doc-src/TutorialI/Types/numerics.tex
author paulson
Thu Jan 04 10:22:33 2001 +0100 (2001-01-04)
changeset 10777 a5a6255748c3
parent 10654 458068404143
child 10779 b0d961105f46
permissions -rw-r--r--
initial material on the Reals
     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 
    34 Literals are available for the types of natural numbers, integers 
    35 and reals and denote integer values of arbitrary size. 
    36 They begin 
    37 with a number sign (\isa{\#}), have an optional minus sign (\isa{-}) and 
    38 then one or more decimal digits. Examples are \isa{\#0}, \isa{\#-3} 
    39 and \isa{\#441223334678}.
    40 
    41 Literals look like constants, but they abbreviate 
    42 terms, representing the number in a two's complement binary notation. 
    43 Isabelle performs arithmetic on literals by rewriting, rather 
    44 than using the hardware arithmetic. In most cases arithmetic 
    45 is fast enough, even for large numbers. The arithmetic operations 
    46 provided for literals are addition, subtraction, multiplication, 
    47 integer division and remainder. 
    48 
    49 \emph{Beware}: the arithmetic operators are 
    50 overloaded, so you must be careful to ensure that each numeric 
    51 expression refers to a specific type, if necessary by inserting 
    52 type constraints.  Here is an example of what can go wrong:
    53 \begin{isabelle}
    54 \isacommand{lemma}\ "\#2\ *\ m\ =\ m\ +\ m"
    55 \end{isabelle}
    56 %
    57 Carefully observe how Isabelle displays the subgoal:
    58 \begin{isabelle}
    59 \ 1.\ (\#2::'a)\ *\ m\ =\ m\ +\ m
    60 \end{isabelle}
    61 The type \isa{'a} given for the literal \isa{\#2} warns us that no numeric
    62 type has been specified.  The problem is underspecified.  Given a type
    63 constraint such as \isa{nat}, \isa{int} or \isa{real}, it becomes trivial.
    64 
    65 
    66 \subsection{The type of natural numbers, {\tt\slshape nat}}
    67 
    68 This type requires no introduction: we have been using it from the
    69 start.  Hundreds of useful lemmas about arithmetic on type \isa{nat} are
    70 proved in the theories \isa{Nat}, \isa{NatArith} and \isa{Divides}.  Only
    71 in exceptional circumstances should you resort to induction.
    72 
    73 \subsubsection{Literals}
    74 The notational options for the natural numbers can be confusing. The 
    75 constant \isa{0} is overloaded to serve as the neutral value 
    76 in a variety of additive types. The symbols \isa{1} and \isa{2} are 
    77 not constants but abbreviations for \isa{Suc 0} and \isa{Suc(Suc 0)},
    78 respectively. The literals \isa{\#0}, \isa{\#1} and \isa{\#2}  are
    79 entirely different from \isa{0}, \isa{1} and \isa{2}. You  will
    80 sometimes prefer one notation to the other. Literals are  obviously
    81 necessary to express large values, while \isa{0} and \isa{Suc}  are
    82 needed in order to match many theorems, including the rewrite  rules for
    83 primitive recursive functions. The following default  simplification rules
    84 replace small literals by zero and successor: 
    85 \begin{isabelle}
    86 \#0\ =\ 0
    87 \rulename{numeral_0_eq_0}\isanewline
    88 \#1\ =\ 1
    89 \rulename{numeral_1_eq_1}\isanewline
    90 \#2\ +\ n\ =\ Suc\ (Suc\ n)
    91 \rulename{add_2_eq_Suc}\isanewline
    92 n\ +\ \#2\ =\ Suc\ (Suc\ n)
    93 \rulename{add_2_eq_Suc'}
    94 \end{isabelle}
    95 In special circumstances, you may wish to remove or reorient 
    96 these rules. 
    97 
    98 \subsubsection{Typical lemmas}
    99 Inequalities involving addition and subtraction alone can be proved
   100 automatically.  Lemmas such as these can be used to prove inequalities
   101 involving multiplication and division:
   102 \begin{isabelle}
   103 \isasymlbrakk i\ \isasymle \ j;\ k\ \isasymle \ l\isasymrbrakk \ \isasymLongrightarrow \ i\ *\ k\ \isasymle \ j\ *\ l%
   104 \rulename{mult_le_mono}\isanewline
   105 \isasymlbrakk i\ <\ j;\ 0\ <\ k\isasymrbrakk \ \isasymLongrightarrow \ i\
   106 *\ k\ <\ j\ *\ k%
   107 \rulename{mult_less_mono1}\isanewline
   108 m\ \isasymle \ n\ \isasymLongrightarrow \ m\ div\ k\ \isasymle \ n\ div\ k%
   109 \rulename{div_le_mono}
   110 \end{isabelle}
   111 %
   112 Various distributive laws concerning multiplication are available:
   113 \begin{isabelle}
   114 (m\ +\ n)\ *\ k\ =\ m\ *\ k\ +\ n\ *\ k%
   115 \rulename{add_mult_distrib}\isanewline
   116 (m\ -\ n)\ *\ k\ =\ m\ *\ k\ -\ n\ *\ k%
   117 \rulename{diff_mult_distrib}\isanewline
   118 (m\ mod\ n)\ *\ k\ =\ (m\ *\ k)\ mod\ (n\ *\ k)
   119 \rulename{mod_mult_distrib}
   120 \end{isabelle}
   121 
   122 \subsubsection{Division}
   123 The library contains the basic facts about quotient and remainder
   124 (including the analogous equation, \isa{div_if}):
   125 \begin{isabelle}
   126 m\ mod\ n\ =\ (if\ m\ <\ n\ then\ m\ else\ (m\ -\ n)\ mod\ n)
   127 \rulename{mod_if}\isanewline
   128 m\ div\ n\ *\ n\ +\ m\ mod\ n\ =\ m%
   129 \rulename{mod_div_equality}
   130 \end{isabelle}
   131 
   132 Many less obvious facts about quotient and remainder are also provided. 
   133 Here is a selection:
   134 \begin{isabelle}
   135 a\ *\ b\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   136 \rulename{div_mult1_eq}\isanewline
   137 a\ *\ b\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   138 \rulename{mod_mult1_eq}\isanewline
   139 a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   140 \rulename{div_mult2_eq}\isanewline
   141 a\ mod\ (b*c)\ =\ b * (a\ div\ b\ mod\ c)\ +\ a\ mod\ b%
   142 \rulename{mod_mult2_eq}\isanewline
   143 0\ <\ c\ \isasymLongrightarrow \ (c\ *\ a)\ div\ (c\ *\ b)\ =\ a\ div\ b%
   144 \rulename{div_mult_mult1}
   145 \end{isabelle}
   146 
   147 Surprisingly few of these results depend upon the
   148 divisors' being nonzero.  Isabelle/HOL defines division by zero:
   149 \begin{isabelle}
   150 a\ div\ 0\ =\ 0
   151 \rulename{DIVISION_BY_ZERO_DIV}\isanewline
   152 a\ mod\ 0\ =\ a%
   153 \rulename{DIVISION_BY_ZERO_MOD}
   154 \end{isabelle}
   155 As a concession to convention, these equations are not installed as default
   156 simplification rules but are merely used to remove nonzero-divisor
   157 hypotheses by case analysis.  In \isa{div_mult_mult1} above, one of
   158 the two divisors (namely~\isa{c}) must be still be nonzero.
   159 
   160 The \textbf{divides} relation has the standard definition, which
   161 is overloaded over all numeric types: 
   162 \begin{isabelle}
   163 m\ dvd\ n\ \isasymequiv\ {\isasymexists}k.\ n\ =\ m\ *\ k
   164 \rulename{dvd_def}
   165 \end{isabelle}
   166 %
   167 Section~\ref{sec:proving-euclid} discusses proofs involving this
   168 relation.  Here are some of the facts proved about it:
   169 \begin{isabelle}
   170 \isasymlbrakk m\ dvd\ n;\ n\ dvd\ m\isasymrbrakk \ \isasymLongrightarrow \ m\ =\ n%
   171 \rulename{dvd_anti_sym}\isanewline
   172 \isasymlbrakk k\ dvd\ m;\ k\ dvd\ n\isasymrbrakk \ \isasymLongrightarrow \ k\ dvd\ (m\ +\ n)
   173 \rulename{dvd_add}
   174 \end{isabelle}
   175 
   176 \subsubsection{Simplifier tricks}
   177 The rule \isa{diff_mult_distrib} shown above is one of the few facts
   178 about \isa{m\ -\ n} that is not subject to
   179 the condition \isa{n\ \isasymle \  m}.  Natural number subtraction has few
   180 nice properties; often it is best to remove it from a subgoal
   181 using this split rule:
   182 \begin{isabelle}
   183 P(a-b)\ =\ ((a<b\ \isasymlongrightarrow \ P\
   184 0)\ \isasymand \ (\isasymforall d.\ a\ =\ b+d\ \isasymlongrightarrow \ P\
   185 d))
   186 \rulename{nat_diff_split}
   187 \end{isabelle}
   188 For example, it proves the following fact, which lies outside the scope of
   189 linear arithmetic:
   190 \begin{isabelle}
   191 \isacommand{lemma}\ "(n-1)*(n+1)\ =\ n*n\ -\ 1"\isanewline
   192 \isacommand{apply}\ (simp\ split:\ nat_diff_split)\isanewline
   193 \isacommand{done}
   194 \end{isabelle}
   195 
   196 Suppose that two expressions are equal, differing only in 
   197 associativity and commutativity of addition.  Simplifying with the
   198 following equations sorts the terms and groups them to the right, making
   199 the two expressions identical:
   200 \begin{isabelle}
   201 m\ +\ n\ +\ k\ =\ m\ +\ (n\ +\ k)
   202 \rulename{add_assoc}\isanewline
   203 m\ +\ n\ =\ n\ +\ m%
   204 \rulename{add_commute}\isanewline
   205 x\ +\ (y\ +\ z)\ =\ y\ +\ (x\
   206 +\ z)
   207 \rulename{add_left_commute}
   208 \end{isabelle}
   209 The name \isa{add_ac} refers to the list of all three theorems, similarly
   210 there is \isa{mult_ac}.  Here is an example of the sorting effect.  Start
   211 with this goal:
   212 \begin{isabelle}
   213 \ 1.\ Suc\ (i\ +\ j\ *\ l\ *\ k\ +\ m\ *\ n)\ =\
   214 f\ (n\ *\ m\ +\ i\ +\ k\ *\ j\ *\ l)
   215 \end{isabelle}
   216 %
   217 Simplify using  \isa{add_ac} and \isa{mult_ac}:
   218 \begin{isabelle}
   219 \isacommand{apply}\ (simp\ add:\ add_ac\ mult_ac)
   220 \end{isabelle}
   221 %
   222 Here is the resulting subgoal:
   223 \begin{isabelle}
   224 \ 1.\ Suc\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))\
   225 =\ f\ (i\ +\ (m\ *\ n\ +\ j\ *\ (k\ *\ l)))%
   226 \end{isabelle}
   227 
   228 
   229 \subsection{The type of integers, {\tt\slshape int}}
   230 
   231 Reasoning methods resemble those for the natural numbers, but
   232 induction and the constant \isa{Suc} are not available.
   233 
   234 Concerning simplifier tricks, we have no need to eliminate subtraction (it
   235 is well-behaved), but the simplifier can sort the operands of integer
   236 operators.  The name \isa{zadd_ac} refers to the associativity and
   237 commutativity theorems for integer addition, while \isa{zmult_ac} has the
   238 analogous theorems for multiplication.  The prefix~\isa{z} in many theorem
   239 names recalls the use of $\Bbb{Z}$ to denote the set of integers.
   240 
   241 For division and remainder, the treatment of negative divisors follows
   242 traditional mathematical practice: the sign of the remainder follows that
   243 of the divisor:
   244 \begin{isabelle}
   245 \#0\ <\ b\ \isasymLongrightarrow \ \#0\ \isasymle \ a\ mod\ b%
   246 \rulename{pos_mod_sign}\isanewline
   247 \#0\ <\ b\ \isasymLongrightarrow \ a\ mod\ b\ <\ b%
   248 \rulename{pos_mod_bound}\isanewline
   249 b\ <\ \#0\ \isasymLongrightarrow \ a\ mod\ b\ \isasymle \ \#0
   250 \rulename{neg_mod_sign}\isanewline
   251 b\ <\ \#0\ \isasymLongrightarrow \ b\ <\ a\ mod\ b%
   252 \rulename{neg_mod_bound}
   253 \end{isabelle}
   254 ML treats negative divisors in the same way, but most computer hardware
   255 treats signed operands using the same rules as for multiplication.
   256 
   257 The library provides many lemmas for proving inequalities involving integer
   258 multiplication and division, similar to those shown above for
   259 type~\isa{nat}.  The absolute value function \isa{abs} is
   260 defined for the integers; we have for example the obvious law
   261 \begin{isabelle}
   262 \isasymbar x\ *\ y\isasymbar \ =\ \isasymbar x\isasymbar \ *\ \isasymbar y\isasymbar 
   263 \rulename{abs_mult}
   264 \end{isabelle}
   265 
   266 Again, many facts about quotients and remainders are provided:
   267 \begin{isabelle}
   268 (a\ +\ b)\ div\ c\ =\isanewline
   269 a\ div\ c\ +\ b\ div\ c\ +\ (a\ mod\ c\ +\ b\ mod\ c)\ div\ c%
   270 \rulename{zdiv_zadd1_eq}
   271 \par\smallskip
   272 (a\ +\ b)\ mod\ c\ =\ (a\ mod\ c\ +\ b\ mod\ c)\ mod\ c%
   273 \rulename{zmod_zadd1_eq}
   274 \end{isabelle}
   275 
   276 \begin{isabelle}
   277 (a\ *\ b)\ div\ c\ =\ a\ *\ (b\ div\ c)\ +\ a\ *\ (b\ mod\ c)\ div\ c%
   278 \rulename{zdiv_zmult1_eq}\isanewline
   279 (a\ *\ b)\ mod\ c\ =\ a\ *\ (b\ mod\ c)\ mod\ c%
   280 \rulename{zmod_zmult1_eq}
   281 \end{isabelle}
   282 
   283 \begin{isabelle}
   284 \#0\ <\ c\ \isasymLongrightarrow \ a\ div\ (b*c)\ =\ a\ div\ b\ div\ c%
   285 \rulename{zdiv_zmult2_eq}\isanewline
   286 \#0\ <\ c\ \isasymLongrightarrow \ a\ mod\ (b*c)\ =\ b*(a\ div\ b\ mod\
   287 c)\ +\ a\ mod\ b%
   288 \rulename{zmod_zmult2_eq}
   289 \end{isabelle}
   290 The last two differ from their natural number analogues by requiring
   291 \isa{c} to be positive.  Since division by zero yields zero, we could allow
   292 \isa{c} to be zero.  However, \isa{c} cannot be negative: a counterexample
   293 is
   294 $\isa{a} = 7$, $\isa{b} = 2$ and $\isa{c} = -3$, when the left-hand side of
   295 \isa{zdiv_zmult2_eq} is $-2$ while the right-hand side is $-1$.
   296 
   297 
   298 \subsection{The type of real numbers, {\tt\slshape real}}
   299 
   300 The real numbers enjoy two significant properties that the integers lack. 
   301 They are
   302 \textbf{dense}: between every two distinct real numbers there lies another.
   303 This property follows from the division laws, since if $x<y$ then between
   304 them lies $(x+y)/2$.  The second property is that they are
   305 \textbf{complete}: every set of reals that is bounded above has a least
   306 upper bound.  Completeness distinguishes the reals from the rationals, for
   307 which the set $\{x\mid x^2<2\}$ has no least upper bound.  (It could only be
   308 $\surd2$, which is irrational.)
   309 
   310 The formalization of completeness is long and technical.  Rather than
   311 reproducing it here, we refer you to the theory \texttt{RComplete} in
   312 directory \texttt{Real}.
   313 
   314 Density is trivial to express:
   315 \begin{isabelle}
   316 x\ <\ y\ \isasymLongrightarrow \ \isasymexists r.\ x\ <\ r\ \isasymand \ r\ <\ y%
   317 \rulename{real_dense}
   318 \end{isabelle}
   319 
   320 Here is a selection of rules about the division operator.  The following
   321 are installed as default simplification rules in order to express
   322 combinations of products and quotients as rational expressions:
   323 \begin{isabelle}
   324 x\ *\ (y\ /\ z)\ =\ x\ *\ y\ /\ z%
   325 \rulename{real_times_divide1_eq}\isanewline
   326 y\ /\ z\ *\ x\ =\ y\ *\ x\ /\ z%
   327 \rulename{real_times_divide2_eq}\isanewline
   328 x\ /\ (y\ /\ z)\ =\ x\ *\ z\ /\ y%
   329 \rulename{real_divide_divide1_eq}\isanewline
   330 x\ /\ y\ /\ z\ =\ x\ /\ (y\ *\ z)
   331 \rulename{real_divide_divide2_eq}
   332 \end{isabelle}
   333 
   334 Signs are extracted from quotients in the hope that complementary terms can
   335 then be cancelled:
   336 \begin{isabelle}
   337 -\ x\ /\ y\ =\ -\ (x\ /\ y)
   338 \rulename{real_minus_divide_eq}\isanewline
   339 x\ /\ -\ y\ =\ -\ (x\ /\ y)
   340 \rulename{real_divide_minus_eq}
   341 \end{isabelle}
   342 
   343 The following distributive law is available, but it is not installed as a
   344 simplification rule.
   345 \begin{isabelle}
   346 (x\ +\ y)\ /\ z\ =\ x\ /\ z\ +\ y\ /\ z%
   347 \rulename{real_add_divide_distrib}
   348 \end{isabelle}
   349 
   350 As with the other numeric types, the simplifier can sort the operands of
   351 addition and multiplication.  The name \isa{real_add_ac} refers to the
   352 associativity and commutativity theorems for addition, while similarly
   353 \isa{real_mult_ac} contains those properties for multiplication. 
   354 
   355 The absolute value function \isa{abs} is
   356 defined for the reals, along with many theorems such as this one about
   357 exponentiation:
   358 \begin{isabelle}
   359 \isasymbar r\isasymbar \ \isacharcircum \ n\ =\ \isasymbar r\ \isacharcircum \ n\isasymbar 
   360 \rulename{realpow_abs}
   361 \end{isabelle}
   362 
   363 \emph{Note}: Type \isa{real} is only available in the logic HOL-Real, which
   364 is  HOL extended with the rather substantial development of the real
   365 numbers.  Base your theory upon theory \isa{Real}, not the usual \isa{Main}.
   366 
   367 Also distributed with Isabelle is HOL-Hyperreal,
   368 whose theory \isa{Hyperreal} defines the type \isa{hypreal} of non-standard
   369 reals.  These
   370 \textbf{hyperreals} include infinitesimals, which represent infinitely
   371 small and infinitely large quantities; they facilitate proofs
   372 about limits, differentiation and integration.  The development defines an
   373 infinitely large number, \isa{omega} and an infinitely small positive
   374 number, \isa{epsilon}.  Also available is the approximates relation,
   375 written $\approx$.