doc-src/TutorialI/Types/numerics.tex
author paulson
Mon Jan 12 16:51:45 2004 +0100 (2004-01-12)
changeset 14353 79f9fbef9106
parent 14295 7f115e5c5de4
child 14400 6069098854b9
permissions -rw-r--r--
Added lemmas to Ring_and_Field with slightly modified simplification rules

Deleted some little-used integer theorems, replacing them by the generic ones
in Ring_and_Field

Consolidated integer powers
     1 % $Id$
     2 
     3 \section{Numbers}
     4 \label{sec:numbers}
     5 
     6 \index{numbers|(}%
     7 Until now, our numerical examples have used the type of \textbf{natural
     8 numbers},
     9 \isa{nat}.  This is a recursive datatype generated by the constructors
    10 zero  and successor, so it works well with inductive proofs and primitive
    11 recursive function definitions.  HOL also provides the type
    12 \isa{int} of \textbf{integers}, which lack induction but support true
    13 subtraction.  The integers are preferable to the natural numbers for reasoning about
    14 complicated arithmetic expressions, even for some expressions whose
    15 value is non-negative.  The logic HOL-Complex also has the types
    16 \isa{real} and \isa{complex}: the real and complex numbers.  Isabelle has no 
    17 subtyping,  so the numeric
    18 types are distinct and there are functions to convert between them.
    19 Fortunately most numeric operations are overloaded: the same symbol can be
    20 used at all numeric types. Table~\ref{tab:overloading} in the appendix
    21 shows the most important operations, together with the priorities of the
    22 infix symbols.
    23 
    24 \index{linear arithmetic}%
    25 Many theorems involving numeric types can be proved automatically by
    26 Isabelle's arithmetic decision procedure, the method
    27 \methdx{arith}.  Linear arithmetic comprises addition, subtraction
    28 and multiplication by constant factors; subterms involving other operators
    29 are regarded as variables.  The procedure can be slow, especially if the
    30 subgoal to be proved involves subtraction over type \isa{nat}, which 
    31 causes case splits.  On types \isa{nat} and \isa{int}, \methdx{arith}
    32 can deal with quantifiers (this is known as ``Presburger Arithmetic''),
    33 whereas on type \isa{real} it cannot.
    34 
    35 The simplifier reduces arithmetic expressions in other
    36 ways, such as dividing through by common factors.  For problems that lie
    37 outside the scope of automation, HOL provides hundreds of
    38 theorems about multiplication, division, etc., that can be brought to
    39 bear.  You can locate them using Proof General's Find
    40 button.  A few lemmas are given below to show what
    41 is available.
    42 
    43 \subsection{Numeric Literals}
    44 \label{sec:numerals}
    45 
    46 \index{numeric literals|(}%
    47 The constants \cdx{0} and \cdx{1} are overloaded.  They denote zero and one,
    48 respectively, for all numeric types.  Other values are expressed by numeric
    49 literals, which consist of one or more decimal digits optionally preceeded by
    50 a minus sign (\isa{-}).  Examples are \isa{2}, \isa{-3} and
    51 \isa{441223334678}.  Literals are available for the types of natural numbers,
    52 integers and reals; they denote integer values of arbitrary size.
    53 
    54 Literals look like constants, but they abbreviate 
    55 terms representing the number in a two's complement binary notation. 
    56 Isabelle performs arithmetic on literals by rewriting rather 
    57 than using the hardware arithmetic. In most cases arithmetic 
    58 is fast enough, even for large numbers. The arithmetic operations 
    59 provided for literals include addition, subtraction, multiplication, 
    60 integer division and remainder.  Fractions of literals (expressed using
    61 division) are reduced to lowest terms.
    62 
    63 \begin{warn}\index{overloading!and arithmetic}
    64 The arithmetic operators are 
    65 overloaded, so you must be careful to ensure that each numeric 
    66 expression refers to a specific type, if necessary by inserting 
    67 type constraints.  Here is an example of what can go wrong:
    68 \par
    69 \begin{isabelle}
    70 \isacommand{lemma}\ "2\ *\ m\ =\ m\ +\ m"
    71 \end{isabelle}
    72 %
    73 Carefully observe how Isabelle displays the subgoal:
    74 \begin{isabelle}
    75 \ 1.\ (2::'a)\ *\ m\ =\ m\ +\ m
    76 \end{isabelle}
    77 The type \isa{'a} given for the literal \isa{2} warns us that no numeric
    78 type has been specified.  The problem is underspecified.  Given a type
    79 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.
    80 \end{warn}
    81 
    82 \begin{warn}
    83 \index{recdef@\isacommand {recdef} (command)!and numeric literals}  
    84 Numeric literals are not constructors and therefore
    85 must not be used in patterns.  For example, this declaration is
    86 rejected:
    87 \begin{isabelle}
    88 \isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline
    89 "h\ 3\ =\ 2"\isanewline
    90 "h\ i\ \ =\ i"
    91 \end{isabelle}
    92 
    93 You should use a conditional expression instead:
    94 \begin{isabelle}
    95 "h\ i\ =\ (if\ i\ =\ 3\ then\ 2\ else\ i)"
    96 \end{isabelle}
    97 \index{numeric literals|)}
    98 \end{warn}
    99 
   100 
   101 
   102 \subsection{The Type of Natural Numbers, {\tt\slshape nat}}
   103 
   104 \index{natural numbers|(}\index{*nat (type)|(}%
   105 This type requires no introduction: we have been using it from the
   106 beginning.  Hundreds of theorems about the natural numbers are
   107 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}.  Only
   108 in exceptional circumstances should you resort to induction.
   109 
   110 \subsubsection{Literals}
   111 \index{numeric literals!for type \protect\isa{nat}}%
   112 The notational options for the natural  numbers are confusing.  Recall that an
   113 overloaded constant can be defined independently for each type; the definition
   114 of \cdx{1} for type \isa{nat} is
   115 \begin{isabelle}
   116 1\ \isasymequiv\ Suc\ 0
   117 \rulename{One_nat_def}
   118 \end{isabelle}
   119 This is installed as a simplification rule, so the simplifier will replace
   120 every occurrence of \isa{1::nat} by \isa{Suc\ 0}.  Literals are obviously
   121 better than nested \isa{Suc}s at expressing large values.  But many theorems,
   122 including the rewrite rules for primitive recursive functions, can only be
   123 applied to terms of the form \isa{Suc\ $n$}.
   124 
   125 The following default  simplification rules replace
   126 small literals by zero and successor: 
   127 \begin{isabelle}
   128 2\ +\ n\ =\ Suc\ (Suc\ n)
   129 \rulename{add_2_eq_Suc}\isanewline
   130 n\ +\ 2\ =\ Suc\ (Suc\ n)
   131 \rulename{add_2_eq_Suc'}
   132 \end{isabelle}
   133 It is less easy to transform \isa{100} into \isa{Suc\ 99} (for example), and
   134 the simplifier will normally reverse this transformation.  Novices should
   135 express natural numbers using \isa{0} and \isa{Suc} only.
   136 
   137 \subsubsection{Typical lemmas}
   138 Inequalities involving addition and subtraction alone can be proved
   139 automatically.  Lemmas such as these can be used to prove inequalities
   140 involving multiplication and division:
   141 \begin{isabelle}
   142 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%
   143 \rulename{mult_le_mono}\isanewline
   144 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\
   145 *\ k\ <\ j\ *\ k%
   146 \rulename{mult_less_mono1}\isanewline
   147 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%
   148 \rulename{div_le_mono}
   149 \end{isabelle}
   150 %
   151 Various distributive laws concerning multiplication are available:
   152 \begin{isabelle}
   153 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%
   154 \rulenamedx{add_mult_distrib}\isanewline
   155 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%
   156 \rulenamedx{diff_mult_distrib}\isanewline
   157 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)
   158 \rulenamedx{mod_mult_distrib}
   159 \end{isabelle}
   160 
   161 \subsubsection{Division}
   162 \index{division!for type \protect\isa{nat}}%
   163 The infix operators \isa{div} and \isa{mod} are overloaded.
   164 Isabelle/HOL provides the basic facts about quotient and remainder
   165 on the natural numbers:
   166 \begin{isabelle}
   167 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)
   168 \rulename{mod_if}\isanewline
   169 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%
   170 \rulenamedx{mod_div_equality}
   171 \end{isabelle}
   172 
   173 Many less obvious facts about quotient and remainder are also provided. 
   174 Here is a selection:
   175 \begin{isabelle}
   176 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   177 \rulename{div_mult1_eq}\isanewline
   178 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   179 \rulename{mod_mult1_eq}\isanewline
   180 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   181 \rulename{div_mult2_eq}\isanewline
   182 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%
   183 \rulename{mod_mult2_eq}\isanewline
   184 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%
   185 \rulename{div_mult_mult1}
   186 \end{isabelle}
   187 
   188 Surprisingly few of these results depend upon the
   189 divisors' being nonzero.
   190 \index{division!by zero}%
   191 That is because division by
   192 zero yields zero:
   193 \begin{isabelle}
   194 a\ div\ 0\ =\ 0
   195 \rulename{DIVISION_BY_ZERO_DIV}\isanewline
   196 a\ mod\ 0\ =\ a%
   197 \rulename{DIVISION_BY_ZERO_MOD}
   198 \end{isabelle}
   199 As a concession to convention, these equations are not installed as default
   200 simplification rules.  In \isa{div_mult_mult1} above, one of
   201 the two divisors (namely~\isa{c}) must still be nonzero.
   202 
   203 The \textbf{divides} relation\index{divides relation}
   204 has the standard definition, which
   205 is overloaded over all numeric types: 
   206 \begin{isabelle}
   207 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k
   208 \rulenamedx{dvd_def}
   209 \end{isabelle}
   210 %
   211 Section~\ref{sec:proving-euclid} discusses proofs involving this
   212 relation.  Here are some of the facts proved about it:
   213 \begin{isabelle}
   214 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%
   215 \rulenamedx{dvd_anti_sym}\isanewline
   216 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)
   217 \rulenamedx{dvd_add}
   218 \end{isabelle}
   219 
   220 \subsubsection{Simplifier Tricks}
   221 The rule \isa{diff_mult_distrib} shown above is one of the few facts
   222 about \isa{m\ -\ n} that is not subject to
   223 the condition \isa{n\ \isasymle \  m}.  Natural number subtraction has few
   224 nice properties; often you should remove it by simplifying with this split
   225 rule:
   226 \begin{isabelle}
   227 P(a-b)\ =\ ((a<b\ \isasymlongrightarrow \ P\
   228 0)\ \isasymand \ (\isasymforall d.\ a\ =\ b+d\ \isasymlongrightarrow \ P\
   229 d))
   230 \rulename{nat_diff_split}
   231 \end{isabelle}
   232 For example, splitting helps to prove the following fact:
   233 \begin{isabelle}
   234 \isacommand{lemma}\ "(n\ -\ 2)\ *\ (n\ +\ 2)\ =\ n\ *\ n\ -\ (4::nat)"\isanewline
   235 \isacommand{apply}\ (simp\ split:\ nat_diff_split,\ clarify)\isanewline
   236 \ 1.\ \isasymAnd d.\ \isasymlbrakk n\ <\ 2;\ n\ *\ n\ =\ 4\ +\ d\isasymrbrakk \ \isasymLongrightarrow \ d\ =\ 0
   237 \end{isabelle}
   238 The result lies outside the scope of linear arithmetic, but
   239  it is easily found
   240 if we explicitly split \isa{n<2} as \isa{n=0} or \isa{n=1}:
   241 \begin{isabelle}
   242 \isacommand{apply}\ (subgoal_tac\ "n=0\ |\ n=1",\ force,\ arith)\isanewline
   243 \isacommand{done}
   244 \end{isabelle}
   245 
   246 Suppose that two expressions are equal, differing only in 
   247 associativity and commutativity of addition.  Simplifying with the
   248 following equations sorts the terms and groups them to the right, making
   249 the two expressions identical:
   250 \begin{isabelle}
   251 m\ +\ n\ +\ k\ =\ m\ +\ (n\ +\ k)
   252 \rulenamedx{add_assoc}\isanewline
   253 m\ +\ n\ =\ n\ +\ m%
   254 \rulenamedx{add_commute}\isanewline
   255 x\ +\ (y\ +\ z)\ =\ y\ +\ (x\
   256 +\ z)
   257 \rulename{add_left_commute}
   258 \end{isabelle}
   259 The name \isa{add_ac}\index{*add_ac (theorems)} 
   260 refers to the list of all three theorems; similarly
   261 there is \isa{mult_ac}.\index{*mult_ac (theorems)} 
   262 Here is an example of the sorting effect.  Start
   263 with this goal:
   264 \begin{isabelle}
   265 \ 1.\ Suc\ (i\ +\ j\ *\ l\ *\ k\ +\ m\ *\ n)\ =\
   266 f\ (n\ *\ m\ +\ i\ +\ k\ *\ j\ *\ l)
   267 \end{isabelle}
   268 %
   269 Simplify using  \isa{add_ac} and \isa{mult_ac}:
   270 \begin{isabelle}
   271 \isacommand{apply}\ (simp\ add:\ add_ac\ mult_ac)
   272 \end{isabelle}
   273 %
   274 Here is the resulting subgoal:
   275 \begin{isabelle}
   276 \ 1.\ Suc\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))\
   277 =\ f\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))%
   278 \end{isabelle}%
   279 \index{natural numbers|)}\index{*nat (type)|)}
   280 
   281 
   282 
   283 \subsection{The Type of Integers, {\tt\slshape int}}
   284 
   285 \index{integers|(}\index{*int (type)|(}%
   286 Reasoning methods resemble those for the natural numbers, but induction and
   287 the constant \isa{Suc} are not available.  HOL provides many lemmas
   288 for proving inequalities involving integer multiplication and division,
   289 similar to those shown above for type~\isa{nat}.  
   290 
   291 The \rmindex{absolute value} function \cdx{abs} is overloaded for the numeric types.
   292 It is defined for the integers; we have for example the obvious law
   293 \begin{isabelle}
   294 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar 
   295 \rulename{abs_mult}
   296 \end{isabelle}
   297 
   298 \begin{warn}
   299 The absolute value bars shown above cannot be typed on a keyboard.  They
   300 can be entered using the X-symbol package.  In \textsc{ascii}, type \isa{abs x} to
   301 get \isa{\isasymbar x\isasymbar}.
   302 \end{warn}
   303 
   304 The \isa{arith} method can prove facts about \isa{abs} automatically, 
   305 though as it does so by case analysis, the cost can be exponential.
   306 \begin{isabelle}
   307 \isacommand{lemma}\ "abs\ (x+y)\ \isasymle \ abs\ x\ +\ abs\ (y\ ::\ int)"\isanewline
   308 \isacommand{by}\ arith
   309 \end{isabelle}
   310 
   311 Concerning simplifier tricks, we have no need to eliminate subtraction: it
   312 is well-behaved.  As with the natural numbers, the simplifier can sort the
   313 operands of sums and products.  The name \isa{zadd_ac}\index{*zadd_ac (theorems)}
   314 refers to the
   315 associativity and commutativity theorems for integer addition, while
   316 \isa{zmult_ac}\index{*zmult_ac (theorems)}
   317 has the analogous theorems for multiplication.  The
   318 prefix~\isa{z} in many theorem names recalls the use of $\mathbb{Z}$ to
   319 denote the set of integers.
   320 
   321 For division and remainder,\index{division!by negative numbers}
   322 the treatment of negative divisors follows
   323 mathematical practice: the sign of the remainder follows that
   324 of the divisor:
   325 \begin{isabelle}
   326 0\ <\ b\ \isasymLongrightarrow \ 0\ \isasymle \ a\ mod\ b%
   327 \rulename{pos_mod_sign}\isanewline
   328 0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%
   329 \rulename{pos_mod_bound}\isanewline
   330 b\ <\ 0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ 0
   331 \rulename{neg_mod_sign}\isanewline
   332 b\ <\ 0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%
   333 \rulename{neg_mod_bound}
   334 \end{isabelle}
   335 ML treats negative divisors in the same way, but most computer hardware
   336 treats signed operands using the same rules as for multiplication.
   337 Many facts about quotients and remainders are provided:
   338 \begin{isabelle}
   339 (a\ +\ b)\ div\ c\ =\isanewline
   340 a\ div\ c\ +\ b\ div\ c\ +\ (a\ mod\ c\ +\ b\ mod\ c)\ div\ c%
   341 \rulename{zdiv_zadd1_eq}
   342 \par\smallskip
   343 (a\ +\ b)\ mod\ c\ =\ (a\ mod\ c\ +\ b\ mod\ c)\ mod\ c%
   344 \rulename{zmod_zadd1_eq}
   345 \end{isabelle}
   346 
   347 \begin{isabelle}
   348 (a\ *\ b)\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   349 \rulename{zdiv_zmult1_eq}\isanewline
   350 (a\ *\ b)\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   351 \rulename{zmod_zmult1_eq}
   352 \end{isabelle}
   353 
   354 \begin{isabelle}
   355 0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   356 \rulename{zdiv_zmult2_eq}\isanewline
   357 0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\
   358 c)\ +\ a\ mod\ b%
   359 \rulename{zmod_zmult2_eq}
   360 \end{isabelle}
   361 The last two differ from their natural number analogues by requiring
   362 \isa{c} to be positive.  Since division by zero yields zero, we could allow
   363 \isa{c} to be zero.  However, \isa{c} cannot be negative: a counterexample
   364 is
   365 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of
   366 \isa{zdiv_zmult2_eq} is $-2$ while the right-hand side is~$-1$.%
   367 \index{integers|)}\index{*int (type)|)}
   368 
   369 Induction is less important for integers than it is for the natural numbers, but it can be valuable if the range of integers has a lower or upper bound.  There are four rules for integer induction, corresponding to the possible relations of the bound ($\geq$, $>$, $\leq$ and $<$):
   370 \begin{isabelle}
   371 \isasymlbrakk k\ \isasymle \ i;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk k\ \isasymle \ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
   372 \rulename{int_ge_induct}\isanewline
   373 \isasymlbrakk k\ <\ i;\ P(k+1);\ \isasymAnd i.\ \isasymlbrakk k\ <\ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
   374 \rulename{int_gr_induct}\isanewline
   375 \isasymlbrakk i\ \isasymle \ k;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk i\ \isasymle \ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
   376 \rulename{int_le_induct}\isanewline
   377 \isasymlbrakk i\ <\ k;\ P(k-1);\ \isasymAnd i.\ \isasymlbrakk i\ <\ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%
   378 \rulename{int_less_induct}
   379 \end{isabelle}
   380 
   381 
   382 \subsection{The Type of Real Numbers, {\tt\slshape real}}
   383 
   384 \index{real numbers|(}\index{*real (type)|(}%
   385 The real numbers enjoy two significant properties that the integers lack. 
   386 They are
   387 \textbf{dense}: between every two distinct real numbers there lies another.
   388 This property follows from the division laws, since if $x<y$ then between
   389 them lies $(x+y)/2$.  The second property is that they are
   390 \textbf{complete}: every set of reals that is bounded above has a least
   391 upper bound.  Completeness distinguishes the reals from the rationals, for
   392 which the set $\{x\mid x^2<2\}$ has no least upper bound.  (It could only be
   393 $\surd2$, which is irrational.)
   394 The formalization of completeness is complicated; rather than
   395 reproducing it here, we refer you to the theory \texttt{RComplete} in
   396 directory \texttt{Real}.
   397 Density, however, is trivial to express:
   398 \begin{isabelle}
   399 x\ <\ y\ \isasymLongrightarrow \ \isasymexists r.\ x\ <\ r\ \isasymand \ r\ <\ y%
   400 \rulename{dense}
   401 \end{isabelle}
   402 
   403 Here is a selection of rules about the division operator.  The following
   404 are installed as default simplification rules in order to express
   405 combinations of products and quotients as rational expressions:
   406 \begin{isabelle}
   407 a\ *\ (b\ /\ c)\ =\ a\ *\ b\ /\ c
   408 \rulename{times_divide_eq_right}\isanewline
   409 b\ /\ c\ *\ a\ =\ b\ *\ a\ /\ c
   410 \rulename{times_divide_eq_left}\isanewline
   411 a\ /\ (b\ /\ c)\ =\ a\ *\ c\ /\ b
   412 \rulename{divide_divide_eq_right}\isanewline
   413 a\ /\ b\ /\ c\ =\ a\ /\ (b\ *\ c)
   414 \rulename{divide_divide_eq_left}
   415 \end{isabelle}
   416 
   417 Signs are extracted from quotients in the hope that complementary terms can
   418 then be cancelled:
   419 \begin{isabelle}
   420 -\ (a\ /\ b)\ =\ -\ a\ /\ b
   421 \rulename{minus_divide_left}\isanewline
   422 -\ (a\ /\ b)\ =\ a\ /\ -\ b
   423 \rulename{minus_divide_right}
   424 \end{isabelle}
   425 
   426 The following distributive law is available, but it is not installed as a
   427 simplification rule.
   428 \begin{isabelle}
   429 (a\ +\ b)\ /\ c\ =\ a\ /\ c\ +\ b\ /\ c%
   430 \rulename{add_divide_distrib}
   431 \end{isabelle}
   432 
   433 As with the other numeric types, the simplifier can sort the operands of
   434 addition and multiplication.  The name \isa{real_add_ac} refers to the
   435 associativity and commutativity theorems for addition, while similarly
   436 \isa{real_mult_ac} contains those properties for multiplication. 
   437 
   438 The absolute value function \isa{abs} is
   439 defined for the reals, along with many theorems such as this one about
   440 exponentiation:
   441 \begin{isabelle}
   442 \isasymbar a\ \isacharcircum \ n\isasymbar\ =\ 
   443 \isasymbar a\isasymbar \ \isacharcircum \ n
   444 \rulename{power_abs}
   445 \end{isabelle}
   446 
   447 Numeric literals\index{numeric literals!for type \protect\isa{real}}
   448 for type \isa{real} have the same syntax as those for type
   449 \isa{int} and only express integral values.  Fractions expressed
   450 using the division operator are automatically simplified to lowest terms:
   451 \begin{isabelle}
   452 \ 1.\ P\ ((3\ /\ 4)\ *\ (8\ /\ 15))\isanewline
   453 \isacommand{apply} simp\isanewline
   454 \ 1.\ P\ (2\ /\ 5)
   455 \end{isabelle}
   456 Exponentiation can express floating-point values such as
   457 \isa{2 * 10\isacharcircum6}, but at present no special simplification
   458 is performed.
   459 
   460 
   461 \begin{warn}
   462 Type \isa{real} is only available in the logic HOL-Complex, which
   463 is  HOL extended with a definitional development of the real and complex
   464 numbers.  Base your theory upon theory
   465 \thydx{Complex_Main}, not the usual \isa{Main}.%
   466 \index{real numbers|)}\index{*real (type)|)}
   467 Launch Isabelle using the command 
   468 \begin{verbatim}
   469 Isabelle -l HOL-Complex
   470 \end{verbatim}
   471 \end{warn}
   472 
   473 Also available in HOL-Complex is the
   474 theory \isa{Hyperreal}, which define the type \tydx{hypreal} of 
   475 \rmindex{non-standard reals}.  These
   476 \textbf{hyperreals} include infinitesimals, which represent infinitely
   477 small and infinitely large quantities; they facilitate proofs
   478 about limits, differentiation and integration~\cite{fleuriot-jcm}.  The
   479 development defines an infinitely large number, \isa{omega} and an
   480 infinitely small positive number, \isa{epsilon}.  The 
   481 relation $x\approx y$ means ``$x$ is infinitely close to~$y$.''
   482 Theory \isa{Hyperreal} also defines transcendental functions such as sine,
   483 cosine, exponential and logarithm --- even the versions for type
   484 \isa{real}, because they are defined using nonstandard limits.%
   485 \index{numbers|)}