doc-src/TutorialI/Types/numerics.tex
 author paulson Thu May 08 12:52:15 2003 +0200 (2003-05-08) changeset 13979 4c3a638828b9 parent 13750 b5cd10cb106b child 13983 afc0dadddaa4 permissions -rw-r--r--
HOL-Complex
     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.

    32

    33 The simplifier reduces arithmetic expressions in other

    34 ways, such as dividing through by common factors.  For problems that lie

    35 outside the scope of automation, HOL provides hundreds of

    36 theorems about multiplication, division, etc., that can be brought to

    37 bear.  You can locate them using Proof General's Find

    38 button.  A few lemmas are given below to show what

    39 is available.

    40

    41 \subsection{Numeric Literals}

    42 \label{sec:numerals}

    43

    44 \index{numeric literals|(}%

    45 The constants \cdx{0} and \cdx{1} are overloaded.  They denote zero and one,

    46 respectively, for all numeric types.  Other values are expressed by numeric

    47 literals, which consist of one or more decimal digits optionally preceeded by

    48 a minus sign (\isa{-}).  Examples are \isa{2}, \isa{-3} and

    49 \isa{441223334678}.  Literals are available for the types of natural numbers,

    50 integers and reals; they denote integer values of arbitrary size.

    51

    52 Literals look like constants, but they abbreviate

    53 terms representing the number in a two's complement binary notation.

    54 Isabelle performs arithmetic on literals by rewriting rather

    55 than using the hardware arithmetic. In most cases arithmetic

    56 is fast enough, even for large numbers. The arithmetic operations

    57 provided for literals include addition, subtraction, multiplication,

    58 integer division and remainder.  Fractions of literals (expressed using

    59 division) are reduced to lowest terms.

    60

    61 \begin{warn}\index{overloading!and arithmetic}

    62 The arithmetic operators are

    63 overloaded, so you must be careful to ensure that each numeric

    64 expression refers to a specific type, if necessary by inserting

    65 type constraints.  Here is an example of what can go wrong:

    66 \par

    67 \begin{isabelle}

    68 \isacommand{lemma}\ "2\ *\ m\ =\ m\ +\ m"

    69 \end{isabelle}

    70 %

    71 Carefully observe how Isabelle displays the subgoal:

    72 \begin{isabelle}

    73 \ 1.\ (2::'a)\ *\ m\ =\ m\ +\ m

    74 \end{isabelle}

    75 The type \isa{'a} given for the literal \isa{2} warns us that no numeric

    76 type has been specified.  The problem is underspecified.  Given a type

    77 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.

    78 \end{warn}

    79

    80 \begin{warn}

    81 \index{recdef@\isacommand {recdef} (command)!and numeric literals}

    82 Numeric literals are not constructors and therefore

    83 must not be used in patterns.  For example, this declaration is

    84 rejected:

    85 \begin{isabelle}

    86 \isacommand{recdef}\ h\ "\isacharbraceleft \isacharbraceright "\isanewline

    87 "h\ 3\ =\ 2"\isanewline

    88 "h\ i\ \ =\ i"

    89 \end{isabelle}

    90

    91 You should use a conditional expression instead:

    92 \begin{isabelle}

    93 "h\ i\ =\ (if\ i\ =\ 3\ then\ 2\ else\ i)"

    94 \end{isabelle}

    95 \index{numeric literals|)}

    96 \end{warn}

    97

    98

    99

   100 \subsection{The Type of Natural Numbers, {\tt\slshape nat}}

   101

   102 \index{natural numbers|(}\index{*nat (type)|(}%

   103 This type requires no introduction: we have been using it from the

   104 beginning.  Hundreds of theorems about the natural numbers are

   105 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}.  Only

   106 in exceptional circumstances should you resort to induction.

   107

   108 \subsubsection{Literals}

   109 \index{numeric literals!for type \protect\isa{nat}}%

   110 The notational options for the natural  numbers are confusing.  Recall that an

   111 overloaded constant can be defined independently for each type; the definition

   112 of \cdx{1} for type \isa{nat} is

   113 \begin{isabelle}

   114 1\ \isasymequiv\ Suc\ 0

   115 \rulename{One_nat_def}

   116 \end{isabelle}

   117 This is installed as a simplification rule, so the simplifier will replace

   118 every occurrence of \isa{1::nat} by \isa{Suc\ 0}.  Literals are obviously

   119 better than nested \isa{Suc}s at expressing large values.  But many theorems,

   120 including the rewrite rules for primitive recursive functions, can only be

   121 applied to terms of the form \isa{Suc\ $n$}.

   122

   123 The following default  simplification rules replace

   124 small literals by zero and successor:

   125 \begin{isabelle}

   126 2\ +\ n\ =\ Suc\ (Suc\ n)

   127 \rulename{add_2_eq_Suc}\isanewline

   128 n\ +\ 2\ =\ Suc\ (Suc\ n)

   129 \rulename{add_2_eq_Suc'}

   130 \end{isabelle}

   131 It is less easy to transform \isa{100} into \isa{Suc\ 99} (for example), and

   132 the simplifier will normally reverse this transformation.  Novices should

   133 express natural numbers using \isa{0} and \isa{Suc} only.

   134

   135 \subsubsection{Typical lemmas}

   136 Inequalities involving addition and subtraction alone can be proved

   137 automatically.  Lemmas such as these can be used to prove inequalities

   138 involving multiplication and division:

   139 \begin{isabelle}

   140 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%

   141 \rulename{mult_le_mono}\isanewline

   142 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\

   143 *\ k\ <\ j\ *\ k%

   144 \rulename{mult_less_mono1}\isanewline

   145 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%

   146 \rulename{div_le_mono}

   147 \end{isabelle}

   148 %

   149 Various distributive laws concerning multiplication are available:

   150 \begin{isabelle}

   151 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%

   152 \rulenamedx{add_mult_distrib}\isanewline

   153 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%

   154 \rulenamedx{diff_mult_distrib}\isanewline

   155 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)

   156 \rulenamedx{mod_mult_distrib}

   157 \end{isabelle}

   158

   159 \subsubsection{Division}

   160 \index{division!for type \protect\isa{nat}}%

   161 The infix operators \isa{div} and \isa{mod} are overloaded.

   162 Isabelle/HOL provides the basic facts about quotient and remainder

   163 on the natural numbers:

   164 \begin{isabelle}

   165 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)

   166 \rulename{mod_if}\isanewline

   167 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%

   168 \rulenamedx{mod_div_equality}

   169 \end{isabelle}

   170

   171 Many less obvious facts about quotient and remainder are also provided.

   172 Here is a selection:

   173 \begin{isabelle}

   174 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%

   175 \rulename{div_mult1_eq}\isanewline

   176 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%

   177 \rulename{mod_mult1_eq}\isanewline

   178 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%

   179 \rulename{div_mult2_eq}\isanewline

   180 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%

   181 \rulename{mod_mult2_eq}\isanewline

   182 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%

   183 \rulename{div_mult_mult1}

   184 \end{isabelle}

   185

   186 Surprisingly few of these results depend upon the

   187 divisors' being nonzero.

   188 \index{division!by zero}%

   189 That is because division by

   190 zero yields zero:

   191 \begin{isabelle}

   192 a\ div\ 0\ =\ 0

   193 \rulename{DIVISION_BY_ZERO_DIV}\isanewline

   194 a\ mod\ 0\ =\ a%

   195 \rulename{DIVISION_BY_ZERO_MOD}

   196 \end{isabelle}

   197 As a concession to convention, these equations are not installed as default

   198 simplification rules.  In \isa{div_mult_mult1} above, one of

   199 the two divisors (namely~\isa{c}) must still be nonzero.

   200

   201 The \textbf{divides} relation\index{divides relation}

   202 has the standard definition, which

   203 is overloaded over all numeric types:

   204 \begin{isabelle}

   205 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k

   206 \rulenamedx{dvd_def}

   207 \end{isabelle}

   208 %

   209 Section~\ref{sec:proving-euclid} discusses proofs involving this

   210 relation.  Here are some of the facts proved about it:

   211 \begin{isabelle}

   212 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%

   213 \rulenamedx{dvd_anti_sym}\isanewline

   214 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)

   215 \rulenamedx{dvd_add}

   216 \end{isabelle}

   217

   218 \subsubsection{Simplifier Tricks}

   219 The rule \isa{diff_mult_distrib} shown above is one of the few facts

   220 about \isa{m\ -\ n} that is not subject to

   221 the condition \isa{n\ \isasymle \  m}.  Natural number subtraction has few

   222 nice properties; often you should remove it by simplifying with this split

   223 rule:

   224 \begin{isabelle}

   225 P(a-b)\ =\ ((a<b\ \isasymlongrightarrow \ P\

   226 0)\ \isasymand \ (\isasymforall d.\ a\ =\ b+d\ \isasymlongrightarrow \ P\

   227 d))

   228 \rulename{nat_diff_split}

   229 \end{isabelle}

   230 For example, splitting helps to prove the following fact:

   231 \begin{isabelle}

   232 \isacommand{lemma}\ "(n\ -\ 2)\ *\ (n\ +\ 2)\ =\ n\ *\ n\ -\ (4::nat)"\isanewline

   233 \isacommand{apply}\ (simp\ split:\ nat_diff_split,\ clarify)\isanewline

   234 \ 1.\ \isasymAnd d.\ \isasymlbrakk n\ <\ 2;\ n\ *\ n\ =\ 4\ +\ d\isasymrbrakk \ \isasymLongrightarrow \ d\ =\ 0

   235 \end{isabelle}

   236 The result lies outside the scope of linear arithmetic, but

   237  it is easily found

   238 if we explicitly split \isa{n<2} as \isa{n=0} or \isa{n=1}:

   239 \begin{isabelle}

   240 \isacommand{apply}\ (subgoal_tac\ "n=0\ |\ n=1",\ force,\ arith)\isanewline

   241 \isacommand{done}

   242 \end{isabelle}

   243

   244 Suppose that two expressions are equal, differing only in

   245 associativity and commutativity of addition.  Simplifying with the

   246 following equations sorts the terms and groups them to the right, making

   247 the two expressions identical:

   248 \begin{isabelle}

   249 m\ +\ n\ +\ k\ =\ m\ +\ (n\ +\ k)

   250 \rulenamedx{add_assoc}\isanewline

   251 m\ +\ n\ =\ n\ +\ m%

   252 \rulenamedx{add_commute}\isanewline

   253 x\ +\ (y\ +\ z)\ =\ y\ +\ (x\

   254 +\ z)

   255 \rulename{add_left_commute}

   256 \end{isabelle}

   257 The name \isa{add_ac}\index{*add_ac (theorems)}

   258 refers to the list of all three theorems; similarly

   259 there is \isa{mult_ac}.\index{*mult_ac (theorems)}

   260 Here is an example of the sorting effect.  Start

   261 with this goal:

   262 \begin{isabelle}

   263 \ 1.\ Suc\ (i\ +\ j\ *\ l\ *\ k\ +\ m\ *\ n)\ =\

   264 f\ (n\ *\ m\ +\ i\ +\ k\ *\ j\ *\ l)

   265 \end{isabelle}

   266 %

   267 Simplify using  \isa{add_ac} and \isa{mult_ac}:

   268 \begin{isabelle}

   269 \isacommand{apply}\ (simp\ add:\ add_ac\ mult_ac)

   270 \end{isabelle}

   271 %

   272 Here is the resulting subgoal:

   273 \begin{isabelle}

   274 \ 1.\ Suc\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))\

   275 =\ f\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))%

   276 \end{isabelle}%

   277 \index{natural numbers|)}\index{*nat (type)|)}

   278

   279

   280

   281 \subsection{The Type of Integers, {\tt\slshape int}}

   282

   283 \index{integers|(}\index{*int (type)|(}%

   284 Reasoning methods resemble those for the natural numbers, but induction and

   285 the constant \isa{Suc} are not available.  HOL provides many lemmas

   286 for proving inequalities involving integer multiplication and division,

   287 similar to those shown above for type~\isa{nat}.

   288

   289 The \rmindex{absolute value} function \cdx{abs} is overloaded for the numeric types.

   290 It is defined for the integers; we have for example the obvious law

   291 \begin{isabelle}

   292 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar

   293 \rulename{abs_mult}

   294 \end{isabelle}

   295

   296 \begin{warn}

   297 The absolute value bars shown above cannot be typed on a keyboard.  They

   298 can be entered using the X-symbol package.  In \textsc{ascii}, type \isa{abs x} to

   299 get \isa{\isasymbar x\isasymbar}.

   300 \end{warn}

   301

   302 The \isa{arith} method can prove facts about \isa{abs} automatically,

   303 though as it does so by case analysis, the cost can be exponential.

   304 \begin{isabelle}

   305 \isacommand{lemma}\ "abs\ (x+y)\ \isasymle \ abs\ x\ +\ abs\ (y\ ::\ int)"\isanewline

   306 \isacommand{by}\ arith

   307 \end{isabelle}

   308

   309 Concerning simplifier tricks, we have no need to eliminate subtraction: it

   310 is well-behaved.  As with the natural numbers, the simplifier can sort the

   311 operands of sums and products.  The name \isa{zadd_ac}\index{*zadd_ac (theorems)}

   312 refers to the

   313 associativity and commutativity theorems for integer addition, while

   314 \isa{zmult_ac}\index{*zmult_ac (theorems)}

   315 has the analogous theorems for multiplication.  The

   316 prefix~\isa{z} in many theorem names recalls the use of $\mathbb{Z}$ to

   317 denote the set of integers.

   318

   319 For division and remainder,\index{division!by negative numbers}

   320 the treatment of negative divisors follows

   321 mathematical practice: the sign of the remainder follows that

   322 of the divisor:

   323 \begin{isabelle}

   324 0\ <\ b\ \isasymLongrightarrow \ 0\ \isasymle \ a\ mod\ b%

   325 \rulename{pos_mod_sign}\isanewline

   326 0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%

   327 \rulename{pos_mod_bound}\isanewline

   328 b\ <\ 0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ 0

   329 \rulename{neg_mod_sign}\isanewline

   330 b\ <\ 0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%

   331 \rulename{neg_mod_bound}

   332 \end{isabelle}

   333 ML treats negative divisors in the same way, but most computer hardware

   334 treats signed operands using the same rules as for multiplication.

   335 Many facts about quotients and remainders are provided:

   336 \begin{isabelle}

   337 (a\ +\ b)\ div\ c\ =\isanewline

   338 a\ div\ c\ +\ b\ div\ c\ +\ (a\ mod\ c\ +\ b\ mod\ c)\ div\ c%

   339 \rulename{zdiv_zadd1_eq}

   340 \par\smallskip

   341 (a\ +\ b)\ mod\ c\ =\ (a\ mod\ c\ +\ b\ mod\ c)\ mod\ c%

   342 \rulename{zmod_zadd1_eq}

   343 \end{isabelle}

   344

   345 \begin{isabelle}

   346 (a\ *\ b)\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%

   347 \rulename{zdiv_zmult1_eq}\isanewline

   348 (a\ *\ b)\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%

   349 \rulename{zmod_zmult1_eq}

   350 \end{isabelle}

   351

   352 \begin{isabelle}

   353 0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%

   354 \rulename{zdiv_zmult2_eq}\isanewline

   355 0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\

   356 c)\ +\ a\ mod\ b%

   357 \rulename{zmod_zmult2_eq}

   358 \end{isabelle}

   359 The last two differ from their natural number analogues by requiring

   360 \isa{c} to be positive.  Since division by zero yields zero, we could allow

   361 \isa{c} to be zero.  However, \isa{c} cannot be negative: a counterexample

   362 is

   363 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of

   364 \isa{zdiv_zmult2_eq} is $-2$ while the right-hand side is~$-1$.%

   365 \index{integers|)}\index{*int (type)|)}

   366

   367 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 $<$):

   368 \begin{isabelle}

   369 \isasymlbrakk k\ \isasymle \ i;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk k\ \isasymle \ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%

   370 \rulename{int_ge_induct}\isanewline

   371 \isasymlbrakk k\ <\ i;\ P(k+1);\ \isasymAnd i.\ \isasymlbrakk k\ <\ i;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i+1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%

   372 \rulename{int_gr_induct}\isanewline

   373 \isasymlbrakk i\ \isasymle \ k;\ P\ k;\ \isasymAnd i.\ \isasymlbrakk i\ \isasymle \ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%

   374 \rulename{int_le_induct}\isanewline

   375 \isasymlbrakk i\ <\ k;\ P(k-1);\ \isasymAnd i.\ \isasymlbrakk i\ <\ k;\ P\ i\isasymrbrakk \ \isasymLongrightarrow \ P(i-1)\isasymrbrakk \ \isasymLongrightarrow \ P\ i%

   376 \rulename{int_less_induct}

   377 \end{isabelle}

   378

   379

   380 \subsection{The Type of Real Numbers, {\tt\slshape real}}

   381

   382 \index{real numbers|(}\index{*real (type)|(}%

   383 The real numbers enjoy two significant properties that the integers lack.

   384 They are

   385 \textbf{dense}: between every two distinct real numbers there lies another.

   386 This property follows from the division laws, since if $x<y$ then between

   387 them lies $(x+y)/2$.  The second property is that they are

   388 \textbf{complete}: every set of reals that is bounded above has a least

   389 upper bound.  Completeness distinguishes the reals from the rationals, for

   390 which the set $\{x\mid x^2<2\}$ has no least upper bound.  (It could only be

   391 $\surd2$, which is irrational.)

   392 The formalization of completeness is complicated; rather than

   393 reproducing it here, we refer you to the theory \texttt{RComplete} in

   394 directory \texttt{Real}.

   395 Density, however, is trivial to express:

   396 \begin{isabelle}

   397 x\ <\ y\ \isasymLongrightarrow \ \isasymexists r.\ x\ <\ r\ \isasymand \ r\ <\ y%

   398 \rulename{real_dense}

   399 \end{isabelle}

   400

   401 Here is a selection of rules about the division operator.  The following

   402 are installed as default simplification rules in order to express

   403 combinations of products and quotients as rational expressions:

   404 \begin{isabelle}

   405 x\ *\ (y\ /\ z)\ =\ x\ *\ y\ /\ z

   406 \rulename{real_times_divide1_eq}\isanewline

   407 y\ /\ z\ *\ x\ =\ y\ *\ x\ /\ z

   408 \rulename{real_times_divide2_eq}\isanewline

   409 x\ /\ (y\ /\ z)\ =\ x\ *\ z\ /\ y

   410 \rulename{real_divide_divide1_eq}\isanewline

   411 x\ /\ y\ /\ z\ =\ x\ /\ (y\ *\ z)

   412 \rulename{real_divide_divide2_eq}

   413 \end{isabelle}

   414

   415 Signs are extracted from quotients in the hope that complementary terms can

   416 then be cancelled:

   417 \begin{isabelle}

   418 -\ x\ /\ y\ =\ -\ (x\ /\ y)

   419 \rulename{real_minus_divide_eq}\isanewline

   420 x\ /\ -\ y\ =\ -\ (x\ /\ y)

   421 \rulename{real_divide_minus_eq}

   422 \end{isabelle}

   423

   424 The following distributive law is available, but it is not installed as a

   425 simplification rule.

   426 \begin{isabelle}

   427 (x\ +\ y)\ /\ z\ =\ x\ /\ z\ +\ y\ /\ z%

   428 \rulename{real_add_divide_distrib}

   429 \end{isabelle}

   430

   431 As with the other numeric types, the simplifier can sort the operands of

   432 addition and multiplication.  The name \isa{real_add_ac} refers to the

   433 associativity and commutativity theorems for addition, while similarly

   434 \isa{real_mult_ac} contains those properties for multiplication.

   435

   436 The absolute value function \isa{abs} is

   437 defined for the reals, along with many theorems such as this one about

   438 exponentiation:

   439 \begin{isabelle}

   440 \isasymbar r\ \isacharcircum \ n\isasymbar\ =\

   441 \isasymbar r\isasymbar \ \isacharcircum \ n

   442 \rulename{realpow_abs}

   443 \end{isabelle}

   444

   445 Numeric literals\index{numeric literals!for type \protect\isa{real}}

   446 for type \isa{real} have the same syntax as those for type

   447 \isa{int} and only express integral values.  Fractions expressed

   448 using the division operator are automatically simplified to lowest terms:

   449 \begin{isabelle}

   450 \ 1.\ P\ ((3\ /\ 4)\ *\ (8\ /\ 15))\isanewline

   451 \isacommand{apply} simp\isanewline

   452 \ 1.\ P\ (2\ /\ 5)

   453 \end{isabelle}

   454 Exponentiation can express floating-point values such as

   455 \isa{2 * 10\isacharcircum6}, but at present no special simplification

   456 is performed.

   457

   458

   459 \begin{warn}

   460 Type \isa{real} is only available in the logic HOL-Real, which

   461 is  HOL extended with a definitional development of the real

   462 numbers.  Base your theory upon theory

   463 \thydx{Real}, not the usual \isa{Main}.%

   464 \index{real numbers|)}\index{*real (type)|)}

   465 Launch Isabelle using the command

   466 \begin{verbatim}

   467 Isabelle -l HOL-Real

   468 \end{verbatim}

   469 \end{warn}

   470

   471 Also distributed with Isabelle is HOL-Hyperreal,

   472 whose theory \isa{Hyperreal} defines the type \tydx{hypreal} of

   473 \rmindex{non-standard reals}.  These

   474 \textbf{hyperreals} include infinitesimals, which represent infinitely

   475 small and infinitely large quantities; they facilitate proofs

   476 about limits, differentiation and integration~\cite{fleuriot-jcm}.  The

   477 development defines an infinitely large number, \isa{omega} and an

   478 infinitely small positive number, \isa{epsilon}.  The

   479 relation $x\approx y$ means $x$ is infinitely close to~$y$.''

   480 Theory \isa{Hyperreal} also defines transcendental functions such as sine,

   481 cosine, exponential and logarithm --- even the versions for type

   482 \isa{real}, because they are defined using nonstandard limits.%

   483 \index{numbers|)}