doc-src/TutorialI/Types/numerics.tex
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     1 Our examples until now have used the type of \textbf{natural numbers},

     2 \isa{nat}.  This is a recursive datatype generated by the constructors

     3 zero  and successor, so it works well with inductive proofs and primitive

     4 recursive function definitions. Isabelle/HOL also has the type \isa{int} of

     5 \textbf{integers}, which gives up induction in exchange  for proper

     6 subtraction. The logic HOL-Real also has the type \isa{real} of real

     7 numbers.  Isabelle has no subtyping,  so the numeric types are distinct and

     8 there are  functions to convert between them.

     9

    10 The integers are preferable to the natural  numbers for reasoning about

    11 complicated arithmetic expressions. For  example, a termination proof

    12 typically involves an integer metric  that is shown to decrease at each

    13 loop iteration. Even if the  metric cannot become negative, proofs about it

    14 are usually easier  if the integers are used rather than the natural

    15 numbers.

    16

    17 Many theorems involving numeric types can be proved automatically by

    18 Isabelle's arithmetic decision procedure, the method

    19 \isa{arith}.  Linear arithmetic comprises addition, subtraction

    20 and multiplication by constant factors; subterms involving other operators

    21 are regarded as variables.  The procedure can be slow, especially if the

    22 subgoal to be proved involves subtraction over type \isa{nat}, which

    23 causes case splits.

    24

    25 The simplifier reduces arithmetic expressions in other

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

    27 outside the scope of automation, the library has hundreds of

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

    29 bear.  You can find find them by browsing the library.  Some

    30 useful lemmas are shown below.

    31

    32 \subsection{Numeric Literals}

    33 \label{sec:numerals}

    34

    35 Literals are available for the types of natural numbers, integers

    36 and reals and denote integer values of arbitrary size.

    37 They begin

    38 with a number sign (\isa{\#}), have an optional minus sign (\isa{-}) and

    39 then one or more decimal digits. Examples are \isa{\#0}, \isa{\#-3}

    40 and \isa{\#441223334678}.

    41

    42 Literals look like constants, but they abbreviate

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

    44 Isabelle performs arithmetic on literals by rewriting, rather

    45 than using the hardware arithmetic. In most cases arithmetic

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

    47 provided for literals are addition, subtraction, multiplication,

    48 integer division and remainder.

    49

    50 \emph{Beware}: the arithmetic operators are

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

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

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

    54 \begin{isabelle}

    55 \isacommand{lemma}\ "\#2\ *\ m\ =\ m\ +\ m"

    56 \end{isabelle}

    57 %

    58 Carefully observe how Isabelle displays the subgoal:

    59 \begin{isabelle}

    60 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m

    61 \end{isabelle}

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

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

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

    65

    66

    67 \subsection{The type of natural numbers, {\tt\slshape nat}}

    68

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

    70 start.  Hundreds of useful lemmas about arithmetic on type \isa{nat} are

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

    72 in exceptional circumstances should you resort to induction.

    73

    74 \subsubsection{Literals}

    75 The notational options for the natural numbers can be confusing. The

    76 constant \isa{0} is overloaded to serve as the neutral value

    77 in a variety of additive types. The symbols \isa{1} and \isa{2} are

    78 not constants but abbreviations for \isa{Suc 0} and \isa{Suc(Suc 0)},

    79 respectively. The literals \isa{\#0}, \isa{\#1} and \isa{\#2}  are

    80 entirely different from \isa{0}, \isa{1} and \isa{2}. You  will

    81 sometimes prefer one notation to the other. Literals are  obviously

    82 necessary to express large values, while \isa{0} and \isa{Suc}  are

    83 needed in order to match many theorems, including the rewrite  rules for

    84 primitive recursive functions. The following default  simplification rules

    85 replace small literals by zero and successor:

    86 \begin{isabelle}

    87 \#0\ =\ 0

    88 \rulename{numeral_0_eq_0}\isanewline

    89 \#1\ =\ 1

    90 \rulename{numeral_1_eq_1}\isanewline

    91 \#2\ +\ n\ =\ Suc\ (Suc\ n)

    92 \rulename{add_2_eq_Suc}\isanewline

    93 n\ +\ \#2\ =\ Suc\ (Suc\ n)

    94 \rulename{add_2_eq_Suc'}

    95 \end{isabelle}

    96 In special circumstances, you may wish to remove or reorient

    97 these rules.

    98

    99 \subsubsection{Typical lemmas}

   100 Inequalities involving addition and subtraction alone can be proved

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

   102 involving multiplication and division:

   103 \begin{isabelle}

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

   105 \rulename{mult_le_mono}\isanewline

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

   107 *\ k\ <\ j\ *\ k%

   108 \rulename{mult_less_mono1}\isanewline

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

   110 \rulename{div_le_mono}

   111 \end{isabelle}

   112 %

   113 Various distributive laws concerning multiplication are available:

   114 \begin{isabelle}

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

   116 \rulename{add_mult_distrib}\isanewline

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

   118 \rulename{diff_mult_distrib}\isanewline

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

   120 \rulename{mod_mult_distrib}

   121 \end{isabelle}

   122

   123 \subsubsection{Division}

   124 The library contains the basic facts about quotient and remainder

   125 (including the analogous equation, \isa{div_if}):

   126 \begin{isabelle}

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

   128 \rulename{mod_if}\isanewline

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

   130 \rulename{mod_div_equality}

   131 \end{isabelle}

   132

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

   134 Here is a selection:

   135 \begin{isabelle}

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

   137 \rulename{div_mult1_eq}\isanewline

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

   139 \rulename{mod_mult1_eq}\isanewline

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

   141 \rulename{div_mult2_eq}\isanewline

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

   143 \rulename{mod_mult2_eq}\isanewline

   144 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%

   145 \rulename{div_mult_mult1}

   146 \end{isabelle}

   147

   148 Surprisingly few of these results depend upon the

   149 divisors' being nonzero.  Isabelle/HOL defines division by zero:

   150 \begin{isabelle}

   151 a\ div\ 0\ =\ 0

   152 \rulename{DIVISION_BY_ZERO_DIV}\isanewline

   153 a\ mod\ 0\ =\ a%

   154 \rulename{DIVISION_BY_ZERO_MOD}

   155 \end{isabelle}

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

   157 simplification rules but are merely used to remove nonzero-divisor

   158 hypotheses by case analysis.  In \isa{div_mult_mult1} above, one of

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

   160

   161 The \textbf{divides} relation has the standard definition, which

   162 is overloaded over all numeric types:

   163 \begin{isabelle}

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

   165 \rulename{dvd_def}

   166 \end{isabelle}

   167 %

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

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

   170 \begin{isabelle}

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

   172 \rulename{dvd_anti_sym}\isanewline

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

   174 \rulename{dvd_add}

   175 \end{isabelle}

   176

   177 \subsubsection{Simplifier tricks}

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

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

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

   181 nice properties; often it is best to remove it from a subgoal

   182 using this split rule:

   183 \begin{isabelle}

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

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

   186 d))

   187 \rulename{nat_diff_split}

   188 \end{isabelle}

   189 For example, it proves the following fact, which lies outside the scope of

   190 linear arithmetic:

   191 \begin{isabelle}

   192 \isacommand{lemma}\ "(n-1)*(n+1)\ =\ n*n\ -\ 1"\isanewline

   193 \isacommand{apply}\ (simp\ split:\ nat_diff_split)\isanewline

   194 \isacommand{done}

   195 \end{isabelle}

   196

   197 Suppose that two expressions are equal, differing only in

   198 associativity and commutativity of addition.  Simplifying with the

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

   200 the two expressions identical:

   201 \begin{isabelle}

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

   203 \rulename{add_assoc}\isanewline

   204 m\ +\ n\ =\ n\ +\ m%

   205 \rulename{add_commute}\isanewline

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

   207 +\ z)

   208 \rulename{add_left_commute}

   209 \end{isabelle}

   210 The name \isa{add_ac} refers to the list of all three theorems, similarly

   211 there is \isa{mult_ac}.  Here is an example of the sorting effect.  Start

   212 with this goal:

   213 \begin{isabelle}

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

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

   216 \end{isabelle}

   217 %

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

   219 \begin{isabelle}

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

   221 \end{isabelle}

   222 %

   223 Here is the resulting subgoal:

   224 \begin{isabelle}

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

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

   227 \end{isabelle}

   228

   229

   230 \subsection{The type of integers, {\tt\slshape int}}

   231

   232 Reasoning methods resemble those for the natural numbers, but

   233 induction and the constant \isa{Suc} are not available.

   234

   235 Concerning simplifier tricks, we have no need to eliminate subtraction (it

   236 is well-behaved), but the simplifier can sort the operands of integer

   237 operators.  The name \isa{zadd_ac} refers to the associativity and

   238 commutativity theorems for integer addition, while \isa{zmult_ac} has the

   239 analogous theorems for multiplication.  The prefix~\isa{z} in many theorem

   240 names recalls the use of $\Bbb{Z}$ to denote the set of integers.

   241

   242 For division and remainder, the treatment of negative divisors follows

   243 traditional mathematical practice: the sign of the remainder follows that

   244 of the divisor:

   245 \begin{isabelle}

   246 \#0\ <\ b\ \isasymLongrightarrow \ \#0\ \isasymle \ a\ mod\ b%

   247 \rulename{pos_mod_sign}\isanewline

   248 \#0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%

   249 \rulename{pos_mod_bound}\isanewline

   250 b\ <\ \#0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ \#0

   251 \rulename{neg_mod_sign}\isanewline

   252 b\ <\ \#0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%

   253 \rulename{neg_mod_bound}

   254 \end{isabelle}

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

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

   257

   258 The library provides many lemmas for proving inequalities involving integer

   259 multiplication and division, similar to those shown above for

   260 type~\isa{nat}.  The absolute value function \isa{abs} is

   261 defined for the integers; we have for example the obvious law

   262 \begin{isabelle}

   263 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar

   264 \rulename{abs_mult}

   265 \end{isabelle}

   266

   267 Again, many facts about quotients and remainders are provided:

   268 \begin{isabelle}

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

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

   271 \rulename{zdiv_zadd1_eq}

   272 \par\smallskip

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

   274 \rulename{zmod_zadd1_eq}

   275 \end{isabelle}

   276

   277 \begin{isabelle}

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

   279 \rulename{zdiv_zmult1_eq}\isanewline

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

   281 \rulename{zmod_zmult1_eq}

   282 \end{isabelle}

   283

   284 \begin{isabelle}

   285 \#0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%

   286 \rulename{zdiv_zmult2_eq}\isanewline

   287 \#0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\

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

   289 \rulename{zmod_zmult2_eq}

   290 \end{isabelle}

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

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

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

   294 is

   295 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of

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

   297

   298

   299 \subsection{The type of real numbers, {\tt\slshape real}}

   300

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

   302 They are

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

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

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

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

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

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

   309 $\surd2$, which is irrational.)

   310

   311 The formalization of completeness is long and technical.  Rather than

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

   313 directory \texttt{Real}.

   314

   315 Density is trivial to express:

   316 \begin{isabelle}

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

   318 \rulename{real_dense}

   319 \end{isabelle}

   320

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

   322 are installed as default simplification rules in order to express

   323 combinations of products and quotients as rational expressions:

   324 \begin{isabelle}

   325 x\ *\ (y\ /\ z)\ =\ x\ *\ y\ /\ z%

   326 \rulename{real_times_divide1_eq}\isanewline

   327 y\ /\ z\ *\ x\ =\ y\ *\ x\ /\ z%

   328 \rulename{real_times_divide2_eq}\isanewline

   329 x\ /\ (y\ /\ z)\ =\ x\ *\ z\ /\ y%

   330 \rulename{real_divide_divide1_eq}\isanewline

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

   332 \rulename{real_divide_divide2_eq}

   333 \end{isabelle}

   334

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

   336 then be cancelled:

   337 \begin{isabelle}

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

   339 \rulename{real_minus_divide_eq}\isanewline

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

   341 \rulename{real_divide_minus_eq}

   342 \end{isabelle}

   343

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

   345 simplification rule.

   346 \begin{isabelle}

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

   348 \rulename{real_add_divide_distrib}

   349 \end{isabelle}

   350

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

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

   353 associativity and commutativity theorems for addition, while similarly

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

   355

   356 The absolute value function \isa{abs} is

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

   358 exponentiation:

   359 \begin{isabelle}

   360 \isasymbar r\isasymbar \ \isacharcircum \ n\ =\ \isasymbar r\ \isacharcircum \ n\isasymbar

   361 \rulename{realpow_abs}

   362 \end{isabelle}

   363

   364 \emph{Note}: Type \isa{real} is only available in the logic HOL-Real, which

   365 is  HOL extended with the rather substantial development of the real

   366 numbers.  Base your theory upon theory \isa{Real}, not the usual \isa{Main}.

   367

   368 Also distributed with Isabelle is HOL-Hyperreal,

   369 whose theory \isa{Hyperreal} defines the type \isa{hypreal} of non-standard

   370 reals.  These

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

   372 small and infinitely large quantities; they facilitate proofs

   373 about limits, differentiation and integration.  The development defines an

   374 infinitely large number, \isa{omega} and an infinitely small positive

   375 number, \isa{epsilon}.  Also available is the approximates relation,

   376 written $\approx$.