_0}\isanewline  paulson@10594  114 \#1\ =\ 1  paulson@10594  115 \rulename{numeral_1_eq_1}\isanewline  paulson@10594  116 \#2\ +\ n\ =\ Suc\ (Suc\ n)  paulson@10594  117 \rulename{add_2_eq_Suc}\isanewline  paulson@10594  118 n\ +\ \#2\ =\ Suc\ (Suc\ n)  paulson@10594  119 \rulename{add_2_eq_Suc'}  paulson@10594  120 \end{isabelle}  paulson@10594  121 In special circumstances, you may wish to remove or reorient  paulson@10594  122 these rules.  paulson@10594  123 paulson@10594  124 \subsubsection{Typical lemmas}  paulson@10594  125 Inequalities involving addition and subtraction alone can be proved  paulson@10594  126 automatically. Lemmas such as these can be used to prove inequalities  paulson@10594  127 involving multiplication and division:  paulson@10594  128 \begin{isabelle}  paulson@10594  129 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%  paulson@10594  130 \rulename{mult_le_mono}\isanewline  paulson@10594  131 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\  paulson@10594  132 *\ k\ <\ j\ *\ k%  paulson@10594  133 \rulename{mult_less_mono1}\isanewline  paulson@10594  134 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%  paulson@10594  135 \rulename{div_le_mono}  paulson@10594  136 \end{isabelle}  paulson@10594  137 %  paulson@10594  138 Various distributive laws concerning multiplication are available:  paulson@10594  139 \begin{isabelle}  paulson@10594  140 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%  paulson@10594  141 \rulename{add_mult_distrib}\isanewline  paulson@10594  142 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%  paulson@10594  143 \rulename{diff_mult_distrib}\isanewline  paulson@10594  144 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)  paulson@10594  145 \rulename{mod_mult_distrib}  paulson@10594  146 \end{isabelle}  paulson@10594  147 paulson@10594  148 \subsubsection{Division}  paulson@10881  149 The infix operators \isa{div} and \isa{mod} are overloaded.  paulson@10881  150 Isabelle/HOL provides the basic facts about quotient and remainder  paulson@10881  151 on the natural numbers:  paulson@10594  152 \begin{isabelle}  paulson@10594  153 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)  paulson@10594  154 \rulename{mod_if}\isanewline  paulson@10594  155 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%  paulson@10594  156 \rulename{mod_div_equality}  paulson@10594  157 \end{isabelle}  paulson@10594  158 paulson@10594  159 Many less obvious facts about quotient and remainder are also provided.  paulson@10594  160 Here is a selection:  paulson@10594  161 \begin{isabelle}  paulson@10594  162 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%  paulson@10594  163 \rulename{div_mult1_eq}\isanewline  paulson@10594  164 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%  paulson@10594  165 \rulename{mod_mult1_eq}\isanewline  paulson@10594  166 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%  paulson@10594  167 \rulename{div_mult2_eq}\isanewline  paulson@10594  168 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%  paulson@10594  169 \rulename{mod_mult2_eq}\isanewline  paulson@10594  170 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%  paulson@10594  171 \rulename{div_mult_mult1}  paulson@10594  172 \end{isabelle}  paulson@10594  173 paulson@10594  174 Surprisingly few of these results depend upon the  paulson@10794  175 divisors' being nonzero. That is because division by  paulson@10794  176 zero yields zero:  paulson@10594  177 \begin{isabelle}  paulson@10594  178 a\ div\ 0\ =\ 0  paulson@10594  179 \rulename{DIVISION_BY_ZERO_DIV}\isanewline  paulson@10594  180 a\ mod\ 0\ =\ a%  paulson@10594  181 \rulename{DIVISION_BY_ZERO_MOD}  paulson@10594  182 \end{isabelle}  paulson@10594  183 As a concession to convention, these equations are not installed as default  paulson@10594  184 simplification rules but are merely used to remove nonzero-divisor  paulson@10594  185 hypotheses by case analysis. In \isa{div_mult_mult1} above, one of  paulson@10594  186 the two divisors (namely~\isa{c}) must be still be nonzero.  paulson@10594  187 paulson@10594  188 The \textbf{divides} relation has the standard definition, which  paulson@10594  189 is overloaded over all numeric types:  paulson@10594  190 \begin{isabelle}  paulson@10594  191 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k  paulson@10594  192 \rulename{dvd_def}  paulson@10594  193 \end{isabelle}  paulson@10594  194 %  paulson@10594  195 Section~\ref{sec:proving-euclid} discusses proofs involving this  paulson@10594  196 relation. Here are some of the facts proved about it:  paulson@10594  197 \begin{isabelle}  paulson@10594  198 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%  paulson@10594  199 \rulename{dvd_anti_sym}\isanewline  paulson@10594  200 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)  paulson@10594  201 \rulename{dvd_add}  paulson@10594  202 \end{isabelle}  paulson@10594  203 paulson@10594  204 \subsubsection{Simplifier tricks}  paulson@10594  205 The rule \isa{diff_mult_distrib} shown above is one of the few facts  paulson@10594  206 about \isa{m\ -\ n} that is not subject to  paulson@10594  207 the condition \isa{n\ \isasymle \ m}. Natural number subtraction has few  paulson@10794  208 nice properties; often you should remove it by simplifying with this split  paulson@10794  209 rule:  paulson@10594  210 \begin{isabelle}  paulson@10594  211 `P(a-b)\ =\ ((a